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ContentsArticles

On a Heuristic Point of View about the Creation and Conversion of Light 1On the Electrodynamics of Moving Bodies 10The Development of Our Views on the Composition and Essence of Radiation 11The Field Equations of Gravitation 19The Foundation of the Generalised Theory of Relativity 22Dialog about Objections against the Theory of Relativity 56Time, Space, and Gravitation 62A Brief Outline of the Development of the Theory of Relativity 64Ether and the Theory of Relativity 68The Bad Nauheim Debate 73Geometry and Experience 82The Meaning of Relativity 90On the Relative Motion of the Earth and the Luminiferous Ether 90

ReferencesArticle Sources and Contributors 91Image Sources, Licenses and Contributors 92

Article LicensesLicense 93

On a Heuristic Point of View about the Creation and Conversion of Light 1

On a Heuristic Point of View about the Creationand Conversion of Light

On a Heuristic Point of View about the Creation and Conversion of Light (1905) by Albert Einstein, translated by Wikisource

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Maxwell's theory of electromagnetic processes in so-called empty space differs in a profound, essential way from thecurrent theoretical models of gases and other matter. On the one hand, we consider the state of a material body to bedetermined completely by the positions and velocities of a finite number of atoms and electrons, albeit a very largenumber. By contrast, the electromagnetic state of a region of space is described by continuous functions and, hence,cannot be determined exactly by any finite number of variables. Thus, according to Maxwell's theory, the energy ofpurely electromagnetic phenomena (such as light) should be represented by a continuous function of space. Bycontrast, the energy of a material body should be represented by a discrete sum over the atoms and electrons; hence,the energy of a material body cannot be divided into arbitrarily many, arbitrarily small components. However,according to Maxwell's theory (or, indeed, any wave theory), the energy of a light wave emitted from a point sourceis distributed continuously over an ever larger volume.The wave theory of light with its continuous spatial functions has proven to be an excellent model of purely opticalphenomena and presumably will never be replaced by another theory. Nevertheless, we should consider that opticalexperiments observe only time-averaged values, rather than instantaneous values. Hence, despite the perfectagreement of Maxwell's theory with experiment, the use of continuous spatial functions to describe light may lead tocontradictions with experiments, especially when applied to the generation and transformation of light.In particular, black body radiation, photoluminescence, generation of cathode rays from ultraviolet light and otherphenomena associated with the generation and transformation of light seem better modeled by assuming that theenergy of light is distributed discontinuously in space. According to this picture, the energy of a light wave emittedfrom a point source is not spread continuously over ever larger volumes, but consists of a finite number of energyquanta that are spatially localized at points of space, move without dividing and are absorbed or generated only as awhole.Subsequently, I wish to explain the reasoning and supporting evidence that led me to this picture of light, in the hopethat some researchers may find it useful for their experiments.

A certain problem concerning the theory of "black body radiation".We begin by applying Maxwell's theory of light and electrons to the following situation. Let there be a cavity withperfectly reflecting walls, filled with a number of freely moving electrons and gas molecules that interact viaconservative forces whenever they come close, i.e., that collide with each other just as gas molecules in the kinetictheory of gases.[1] In addition, let there be a number of electrons bound to spatially well-separated points by restoringforces that increase linearly with separation. These electrons also interact with the free molecules and electrons byconservative potentials when they approach very closely. We denote these electrons, which are bound at points ofspace, as "resonators", since they absorb and emit electromagnetic waves of a particular period.According to the present theory of the generation of light, the radiation in the cavity must be identical to black bodyradiation (which may be found by assuming Maxwell's theory and dynamic equilibrium), at least if one assumes thatresonators exist for every frequency under consideration.

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Initially, let us neglect the radiation absorbed and emitted by the resonators and focus instead on the requirement ofthermal equilibrium and its implications for the interaction (collisions) between molecules and electrons. Accordingto the kinetic theory of gases, dynamic equilibrium requires that the average kinetic energy of a resonator equal theaverage kinetic energy of a freely moving gas molecule. Decomposing the motion of a resonator electron into threemutually perpendicular oscillations, we find that the average energy of such a linear oscillation is

where R is the absolute gas constant, N is the number of "real molecules" in a gram equivalent and T is the absolutetemperature. Because of the time averages of the kinetic and potential energy, the energy is ⅔ as large as thekinetic energy of a single free gas molecule. Even if something (such as radiative processes) causes thetime-averaged energy of a resonator to deviate from the value , collisions with the free electrons and gasmolecules will return its average energy to by absorbing or releasing energy. Hence, in this situation, dynamicequilibrium can only exist when every resonator has an average energy .We apply a similar consideration now to the interaction between the resonators and the ambient radiation within thecavity. For this case, Planck has derived the necessary condition for dynamic equilibrium [2]; treating the radiation asa completely random process.[3]

He found:

Here, is the average energy of a resonator of eigenfrequency ν (per oscillatory component), L is the speed oflight, ν is the frequency, and ρνdν is the energy density of the cavity radiation of frequency between ν and ν + dν.If the net radiative energy of frequency ν is not to continually increase or decrease, the following equality must hold

or, equivalently,

This condition for dynamic equilibrium not only lacks agreement with experiment, it also eliminates any possibilityfor equilibrium between matter and aether. The wider the range of frequencies of the resonators is chosen the biggerthe radiation energy in the space becomes, and in the limit we obtain:

Planck's Derivation of the Fundamental QuantumIn the next section we want to show that the determination that Mr. Planck gave of the elementary quanta is to someextent independent of the "black body radiation" theory that he created.The Formula by Planck [4] for ρν that suffices for all experiments so far goes

where


In the limit of large values of T/ν, that is for large wavelengths and radiation densities this formula approaches theform:

One recognizes that this formula is the same as the one that was derived from Maxwell theory and electron theory.Equating the coefficients of the formula's:

or

that is, a hydrogen atom weighs 1/N gram = 1.62·10-24g. This is precisely the value found by Mr. Planck, which is insatisfactory agreement with values obtained in other ways.This brings us to the conclusion: the larger the energy density and the wavelength of radiation the more suitable thetheoretical basis that we used; but for small wavelengths and low radiation densities the basis fails completely.In the following the "black body radiation" is to be considered in terms of what is experienced, without forming apicture of the creation and propagation of the radiation.

The Entropy of RadiationThe following discussion is contained in a famous work of Mr. Wien, and is only included here for the sake ofcompleteness.Let there be radiation taking up volume v. We assume that the observable properties of the radiation are determinedcompletely when the radiation densities ρ(ν) are given for all frequencies. [5] Since we can regard radiations ofdifferent frequency as separable without doing work or transferring heat the entropy of the radiation can beexpressed in the form

where φ is a function of the variables ρ and ν. φ can be reduced to a function of only one variable by expressing thatthe entropy of radiation between reflecting walls is not changed by adiabatic compression. We won't go into thathowever, but investigate right away how the function φ can be obtained from the radiation law of the black body.In the case of "black body radiation" ρ is such a function of ν that for a given energy the entropy is a maximum, thatis, that

When

From this it follows that for any choice of δρ as function of ν

Where λ is independent of ν. Thus is independent of νFor the temperature increase of dT of a black body radiation of volume v = 1 the following equation is valid:


or, since is independent of ν:

Since dE is equal to the transferred heat, and the process is reversible we also have:

Equating formulas gives:

This is the black body radiation law. So it's possible to determine the black body radiation from the function φ.Conversely, through integration one can obtain φ from the black body radiation law keeping in mind that φ vanishesfor ρ = 0.

Limiting law for the entropy of monochromatic radiation at low radiationdensityAdmittedly, the observations of "black body radiation" so far indicate that the law that Mr. Wien originally devisedfor the "black body radiation"

is not exactly valid. However, for large values of ν/T experiment completely confirms the law. We shall base ourcalculations on this formula, keeping in mind that the results will be valid within certain limitations only.First, we get from this equation:

and then, using the relation obtained in the preceding section:

Let there be a radiation of energy E, with a frequency between ν and ν + dν. Let the radiation extend over volume v.The entropy of this radiation is:

We will limit ourselves to investigating the dependency of the radiation's entropy on the volume that is occupied. Letthe entropy of the radiation be called S0 when it occupies the volume v0, then we get:

This equation shows that the entropy of monochromatic radiation of sufficiently low density varies with volumeaccording to the same law as the entropy of an ideal gas or that of a dilute solution. In the following the equation justfound will be interpreted in terms of the principle introduced by Mr. Boltzmann that says that the entropy of a systemis a function of the probability of its state.


Molecular Theoretical investigation of the Volume Dependence of the Entropyof Gases and Dilute SolutionsIn calculating Entropy on the grounds of molecular theory the word "probability" is often used in a meaning that isn'tcovered by the definition in probability theory. Especially the "cases of equal probability" are often set byhypothesis, where the applied theoretical representation is sufficiently definite to deduce probabilities without fixingthem by hypothesis. I will show in a separate work that in considerations of thermal processes one obtains acomplete result with the so-called "statistical probability". This way I hope to remove a logical difficulty that is inthe way of fully implementing Boltzmann's principle. Here however only its general formulation and application inquite specific cases will be given.When it's meaningful to talk about the probability of a state of a system, and additionally every increase of entropycan be described as a transition to a more probable state, the entropy S1 of a system is a function of the probabilityW1 of its instantaneous state. In the case of two systems S1 and S2, one can state:

If one considers these systems as a single system with entropy S and probability W, then:

and

The latter equation expresses that the states of the two systems are independent.From these equations it follows:

and hence finally

The quantity C is also a universal constant; it follows from kinetic gas theory, where the constants R and N have thesame meaning as above. Denoting the entropy at a particular starting state as S0, and the relative probability of a statewith entropy S as W we have in general:

We now consider the following special case. Let a number (n) of movable points (for example molecules) be presentin a volume v0, these points will be the subject of our considerations. Other than these, arbitrarily many othermovable points can be present. As to the law that describes how the considered points move around in the space theonly assumption is that no part of the space (and no direction) is favored over others. The number of the(first-mentioned) points that we are considering is so small that mutual interactions are negligible.The system considered, which can be for example an ideal gas or a diluted solution, has a certain entropy. We take apart of the volume v0 with a size of v and we think of all n movable points displaced to that volume v, with otherwiseno change of the system. Clearly this state has another entropy (S), and here we want to determine that entropydifference with the help of Boltzmann's principle.We ask: how large is the probability of the last-mentioned state relative to the original state? Or, what is theprobability that at some point in time all n independently moving points in a volume v0 have by chance ended up inthe volume v?For this probability, which is a "statistical probability" one obtains the value:


one derives from this, applying Boltzmann's principle:

It's noteworthy that for this derivation, from which the Boyle-Gay-Lussac law and the identical law of osmoticpressure can be easily derived thermodynamically [6], there is no need to make any assumption regarding the way themolucules move.

Interpretation of the Volume Dependence of the Entropy of MonochromaticRadiation using Boltzmann's PrincipleIn paragraph 4 we found for the dependence of Entropy of the monochromatic radiation on volume the expression:

This formula can be recast as follows:

Comparing this with the general formula that expresses Boltzmann's principle

we arrive at the following conclusion:If monochromatic radiation of frequency ν and energy E is enclosed (by reflecting walls) in the volume v0, then theprobability that at an arbitrary point in time all of the radiation energy located in a part v of the volume v0 is:

Subsequently we conclude:In terms of heat theory monochromatic radiation of low density (within the realm of validity of Wien's radiationformula) behaves as if it consisted of independent energy quanta of the magnitude Rβν/N.We also want to compare the average magnitude of the energy quanta of the "black body radiation" with the meanaverage energy of the center-of-mass-motion of a molecule at the same temperature. The latter is 3/2(R/N)T, and forthe average energy of the Energy quanta Wien's formula gives:

The fact that monochromatic radiation (of sufficiently low density) behaves as regards to dependency of entropy onvolume like a discontinuous medium that consists of energy quanta of magnitude Rβν/N suggests we shouldinvestigate whether the laws of generation and transformation of light are what they must be if light consisted ofsuch energy quanta. In the following we will address that question.


Stokes' RuleLet monochromatic light be transformed by photoluminence into light of another frequency, and let it be assumedthat according to the result just obtained the generating as well as the generated light consists of energy quanta ofmagnitude (R/N)βν, where ν is the corresponding frequency. The transformation process can then be interpreted asfollows. Each generating energy quantum of frequency ν1 is absorbed and generates—at least with sufficiently smalldensity of the generating energy quanta—by itself a light quantum of of frequency ν2; possibly other light quanta offrequency ν3, ν4 etc. as well as other form of energy (e.g heat) can be generated simultaneously. Through whichintermedia processes the final result comes about is immaterial. If the photoluminescing substance isn't a continuoussource of energy it follows from the energy principle that the energy of the generated energy quanta are not largerthan the generating light quanta; therefore the following relation must hold:

or

As is well known this is Stokes' rule.Especially noteworthy is that with weak illumination the amount of generated light must, other circumstances beingequal, be proportional to the amount of exciting light, since every incident energy quantum will cause oneelementary process of the above indicated kind, independent of the action of other exciting energy quanta. Inparticular there will be no lower limit of the intensity of the exciting light below which the light would be incapableof exciting light.According to the way the understanding of the phenomena is laid down here deviations from Stokes' rule areconceivable in the following cases:1.1. When the number of energy quanta per unit of volume that are simultaneously involved in the transformation is

so large that the energy quantum of the generated light can receive the energy of several exciting energy quanta.2.2. When the generating (or generated) light does not have the energy characteristics of "black body radiation" that is

in the realm of validity of Wien's law, when for instance the exciting light is generated by a body of such hightemperature that for the wavelengths considered Wien's law is no longer valid.

The last mentioned possibility merits special attention. According to the developed understanding it cannot beexcluded that a "non-Wienian radiation", even in high dilution, would behave energetically differently from a "blackbody radiation" within the validity range of Wien's law.

On the Generation of Cathode Rays by Illumination of Solid BodiesThe usual understanding, that the energy of light is distributed over the space through which it travels in acontinuous way encounters extraordinarily large difficulties in attempts to explain photo-electric phenomena, as hasbeen presented in the groundbreaking article by Mr. Lenard. [7].According to the understanding that the exciting light consists of energy quanta of energy (R/N)βν the generation ofcathode rays by light can be conceived as follows. Quanta of energy penetrate the surface layer of the solid, and theirenergy is transformed, at least partially, in kinetic energy of electrons. The simplest picture is one where the lightquantum gives its entire energy to a single electron; we assume that this will occur. However, it must not beexcluded that electrons accept the energy of light quanta only partially. An electron that has been loaded with kineticenergy will have lost some of its energy when it arrives at the surface. Other than that we must assume that onleaving the solid every electron must do an amount of work P (characteristic of that solid). Electrons residing right atthe surface, excited at right angles to it, will leave the solid with the largest normal velocity. The kinetic energy ofsuch electrons is


If the body is charged to a positive potential Π and surrounded by conductors with potential zero and Π is justenough to prevent loss of electricity by the body, then we must have:

where ε is the electrical mass of the electron, or

where E is the charge of one gram equivalent of a single-valued ion and P' is the potentel of this amount of negativeelectricity with respect to this body. [8]

If we set E = 9.6·103, then Π·10-8 is the potential in volts that the body will attain when it is irradiated in vacuum.To see now whether the derived relation agrees with experiment to within an order of magnitude we set P' = 0,ν = 1.03·1015 (corresponding to the ultraviolet limit of the solar spectrum), and β = 4.866·10-11. We obtainΠ·107 = 4.3 Volt, which agrees to within an order of magnitude with the results of Mr. Lenard. [9]

If the formula derived is correct, then Π, as a function of frequency of the excited light represented in Cartesiancoordinates, must be a straight line, whose inclination is independent from the nature of the substance investigated.As far as I can see no contradiction exists between our understanding and the properties of photo-electric actionobserved by Mr. Lenard. If each energy quantum of the exciting light releases its energy independently from allothers to the electrons, the distribution of velocities of the electrons, which means the quality of the generatedcathode radiation, will be independent of the intensity of the exciting light; the number of electrons that exits thebody, on the other hand, will, in otherwise equal circumstances, be proportional to the intensity of the exciting light.[10]

We expect that limits of validity of these rules will be similar in nature to the expected deviations from Stokes' rule.In the preceding it has been assumed that the energy of at least some of the energy quanta of the generating light istransferred completely to a single electron. If one does not start with that natural supposition then instead of theabove equation one obtains:

For cathode-luminescence, which constitutes the inverse process of the one just examined, on obtains by way ofanalogous consideration:

For the materials investigated by Mr. Lenard PE is always significantly larger than Rβν, as the voltage that thecathode rays have had to traverse to generate even visible light is in some cases several hundred, in other casesthousands of volts. [11]

Ionization of Gases by Ultraviolet LightWe have to assume that in ionization of a gas by ultraviolet light always one absorbed light energy quantum is usedfor the ionization of just one gas molecule. Firstly it follows that the ionization energy (that is, the theoreticallynecessary energy to ionize) of a molecule cannot be larger than the energy of an absorbed light energy quantum.Taking J as the (theoretical) ionization energy per gram equivalent, we have:

According to Lenard's measurements for air the largest wavelength that has an effect is about 1.9·10-5 cm, so


An upper limit for the ionization energy can also be obtained from the ionization voltage in rarefied gases.According to Stark [12] the smallest measured ionization voltage (for platinum anodes) is for air about 10 volt. [13]

We have thus for J an upper limit 9.6·1012, which is nearly the same as the one just found. There is anotherconsequence that in my mind is very important to verify. If every light energy quantum ionizes one molecule thenthe following relation must exist between the absorbed quantity of light L and the number j of thereby ionized grammolecules:

If our understanding reflects reality this relation must hold for every gas that (at the particular frequency) has noabsorption that isn't accompanied by ionization.Bern, march 17, 1905

[1][1] This assumption is equivalent to the condition that the mean kinetic energies of gas molecules and electrons are equal to each other whenthere is thermal equilibrium. As is known, using this condition Mr. Drude has theoretically derived the relation between thermal and electricconductivity of metals.

[2] M. Planck, Ann. d. Phys. 1 p.99. 1900.[3] This condition can be formulated as follows. We expand the Z-component of the electric force (Z) in a given point in the space between the

time coordinates of t=0 and t=T (where T is a large amount of time compared to all the vibration periods considered) in a Fourier series

UNIQ-math-0-64bee9a14e9badfe-QINU

where and .

Performing this expansion arbitrarily often with arbitrarily chosen initial times yields a range of differentcombinations for the quantities Aν and αν. Then for the frequencies of the different combinations of the quantities Aνand αν there are the (statistical) probabilities dW of the form:

The radiation is then as unordered as imaginable, if

That is if the probability of a particular value of A and α respectively is independent of the value of other values of Aand x respectively. The more closely the demand is satisfied that the separate pairs of values Aν and αν depend on theemission and absorption process of separate resonators, the more closely will the examined case be one of being asunordered as imaginable.[4][4] M. Planck, Ann. d. Phys. 4. p.561. 1901.[5][5] This is an arbitrary assumption. The natural course of action is to stay with this simplest assumption until experiment forces us to abandon it.[6] If E is the energy of the system, then one obtains:

;therefore

[7][7] P. Lenard, Ann. d. Phys. 8. p.169 u. 170. 1902.[8] If one assumes that in order to release an electron from a neutral molecule light must do a certain amount of work then one doesn't have to

change the derived relation; one only has to think of P' as the sum of two terms.[9][9] P. Lenard, Ann. d. Phys. 8. p165. u. 184 Taf. I, Fig.2 1902.[10][10] P. Lenard, l. c. p.150 und p. 166-168.[11] P. Lenard, Ann. d. Phys. 12. p.469. 1903.[12][12] J. Stark, Die Elektricität in Gasen p. 57. Leipzig 1902.[13][13] within the gas the ionization voltage for negative ions is nonetheless five times larger


This is a translation and has a separate copyright status from the original text. The license for the translation applies to this edition only.

Original:This work is in the public domain in the United States because it was published before January 1,1923. It may be copyrighted outside the U.S. (see Help:Public domain).

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On the Electrodynamics of Moving Bodies

On the Electrodynamics of Moving Bodies (1920) by Albert Einstein, translated by Meghnad Saha and Wikisource

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German original: Einstein, Albert (1905), "Zur Elektrodynamik bewegter Körper", Annalen der Physik 322 (10):891–921. (Received June 30, 1905; published September 26, 1905). See also the 1923 edition.• Saha's translation: The Principle of Relativity: Original Papers by A. Einstein and H. Minkowski, University of

Calcutta, 1920, pp. 1-34, Online [1].• In this Wikisource edition, Saha's notation was replaced by Einstein's original notation. Also many passages were

re-written and translated from the German original. (See Saha's original for comparison).

[1] http:/ / www. archive. org/ details/ principleofrelat00eins


Original:This work is in the public domain in the United States because it was published before January 1, 1923.

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Translation:This work is in the public domain in the United States because it was published before January 1, 1923.


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The Development of Our Views on the Composition and Essence of Radiation 11

The Development of Our Views on theComposition and Essence of Radiation

The Development of Our Views on the Composition and Essence of Radiation (1909) by Albert Einstein, translated by Wikisource

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A paper by Einstein reviewing the historical development of the wave and particle models of light and suggesting(for the first time) that light is simultaneously a wave and a particle. Einstein states clearly that the energy andmomentum of light are concentrated in particles, and that the processes of emission and absorption should bemodeled as inverse elementary processes. Einstein also shows that the particle and wave components of lightproduce independent pressure fluctuations in blackbody radiation. Contrary to modern physics, he speculates that thephoton might carry with itself an oscillating field, a concept that may have led de Broglie to the "pilot wave" idea.This is a translation of the 1909 paper, "über die Entwicklung unserer Anschauungen über das Wesen und dieKonstitution der Strahlung"When light was shown to exhibit interference and diffraction, it seemed almost certain that light should beconsidered a wave. Since light can also propagate through empty space, one had to imagine a strange substance, anether, that mediated the propagation of light waves. Since light also propagates in material objects, one had toassume that this ether was also present in material objects, and was chiefly responsible for the propagation of light inmaterial objects. The existence of the ether seemed beyond doubt. In the first volume of Chwolson's excellentphysics textbook, he states in the introduction to ether, "The hypothesis of this one agent's existence isextraordinarily close to certainty."Today, however, we regard the ether hypothesis as obsolete. A large body of facts shows undeniably that light hascertain fundamental properties that are better explained by Newton's emission theory of light than by the oscillationtheory. For this reason, I believe that the next phase in the development of theoretical physics will bring us a theoryof light that can be considered a fusion of the oscillation and emission theories. The purpose of the followingremarks is to justify this belief and to show that a profound change in our views on the composition and essence oflight is imperative.The greatest advance in theoretical optics since the introduction of the oscillation theory was Maxwell's brilliantdiscovery that light can be understood as an electromagnetic process. This theory replaces mechanical quantities (thedeformation and velocity of the ether's parts) with the electromagnetic state of the ether and the material underconsideration. It reduces optical problems to electromagnetic problems. As the electromagnetic theory developed, itbecame ever less of a concern whether the electromagnetic processes could be explained by mechanical processes.One became used to treating electric and magnetic fields as fundamental concepts that did not require a mechanicalinterpretation.The introduction of the electromagnetic theory simplified the elements of theoretical optics and reduced the numberof arbitrary hypotheses. The old question about the oscillation direction of polarized light became irrelevant. Thedifficulties concerning the boundary conditions between two media were resolved using the theory's fundamentalprinciples. An arbitrary hypothesis was no longer needed to eliminate longitudinal light waves. A consequence of thetheory was the pressure of light, which plays such an important role in radiation theory and which has just recentlybeen confirmed experimentally. I don't want to make an exhaustive list of well-known accomplishments, but ratherconcentrate on one main point, on which the electromagentic and the mechanical theories of light agree — or rather,seem to agree.



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In both theories, light is esentially an embodiment of the state of a hypothetical medium, the ether, which existseverywhere, even in the absence of light. It was therefore assumed that motions of this medium would influenceoptical and electromagnetic phenomena. The search for laws describing this influence caused a change in the basicideas about the nature of radiation. Let us briefly examine the progression of this change.The main outstanding question was the following: does the ether participate in the motions of matter, or does theether inside moving matter move differently or, perhaps, does the ether ignore the motions of matter and remainforever at rest? To decide this question, Fizeau performed an important interference experiment, based on thefollowing line of reasoning. Assume that light propagates with speed in a certain object, if the object is at rest. Ifthe object, when moved, takes its ether along with it, the light will propagate in the same way as when the object wasat rest. Therefore, the speed of propagation in the object will again be . However, taken absolutely, i.e., relative toan observer not moving with the object, the speed of propagation will be the geometric sum of and the velocity of the object. If the motion and propagation are along the same axis and have the same sense, the is simply thesum of the two speeds

This is a consequence of assuming that the ether participates in the motion of its object.To test whether this prediction was true, Fizeau let two coherent, monochromatic beams pass axially each throughtheir own water-filled pipe and then interfere with one another. Then he set the water in the pipes moving, one in thedirection of the light's propagation and the other opposite to it. In this way, a shift of the interference pattern wasproduced, from which Fizeau could derive the influence of the object's velocity on the absolute velocity.As is well-known, the change is smaller than predicted by the hypothesis of complete participation, although thesense of the change is as expected. Expressed mathematically,

where is always smaller than one. Ignoring dispersion,

This experiment demonstrated that matter does not completely carry along its ether but, in general, the ether ismoving relative to matter. Now the Earth is a material object, which moves in different directions over the course ofa year relative to the solar system. The ether in our laboratories was assumed to not participate in the Earth's motioncompletely, just as the ether did not participate in the water's motion completely in Fizeau's experiment. Thus, theconclusion was that the ether was moving relative to our instruments, and that this relative motion changed over thecourse of a day and of a year. This relative motion was expected to produce a visible anisotropy of space, i.e., opticalphenomena were expected to depend on the orientation of the apparatus. The most diverse experiments wereperformed without detecting the expected dependence of phenomena on orientation.This contradiction was chiefly eliminated by the pioneering work of H. A. Lorentz in 1895. Lorentz showed that ifthe ether were taken to be at rest and did not participate at all in the motions of matter, no other hypotheses werenecessary to arrive at a theory that did justice to almost all of the phenomena. In particular, Fizeau's experimentswere explained, as well as the negative results of the above-mentioned attempts to detect the Earth's motion relativeto the ether. Only one experiment seemed incompatible with Lorentz's theory, namely, the interference experiment ofMichelson and Morley.According to Lorentz's theory, a uniform translational motion of the apparatus of optical experiments does not affectlight's progress, if we ignore second- and higher-order terms of the quotient (speed of apparatus)/(speed of light).The Michelson and Morley interference experiment showed that, in a special case, second-order terms also cannot bedetected, although they were expected from the standpoint of the ether-at-rest theory. To include this experiment inthe theory, Lorentz and FitzGerald introduced the postulate that all objects, including the parts of Michelson andMorley's experimental set-up, changed their form in a certain way, if they moved relative to the ether.


This state of affairs was very unsatisfying. The only useful and fundamentally basic theory was that of Lorentz,which depended on a completely immobile ether. The Earth had to be seen as moving relative to this ether. But everyexperiment designed to demonstrate this ether had a negative result, so that one was driven to a very strangehypothesis to understand why such a relative motion was not detectable.Michelson's experiment suggests the axiom that all phenomena obey the same laws relative to the Earth's referenceframe or, more generally, relative to any reference frame in unaccelerated motion. For brevity, let us call thispostulate the relativity principle. Before we tackle the problem of whether it is possible to maintain the relativityprinciple, let us briefly consider what happens to the ether hypothesis, if we maintain the relativity principle.The foundation of the ether hypothesis is the experimentally based assumption that the ether is at rest. The relativityprinciple states that all natural laws that hold in a reference frame moving uniformly relative to the ether areidentical with those that hold in , a reference frame at rest relative to the ether. If that is so, we can just as wellimagine the ether is at rest relative to , not . It is completely unnatural to distinguish the two referenceframes and by introducing an ether that is at rest in one. A satisfying theory can only be reached if wedispense with the ether hypothesis. Then the electromagnetic fields that make up light no longer appear as a state of ahypothetical medium, but rather as independent entities that the light source gives off, just as in Newton's emissiontheory of light. As in that theory, space that is free of matter and radiation is truly empty.Superficial consideration suggests that the essential parts of Lorentz's theory cannot be reconciled with the relativityprinciple. According to Lorentz's theory, if a light beam propagates through space, it does so with a speed in theresting frame of the ether, independently of the state of motion of the emitting object. Let's call this theconstancy of the speed of light principle. The theorem of the addition of speeds states that the same light beam willnot propagate at speed in a different frame moving uniformly relative to the ether. The laws of propagationthus seem to be different in the two frames and, hence, the relativity principle seems to be incompatible with thelaws governing light's propagation.However, the theorem of the addition of speeds rests on arbitrary axioms. It presupposes that information about timeand the form of moving objects has meaning independent of the motion of the moving reference frame. But one canconvince oneself that the definitions of time and the form of moving objects require the introduction of clocks at restin the reference frame under consideration. These concepts must be defined for each reference frame, and it is notself-evident that these definitions will produce the same time values in two frames and moving relative toone another. Similarly, it cannot be said a priori that statements about the form of objects in will also be valid in

.Hence, the hitherto prevailing transformation equations in passing from one frame to another moving relative to itrest on arbitrary assumptions. If these are abandoned, the essence of Lorentz's theory or, more generally, the"constancy of the speed of light" principle can be reconciled with the relativity principle. These two principles leadto certain unambiguous transformation equations characterized by the identity

for an appropriate choice of initial origins. In this equation, is the speed of light in vacuo, are thespace-time coordinates in , and are those in .This path leads to the so-called relativity theory. I only wish to bring in one of its consequences, for it brings with itcertain modifications of the fundamental ideas of physics. It turns out that the inertial mass of an object decreases by

when that object emits radiation of energy . This can be derived as follows.We consider a free object at rest that emits the same amount of radiative energy in two opposing directions. In doingso, it remains at rest. Let the object's energy prior to emission be denoted , and its energy after emission andlet be the energy of the emitted radiation. By the energy principle, we have


Now consider the object and the same emitted radiation from a reference frame moving with velocity relative to theobject. Relativity theory gives us the means of calculating the energy emitted in the new reference frame. Oneobtains the value

Since the conservation of energy principle must also hold in the new reference frame, one obtains analogously

Subtracting and ignoring fourth- and higher-order terms in , we get

But is the object's kinetic energy before the light emission and is its kinetic energy after the lightemission. If we call its mass before emission and its mass after emission then, by ignoring terms higher thansecond order, we can write

or

Thus, the inertial mass of an object is diminished by the emission of light. The energy given up was part of the massof the object. One can further conclude that every absorption or release of energy brings with it an increase ordecrease in the mass of the object under consideration. Energy and mass seem to be just as equivalent as heat andmechanical energy.Relativity theory has changed our views on light. Light is conceived not as a manifestation of the state of somehypothetical medium, but rather as an independent entity like matter. Moreover, this theory shares with thecorpuscular theory of light the unusual property that light carries inertial mass from the emitting to the absorbingobject. Relativity theory does not alter our conception of radiation's structure; in particular, it does not affect thedistribution of energy in radiation-filled space. Nevertheless, with respect to this question, I believe that we stand atthe beginning of a development of the greatest importance that cannot yet be surveyed. The statements that followare largely my personal opinion, or the results of considerations that have not yet been checked enough by others. If Ipresent them here in spite of their uncertainty, the reason is not an excessive faith in my own views, but rather thehope to induce one or another of you to deal with the questions considered.Even without delving deeply into theory, one notices that our theory of light cannot explain certain fundamentalproperties of phenomena associated with light. Why does the color of light, and not its intensity, determine whether acertain photochemical reaction occurs? Why is light of short wavelength generally more effective chemically thanlight of longer wavelength? Why is the speed of photoelectrically produced cathode rays independent of the light'sintensity? Why are higher temperatures (and, thus, higher molecular energies) required to add a short-wavelengthcomponent to the radiation emitted by an object?The oscillation theory, in its present formulation, gives no answers to these questions. In particular, it is completelyincomprehensible why cathode rays produced photoelectrically or by X-rays acquire such a considerable velocityindependent of the light's intensity. The appearance of such great amounts of energy in molecular entities under theinfluence of a light source in which the energy is distributed so thinly (as we must assume for light radiation andX-rays, given the oscillation theory) drove competent physicists to take refuge in a rather far-out hypothesis. Theyassumed that light played merely a releasing role in the process, and that the molecular energies produced were of aradioactive nature. Since this hypothesis has already been abandoned, I won't bring any arguments against it.


The fundamental property of the oscillation theory that engenders these difficulties seems to me the following. In thekinetic theory of molecules, for every process in which only a few elementary particles participate (e.g., molecularcollisions), the inverse process also exists. But that is not the case for the elementary processes of radiation.According to our prevailing theory, an oscillating ion generates a spherical wave that propagates outwards. Theinverse process does not exist as an elementary process. A converging spherical wave is mathematically possible, tobe sure; but to approach its realization requires a vast number of emitting entities. The elementary process ofemission is not invertible. In this, I believe, our oscillation theory does not hit the mark. Newton's emission theory oflight seems to contain more truth with respect to this point than the oscillation theory since, first of all, the energygiven to a light particle is not scattered over infinite space, but remains available for an elementary process ofabsorption.Consider the laws governing the production of secondary cathode radiation by X-rays. If primary cathode raysimpinge on a metal plate P1, they produce X-rays. If these X-rays impinge on a second metal plate P2, cathode raysare again produced whose speed is of the same order as that of the primary cathode rays. As far as we know today,the speed of the secondary cathode rays depends neither on the distance between P1 and P2, nor on the intensity ofthe primary cathode rays, but rather entirely on the speed of the primary cathode rays. Let's assume that this isstrictly true. What would happen if we reduced the intensity of the primary cathode rays or the size of P1 on whichthey fall, so that the impact of an electron of the primary cathode rays can be considered an isolated process? If theabove is really true then, because of the independence of the secondary cathode rays' speed on the primary cathoderays' intensity, we must assume that an electron impinging on P1 will either cause no electrons to be produced at P2,or else a secondary emission of an electron whose speed is of the same order as that of the initial electron impingingon P1. In other words, the elementary process of radiation seems to occur in such a way that it does not scatter theenergy of the primary electron in a spherical wave propagating in every direction, as the oscillation theory demands.Rather, at least a large part of this energy seems to be available at some place on P2, or somewhere else. Theelementary process of the emission of radiation appears to be directional. Moreover, one has the impression that theproduction of X-rays at P1 and the production of secondary cathode rays at P2 are essentially inverse processes.Therefore, the composition of radiation seems to be different from what our oscillation theory predicts. The theory ofthermal radiation has given important clues about this, mostly by the theory on which Planck based his radiationformula. Since I cannot assume that everyone is familiar with this theory, I will cover its essential points briefly.A radiation of definite composition occupies the interior of a cavity of temperature , and is independent of thecavity's material composition. The cavity contains an energy density for frequencies between and . Finding as a function of and poses a problem. If an electric resonator of eigenfrequency and negligibledamping occupies the cavity, the time average of the resonator's energy as a function of can be calculatedfrom the electromagnetic theory of radiation. The problem is thereby reduced to that of determining as a functionof . However, the latter problem can also be reduced to the following. Let the cavity contain very manyresonators of frequency ; how does the entropy of the system depend on its energy?To resolve this question, Planck utilized the general relationship between entropy and the probability of a state, asderived by Boltzmann from his investigations in the theory of gases. In general

where is a universal constant, and is the probability of the state under consideration. This probability ismeasured by the number of "configurations", a number that counts the number of ways the state under considerationcan be realized. In the case above, the state of the resonator system is defined by its total energy, so the question tobe answered reads: how many ways can a given total energy be distributed among resonators? To find this out,Planck divided the total energy into parts of equal energy . A configuration is determined by the number of parts

allotted to each resonator. The number of such configurations that result in the given total energy is calculated andset equal to .


From Wien's shift law, which can be derived from thermodynamic principles, Planck concluded that must be setequal to , where is independent of . In this way, he found his radiation formula, which agrees with all ofour experience hitherto

It might seem that, in accordance with this derivation, Planck's radiation formula follows from the presentelectromagnetic theory. This is not the case, for the following reason. The number of configurations, of which wewere just speaking, can be thought of as expressing the multiplicity of the distribution possibilities of the total energyamong resonators, if every imaginable distribution of energy approached within some approximation of thecalculated number of configurations . This requires that the energy be small compared to average resonatorenergy for all . But simple calculation shows that, for a wavelength of 0.5 μm and an absolute temperature = 1700 K, is not only not small compared to one, but is very big compared to one; the value is approximately

. This numerical example shows that the counting of the states must have gone awry, if the resonator'senergy can only assume the value zero or times its average energy (or a multiple thereof). Clearly, insuch a process, only a vanishingly small part of those energy distributions, which must be possible according thefundamental principles of the theory, are drawn upon to determine the entropy. Therefore, according to thefundamental principles of the theory, the number of configurations is not an expression for the probability inBoltzmann's sense. In my opinion, accepting Planck's theory means denying the precepts of our radiation theory.I have already attempted to show that the present principles of the theory of radiation must be abandoned. In anyevent, it is unthinkable to reject Planck's theory because it does not fit those fundamental principles. This theory hasled to a determination of the elementary quanta, which has been splendidly confirmed by the most recentmeasurements on alpha-particles. For the elementary quantum of electricity, Rutherford and Geiger obtained themean value , Regener , while Planck, using his radiation theory, determined theintermediate value from the constants of the radiation formula.Planck's theory leads to the following conjecture. If it is really true that a radiative resonator can only assume energyvalues that are multiples of , the obvious assumption is that the emission and absorption of light occurs only inthese energy quantities. On the basis of this hypothesis, the light-quanta hypothesis, the questions raised aboveabout the emission and absorption of light can be answered. As far as we know, the quantitative consequences of thislight-quanta hypothesis are confirmed. This provokes the following question. Is it not thinkable that Planck'sradiation formula is correct, but that another derivation could be found that does not rest on such a seeminglymonstrous assumption as Planck's theory? Is it not possible to replace the light-quanta hypothesis with anotherassumption, with which one could do justice to known phenomena? If it is necessary to modify the theory's elements,couldn't one keep the propagation laws intact, and only change the conceptions of the elementary processes ofemission and absorption?To arrive at a certain answer to this question, let us proceed in the opposite direction of Planck in his radiationtheory. Let us view Planck's radiation formula as correct, and ask ourselves whether something concerning thecomposition of radiation can be derived from it. Of two considerations that I have carried out, I wish to sketch onefor you, which seems especially convincing to me because it can be imagined so clearly.Let there be an ideal gas inside a cavity, as well as a solid plate that is free to move perpendicularly to its plane. Because of the irregularity of the collisions between the gas molecules and the plate, the latter is moved such that its average kinetic energy is one-third of the kinetic energy of a monatomic gas molecule. (This follows from statistical mechanics.) Besides this gas (which we can imagine as consisting of only a few molecules), we assume that there is also thermal radiation at the gas temperature. This will be the case if the walls of the cavity are also at the same temperature , do not let radiation pass through and are not completely reflective. Furthermore, we assume that our plate is completely reflective on both sides. In this situation, both the gas and the radiation will affect the plate. If the plate is at rest, the pressures are equal. If the plate is moved, however, the pressure on the forward side pushing back


is greater than its counterpart pushing in the opposite direction. Hence, there will be a net force that opposes themotion of the plate, and increases with the speed of the plate. Let us call this force the "radiation friction".If we assume for the moment that we have taken into account all the radiation's mechanical influence, we cansummarize as follows. Collisions with gas molecules at irregular intervals give the plate irregular momentum. Thespeed of the plate decreases continuously between two such collisions, due to the radiation friction, which transformsthe kinetic energy of the plate into radiative energy. As a result, the energy of the gas molecules would becontinuously transformed into the energy of radiation, until all the available energy had turned into energy ofradiation. There would be no equilibrium between gas and radiation.This argument is fallacious because, similar to gaseous pressure, the radiation pressure on the plate cannot beconsidered constant in time and free of irregular variations. To allow thermal equilibrium, the variations in theradiation presure must be such that, on average, it compensates for the plate's loss of speed due to radiation friction.Remember that the average kinetic energy of the plate is one-third that of a monatomic gas molecule. Given theradiation law, the radiation friction can be calculated and, thence, the average amount of momentum the plate mustreceive from variations in the light-pressure to maintain a statistical equilibrium.The argument becomes even more interesting if we choose a plate that completely reflects only frequencies between

and , and is transparent to all other radiation. This gives the variations in the radiation pressure for thatfrequency band. I merely state the result here. Let be the magnitude of the motion communicated to the plate intime by irregular variations in the radiation pressure. The average value of is given by the expression

First of all, the simplicity of this expression is noteworthy. Planck's theory seems to be the only one that agrees withexperiment to within observational error and yields such a simple expression for the statistical properties of theradiation pressure.In trying to understand this expression, one notices at once that it is the sum of two terms. It is as if two independentcauses were working to produce variations in the radiation pressure. One can conclude from the fact that isproportional to that the pressure variations for two neighboring regions are completely independent of each other,if the regions have dimensions large compared to the wavelength of the reflection frequency . The second term ofthe expression for can be explained by the oscillation theory. According to that theory, light rays of slightlydifferent direction, frequency and polarization interfere with one another; variations in the radiation presurecorrespond to uncorrelated occurrences of interference in the whole. Simple dimensional analysis shows that thisvariation must be of the form of the second term of our formula. Clearly, the oscillatory structure of radiation doesindeed give rise to variations in the radiation pressure, as predicted.How can the first term be explained? It is by no means negligible; it is completely dominant in the regime whereWien's radiation formula holds. For a wavelength of 0.5 μm and a temperature = 1700 K, this term isapproximately times larger than the second. It turns out that the first term of our formula results fromassuming that radiation consists of localized groupings of energy that are reflected and move through spaceindependently of one another — an idea presented by the most primitive picture of the light-quanta hypothesis.Therefore, I believe one must conclude the following from the above formula derived from Planck's radiationformula: In addition to the spatial irregularities in the distribution of radiation's energy that arise from theoscillation theory, there are also other irregularities in the same spatial distribution that completely dominate thefirst-mentioned irregularities when the energy density of the radiation is small. I add that another argumentinvolving the spatial distribution of the energy gives exactly the same results as those given above.As far as I know, no mathematical theory has been advanced that does justice to both its oscillatory structure and its quantum structure, which we derived from the first term of the above formula. The difficulty lies in the fact that the variational properties of radiation, as expressed in the above formula, offer few reference points for setting up such a


theory. One might imagine a situation in which diffraction and interference were still unknown, but one knew thatthe average magnitude of the irregular variations of the radiation pressure was determined by the second term of theabove equation, where is a parameter of unknown meaning that determines the color. On this basis, who would haveenough imagination to construct an oscillatory theory of light?Anyway, this conception seems to me the most natural: that the manifestation of light's electromagnetic waves isconstrained at singularity points, like the manifestation of electrostatic fields in the theory of the electron. It cannotbe ruled out that, in such a theory, the entire energy of the electromagnetic field could be viewed as localized in thesesingularities, just like the old theory of action-at-a-distance. I imagine to myself, each such singular point surroundedby a field that has essentially the same character as a plane wave, and whose amplitude decreases with the distancebetween the singular points. If many such singularities are separated by a distance small with respect to thedimensions of the field of one singular point, their fields will be superimposed, and will form in their totality anoscillating field that is only slightly different from the oscillating field in our present electromagnetic theory of light.Of course, it need not be emphasized that such a picture is worthless unless it leads to an exact theory. I only wishedto illustrate that the two structural properties of radiation according to Planck's formula (oscillation structure andquantum structure) should not be considered incompatible with one another.












The Field Equations of Gravitation 19

The Field Equations of Gravitation

The Field Equations of Gravitation (1915) by Albert Einstein, translated by Wikisource


In German: Die Feldgleichungen der Gravitation, Preussische Akademie der Wissenschaften, Sitzungsberichte,1915 (part 2), 844–847, Internet Archive [1], Hathi Trust [2]

Session from November 25, 1915; published December 2, 1915.

The Field Equations of GravitationBy A. Einstein

I have shown in two recently published reports,[3] how one can arrive at field equations of gravitation, that are inagreement with the postulate of general relativity, i.e. which in their general form are covariant in respect to arbitrarysubstitutions of space-time variables.The line of development was as follows. At first I found equations, that contain Newton's theory as approximationand that are covariant in respect to arbitrary substitutions of the determinant 1. Afterwards I found, that thoseequations in general correspond to covariant ones, if the scalar of the energy tensor of "matter" vanishes. Thecoordinate system had to be specialized in accordance with the simple rule, that is made to 1, whereby theequations of the theory experience an eminent simplification. In the course of this, however, one had to introduce thehypothesis, that the scalar of the energy tensor of matter vanishes.Recently I find now, that one is able to dispense with hypothesis concerning the energy tensor of matter, if one fillsin the energy tensor of matter into the field equations in a somehow different way than it was done in my two earlierreports. The field equations for vacuum, upon which I based the explanation of the perihelion motion of mercury,remain untouched by this modification. I give the complete consideration again at this place, so that the reader is notforced to uninterruptedly consultate the earlier reports.From the well known Riemannian covariant of fourth rank, the following covariant of second rank is derived:

(1)

(1a)

(1b)

We obtain the ten general covariant equations of the gravitational field in spaces, in which "matter" is absent, byputting

(2)

These equations can be formed in a simpler way, when one choses the reference system so that . Thenvanishes due to (1b), so that one obtains instead of (2)



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(3)

(3a)

Here we put

(4)

which magnitudes we will denote as the "components" of the gravitational field.If "matter" exists in the considered space, then its energy tensor appears on the right hand side of (2) or (3). We put

(2a)

where we put

(5)

is the scalar of the energy tensor of "matter", the right hand side of (2a) is a tensor. If we specialize the coordinatesystem in the ordinary way again, then we obtain instead of (2a) the equivalent equations

(6)

(3a)

Like always we assume, that the divergence of the energy tensor of matter vanishes in the sense of the generaldifferential calculus (Momentum-Energy theorem). When specializing the coordinate choice in accordance with (3a),it follows from it, that the shall fulfill the conditions

(7)

or

(7a)

If one multiplies (6) by and sums over and , then one obtains[4] in respect to (7) and in respect to the

relation following from (3a)

the conservation law for matter and the gravitational field together in the form

(8)

where (the "energy tensor" of the gravitational field) is given by


(8a)

The reasons that drove me to the introduction of the second member on the right-hand side of (2a) and (6), becomeclear from the following considerations, that are completely analogous to those given at the place just mentioned (p.785).

If we multiply (6) by and sum over the indices and , then we obtain after simple calculation

(9)

where corresponding to (5) it is put for abbreviation

(8b)

Note, that it follows from the additional term, that in (9) the energy tensor of the gravitational field occurs besidesthat of matter in the same way, which is not the case in equations (21) l.c..Furthermore one derives instead of equation (22) l.c., in the way as it is given there by the aid of the energy equation,the relations:

(10)

From our additional term it follows, that these equations contain no new condition in respect to (9), so thatconcerning the energy tensor of matter, no other presupposition has to be made than the one, that it has to be inagreement with the momentum-energy theorem.By that, the general theory of relativity as a logical building is eventually finished. The relativity postulate in itsgeneral form that makes the space-time coordinates to physically meaningless parameters, is directed with stringentnecessity to a very specific theory of gravitation that explains the perihelion motion of mercury. However, thegeneral relativity postulate offers nothing new about the essence of the other natural processes, which wasn't alreadytaught by the special theory of relativity. My opinion regarding this issue, recently expressed at this place, waserroneous. Any physical theory equivalent to the special theory of relativity, can be included in the general theory ofrelativity by means of the absolute differential calculus, without that the latter gives any criterion for theadmissibility of that theory.

[1] http:/ / www. archive. org/ details/ sitzungsberichte1915deut[2] http:/ / hdl. handle. net/ 2027/ mdp. 39015009166318[3][3] Sitzungsber. XLIV, p. 778 and XLVI, p. 799, 1915[4][4] Concerning the derivation see Sitzungsber. XLIV, 1915, p. 784/785. For the following, I request the reader to use the derivations given on p.

785 for comparison.

This work is in the public domain in the United States because it was published before January 1, 1923.

The author died in 1955, so this work is also in the public domain in countries and areas where the copyright term is the author's lifeplus 50 years or less. This work may also be in the public domain in countries and areas with longer native copyright terms that applythe rule of the shorter term to foreign works.

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The Foundation of the Generalised Theory of Relativity 22

The Foundation of the Generalised Theory ofRelativity

The Foundation of the Generalised Theory of Relativity (1916) by Albert Einstein, translated by Satyendra Nath Bose


German original: Die Grundlage der allgemeinen Relativitätstheorie, Annalen der Physik 354 (7), 769-822, Online[1]

Translation: The Principle of Relativity: Original Papers by A. Einstein and H. Minkowski, University of Calcutta,1920, pp. 89-163, Online [1].•• In this Wikisource edition of Bose's translation, his notation was replaced by Einstein's original notation. Also

some slight inaccuracies were corrected, and the omitted references were included and translated from theGerman original.

The Foundation of the Generalised Theory of RelativityBy A. Einstein.

The theory which is sketched in the following pages forms the most wide-going generalization conceivable of whatis at present known as "the theory of Relativity;" this latter theory I differentiate from the former "Special Relativitytheory," and suppose it to be known. The generalization of the Relativity theory has been made much easier throughthe form given to the special Relativity theory by Minkowski, which mathematician was the first to recognize clearlythe formal equivalence of the space like and time-like co-ordinates, and who made use of it in the building up of thetheory. The mathematical apparatus useful for the general relativity theory, lay already complete in the "AbsoluteDifferential Calculus", which were based on the researches of Gauss, Riemann and Christoffel on the non-Euclideanmanifold, and which have been shaped into a system by Ricci and Levi-Civita, and already applied to the problemsof theoretical physics. I have in part B of this communication developed in the simplest and clearest manner, all thesupposed mathematical auxiliaries, not known to Physicists, which will be useful for our purpose, so that, a study ofthe mathematical literature is not necessary for an understanding of this paper. Finally in this place I thank my friendGrossmann, by whose help I was not only spared the study of the mathematical literature pertinent to this subject, butwho also aided me in the researches on the field equations of gravitation.

A. Principal considerations about the Postulate of Relativity.

§ 1. Remarks on the Special Relativity Theory.The special relativity theory rests on the following postulate which also holds valid for the Galileo-Newtonianmechanics.If a co-ordinate system K be so chosen that when referred to it the physical laws hold in their simplest forms, theselaws would be also valid when referred to another system of co-ordinates which is subjected to an uniformtranslational motion relative to K. We call this postulate "The Special Relativity Principle." By the word special, it issignified that the principle is limited to the case, when has uniform translatory motion with reference to K, butthe equivalence of K and does not extend to the case of non-uniform motion of relative to K.The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postulate,but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the specialrelativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lorentz transformation,


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with all the relations between moving rigid bodies and clocks.The modification which the theory of space and time has undergone through the special relativity theory, is indeed aprofound one, but a weightier point remains untouched. According to the special relativity theory, the theorems ofgeometry are to be looked upon as the laws about any possible relative positions of solid bodies at rest, and moregenerally the theorems of kinematics, as theorems which describe the relation between measurable bodies andclocks. Consider two material points of a solid body at rest; then according to these conceptions there corresponds tothese points a wholly definite extent of length, independent of kind, position, orientation and time of the body.Similarly let us consider two positions of the pointers of a clock which is at rest with reference to a co-ordinatesystem; then to these positions, there always corresponds a time-interval of a definite length, independent of time andplace. It would be soon shown that the general relativity theory can not hold fast to this simple physical significanceof space and time.

§ 2. About the reasons which explain the extension of the relativity-postulate.To the classical mechanics (no less than) to the special relativity theory, is attached an epistemological defect, whichwas perhaps first clearly pointed out by E. Mach. We shall illustrate it by the following example; Let two fluidbodies of equal kind and magnitude swim freely in space at such a great distance from one another (and from allother masses) that only that sort of gravitational forces are to be taken into account which the parts of any of thesebodies exert upon each other. The distance of the bodies from one another is invariable. The relative motion of thedifferent parts of each body is not to occur. But each mass is seen to rotate by an observer at rest relative to the othermass round the connecting line of the masses with a constant angular velocity (definite relative motion for both themasses). Now let us think that the surfaces of both the bodies ( and ) are measured with the help ofmeasuring rods (relatively at rest); it is then found that the surface of is a sphere and the surface of the other is anellipsoid of rotation. We now ask, why is this difference between the two bodies? An answer to this question canonly then be regarded as satisfactory[2] from the epistemological standpoint when the thing adduced as the cause isan observable fact of experience. The law of causality has the sense of a definite statement about the world ofexperience only when observable facts alone appear as causes and effects.The Newtonian mechanics does not give to this question any satisfactory answer. For example, it says:— The lawsof mechanics hold true for a space relative to which the body is at rest, not however for a space relative towhich is at rest.The Galiliean space, which is here introduced is however only a purely imaginary cause, not an observable thing. Itis thus clear that the Newtonian mechanics does not, in the case treated here, actually fulfil the requirements ofcausality, but produces on the mind a fictitious complacency, in that it makes responsible a wholly imaginary cause

for the different behaviours of the bodies and which are actually observable.A satisfactory explanation to the question put forward above can only be thus given:— that the physical systemcomposed of and shows for itself alone no conceivable cause to which the different behaviour of and can be attributed. The cause must thus lie outside the system. We are therefore led to the conception that the generallaws of motion which determine specially the forms of and must be of such a kind, that the mechanicalbehaviour of and must be essentially conditioned by the distant masses, which we had not brought into thesystem considered. These distant masses, (and their relative motion as regards the bodies under consideration) arethen to be looked upon as the seat of the principal observable causes for the different behaviours of the bodies underconsideration. They take the place of the imaginary cause . Among all the conceivable spaces and etc.moving in any manner relative to one another, there is a priori, no one set which can be regarded as affording greateradvantages, against which the objection which was already raised from the standpoint of the theory of knowledgecannot be again revived. The laws of physics must be so constituted that they should remain valid for any system ofco-ordinates moving in any manner. We thus arrive at an extension of the relativity postulate.


Besides this momentous epistemological argument, there is also a well-known physical fact which speaks in favourof an extension of the relativity theory. Let there be a Galiliean co-ordinate system K relative to which (at least in thefour-dimensional region considered) a mass at a sufficient distance from other masses move uniformly in a line. Let

be a second co-ordinate system which has a uniformly accelerated motion relative to K. Relative to anymass at a sufficiently great distance experiences an accelerated motion such that its acceleration and its direction ofacceleration is independent of its material composition and its physical conditions.Can any observer, at rest relative to , then conclude that he is in an actually accelerated reference-system? Thisis to be answered in the negative; the above-named behaviour of the freely moving masses relative to can beexplained in as good a manner in the following way. The reference-system has no acceleration. In thespace-time region considered there is a gravitation-field which generates the accelerated motion relative to .This conception is feasible, because to us the experience of the existence of a field of force (namely the gravitationfield) has shown that it possesses the remarkable property of imparting the same acceleration to all bodies.[3] Themechanical behaviour of the bodies relative to is the same as experience would expect of them with reference tosystems which we assume from habit as stationary; thus it explains why from the physical stand-point it can beassumed that the systems K and can both with the same legitimacy be taken as at rest, that is, they will beequivalent as systems of reference for a description of physical phenomena.From these discussions we see, that the working out of the general relativity theory must, at the same time, lead to atheory of gravitation; for we can "create" a gravitational field by a simple variation of the co-ordinate system. Alsowe see immediately that the principle of the constancy of light-velocity must be modified, for we recognise easilythat the path of a ray of light with reference to must be, in general, curved, when light travels with a definite andconstant velocity in a straight line with reference to K.

§ 3. The time-space continuum. Requirements of the general Co-variance for the equationsexpressing the laws of Nature in general.In the classical mechanics as well as in the special relativity theory, the co-ordinates of time and space have animmediate physical significance; when we say that any arbitrary point has as its co-ordinate, it signifies thatthe projection of the point-event on the -axis ascertained by means of a solid rod according to the rules ofEuclidean Geometry is reached when a definite measuring rod, the unit rod, can be carried times from the originof co-ordinates along the axis. A point having as the co-ordinate signifies that a unit clock whichis adjusted to be at rest relative to the system of co-ordinates, and coinciding in its spatial position, with thepoint-event and set according to some definite standard has gone over periods before the occurrence of thepoint-event.[4]

This conception of time and space is continually present in the mind of the physicist, though often in an unconsciousway, as is clearly recognised from the role which this conception has played in physical measurements. Thisconception must also appear to the reader to be lying at the basis of the second consideration of the last paragraphand imparting a sense to these conceptions. But we wish to show that we are to abandon it and in general to replace itby more general conceptions in order to be able to work out thoroughly the postulate of general relativity,— the caseof special relativity appearing as a limiting case when there is no gravitation.We introduce in a space, which is free from Gravitation-field, a Galiliean Co-ordinate System K(x, y, z, t) and also, another system K'(y', y', z', t') rotating uniformly relative to K. The origin of both the systems as well as their Z-axes might continue to coincide. We will show that for a space-time measurement in the system , the above established rules for the physical significance of time and space can not be maintained. On grounds of symmetry it is clear that a circle round the origin in the X-Y plane of K, can also be looked upon as a circle in the - plane of

. Let us now think of measuring the circumference and the diameter of these circles, with a unit measuring rod (infinitely small compared with the radius) and take the quotient of both the results of measurement. If this experiment be carried out with a measuring rod at rest relatively to the Galiliean system K we would get π, as the


quotient. The result of measurement with a rod relatively at rest as regards would be a number which is greater thanπ. This can be seen easily when we regard the whole measurement-process from the system K and remember that therod placed on the periphery suffers a Lorentz-contraction, not however when the rod is placed along the radius.Euclidean Geometry therefore does not hold for the system ; the above fixed conceptions of co-ordinates whichassume the validity of Euclidean Geometry fail with regard to the system . We cannot similarly introduce in a timecorresponding to physical requirements, which will be shown by all similarly prepared clocks at rest relative to thesystem . In order to see this we suppose that two similarly made clocks are arranged one at the centre and one at theperiphery of the circle, and considered from the stationary system K. According to the well-known results of thespecial relativity theory it follows — (as viewed from K) — that the clock placed at the periphery will go slowerthan the second one which is at rest. The observer at the common origin of co-ordinates who is able to see the clockat the periphery by means of light will see the clock at the periphery going slower than the clock beside him. Sincehe cannot allow the velocity of light to depend explicitly upon the time in the way under consideration he willinterpret his observation by saying that the clock on the periphery "actually" goes slower than the clock at the origin.He cannot therefore do otherwise than define time in such a way that the rate of going of a clock depends on itsposition.We therefore arrive at this result. In the general relativity theory time and space magnitudes cannot be so defined thatthe difference in spatial co-ordinates can be immediately measured by the unit-measuring rod, and time-likeco-ordinate difference with the aid of a normal clock.The means hitherto at our disposal, for placing our co-ordinate system in the time-space continuum, in a definiteway, therefore completely fail and it appears that there is no other way which will enable us to fit the co-ordinatesystem to the four-dimensional world in such a way, that by it we can expect to get a specially simple formulation ofthe laws of Nature. So that nothing remains for us but to regard all conceivable[5] co-ordinate systems as equallysuitable for the description of natural phenomena. This amounts to the following law:—That in general, Laws of Nature are expressed by means of equations which are valid for all co-ordinate systems,that is, which are covariant for all possible transformations. It is clear that a physics which satisfies this postulatewill be unobjectionable from the standpoint of the general relativity postulate. Because among all substitutions thereare, in every case, contained those, which correspond to all relative motions of the co-ordinate system (in threedimensions). This condition of general covariance which takes away the last remnants of physical objectivity fromspace and time, is a natural requirement, as seen from the following considerations. All our well-substantiatedspace-time propositions amount to the determination of space-time coincidences. If, for example, the event consistedin the motion of material points, then, for this last case, nothing else are really observable except the encountersbetween two or more of these material points. The results of our measurements are nothing else than well-provedtheorems about such coincidences of material points, of our measuring rods with other material points, coincidencesbetween the hands of a clock, dial-marks and point-events occurring at the same position and at the same time.The introduction of a system of co-ordinates serves no other purpose than an easy description of totality of suchcoincidences. We fit to the world our space-time variables such that to any and every point-eventcorresponds a system of values of . Two coincident point-events correspond to the same value of thevariables ; i.e., the coincidence is characterised by the equality of the co-ordinates. If we now introduceany four functions as coordinates, so that there is an unique correspondence between them, theequality of all the four co-ordinates in the new system will still be the expression of the space-time coincidence oftwo material points. As the purpose of all physical laws is to allow us to remember such coincidences, there is apriori no reason present, to prefer a certain co-ordinate system to another; i.e., we get the condition of generalcovariance.


§ 4. Relation of four co-ordinates to spatial and temporal measurements. Analyticalexpression for the Gravitation-field.I am not trying in this communication to deduce the general Relativity-theory as the simplest logical system possible,with a minimum of axioms. But it is my chief aim to develop the theory in such a manner that the reader perceivesthe psychological naturalness of the way proposed, and the fundamental assumptions appear to be most reasonableaccording to the light of experience. In this sense, we shall now introduce the following supposition; that for aninfinitely small four-dimensional region, the relativity theory is valid in the special sense when the axes are suitablychosen.The nature of acceleration of an infinitely small (positional) co-ordinate system is hereby to be so chosen, that thegravitational field does not appear; this is possible for an infinitely small region. are the spatialco-ordinates; is the corresponding time-co-ordinate measured[6] by some suitable measuring clock. Thesecoordinates have, with a given orientation of the system, an immediate physical significance in the sense of thespecial relativity theory (when we take a rigid rod as our unit of measure), The expression

(1)

had then, according to the special relativity theory, a value which may be obtained by space-time measurement, andwhich is independent of the orientation of the local co-ordinate system. Let us take ds as the magnitude of theline-element belonging to two infinitely near points in the four-dimensional region. If ds² belonging to the element

be positive we call it with Minkowski, time-like, and in the contrary case space-like.To the line-element considered, i.e., to both the infinitely near point-events belong also definite differentials

, of the four-dimensional co-ordinates of any chosen system of reference. If there be also a localsystem of the above kind given for the case under consideration, would then be represented by definite linearhomogeneous expressions of

(2)

If we substitute the expression in (1) we get

(3)

where will be functions of , but will no longer depend upon the orientation and motion of the "local"co-ordinates; for ds² is a definite magnitude belonging to two point-events infinitely near in space and time and canbe got by measurements with rods and clocks. The are hereto be so chosen, that ; the summation isto be extended over all values of σ and τ, so that the sum is to be extended, over 4×4 terms, of which 12 are equal inpairs.From the method adopted here, the case of the usual relativity theory comes out when owing to the special behaviourof in a finite region it is possible to choose the system of co-ordinates in such a way that assumes constantvalues —

(4)


We would afterwards see that the choice of such a system of co-ordinates for a finite region is in general notpossible.From the considerations in § 2 and § 3 it is clear, that from the physical stand-point the quantities are to belooked upon as magnitudes which describe the gravitation-field with reference to the chosen system of axes. Weassume firstly, that in a certain four-dimensional region considered, the special relativity theory is true for someparticular choice of co-ordinates. The then have the values given in (4). A free material point moves withreference to such a system uniformly in a straight line. If we now introduce, by any substitution, the space-timeco-ordinates , then in the new system g are no longer constants, but functions of space and time. Atthe same time, the motion of a free point-mass in the new co-ordinates, will appear as curvilinear, and not uniform,in which the law of motion, will be independent of the nature of the moving mass-points. We can thus signify thismotion as one under the influence of a gravitation field. We see that the appearance of a gravitation-field isconnected with space-time variability of . In the general case, we can not by any suitable choice of axes, makespecial relativity theory valid throughout any finite region. We thus deduce the conception that describe thegravitational field. According to the general relativity theory, gravitation thus plays an exceptional role asdistinguished from the others, specially the electromagnetic forces, in as much as the 10 functions representinggravitation, define immediately the metrical properties of the four-dimensional region.

B. Mathematical Auxiliaries for Establishing the General CovariantEquations.We have seen before that the general relativity-postulate leads to the condition that the system of equations forPhysics, must be Covariants for any possible substitution of co-ordinates ; we have now to see how suchgeneral covariant equations can be obtained. We shall now turn our attention to these purely mathematicalpropositions. It will be shown that in the solution, the invariant ds, given in equation (3) plays a fundamental role,which we, following Gauss's Theory of Surfaces, style as the "line-element".The fundamental idea of the general covariant theory is this: — With reference to any co-ordinate system, let certainthings ("tensors") be defined by a number of functions of co-ordinates which are called the components of the tensor.There are now certain rules according to which the components can be calculated in a new system of co-ordinates,when these are known for the original system, and when the transformation connecting the two systems is known.The things herefrom designated as Tensors have further the property that the transformation equation of theircomponents are linear and homogeneous; so that all the components in the new system vanish if they are all zero inthe original system. Thus a law of Nature can be formulated by putting all the components of a tensor equal to zeroso that it is a general covariant equation; thus while we seek the laws of formation of the tensors, we also reach themeans of establishing general Covariant laws.

§ 5. Contravariant and covariant Four-vector.

Contravariant Four-vector. The line-element is defined by the four components whose transformation law isexpressed by the equation

(5)

The are expressed as linear and homogeneous function of ; we can look upon the differentials of theco-ordinates as the components of a tensor, which we designate specially as a contravariant Four-vector.Everything which is defined by Four quantities , with reference to a co-ordinate system, and transformsaccording to the same law,


(5a)

we may call a contravariant Four-vector. From (5a), it follows at once that the sums are alsocomponents of a four-vector, when and are so; corresponding relations hold also for all systems afterwardsintroduced as "tensors" (Rule of addition and subtraction of Tensors).Covariant Four-vector. We call four quantities as the components of a covariant four-vector, when for anychoice of the contravariant four-vector

(6)

From this definition follows the law of transformation of the covariant four-vectors. If we substitute in the right bandside of the equation

the expressions

for following from the inversion of the equation (5a) we get

As in the above equation are independent of one another and perfectly arbitrary, it follows that thetransformation law is: —

(7)

Remarks on the simplification of the mode of writing the expressions. A glance at the equations of this paragraph willshow that the indices which appear twice within the sign of summation [for example ν in (5)] are those over whichthe summation is to be made and that only over the indices which appear twice. It is therefore possible, without lossof clearness, to leave off the summation sign; so that we introduce the rule: wherever the index in any term of anexpression appears twice, it is to be summed over all of them except when it is not expressedly said to the contrary.The difference between the covariant and the contravariant four-vector lies in the transformation laws [(7) and (5)].Both the quantities are tensors according to the above general remarks; in it lies its significance. In accordance withRicci and Levi-Civita, the contravariants and covariants are designated by the over and under indices.

§ 6. Tensors of the second and higher ranks.Contravariant tensor: — If we now calculate all the 16 products of the components and , of twocontravariant four-vectors

(8)

will according to (8) and (5a) satisfy the following transformation law.

(9)

We call a thing which, with reference to any reference system is defined by 16 quantities and fulfils the transformation relation (9), a contravariant tensor of the second rank. Not every such tensor can be built from two


four-vectors, (according to 8). But it is easy to show that any 16 quantities , can be represented as the sum of ofproperly chosen four pairs of four-vectors. From it, we can prove in the simplest way all laws which hold true for thetensor of the second rank defined through (9), by proving it only for the special tensor of the type (8).Contravariant Tensor of any rank: — If is clear that corresponding to (8) and (9), we can define contravarianttensors of the 3rd and higher ranks, with , etc. components. Thus it is clear from (8) and (9) that in this sense, wecan look upon contravariant four-vectors, as eontra variant tensors of the first rank.Covariant tensor. If on the other hand, we take the 16 products of the components of two covariant

four-vectors and ,

(10)

for them holds the transformation law

(11)

By means of these transformation laws, the covariant tensor of the second rank is defined. All remarks which wehave already made concerning the contravariant tensors, hold also for covariant tensors.Remark:— It is convenient to treat the scalar (Invariant) either as a contravariant or a covariant tensor of zero rank.Mixed tensor. We can also define a tensor of the second rank of the type

(12)

which is covariant with reference to μ and contravariant with reference to ν. Its transformation law is

(13)

Naturally there are mixed tensors with any number of covariant indices, and with any number of contravariantindices. The covariant and contravariant tensors can be looked upon as special cases of mixed tensors.Symmetrical tensors: — A contravariant or a covariant tensor of the second or higher rank is called symmetricalwhen any two components obtained by the mutual interchange of two indices are equal. The tensor or issymmetrical, when we have for any combination of indices

(14)

or

(14a)

It must be proved that a symmetry so defined is a property independent of the system of reference. It follows in factfrom (9) remembering (14)

Anti-symmetrical tensor. A contravariant or covariant tensor of the 2nd, 3rd or 4th rank is called anti-symmetricalwhen the two components got by mutually interchanging any two indices are equal and opposite. The tensor or

is thus anti-symmetrical when we have

(15)

or


(15a)

Of the 16 components , the four components vanish, the rest are equal and opposite in pairs; so that thereare only 6 numerically different components present (Six-vector).Thus we also see that the anti-symmetrical tensor (3rd rank) has only 4 components numerically different,and the anti-symmetrical tensor only one.Symmetrical tensors of ranks higher than the fourth, do not exist in a continuum of 4 dimensions.

§ 7. Multiplication of Tensors.Outer multiplication of Tensors:— We get from the components of a tensor of rank z and another of a rank , thecomponents of a tensor of rank for which we multiply all the components of the first with all the componentsof the second in pairs. For example, we obtain the tensor T from the tensors A and B of different kinds: —

The proof of the tensor character of T, follows immediately from the expressions (8), (10) or (12), or thetransformation equations (9), (11), (13); equations (8), (10) and (12) are themselves examples of the outermultiplication of tensors of the first rank.Reduction in rank of a mixed Tensor. From every mixed tensor we can get a tensor which is two ranks lower, whenwe put an index of covariant character equal to an index of the contravariant character and sum according to theseindices ("Reduction"). We get for example, out of the mixed tensor of the fourth rank , the mixed tensor of the

second rank

and from it again by reduction, the tensor of the zero rank .The proof that the result of reduction retains a truly tensorial character, follows either from the representation oftensor according to the generalisation of (12) in combination with (6) or out of the generalisation of (13).Inner and mixed multiplication of Tensors. This consists in the combination of outer multiplication with reduction.Examples: — From the covariant tensor of the second rank and the contravariant tensor of the first rank we get by outer multiplication the mixed tensor

Through reduction according to indices , the covariant four vector

is generated.

These we denote as the inner product of the tensor and . Similarly we get from the tensors and through outer multiplication and two-fold reduction the inner product . Through outer multiplication andone-fold reduction we get out of and , the mixed tensor of the second rank . We canfitly call this operation a mixed one; for it is outer with reference to the indices μ and τ, and inner with respect to theindices ν and σ.We now prove a law, which will be often applicable for proving the tensor-character of certain quantities. According to the above representation, is a scalar, when and are tensors. We also remark that when


is an invariant for every choice of the tensor , then has a tensorial character.Proof: — According to the above assumption, for any substitution we have

From the inversion of (9) we have however

Substitution of this in the above equation gives

This can be true, for any choice of only when the term within the bracket vanishes. From which by referring to(11), the theorem at once follows. This law correspondingly holds for tensors of any rank and character. The proof isquite similar. The law can also be put in the following form. If and are any two vectors, and if for everychoice of them the inner product

is a scalar, then is a covariant tensor. The last law holds even when there is the more special formulation, thatwith any arbitrary choice of the four-vector alone the scalar product

is a scalar, in which case we have the additional condition that satisfies the symmetry condition .According to the method given above, we prove the tensor character of , from which on account ofsymmetry follows the tensor-character of . This law can easily be generalized in the case of covariant andcontravariant tensors of any rank.Finally, from what has been proved, we can deduce the following law which can be easily generalized for any kindof tensor: If the quantities form a tensor of the first rank, when is any arbitrarily chosen four-vector,then is a tensor of the second rank. If for example, is any four-vector, then owing to the tensor character of

, the inner product is a scalar, both the four-vectors and being arbitrarily chosen.Hence the proposition follows at once.

§ 8. A few words about the Fundamental Tensor .The covariant fundamental tensor. In the invariant expression of the square of the linear element

plays the role of any arbitrarily chosen contravariant vector, since further , it follows from theconsiderations of the last paragraph that is a symmetrical covariant tensor of the second rank. We call it the"fundamental tensor". Afterwards we shall deduce some properties of this tensor, which will also be true for anytensor of the second rank. But the special role of the fundamental tensor in our Theory, which has its physical basison the particularly exceptional character of gravitation makes it clear that those relations are to be developed whichwill be required only in the case of the fundamental tensor.The contravariant fundamental tensor. If we form from the determinant scheme the minors of and dividethem by the determinant of , we get certain quantities , which as we shall provegenerates a contravariant tensor.According to the well-known law of Determinants

(16)

where is 1, or 0, depending on or . Instead of the above expression for ds² we can also write


or according to (16) also in the form

Now according to the rules of multiplication, of the foregoing paragraph, the magnitudes

forms a covariant four-vector, and in fact (on account of the arbitrary choice of ) any arbitrary four-vector.If we introduce it in our expression, we get

For any choice of the vectors this is scalar, and , according to its definition is a symmetrical thing in σ andτ, so it follows from the above results, that is a contravariant tensor. Out of (16) it also follows that is atensor which we may call the mixed fundamental tensor.Determinant of the fundamental tensor. According to the law of multiplication of determinants, we have

On the other hand we have

So that it follows

(17)

Invariant of volume. We seek first the transformation law for the determinant . According to (11)

From this by applying the law of multiplication twice, we obtain.

or

On the other hand the law of transformation of the volume element

is according to the well-known law of Jacobi.

By multiplication of the two last equations we get

(18)

Instead of , we shall afterwards introduce which has a real value on account of the hyperboliccharacter of the time-space continuum. The invariant is equal in magnitude to the four-dimensionalvolume-element measured with solid rods and clocks, in accordance with the special relativity theory.Remarks on the character of the space-time continuum — Our assumption that in an infinitely small region the special relativity theory holds, leads us to conclude that ds² can always, according to (1) be expressed in real


magnitudes . If we call "natural" volume element , we have thus

(18a)

Should vanish at any point of the four-dimensional continuum it would signify that to a finite co-ordinatevolume at the place corresponds an infinitely small "natural" volume. This can never be the case; so that g can neverchange its sign; we would, according to the special relativity theory assume that g has a finite negative value. It is ahypothesis about the physical nature of the continuum considered, and also a pre-established rule for the choice ofco-ordinates.If however -g remains positive and finite, it is clear that the choice of co-ordinates can be so made that this quantitybecomes equal to one. We would afterwards see that such a limitation of the choice of co-ordinates would produce asignificant simplification in expressions for laws of nature.In place of (18) it follows then simply that

from this it follows, remembering the law of Jacobi,

(19)

With this choice of co-ordinates, only substitutions with determinant 1, are allowable.It would however be erroneous to think that this step signifies a partial renunciation of the general relativitypostulate. We do not seek those laws of nature which are covariants with regard to the transformations having thedeterminant 1, but we ask: what are the general covariant laws of nature? First we get the law, and then we simplifyits expression by a special choice of the system of reference.Building up of new tensors with the help of the fundamental tensor. Through inner, outer and mixed multiplicationsof a tensor with the fundamental tensor, tensors of other kinds and of other ranks can be formed.Example:—

We would point out specially the following combinations:

("complement" to the covariant or contravariant tensors) and,

We can call the reduced tensor related to .Similarly

It is to be remarked that is no other than the complement of , for we have —


§ 9. Equation of the geodetic line (or of point-motion).As the "line element" ds is a definite magnitude independent of the co-ordinate system, we have also between twopoints and of a four dimensional continuum a line for which is an extremum (geodetic line), i.e., one

which has got a significance independent of the choice of co-ordinates.Its equation is

(20)

From this equation, we can in a well-known way deduce 4 total differential equations which define the geodetic line;this deduction is given here for the sake of completeness.Let λ, be a function of the co-ordinates ; this defines a series of surfaces which cut the geodetic line sought-for aswell as all neighbouring lines from to . We can suppose that all such curves are given when the value of itsco-ordinates are given in terms of λ. The sign δ corresponds to a passage from a point of the geodetic curvesought for to a point of the contiguous curve, both lying on the same surface λ.Then (20) can be replaced by

(20a)

But

So we get by the substitution of δw in (0Oa), remembering that

after partial integration,

(20b)

From which it follows, since the choice of is perfectly arbitrary that should vanish; Then

(20c)

are the equations of geodetic line; since along the geodetic line considered we have ds = 0, we can choose theparameter λ, as the length of the arc measured along the geodetic line. Then w = 1, and we would get in place of(20c)

Or by merely changing the notation suitably,


(20d)

where we have put, following Christoffel,

(21)

Multiply finally (2Od) with (outer multiplication with reference to τ, and inner with respect to σ) we get at lastthe final form of the equation of the geodetic line —

(22)

Here we have put, following Christoffel,

(23)

§ 10. Formation of Tensors through Differentiation.Relying on the equation of the geodetic line, we can now easily deduce laws according to which new tensors can beformed from given tensors by differentiation. For this purpose, we would first establish the general covariantdifferential equations. We achieve this through a repeated application of the following simple law. If a certain curvebe given in our continuum whose points are characterised by the arc-distances s measured from a fixed point on thecurve, and if further , be an invariant space function, then is also an invariant. The proof follows fromthe fact that as well as ds, are both invariantsSince

so that

is also an invariant for all curves which go out from a point in the continuum, i.e., for any choice of the vector .From which follows immediately that

(24)

is a covariant four-vector (gradient of ).According to our law, the differential-quotient

taken along any curve is likewise an invariant. Substituting the value of , we get

Here however we can not at once deduce the existence of any tensor. If we however take that the curves along whichwe are differentiating are geodesics, we get from it by replacing according to (22)


Prom the interchangeability of the differentiation with regard to μ and ν, and also according to (23) and (21) we seethat the bracket is symmetrical with respect to μ and ν.

As we can draw a geodetic line in any direction from any point in the continuum, is thus a four-vector,with an arbitrary ratio of components, so that it follows from the results of § 7 that

(25)

is a covariant tensor of the second rank. We have thus got the result that out of the covariant tensor of the first rank

we can get by differentiation a covariant tensor of 2nd rank

(26)

We call the tensor the "extension" of the tensor . Then we can easily show that this combination also leadsto a tensor, when the vector is not representable as a gradient. In order to see this we first remark that

is a covariant four-vector when ψ and are scalars. This is also the case for a sum of four such terms: —

when are scalars. Now it is however clear that every covariant four-vector is representablein the form of .If for example is a four-vector whose components are any given functions of , we have, (with reference tothe chosen co-ordinate system) only to put

in order to arrive at the result that is equal to .In order to prove then that in a tensor when on the right aide of (26) we substitute any covariant four-vector for

we have only to show that this is true for the four-vector . For this latter case, however, a glance on the righthand side of (26) will show that we have only to bring forth the proof for the case when

Now the right hand side of (25) multiplied by ψ is

which has a tensor character. Similarly,


is also a tensor (outer product of two four-vectors). Through addition follows the tensor character of

Thus we get the desired proof for the four-vector,

hence for any four-vectors as shown above. —With the help of the extension of the four-vector, we can easily define "extension" of a covariant tensor of any rank.This is a generalisation of the extension of the four-vector. We confine ourselves to the case of the extension of thetensors of the 2nd rank for which the law of formation can be clearly seen.As already remarked every covariant tensor of the 2nd rank can be represented[7] as a sum of the tensors of the type

. It would therefore be sufficient to deduce the expression of extension, for one such special tensor.According to (26) we have the expressions

are tensors. Through outer multiplication of the first with and the 2nd with we get tensors of the third rank.Their addition gives the tensor of the third rank

(27)

where . The right hand side of (27) is linear and homogeneous with reference to , and its firstdifferential co-efficient so that this law of formation leads to a tensor not only in the case of a tensor of the type

but also in the case of a summation for all such tensors, i.e, in the case of any covariant tensor of the secondrank. We call the extension of the tensor .It is clear that (26) and (24) are only special cases of (27) (extension of the tensors of the first and zero rank). Ingeneral we can get all special laws of formation of tensors from (27) combined with tensor multiplication.

§ 11. Some special cases of Particular Importance.A few auxiliary lemmas concerning the fundamental tensor. We shall first deduce some of the lemmas much usedafterwards. According to the law of differentiation of determinants, we have

(28)

The last form follows from the first when we remember that , and therefore .

consequently

From (28), it follows that

Again, since

we have, by differentiation,


(30)

By mixed multiplication with and respectively we obtain (changing the mode of writing the indices).

(31)

and

(32)

The expression (31) allows a transformation which we shall often use; according to (21)

(33)

If we substitute this in the second of the formula (31), we get, remembering (23),

(34)

By substituting the right-hand side of (34) in (29), we get

(29a)

Divergence of the contravariant four-vector. Let us multiply (26) with the contravariant fundamental tensor (inner multiplication), then by a transformation of the first member, the right-hand side takes the form

According to (31) and (29) the last member can take the form

Both the first members of that expression, and the second member of the expression above cancel each other, sincethe naming of the summation-indices is immaterial. The last member of that can then be united with first expressionabove. If we put

where as well as are vectors which can be arbitrarily chosen, we obtain finally

(35)

This scalar is the Divergence of the contravariant four-vector ."Rotation" of the (covariant) four-vector. The second member in (26) is symmetrical in the indices μ, and ν, Hence

is an anti-symmetrical tensor built up in a very simple manner. We obtain


(36)

Anti-symmetrical Extension of a Six-vector. If we apply the operation (27) on an anti-symmetrical tensor of thesecond rank , and form all the equations arising from the cyclic interchange of the indices , and add allthem, we obtain a tensor of the third rank

(37)

from which it in easy to see that the tensor is anti-symmetrical.

Divergence of the Six-vector. If (27) is multiplied by (mixed multiplication), then a tensor is obtained. Thefirst member of the right hand side of (27) can be written in the form

If we replace by , by and replace in the transformed first member

and

with the help of (34), then from the right-hand side of (27) there arises an expression with seven terms, of which fourcancel. There remains

(38)

This is the expression for the extension of a contravariant tensor of the second rank; extensions can also be formedfor corresponding contravariant tensors of higher and lower ranks.

We remark that in the same way, we can also form the extension of a mixed tensor :

(39)

By the reduction of (38) with reference to the indices β and σ (inner multiplication with ), we get a contravariantfour-vector

On the account of the symmetry of with reference to the indices β, and , the third member of the right

hand side vanishes when is an anti-symmetrical tensor, which we assume here; the second member can betransformed according to (29a); we therefore get

(40)

This is the expression of the divergence of a contravariant six-vector.Divergence of the mixed tensor of the second rank. Let us form the reduction of (39) with reference to the indices αand σ, we obtain remembering (29a)

(41)


If we introduce into the last term the contravariant tensor , it takes the form

If further is symmetrical it is reduced to

If instead of , we introduce in a similar way the symmetrical covariant tensor , thenowing to (31) the last member can take the form

In the symmetrical case treated, (41) can be replaced by either of the forms

(41)

or

(41b)

which we shall have to make use of afterwards.

§ 12. The Riemann-Christoffel Tensor.We now seek only those tensors, which can be obtained from the fundamental tensor by differentiation alone. Itis found easily. We put in (27) instead of any tensor the fundamental tensor and get from it a new tensor,namely the extension of the fundamental tensor. We can easily convince ourselves that this vanishes identically. Weprove it in the following way; we substitute in (27)

i.e., the extension of a four-vector .Thus we get (by slightly changing the indices) the tensor of the third rank

We use these expressions for the formation of the tensor Thereby the following terms in cancel the corresponding terms in ; the first member, the fourth member, as well as the membercorresponding to the last term within the square bracket. These are all symmetrical in σ and τ. The same is true forthe sum of the second and third members. We thus get

(42)


(43)

The essential thing in this result is that on the right hand side of (42) we have only , but not its differentialco-efficients. From the tensor-character of , and from the fact that is an arbitrary four vector, itfollows, on account of the result of §7, that is a tensor (Riemann-Christoffel Tensor).The mathematical significance of this tensor is as follows; when the continuum is so shaped, that there is aco-ordinate system for which the are constants, all vanish.If we choose instead of the original co-ordinate system any new one, so would the referred to this last system beno longer constants. The tensor character of shows us, however, that these components vanish collectivelyalso in any other chosen system of reference. The vanishing of the Riemann Tensor is thus a necessary condition thatfor some choice of the axis-system the can be taken as constants.[8] In our problem it corresponds to the casewhen by a suitable choice of the co-ordinate system, the special relativity theory holds throughout any finite region.By the reduction of (43) with reference to indices to τ and , we get the covariant tensor of the second rank

(44)

Remarks upon the choice of co-ordinates. — It has already been remarked in § 8, with reference to the equation(18a), that the co-ordinates can with advantage be so chosen that . A glance at the equations got in thelast two paragraphs shows that, through such a choice, the law of formation of the tensors suffers a significantsimplification. It is specially true for the tensor , which plays a fundamental role in the theory. By thissimplification, vanishes of itself so that tensor reduces to .I shall give in the following pages all relations in the simplified form, with the above-named specialisation of theco-ordinates. It is then very easy to go back to the general covariant equations, if it appears desirable in any specialcase.

C. The Theory of the Gravitation-Field

§ 13. Equation of motion of a material point in a gravitation-field. Expression for thefield-components of gravitation.A freely moving body not acted on by external forces moves, according to the special relativity theory, along astraight line and uniformly. This also holds for the generalised relativity theory for any part of the four-dimensionalregion, in which the co-ordinates can be, and are, so chosen that have special constant values of theexpression (4).Let us discuss this motion from the stand-point of any arbitrary co-ordinate-system ; it moves with reference to

(as explained in § 2) in a gravitational field. The laws of motion with reference to , follow easily from thefollowing consideration. With reference to , the law of motion is a four-dimensional straight line and thus ageodesic. As a geodetic-line is defined independently of the system of co-ordinates, it would also be the law ofmotion for the motion of the material-point with reference to ; If we put

(45)


we get the motion of the point with reference to given by

(46)

We now make the very simple assumption that this general covariant system of equations defines also the motion ofthe point in the gravitational field, when there exists no reference-system , with reference to which the specialrelativity theory holds throughout a finite region. The assumption seems to us to be all the more legitimate, as (46)contains only the first differentials of , among which there is no relation in the special case when exists.[9]

If vanish, the point moves uniformly and in a straight line; these magnitudes therefore determine the deviationfrom uniformity. They are the components of the gravitational field.

§ 14. The Field-equation of Gravitation in the absence of matter.In the following, we differentiate "gravitation-field" from "matter", in the sense that everything besides thegravitation-field will be signified as matter; therefore the term includes not only "matter" in the usual sense, but alsothe electro-dynamic field. Our next problem is to seek the field-equations of gravitation in the absence of matter. Forthis we apply the same method as employed in the foregoing paragraph for the deduction of the equations of motionfor material points. A special case in which the field-equations sought-for are evidently satisfied is that of the specialrelativity theory in which have certain constant values. This would be the case in a certain finite region withreference to a definite co-ordinate system . With reference to this system, all the components of theRiemann's Tensor [equation 43] vanish. These vanish then also in the region considered, with reference to everyother co-ordinate system.The equations of the gravitation-field free from matter must thus be in every case satisfied when all vanish.But this condition is clearly one which goes too far. For it is clear that the gravitation-field generated by a materialpoint in its own neighbourhood can never be transformed away by any choice of axes, i.e., it cannot be transformedto a case of constant .

Therefore it is clear that, for a gravitational field free from matter, it is desirable that the symmetrical tensors deduced from the tensors should vanish. We thus get 10 equations for 10 quantities , which are fulfilled inthe special case when all vanish.Remembering (44) we see that in absence of matter the field-equations come out as follows; (when referred to thespecial co-ordinate-system chosen.)

(47)

It can also be shown that the choice of these equations is connected with a minimum of arbitrariness. For besides, there is no tensor of the second rank, which can be built out of and their derivatives no higher than the

second, and which is also linear in them.[10]

It will be shown that the equations arising in a purely mathematical way out of the conditions of the generalrelativity, together with equations (46), give us the Newtonian law of attraction as a first approximation, and lead inthe second approximation to the explanation of the perihelion-motion of mercury discovered by Leverrier (theresidual effect which could not be accounted for by the consideration of all sorts of disturbing factors). My view isthat these are convincing proofs of the physical correctness of the theory.


§ 15. Hamiltonian Function for the Gravitation-field. Laws of Impulse and Energy.In order to show that the field equations correspond to the laws of impulse and energy, it is most convenient to writeit in the following Hamiltonian form: —

(47a)

Here the variations vanish at the limits of the finite four-dimensional integration-space considered.It is first necessary to show that the form (47a) is equivalent to equations (47). For this purpose, let us consider H asa function of and

We have at first

But

The terms arising out of the two last terms within the round bracket are of different signs, and change into oneanother by the interchange of the indices μ and β. They cancel each other in the expression for δH, when they aremultiplied by , which is symmetrical with respect to μ and β so that only the first member of the bracketremains for our consideration. Remembering (31), we thus have: —

Therefore

(48)

If we now carry out the variations in (47a), we obtain the system of equations

(47b)

which, owing to the relations (48), coincide with (47), as was required to be proved.

If (47b) is multiplied by , since

and consequently

we obtain the equation


or[11]

(49)

or, owing to the relations (48), the equations (47) and (34),

(50)

It is to be noticed that is not a tensor, so that the equation (49) holds only for systems for which .This equation expresses the laws of conservation of impulse and energy in a gravitation-held. In fact, the integrationof this equation over a three-dimensional volume V leads to the four equations

(49a)

where are the direction-cosines of the inward drawn normal to the surface-element dS in the EuclideanSense. We recognise in this the usual expression for the laws of conservation. We denote the magnitudes as theenergy-components of the gravitation-field.I will now put the equation (47) in a third form which will be very serviceable for a quick realisation of our object.By multiplying the field-equations (47) with , these are obtained in the "mixed" forms. If we remember that

which owing to (34) is equal to

or slightly altering the notation equal to

The third member of this expression cancel with the second member of the field-equations (47). In place of thesecond term of this expression, we can, on account of the relations (50), put

where Therefore in the place of the equations (47), we obtain

(51)


§ 16. General formulation of the field-equation of Gravitation.The field-equations established in the preceding paragraph for spaces free from matter is to be compared with theequation

of the Newtonian theory. We have now to find the equations which wall correspond to Poisson's Equation

( signifies the density of matter) .The special relativity theory has led to the conception that the inertial mass is no other than energy. It can also befully expressed mathematically by a symmetrical tensor of the second rank, the energy-tensor. We have therefore tointroduce in our generalised theory an energy-tensor associated with matter, which like the energy components

of the gravitation-field (equations 49, and 50) have a mixed character but which however can be connected withsymmetrical covariant tensors.[12] The equation (51) teaches us how to introduce the energy-tensor (corresponding tothe density of Poisson's equation) in the field equations of gravitation. If we consider a complete system (forexample the Solar-system) its total mass, as also its total gravitating action, will depend on the total energy of thesystem, ponderable as well as gravitational.This can be expressed, by putting in (51), in place of energy-components of gravitation-field alone the sum of theenergy-components of matter and gravitation, i.e., .We thus get instead of (51), the tensor-equation

(52)

where (Laue's Scalar). These are the general field-equations of gravitation in the mixed form. In place of(47), we get by working backwards the system

(53)

It must be admitted, that this introduction of the energy-tensor of matter cannot be justified by means of theRelativity-Postulate alone; for we have in the foregoing analysis deduced it from the condition that the energy of thegravitation-field should exert gravitating action in the same way as every other kind of energy. The strongest groundfor the choice of the above equation however lies in this, that they lead, as their consequences, to equationsexpressing the conservation of the components of total energy (the impulses and the energy) which exactlycorrespond to the equations (49) and (49a). This shall be shown afterwards.

§ 17. The laws of conservation in the general case.The equations (52) can be easily so transformed that the second member on the right-hand side vanishes. We reduce(52) with reference to the indices μ and σ and subtract the equation so obtained after multiplication with from

(52). We obtain:

(52a)

we operate on it by . Now,


The first and the third member of the round bracket lead to expressions which cancel one another, as can be easilyseen by interchanging the summation-indices α and σ on the one hand, and β and γ on the other. The second term canbe transformed according to (31). So that we get

The second member of the expression on the left-hand side of (52a) leads first to

or

The expression arising out of the last member within the round bracket vanishes according to (29) on account of thechoice of axes. The two others can be taken together and give us on account of (31), the expression

So that remembering (54) we have the identity

(55)

From (55) and (52a) it follows that

(56)

From the field equations of gravitation, it also follows that the conservation-laws of impulse and energy are satisfied.We see it most simply following the same reasoning which lead to equations (49a); only instead of theenergy-components of the gravitational-field, we are to introduce the total energy-components of matter andgravitational field.

§ 18. The Impulse-energy law for matter as a consequence of the field-equations.

If we multiply (53) with , we get in a way similar to § 15, remembering that

vanishes, the equations

or remembering (56)

(57)

A comparison with (41b) shows that these equations for the above choice of co-ordinates asserts nothing but thevanishing of the divergence of the tensor of the energy-components of matter.


Physically the appearance of the second term on the left-hand side shows that for matter alone the law ofconservation of impulse and energy cannot hold; or can only hold when are constants; i.e., when the field ofgravitation vanishes. The second member is an expression for impulse and energy which the gravitation-field exertsper time and per volume upon matter. This comes out clearer when instead of (57) we write it in the Form of (41).

(57a)

The right-hand side expresses the interaction of the energy of the gravitational-field on matter. The field-equations ofgravitation contain thus at the same time 4 conditions which are to be satisfied by all material phenomena. We getthe equations of the material phenomena completely when the latter is characterised by four other differentialequations independent of one another.[13]

D. The "Material" Phenomena.The Mathematical auxiliaries developed under B at once enables us to generalise, according to the generalised theoryof relativity, the physical laws of matter (Hydrodynamics, Maxwell's Electro-dynamics) as they lie alreadyformulated according to the special-relativity-theory. The generalised Relativity Principle leads us to no furtherlimitation of possibilities; but it enables us to know exactly the influence of gravitation on all processes without theintroduction of any new hypothesis.It is owing to this, that as regards the physical nature of matter (in a narrow sense) no definite necessary assumptionsare to be introduced. The question may lie open whether the theories of the electro-magnetic field and thegravitational-field together, will form a sufficient basis fur the theory of matter. The general relativity postulate canteach us no new principle. But by building up the theory it must be shown whether electro-magnetism andgravitation together can achieve what the former alone did not succeed in doing.

§ 19. Euler's equations for frictionless adiabatic liquid.Let p and , be two scalars, of which the first denotes the "pressure" and the last the "density" of the liquid;between them there is a relation. Let the contravariant symmetrical tensor

(58)

be the contravariant energy-tensor of the liquid. To it also belongs the covariant tensor

(58a)

as well as the mixed tensor[14]

(58b)

If we put the right-hand side of (58b) in (57a), we get the general hydrodynamical equations of Euler according tothe generalised relativity theory. This in principle completely solves the problem of motion; for the four equations(57a) together with the given equation between p and , and the equation

are sufficient, with the given values of , for finding out the six unknowns


If are unknown we have also to take the equations (53). There are now 11 equations for finding out 10 functions, so that the number is more than sufficient. Now it is be noticed that the equation (57a) is already contained in

(53), so that the latter only represents (7) independent equations. This indefiniteness is due to the wide freedom inthe choice of co-ordinates, so that mathematically the problem is indefinite in the sense that three of theSpace-functions can be arbitrarily chosen.[15]

§ 20. Maxwell's Electro-Magnetic field-equations for the vacuum.Let be the components of a covariant four-vector, the electro-magnetic potential; from it let us form according to(36) the Components of the covariant six-vector of the electro-magnetic field according to the system ofequations

(59)

From (59) it follows that the system of equations

(60)

is satisfied of which the left-hand side, according to (37), is an anti-symmetrical tensor of the third kind. This system(60) contains essentially four equations, which can be thus written: —

(60a)

This system of equations corresponds to the second system of equations of Maxwell. We see it at once if we put

(61)

Instead of (60a) we can therefore write according to the usual notation of three-dimensional vector-analysis: —

(60b)

The first Maxwellian system is obtained by a generalisation of the form given by Minkowski.

We introduce the contravariant six-vector by the equation

(62)

and also a contravariant four-vector , which is the electrical current-density in vacuum. Then remembering (40)we can establish the system of equations, which remains invariant for any substitution with determinant 1 (accordingto our choice of co-ordinates)

(63)


If we put

(64)

which quantities become equal to , in the case of the special relativity theory, and besides

we get instead of (63)

(63a)

The equations (60), (62) and (63) give thus a generalisation of Maxwell's field-equations in vacuum, which remainstrue in our chosen system of co-ordinates.The energy-components of the electromagnetic field. Let us form the inner-product

(65)

According to (61) its components can be written down in the three-dimensional notation.

(65a)

is a covariant four-vector whose components are equal to the negative impulse and energy which are transferredto the electro-magnetic field per unit of time, and per unit of volume, by the electrical masses. If the electrical massesbe free, that is, under the influence of the electromagnetic field only, then the covariant four-vector will vanish.

In order to get the energy components of the electro-magnetic field, we require only to give to the equation, the form of the equation (57).

From (63) and (65) we get first,

On account of (60) the second member on the right-hand side admits of the transformation —

Owing to symmetry, this expression can also be written in the form

which can also be put in the form

The first of these terms can be written shortly as

and the second after differentiation can be transformed in the form


If we take all the three terms together, we get the relation

(66)

where

(66a)

On account of (30) the equation (66) becomes equivalent to (57) and (57a) when vanishes. Thus are theenergy-components of the electro-magnetic field. With the help of (61) and (64) we can easily show that theenergy-components of the electro-magnetic field, in the case of the special relativity theory, give rise to thewell-known Maxwell-Poynting expressions.We have now deduced the most general laws which the gravitation-field and matter satisfy when we use aco-ordinate system for which . Thereby we achieve an important simplification in all our formulas andcalculations, without renouncing the conditions of general covariance, as we have obtained the equations through aspecialisation of the co-ordinate system from the general covariant-equations. Still the question is not without formalinterest, whether, when the energy-components of the gravitation-field and matter is defined in a generalised mannerwithout any specialisation of co-ordinates, the laws of conservation have the form of the equation (56), and thefield-equations of gravitation hold in the form (52) or (52a); such that on the left-hand side, we have a divergence inthe usual sense, and on the right-hand side, the sum of the energy-components of matter and gravitation. I havefound out that this is indeed the case. But I am of opinion that the communication of my rather comprehensive workon this subject will not pay, for nothing essentially new comes out of it.

E. § 21. Newton's Theory as a First Approximation.We have already mentioned several times that the special relativity theory is to be looked upon as a special case ofthe general, in which have constant values (4).This signifies, according to what has been said before, a total neglect of the influence of gravitation. We get a morerealistic approximation if we consider the case when differ from (4) only by small magnitudes (compared to 1)where we can neglect small quantities of the second and higher orders (first aspect of the approximation.)Further it should be assumed that within the space-time region considered, at infinite distances (using the wordinfinite in a spatial sense) can, by a suitable choice of co-ordinates, tend to the limiting values (4); i.e., we consideronly those gravitational fields which can be regarded as produced by masses distributed over finite regions.We can assume that this approximation should lead to Newton's theory. For it however, it is necessary to treat thefundamental equations from another point of view. Let us consider the motion of a particle according to the equation(46). In the case of the special relativity theory, the components

can take any values; This signifies that any velocity

can appear which is less than the velocity of light in vacuum (v<1). If we finally limit ourselves to the considerationof the case when v is small compared to the velocity of light, it signifies that the components


can be treated as small quantities, whereas is equal to 1, up to the second-order magnitudes (the secondpoint of view for approximation).Now we see that, according to the first view of approximation the magnitudes are all small quantities of at leastthe first order. A glance at (46) will also show, that in this equation according to the second view of approximation,we are only to take into account those terms for which .By limiting ourselves only to terms of the lowest order we get instead of (46), first, the equations : —

where , or by limiting ourselves only to those terms which according to the first stand-point areapproximations of the first order,

If we further assume that the gravitation-field is quasi-static, i.e., it is limited only to the case when the matterproducing the gravitation-field is moving slowly (relative to the velocity of light) we can neglect the differentiationsof the positional co-ordinates on the right-hand side with respect to time, so that we get

(67)

This is the equation of motion of a material point according to Newton's theory, where plays the part ofgravitational potential. The remarkable thing in the result is that in the first-approximation of motion of the materialpointy only the component of the fundamental tensor appears.Let us now turn to the field-equation (53). In this case, we have to remember that the energy-tensor of matter isexclusively defined in a narrow sense by the density of matter, i.e., by the second member on the right-hand sideof 58 [(58a, or 58b)]. If we make the necessary approximations, then all component vanish except

On the left-hand side of (53) the second term is an infinitesimal of the second order, so that the first leads to thefollowing terms in the approximation, which are rather interesting for us

neglecting all differentiations with regard to time, this leads, when , to the expression

The last of the equations (53) thus leads to

(68)

The equations (67) and (68) together, are equivalent to Newton's law of gravitation.For the gravitation-potential we get from (67) and (68) the exp.

(68a)

whereas the Newtonian theory for the chosen unit of time gives


where K denotes usually the gravitation-constant ; equating them we get (69)

(69)

§ 22. Behaviour of measuring rods and clocks in a statical gravitation-field. Curvature oflight-rays. Perihelion-motion of the paths of the Planets.In order to obtain Newton's theory as a first approximation we had to calculate only out of the 10 components ofthe gravitation-potential , for that is the only component which conies in the first approximate equations ofmotion of a material point in a gravitational field. We see however, that the other components of should alsodiffer from the values given in (4) as required by the condition .For a mass-point at the origin of co-ordinates and generating the gravitational field, we get as a first approximationthe symmetrical solution of the equation: —

is 1 or 0, according as or , and r is the quantity

On account of (68a) we have

(70a)

where M denotes the mass generating the field. It is easy to verify that this solution satisfies approximately thefield-equation outside the mass.Let us now investigate the influences which the field of mass M will have upon the metrical properties of the field.Between the lengths and times ds measured "locally" (§ 4) on the one hand, and the differences in co-ordinates on the other, we have the relation

For a unit measuring rod, for example, placed "parallel" to the x-axis, we have to put

then

If the unit measuring rod lies on the x-axis, the first of the equations (70) gives

From both these relations it follows as a first approximation that

(71)


The unit measuring rod appears, when referred to the co-ordinate-system, shortened by the calculated magnitudethrough the presence of the gravitational field, when we place it radially in the field.Similarly we can get its co-ordinate-length in a tangential position, if we put for example

we then get

(71a)

The gravitational field has no influence upon the length of the rod, when we put it tangentially in the field.Thus Euclidean geometry does not hold in the gravitational field even in the first approximation, if we conceive thatone and the same rod independent of its position and its orientation can serve as the measure of the same extension.But a glance at (70a) and (69) shows that the expected difference is much too small to be noticeable in themeasurement of earth's surface.We would further investigate the rate of going of a unit-clock which is placed in a statical gravitational field. Herewe have for a period of the clock

then we have

or

(72)

Therefore the clock goes slowly when it is placed in the neighbourhood of ponderable masses. It follows from thisthat the spectral lines in the light coming to us from the surfaces of big stars should appear shifted towards the redend of the spectrum.[16]

Let us further investigate the path of light-rays in a statical gravitational field. According to the special relativitytheory, the velocity of light is given by the equation

thus also according to the generalised relativity theory it is given by the equation

(73)

If the direction, i.e., the ratio is given, the equation (73) gives the magnitudes

and with it the velocity,

in the sense of the Euclidean Geometry. We can easily see that, with reference to the co-ordinate system, the rays oflight must appear curved in case are not constant.If n be the direction perpendicular to the direction of propagation, we have, from Huygen's principle, that light-rays[taken in the plane (γ,n)] must suffer a curvature .


Let us find out the curvature which a light-ray suffers when it goes by a mass M at a distance Δ from it. If we use theco-ordinate system according to the above scheme, then the total bending B of light-rays (reckoned positive when itis concave to the origin) is given as a sufficient approximation by

while (73) and (70) give

The calculation gives

(74)

A ray of light just grazing the sun would suffer a bending of whereas one coming by Jupiter would have adeviation of about .If we calculate the gravitation-field to a greater order of approximation and with it the corresponding path of amaterial particle of a relatively small (infinitesimal) mass we set a deviation of the following kind from theKepler-Newtonian Laws of Planetary motion. The Ellipse of Planetary motion suffers a slow rotation in the directionof motion, of amount

(75)

In this Formula a signifies the semi-major axis, c the velocity of light, measured in the usual way, e the eccentricity,T the time of revolution in seconds.[17]

The calculation gives for the planet Mercury, a rotation of path of amount per century, correspondingsufficiently to what has been found by astronomers (Leverrier). They found a residual perihelion motion of thisplanet of the given magnitude which can not be explained by the perturbation of the other planets.

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[1] http:/ / www. physik. uni-augsburg. de/ annalen/ history/ einstein-papers/ 1916_49_769-822. pdf[2] Such an epistemological satisfactory answer can of course be physically wrong, if it is in contradiction with other experiences.[3][3] That the gravitational field has this property with great accuracy, was confirmed by by experiment.[4][4] We assume the possibility for stating the "simultaneity" of immediate adjacent events, or more precisely for the immediate space-time

adjacency (coincidence), without giving a definition for this fundamental expression.[5][5] Of certain limitations, that correspond with the requirement of unequivocal correspondence and that of continuity, we don't want to speak

here.[6][6] The unit time must be chosen in a way, so that light velocity in vacuum - measured in the "local" coordinate system - is equal to 1.[7] By outer multiplication of the vectors with the (arbitrarily given) components or 1,0,0,0 arises a tensor with the components

By addition of four tensors of this type we obtain the tensor with arbitrarily given components.[8] The mathematicians have shown, that this condition is sufficient as well.[9] Only between the second (and first) differentials the relations exist according to § 12.[10] Principally, this can only be said of the tensor , where λ is a constant. However, if we set it to 0, then we come back to equations [11] The reason for the introduction of the factor will become clear later.[12] UNIQ-math-7-64bee9a14e9badfe-QINU and shall be symmetrical tensors.[13][13] See for this D. , Nachr. d. K. Gesellsch. d. Wiss. zu Göttingen, Math.-phys. Klasse. p. 3. 1915[14] For a co-moving observer, who uses for an infinitely small region a reference frame in the sense of special relativity theory, the

energy-density is equal to . In that lies the definition of . Thus is not constant for an incompressible fluid.[15] When refusing the coordinate-choice according to , four space-function remain free to choose, in agreement with the four arbitrary

functions, which we can freely use for the coordinate-choice.[16][16] In support of the existence of such an effect we can allude to the spectral observations on fix-stars according to E. . However, a concluding

examination of that consequence is still missing.[17] As regards the calculation I allude to the original papers, A. , Sitzungsber. d. Preuß. Akad. d. Wiss. 47. p. 831. 1915. — K. , Sitzungsber. d.

Preuß. Akad. d. Wiss. 7. p. 189. 1916






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Dialog about Objections against the Theory of Relativity 56

Dialog about Objections against the Theory ofRelativity

Dialog about Objections against the Theory of Relativity (1918) by Albert Einstein, translated by Wikisource


German original: Dialog über Einwände gegen die Relativitätstheorie, Die Naturwissenschaften, 29 November 1918.Critic:Many times already have the likes of mine submitted to journals a wide variety of reservations about the theory ofrelativity; and rarely has one of you relativists [1] provided an answer. We have no wish to dwell on whether thisneglect was due to arrogance, or a sense of weakness, or laziness - maybe it was a particularly effective mixture ofthese afflictions of the soul, and then maybe it wasn't rare for the criticism to reveal that the critic simply had toolittle knowledge of the matter at hand. These issues will - as announced - not be discussed, but first I want to tell youthis: I have come to visit you personally, so that you will not be able to back out, as you did on other occasions. For Ipromise you that I will stay until you have answered all of my questions.So as not to upset you too much, and possibly even make you undertake this business (which you can't avoidanyway) with a certain pleasure, I will say this in comfort. Unlike many of my collegues, I am not so full with thestatus of my guild so as to make me act as a superior being with superhuman insight and certainty (like newspaperjournalists about scientific literature, or playwright-critics). On the contrary, I talk as a human being, since I amaware that it is not rare for criticism to originate from lack of own thoughts. Also I have no wish to - as was latelydone by one of my colleagues - jump on you like a district-attorney and accuse you of theft of intellectual property,or accuse you of equally dishounorable acts. Only the need to contribute to the clarification of several points, onwhich opinions still widely divert, has motivated my assault. However I must request you to grant publication of ourconversation, not in the least because the shortage of paper is not the only shortage that is causing my friend, theeditor of the Berolinensis, to lose sleep.Since I notice your willingness to comply, I will immediately come to business. Ever since the special theory ofrelativity has been formulated, the outcome concerning the slowing influence of motion on the rate of clocks hascontinuously elicited opposition, and - it seems to me - with good reasons. For this outcome seems to lead inevitablyto a contradiction with the foundations of the theory. To make sure we understand each other completely, let thetheory now be presented with sufficient sharpness.Let K be a Galilean coordinate system in the sense of the special theory of relativity, that is, a frame of reference,relative to which isolated, material points move in straight lines and uniformly. Also, let U1 and U2 be two identicalclocks that are free from outside influences. These will run at the same pace when they are in close proximity andalso at any distance from each other, if they are both at rest relative to K. However, if one of the clocks, for exampleU2, is relative to K in a state of uniform translational motion, then according to the special theory of relativity itshould - as perceived from coordinate system K - go at a slower pace than the clock U1 that is at rest relative to K.This result is in itself highly peculiar. Grave doubts arise when one faces the following well-known thoughtexperiment.Let A and B be two distant points of the system K. To render the picture more precise, let A be the origin of K, andB be a point on the positive x-axis. The two clocks are initially at rest at point A. They run at the same pace, and letthe positions of the hands be the same. We now impart to clock U2 a constant velocity in the positive direction of thex-axis, so that it moves towards B. At B we imagine the velocity reversed, so that clock U2 returns to A. As it arrives



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at A, the clock is decelerated so that it is once again at rest relative to U1. Because the change in the position of thehands of the clock, as judged from K, that might occur during the change of velocity, will not exceed a certain value,and because U2 runs slower than U1 during the motion along the length of A B (as judged from K), clock U2 must,if the length A B is sufficiently long, be running behind U1 - do you agree with that?Relativist:Entirely agreed. With regret I have noticed that some authors, who otherwise have a thorough understanding of thetheory of relativity, wanted to avoid this inevitable result.Critic:Now here's the twist. According to the principle of relativity the whole affair should proceed in the same way if it isrepresented in a coordinate system K', that is co-moving with clock U2. Then relative to K' it is clock U1 that ismoving to and fro, with clock U2 remaining at rest. It then follows that at the end U1 should run behind U2, incontradiction with the above result. Surely even the most devoted followers of the theory will not assert that in thecase of two clocks that have been positioned side by side, each one is running behind the other.Relativist:Your last assertion is of course undisputable. However, the reason that that line of argument as a whole is untenableis that according to the special theory of relativity the coordinate systems K and K' are by no means equivalentsystems. Indeed this theory asserts only the equivalence of all Galilean (unaccelerated) coordinate systems, that is,coordinate systems relative to which sufficiently isolated, material points move in straight lines and uniformly. K issuch a coordinate system, but not the system K', that is accelerated from time to time. Therefore, from the result thatafter the motion to and fro the clock U2 is running behind U1, no contradiction can be constructed against theprinciples of the theory.Critic:I acknowledge that you have rendered my objection powerless, but I have to say that I feel convicted by yourargument rather than convinced. Anyway, my objection immediately arises from its ashes when one bases oneself onthe general theory of relativity. For according to this theory, coordinate systems in arbitrary states of motion arequalified, hence the proceedings described earlier can equally well be referred to the coordinate system K' that iscontinuously connected to U2, as to the system K.Relativist:It is certainly correct that from the point of view of the general theory of relativity we can just as well use coordinatesystem K' as coordinate system K. But it is easy to see that the systems K and K' in connection with the examinedproceedings stand by no means on equal footing. While the proceedings as seen from system K can be regarded asabove, a totally different picture presents itself as seen from K', as can be seen from the following comparison:

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K is the reference frame. K' is the reference frame

1. The clock U2 is accelerated by an externalforce along the positive x-axis, until it hasreached velocity v. U1 remains at rest.

1. A gravitational field appears, that is directed towards the negative x-axis. Clock U1 isaccelerated in free fall, until it has reached velocity v. An external force acts upon clock U2,preventing it from being set in motion by the gravitational field. When the clock U1 has reachedvelocity v the gravitational field disappears.

2. U2 moves with constant velocity v up to pointB of the positive x-axis. U1 remains at rest.

2. U1 moves with constant velocity to the point B' on the negative x-axis. U2 remains at rest.

3. Clock U2 is accelerated by an external forceacting in the negative direction of the x-axisuntil it has reached velocity v in the negativex-direction. U1 remains at rest.

3. A homogenous gravitational field appears, that is directed towards the positive x-axis. ClockU1 is accelerated in the direction of the positive x-axis until it has reached the velocity v, thenthe gravitational field disappears again. An external force, acting upon U2 in the negativedirection of the x-axis prevents U2 from being set in motion by the gravitational field.

4. U2 moves with constant velocity in thenegative direction of the x-axis until itapproaches U1. U1 remains at rest.

4. U1 moves with constant velocity v in the direction of the positive x-axis until it approachesU2. U2 remains at rest.

5. An external force brings clock U2 to rest. 5. A gravitational field that is directed towards the negative x-axis appears and brings U1 to ahalt. Then the gravitational field disappears again. An external force keeps U2 in a state of rest.

It should be kept in mind that in the left and in the right section exactly the same proceedings are described, it is justthat the description on the left relates to the coordinate system K, the description on the right relates to the coordinatesystem K'. According to both descriptions the clock U2 is running a certain amount behind clock U1 at the end of theobserved process. When relating to the coordinate system K' the behaviour explains itself as follows: During thepartial processes 2 and 4 the clock U1, going at a velocity v, runs indeed at a slower pace than the resting clock U2.However, this is more than compensated by a faster pace of U1 during partial process 3. According to the generaltheory of relativity, a clock will go faster the higher the gravitational potential of the location where it is located, andduring partial process 3 U2 happens to be located at a higher gravitational potential than U1. The calculation showsthat this speeding ahead constitutes exactly twice as much as the lagging behind during the partial processes 2 and 4.This consideration completely clears up the paradox that you brought up.Critic:I do see that you have cleverly pulled away from the noose, but I would be lying if I would declare myself fullysatisfied. The stumbling stone has not been removed; it has been relocated. You see, your consideration only showsthe connection of the difficulty that was just discussed with another difficulty, that has also often been presented.You have solved the paradox, by taking the influence on the clocks into account of a gravitational field relative to K'.But isn't this gravitational field merely fictitious? Its existence is conjured up by a mere choice of coordinate system.Surely, real gravitational fields are brought forth by mass, and cannot be made to disappear by a suitable choice ofcoordinate system. How are we supposed to believe that a merely fictitious field could have such an influence on thepace of a clock?Relativist:In the first place I must point out that the distinction real - unreal is hardly helpful. In relation to K' the gravitationalfield "exists" in the same sense as any other physical entity that can only be defined with reference to a coordinatesystem, even though it is not present in relation to the system K. No special peculiarity resides here, as can easily beseen from the following example from classical mechanics. Nobody doubts the "reality" of kinetic energy, otherwisethe very reality of energy would have to be denied. But it is clear that the kinetic energy of a body is dependent onthe state of motion of the coordinate system, with a suitable choice of the latter one can arrange for the kineticenergy of the continuous motion of a body to assume a given positive value or the value of zero. In the special casewhere all the masses have a velocity in the same direction and of the same magnitude, a suitable choice of coordinatesystem can adjust the collective kinetic energy to zero. To me it appears that the analogy is complete.


Rather than distinguishing between "real" and "unreal" we want to more clearly distinguish between quantities thatare inherent in the physical system as such (independent from the choice of coordinate system), and quantities thatdepend on the coordinate system. The next step would be to demand that only quantities of the first kind enter thelaws of physics. However, it has been found that this objective cannot be realized in practice, as has already beendemonstrated clearly by the development of classical mechanics. One could for instance consider, and this hasactually been attempted, to enter into the laws of classical mechanics not the coordinates, but instead just thedistances between the material points; a priori one could expect that in this way the goal of the theory of relativitywould be reached most easily. The scientific development has however not confirmed this expectation. She cannotdispense with the coordinate system, and therefore has to use in the coordinates quantities that cannot be construedas results of definite measurements. According to the general theory of relativity the four coordinates of thespace-time continuum are entirely arbitrary choosable parameters, devoid of any independent physical meaning. Thisarbitrariness partially affects also those quantities (field components) that are instrumental in describing the physicalreality. Only certain, generally quite complicated expressions, that are constructed out of field components andcoordinates, correspond to coordinate-independent, measurable (that is, real) quantities. For example, the componentof the gravitational field in a space-time point is still not a quantity that is independent of coordinate choice; thus thegravitational field at a certain place does not correspond to something "physically real", but in connection with otherdata it does. Therefore one can neither say, that the gravitational field in a certain place is something "real', nor that itis "merely fictitious".The circumstance that according to the general theory of relativity the connection between the quantities that occurin the equations and the measurable quantities is much more indirect than in terms of the usual theories, probablyconstitutes the main difficulty that one encounters when studying this theory. Also your last objection was based onthe fact that you did not keep this circumstance constantly in mind.You declared the fields that were called for in the clock example also as merely fictitious, only because the fieldlines of actual gravitational fields are necessarily brought forth by mass; in the discussed examples no mass thatcould bring forth those fields was present. This can be elaborated upon in two ways. Firstly, it is not an a priorinecessity that the particular concept of the Newtonian theory, according to which every gravitational field isconceived as being brought forth by mass, should be retained in the general theory of relativity. This question isinterconnected with the circumstance pointed out previously, that the meaning of the field components is much lessdirectly defined as in the Newtonian theory. Secondly, it cannot be maintained that there are no masses present, thatcan be attributed with bringing forth the fields. To be sure, the accelerated coordinate systems cannot be called uponas real causes for the field, an opinion that a jocular critic saw fit to attribute to me on one occasion. But all the starsthat are in the universe, can be conceived as taking part in bringing forth the gravitational field; because during theaccelerated phases of the coordinate system K' they are accelerated relative to the latter and thereby can induce agravitational field, similar to how electric charges in accelerated motion can induce an electric field. Approximateintegration of the gravitational equations has in fact yielded the result that induction effects must occur when massesare in accelerated motion. This consideration makes it clear that a complete clarification of the questions you haveraised can only be attained if one envisions for the geometric-mechanical constitution of the Universe arepresentation that complies with the theory. I have attempted to do so last year, and I have reached a conception that- to my mind - is completely satisfactory; going into this would however take us too far.Critic: After your last statements it does seem to me that no self-contradiction of the theory of relativity can be deduced from the clock-paradox. Indeed, it now seems not unlikely to me that the theory is free from self-contradiction altogether, but it does not in itself mean that the theory should be considered in earnest. I really don't see why for the sake of some conceptual preference - namely for the concept of relativity - one ought to accept the burden of such gruesome complications and calculational difficulties. In your last answer you yourself have amply demonstrated that they are considerable. For example, would anyone get it in his head to actually use the possibility offered by the theory of relativity to relate the motions of the celestial bodies of the solar system to a geocentric coordinate system


that on top of that is participating in the rotation of the Earth? Would anyone really be allowed to see this coordinatesystem as "at rest" and as equally valid, relative to which the fixed stars are tearing around with tremendous speed?Doesn't such an approach collide head on with common sense, and with the demand of economy of thought? I willnot refrain now from repeating the drastic words that have lately been uttered by Lenard about the subject. Afterdiscussing the special theory of relativity, in which he modeled the "moving" coordinate system with a rolling traincarriage, he said: "Now imagine that this train carriage makes a clearly non-uniform motion. When through theaction of inertia everything inside the train is wrecked, while outside everything remains undamaged, then, in myopinion, no sane mind will draw any other conclusion than that it was the train that changed its motion with a jerk,and not the surroundings. The generalized principle of relativity demands in its simple sense, that in this case also itmust be admitted that possibly it was after all the surroundings that went through the change of velocity, and that theentire accident in the train is just a consequence of this jerk of the environment, transmitted through a 'gravitationaleffect' of the environment on the inside of the train. For the related question why the church tower adjacent to thetrain hasn't toppled over, while it underwent the jerk together with the surroundings - why such consequences of thejerk are so exclusively confined to the train, even though an unambiguous conclusion about the seat of the change ofmotion is supposed not to be had - the principle has as it would seem no satisfying answer for the simple mind."Relativist: There are several reasons that compel us to willingly accept the complications that the theory leads us to. In the first place, it means for a man who maintains consistency of thought a great satisfaction to see that the concept of absolute motion, to which kinematically no meaning can be attributed, does not have to enter physics; it cannot be denied that by avoiding this concept the foundation of physics has gained in consistency. Also the fact of the equality of inertia and weight of a body urgently requires an explanation. Apart from that, physics needs a method to attain an action-by-contact-theory of gravitation. Without an effective confining principle the theorists could hardly attempt the problem, because ever so many theories could be formulated, that satisfy the limited experiences in this area. Embarras de richesse is one of the most malicious opponents to make the theoretician's life miserable. The postulate of relativity reduces the possibilities in such a way that the road that the theory had to go was predetermined. Lastly the secular perihelion motion of the planet Mercury had to be clarified. This perihelion motion was certainly noticed by astronomers, and they were unsuccessful in finding an explanation on the basis of the Newtonian theory. - In asserting the equality of coordinate systems as a matter of principle it is not said that every coordinate system is equally convenient for examining a certain physical system; we see this in classical mechanics also. For example, strictly speaking one cannot say that the Earth moves in an ellipse around the Sun, because that statement presupposes a coordinate system in which the Sun is at rest, while classical mechanics also allows systems relative to which the Sun rectilinearly and uniformly moves. In examining the motion of the Earth nobody will decide to use a coordinate system of the last kind, and neither will anyone decide from considering this example that the coordinate system, whose origin is co-moving with the center of mass of the considered mechanical system, is in principle privileged over other coordinate systems. It is the same in the example you mentioned. Nobody will use a coordinate system that is at rest relative to the planet Earth, because that would be impractical. However as a matter of principle such a theory of relativity is equally valid as any other. The situation, that the fixed stars are circling with tremendous velocities, when one bases an examination on such a coordinate system, does not constitute an argument against the admissibility, but merely against the efficiency of this choice of coordinates, nor does the complicated form of the relative to this coordinate system acting gravitational field, which for example would also have the components that correspond to the centrifugal force. In Mister Lenard's example the situation is similar. In terms of the theory of relativity the case may not be construed in such a way that possibly it is after all the surroundings (of the train) that experienced the change in velocity. We are not dealing here with two different, mutually exclusive hypotheses about the seat of the motion, rather with two ways, equally valid in principle, of representing the same factual situation.[2] For the decision which representation to choose only reasons of efficiency are decisive, not arguments of a principle kind. Just how little merit there is in calling upon the so-called "common sense", is shown by the following counterexample. Lenard himself says that so far no objections could be raised against the validity of


the special principle of relativity (that is the principle of relativity of uniform translational motion of coordinatesystems). The uniformly riding train can equally well be regarded as "at rest", the rails together with the entiresurroundings can be regarded as "uniformly moving". Would the "common sense" of the train driver allow for that?He will point out that it is not the surroundings that he needs to continuously heat and lubricate, it is the locomotive,and consequently it must be in the latter that the result of his labour shows itself.Critic:After this conversation I have to admit that the refutation of your point of view is not as easy as it seemed to meearlier. I do have more objections up my sleeve. But before pestering you with that I want to think over our presentconversation thoroughly. Before we depart, one more question, that does not concern an objection, but that I ask outof pure curiosity: how does the diseased man of theoretical physics fare, the Aether, that many of you have declaredto be definitely dead?Relativist:Its fortunes have taken some turns, and overall one cannot say that it is dead now. Prior to Lorentz it existed as anall-pervasive fluid, as a gas-like fluid, and other than that in the most diverse forms of being, different from author toauthor. With Lorentz it became rigid, and embodied the resting coordinate system, respectively a privileged state ofmotion in the world. According to the special theory of relativity there was no longer a privileged state of motion,this meant a denial of the Aether in this sense of the preceding theories. For if there would be an Aether, then in eachspace-time point there would have to be a particular state of motion, that would have to play a part in optics. There isno such privileged state of motion, as has been taught to us by the special theory of relativity, and that is why there isno Aether in the old sense. The general theory of relativity also does not know a privileged state of motion in a point,that one could vaguely interpret as velocity of an Aether. However, while according to the special theory of relativitya part of space without matter and without electromagnetic field seems to be characterized as absolutely empty, e. g.not characterized by any physical quantities, empty space in this sense has according to the general theory ofrelativity physical qualities which are mathematically characterized by the components of the gravitational potential,that determine the metrical behavior of this part of space as well as its gravitational field. One can quite wellconstrue this circumstance in such a way that one speaks of an Aether, whose state of being is different from point topoint. Only one must take care not to attribute to this Aether properties similar to properties of matter (for exampleevery point a certain velocity).[1][1] Here, "relativist" is to be understood as a supporter of the physical theory of relativity, not the philosophic relativism[2] That the tower doesn't fall over is, according to the second representation, due to the fact that it free-falls with the ground and the entire Earth

in a gravitational field (which is present during the jerk), while the train is kept from free falling by external forces (braking forces). A freefalling body behaves with regard to internal processes as a free floating body, isolated from all external influences.












Time, Space, and Gravitation 62

Time, Space, and Gravitation

Time, Space, and Gravitation (1919) by Albert Einstein


From: Science, 51 (No. 1305); January 2, 1920; pp. 8-10, Online [1]

Originally published in London Times; November 28, 1919.

TIME, SPACE, AND GRAVITATIONAfter the lamentable breach in the former international relations existing among men of science, it is with joy andgratefulness that I accept this opportunity of communication with English astronomers and physicists. It was inaccordance with the high and proud tradition of English science that English scientific men should have given theirtime and labor, and that English institutions should have provided the material means, to test a theory that had beencompleted and published in the country of their enemies in the midst of war. Although investigation of the influenceof the solar gravitational field on rays of light is a purely objective matter, I am none the less very glad to express mypersonal thanks to my English colleagues in this branch of science; for without their aid I should not have obtainedproof of the most vital deduction from my theory.There are several kinds of theory in physics. Most of them are constructive. These attempt to build a picture ofcomplex phenomena out of some relatively simple proposition. The kinetic theory of gases, for instance, attempts torefer to molecular movement the mechanical thermal, and diffusional properties of gases. When we say that weunderstand a group of natural phenomena, we mean that we have found a constructive theory which embraces them.

THEORIES OF PRINCIPLEBut in addition to this most weighty group of theories, there is another group consisting of what I call theories ofprinciple. These employ the analytic, not the synthetic method. Their starting-point and foundation are nothypothetical constituents, but empirically observed general properties of phenomena, principles from whichmathematical formula are deduced of such a kind that they apply to every case which presents itself.Thermodynamics, for instance, starting from the fact that perpetual motion never occurs in ordinary experience,attempts to deduce from this, by analytic processes, a theory which will apply in every case. The merit ofconstructive theories is their comprehensiveness, adaptability, and clarity, that of the theories of principle, theirlogical perfection, and the security of their foundation.The theory of relativity is a theory of principle. To understand it, the principles on which it rests must be grasped.But before stating these it is necessary to point out that the theory of relativity is like a house with two separatestories, the special relativity theory and the general theory of relativity.Since the time of the ancient Greeks it has been well known that in describing the motion of a body we must refer toanother body. The motion of a railway train is described with reference to the ground, of a planet with reference tothe total assemblage of visible fixed stars. In physics the bodies to which motions are spatially referred are termedsystems of coordinates. The laws of mechanics of Galileo and Newton can be formulated only by using a system ofcoordinates.The state of motion of a system of coordinates can not be chosen arbitrarily if the laws of mechanics are to hold good (it must be free from twisting and from acceleration). The system of coordinates employed in mechanics is called an inertia-system. The state of motion of an inertia-system, so far as mechanics are concerned, is not restricted by nature to one condition. The condition in the following proposition suffices: a system of coordinates moving in the same direction and at the same rate as a system of inertia is itself a system of inertia. The special relativity theory is therefore the application of the following proposition to any natural process: "Every law of nature which holds good




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with respect to a coordinate system K must also hold good for any other system K' provided page 9 that K and K' arein uniform movement of translation."The second principle on which the special relativity theory rests is that of the constancy of the velocity of light in avacuum. Light in a vacuum has a definite and constant velocity, independent of the velocity of its source. Physicistsowe their confidence in this proposition to the Maxwell-Lorentz theory of electro-dynamics.The two principles which I have mentioned have received strong experimental confirmation, but do not seem to belogically compatible. The special relativity theory achieved their logical reconciliation by making a change inkinematics, that is to say, in the doctrine of the physical laws of space and time. It became evident that a statement ofthe coincidence of two events could have a meaning only in connection with a system of coordinates, that the massof bodies and the rate of movement of clocks must depend on their state of motion with regard to the coordinates.

THE OLDER PHYSICSBut the older physics, including the laws of motion of Galileo and Newton, clashed with the relativistic kinematicsthat I have indicated. The latter gave origin to certain generalized mathematical conditions with which the laws ofnature would have to conform if the two fundamental principles were compatible. Physics had to be modified. Themost notable change was a new law of motion for (very rapidly) moving mass-points, and this soon came to beverified in the case of electrically-laden particles. The most important result of the special relativity systemconcerned the inert mass of a material system. It became evident that the inertia of such a system must depend on itsenergy-content, so that we were driven to the conception that inert mass was nothing else than latent energy. Thedoctrine of the conservation of mass lost its independence and became merged in the doctrine of conservation ofenergy.The special relativity theory which was simply a systematic extension of the electro dynamics of Maxwell andLorentz, had consequences which reached beyond itself. Must the independence of physical laws with regard to asystem of coordinates be limited to systems of coordinates in uniform movement of translation with regard to oneanother? What has nature to do with the coordinate systems that we propose and with their motions? Although it maybe necessary for our descriptions of nature to employ systems of coordinates that we have selected arbitrarily, thechoice should not be limited in any way so far as their state of motion is concerned. (General theory of relativity.)The application of this general theory of relativity was found to be in conflict with a well-known experiment,according to which it appeared that the weight and the inertia of a body depended oh the same constants (identity ofinert and heavy masses). Consider the case of a system of coordinates which is conceived as being in stable rotationrelative to a system of inertia in the Newtonian sense. The forces which, relatively to this system, are centrifugalmust, in the Newtonian sense, be attributed to inertia. But these centrifugal forces are, like gravitation, proportionalto the mass of the bodies. It is not, then, possible to regard the system of coordinates as at rest, and the centrifugalforces of gravitational? The interpretation seemed obvious, but classical mechanics forbade it.This slight sketch indicates how a generalized theory of relativity must include the laws of gravitation, and actualpursuit of the conception has justified the hope. But the way was harder than was expected, because it contradictedEuclidian geometry. In other words, the laws according to which material bodies are arranged in space do not exactlyagree with the laws of space prescribed by the Euclidian geometry of solids. This is what is meant by the phrase "awarp in space." The fundamental concepts "straight," "plane ," etc., accordingly lose their exact meaning in physics.In the generalized theory of relativity, the doctrine of space and time, kinematics, is no longer one of the absolutefoundations of general physics. The geometrical states of bodies page 10 and the rates of clocks depend in the firstplace on their gravitational fields, which again are produced by the material systems concerned.Thus the new theory of gravitation diverges widely from that of Newton with respect to its basal principle. But inpractical application the two agree so closely that it has been difficult to find cases in which the actual differencescould be subjected to observation. As yet only the following have been suggested:1. The distortion of the oval orbits of planets round the sun (confirmed in the case of the planet Mercury).


2. The deviation of light-rays in a gravitational field (confirmed by the English Solar Eclipse expedition).3. The shifting of spectral lines towards the red end of the spectrum in the case of light coming to us from stars ofappreciable mass (not yet confirmed).The great attraction of the theory is its logical consistency. If any deduction from it should prove untenable, it mustbe given up. A modification of it seems impossible without destruction of the whole.No one must think that Newton's great creation can be overthrown in any real sense by this or by any other theory.His clear and wide ideas will for ever retain their significance as the foundation on which our modern conceptions ofphysics have been built.

Albert Einstein



References[1] http:/ / www. archive. org/ details/ science511920mich

A Brief Outline of the Development of the Theoryof Relativity

A Brief Outline of the Development of the Theory of Relativity (1921) by Albert Einstein, translated by Robert William Lawson


From: Nature, 106 (No. 2677); February 17, 1921; pp. 782-784, Online [1]

A Brief Outline of the Development of the Theory of Relativity.

By Prof. A. Einstein.[Translated by Dr. Robert W. Lawson.]

There is something attractive in presenting the evolution of a sequence of ideas in as brief a form as possible, and yetwith a completeness sufficient to preserve throughout the continuity of development. We shall endeavour to do thisfor the Theory of Relativity, and to show that the whole ascent is composed of small, almost self-evident steps ofthought.The entire development starts off from, and is dominated by, the idea of Faraday and Maxwell, according to whichall physical processes involve a continuity of action (as opposed to action at a distance), or, in the language ofmathematics, they are expressed by partial differential equations. Maxwell succeeded in doing this forelectro-magnetic processes in bodies at rest by means of the conception of the magnetic effect of thevacuum-displacement-current, together with the postulate of the identity of the nature of electro-dynamic fieldsproduced by induction, and the electro-static field.The extension of electro-dynamics to the case of moving bodies fell to the lot of Maxwell's successors. H. Hertzattempted to solve the problem by ascribing to empty space (the aether) quite similar physical properties to thosepossessed by ponderable matter; in particular, like ponderable matter, the aether ought to have at every point a





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A Brief Outline of the Development of the Theory of Relativity 65

definite velocity. As in bodies at rest, electromagnetic or magneto-electric induction ought to be determined by therate of change of the electric or magnetic flow respectively, provided that these velocities of alteration are referred tosurface elements moving with the body. But the theory of Hertz was opposed to the fundamental experiment ofFizeau on the propagation of light in flowing liquids. The most obvious extension of Maxwell's theory to the case ofmoving bodies was incompatible with the results of experiment.At this point, H. A. Lorentz came to the rescue. In view of his unqualified adherence to the atomic theory of matter,Lorentz felt unable to regard the latter as the seat of continuous electromagnetic fields. He thus conceived of thesefields as being conditions of the aether, which was regarded as continuous. Lorentz considered the aether to beintrinsically independent of matter, both from a mechanical and a physical point of view. The aether did not take partin the motions of matter, and a reciprocity between aether and matter could be assumed only in so far as the latterwas considered to be the carrier of attached electrical charges. The great value of the theory of Lorentz lay in the factthat the entire electrodynamics of bodies at rest and of bodies in motion was led back to Maxwell's equations ofempty space. Not only did this theory surpass that of Hertz from the point of view of method, but with its aid H. A.Lorentz was also pre-eminently successful in explaining the experimental facts.The theory appeared to be unsatisfactory only in one point of fundamental importance. It appeared to give preferenceto one system of coordinates of a particular state of motion (at rest relative to the aether) as against all other systemsof co-ordinates in motion with respect to this one. In this point the theory seemed to stand in direct opposition toclassical mechanics, in which all inertial systems which are in uniform motion with respect to each other are equallyjustifiable as systems of co-ordinates (Special Principle of Relativity). In this connection, all experience also in therealm of electro-dynamics (in particular Michelson's experiment) supported the idea of the equivalence of all inertialsystems, i.e. was in favour of the special principle of relativity.The Special Theory of Relativity owes its origin to this difficulty, which, because of its fundamental nature, was feltto be intolerable. This theory originated as the answer to the question: Is the special principle of relativity reallycontradictory to the field equations of Maxwell for empty space? The answer to this question appeared to be in theaffirmative. For if those equations are valid with reference to a system of co-ordinates K, and we introduce a newsystem of co-ordinates K' in conformity with the — to all appearances readily establishable — equations oftransformation

(Galileo transformation),

then Maxwell's field equations are no longer valid in the new co-ordinates (x', y', z', t'). But appearances aredeceptive. A more searching analysis of the physical significance of space and time rendered it evident that theGalileo transformation is founded on arbitrary assumptions, and in particular on the assumption that the statement ofsimultaneity has a meaning which is independent of the state of motion of the system of co-ordinates used. It wasshown that the field equations for vacuo satisfy the special principle of relativity, provided we make use of theequations of transformation stated below:

(Lorentz transformation).

In these equations x, y, z represent the co-ordinates measured with measuring-rods which are at rest with reference tothe system of co-ordinates, and t represents the time measured with suitably adjusted clocks of identical construction,which are in a state of rest. page 783


Now in order that the special principle of relativity may hold, it is necessary that all the equations of physics do notalter their form in the transition from one inertial system to another, when we make use of the Lorentztransformation for the calculation of this change. In the language of mathematics, ail systems of equations thatexpress physical laws must be co-variant with respect to the Lorentz transformation. Thus, from the point of view ofmethod, the special principle of relativity is comparable to Carnot's principle of the impossibility of perpetual motionof the second kind, for, like the latter, it supplies us with a general condition which all natural laws must satisfy.Later, H. Minkowski found a particularly elegant and suggestive expression for this condition of co-variance, onewhich reveals a formal relationship between Euclidean geometry of three dimensions and the space-time continuumof physics.

Euclidean Geometry ofThree Dimensions.

Special Theory ofRelativity.

Corresponding to two neighbouring points in space,there exists a numerical measure (distance ds) whichconforms to the equation

ds² = dx1² + dx2² + dx3²

Corresponding to two neighbouring points in space-time (point events), there exists anumerical measure (distance ds) which conforms to the equation

ds² = dx1² + dx2² + dx3² + dx4²

It is independent of the system of co-ordinates chosen,and can be measured with the unit measuring-rod.

It is independent of the inertial system chosen, and can be measured with the unitmeasuring-rod and a standard clock. x1, x2, x3, are here rectangular co-ordinates, whilst

is the time multiplied by the imaginary unit and by he velocity of light.

The permissible transformations are of such a characterthat the expression for ds² is invariant, i.e. the linearorthogonal transformations are permissible.

The permissible transformations are of such a character that the expression for ds² isinvariant, i.e. those linear orthogonal substitutions are permissible which maintain thesemblance of reality of x1, x2, x3, x4 These substitutions are the Lorentz transformations.

With respect to these transformations, the laws ofEuclidean geometry are invariant.

With respect to these transformations, the laws of physics are invariant.

From this it follows that, in respect of its rôle in the equations of physics, though not with regard to its physicalsignificance, time is equivalent to the space co-ordinates (apart from the relations of reality). From this point of view,physics is, as it were, a Euclidean geometry of four dimensions, or, more correctly, a statics in a four-dimensionalEuclidean continuum.The development of the special theory of relativity consists of two main steps, namely, the adaptation of thespace-time "metrics" to Maxwell's electro-dynamics, and an adaptation of the rest of physics to that alteredspace-time "metrics." The first of these processes yields the relativity of simultaneity, the influence of motion onmeasuring-rods and clocks, a modification of kinematics, and in particular a new theorem of addition of velocities.The second process supplies us with a modification of Newton's law of motion for large velocities, together withinformation of fundamental importance on the nature of inertial mass.It was found that inertia is not a fundamental property of matter, nor, indeed, an irreducible magnitude, but aproperty of energy. If an amount of energy E be given to a body, the inertial mass of the body increases by anamount E/c², where c is the velocity of light in vacuo. On the other hand, a body of mass m is to be regarded as astore of energy of magnitude mc².Furthermore, it was soon found impossible to link up the science of gravitation with the special theory of relativity in a natural manner. In this connection I was struck by the fact that the force of gravitation possesses a fundamental property, which distinguishes it from electro-magnetic forces. All bodies fall in a gravitational field with the same acceleration, or — what is only another formulation of the same fact — the gravitational and inertial masses of a body are numerically equal to each other. This numerical equality suggests identity in character. Can gravitation and inertia be identical? This question leads directly to the General Theory of Relativity. Is it not possible for me to regard the earth as free from rotation, if I conceive of the centrifugal force, which acts on all bodies at rest relatively to the earth, as being a "real" field of gravitation, or part of such a field? If this idea can be carried out, then we shall have proved in very truth the identity of gravitation and inertia. For the same property which is regarded as inertia


from the point of view of a system not taking part in the rotation can be interpreted as gravitation when consideredwith respect to a system that shares the rotation. According to Newton, this interpretation is impossible, because byNewton's law the centrifugal field cannot be regarded as being produced by matter, and because in Newton's theorythere is no place for a "real" field of the "Koriolis-field" type. But perhaps Newton's law of field could be replacedby another that fits in with the field which holds with respect to a "rotating" system of co-ordinates? My convictionof the identity of inertial and gravitational mass aroused within me the feeling of absolute confidence in thecorrectness of this interpretation. In this connection I gained encouragement from the following idea. We are familiarwith the "apparent" fields which are valid relatively page 784 to systems of co-ordinates possessing arbitrary motionwith respect to an inertial system. With the aid of these special fields we should be able to study the law which issatisfied in general by gravitational fields. In this connection we shall have to take account of the fact that theponderable masses will be the determining factor in producing the field, or, according to the fundamental result ofthe special theory of relativity, the energy density — a magnitude having the transformational character of a tensor.On the other hand, considerations based on the metrical results of the special theory of relativity led to the result thatEuclidean metrics can no longer be valid with respect to accelerated systems of co-ordinates. Although it retardedthe progress of the theory several years, this enormous difficulty was mitigated by our knowledge that Euclideanmetrics holds for small domains. As a consequence, the magnitude ds, which was physically defined in the specialtheory of relativity hitherto, retained its significance also in the general theory of relativity. But the co-ordinatesthemselves lost their direct significance, and degenerated simply into numbers with no physical meaning, the solepurpose of which was the numbering of the space-time points. Thus in the general theory of relativity theco-ordinates perform the same function as the Gaussian co-ordinates in the theory of surfaces. A necessaryconsequence of the preceding is that in such general co-ordinates the measurable magnitude ds must be capable ofrepresentation in the form

where the symbols guv are functions of the space-time co-ordinates. From the above it also follows that the nature ofthe space-time variation of the factors guv determines, on one hand the space time metrics, and on the other thegravitational field which governs the mechanical behaviour of material points.The law of the gravitational field is determined mainly by the following conditions: First, it shall be valid for anarbitrary choice of the system of co-ordinates; secondly, it shall be determined by the energy tensor of matter; andthirdly, it shall contain no higher differential coefficients of the factors guv than the second, and must be linear inthese. In this way a law was obtained which, although fundamentally different from Newton's law, corresponded soexactly to the latter in the deductions derivable from it that only very few criteria were to be found on which thetheory could be decisively tested by experiment.The following are some of the important questions which are awaiting solution at the present time. Are electrical andgravitational fields really so different in character that there is no formal unit to which they can be reduced? Dogravitational fields play a part in the constitution of matter, and is the continuum within the atomic nucleus to beregarded as appreciably non-Euclidean? A final question has reference to the cosmological problem. Is inertia to betraced to mutual action with distant masses? And connected with the latter: Is the spatial extent of the universefinite? It is here that my opinion differs from that of Eddington. With Mach, I feel that an affirmative answer isimperative, but for the time being nothing can be proved. Not until a dynamical investigation of the large systems offixed stars has been performed from the point of view of the limits of validity of the Newtonian law of gravitation forimmense regions of space will it perhaps be possible to obtain eventually an exact basis for the solution of thisfascinating question.







References[1] http:/ / www. archive. org/ details/ naturejournal106londuoft

Ether and the Theory of Relativity

Ether and the Theory of Relativity (1922) by Albert Einstein, translated by George Barker Jeffery and Wilfrid Perrett


From: Sidelights on Relativity (1922), pp. 3-24, London: Methuen, Online [1]

German original: Äther und Relativitätstheorie (1920), Berlin: Springer

ETHER AND THE THEORY OF RELATIVITYAn Address delivered on May 5th, 1920,

in the University of LeydenHow does it come about that alongside of the idea of ponderable matter, which is derived by abstraction fromeveryday life, the physicists set the idea of the existence of another kind of matter, the ether? The explanation isprobably to be sought in those phenomena which have given rise to the theory of action at a distance, and in theproperties of light which have led to the undulatory theory. Let us devote a little while to the consideration of thesetwo subjects.Outside of physics we know nothing of action at a distance. When we try to connect cause and effect in theexperiences which natural objects afford us, it seems at first as if there were no other mutual actions than those ofimmediate contact, e.g. the communication of motion by impact, push and pull, heating or inducing combustion bymeans of a flame, etc. It is true that even in everyday experience weight, which is in a sense action at a distance,plays a very important part. But since in daily experience the weight of bodies meets us as something constant,something not linked to any cause which is variable in time or place, we do not in everyday life speculate as to thecause of gravity, and therefore do not become conscious of its character as action at a distance. It was Newton'stheory of gravitation that first assigned a cause for gravity by interpreting it as action at a distance, proceeding frommasses. Newton's theory is probably the greatest stride ever made in the effort towards the causal nexus of naturalphenomena. And yet this theory evoked a lively sense of discomfort among Newton's contemporaries, because itseemed to be in conflict with the principle springing from the rest of experience, that there can be reciprocal action









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Ether and the Theory of Relativity 69

only through contact, and not through immediate action at a distance.It is only with reluctance that man's desire for knowledge endures a dualism of this kind. How was unity to bepreserved in his comprehension of the forces of nature? Either by trying to look upon contact forces as beingthemselves distant forces which admittedly are observable only at a very small distance and this was the road whichNewton's followers, who were entirely under the spell of his doctrine, mostly preferred to take; or by assuming thatthe Newtonian action at a distance is only apparently immediate action at a distance, but in truth is conveyed by amedium permeating space, whether by movements or by elastic deformation of this medium. Thus the endeavourtoward a unified view of the nature of forces leads to the hypothesis of an ether. This hypothesis, to be sure, did notat first bring with it any advance in the theory of gravitation or in physics generally, so that it became customary totreat Newton's law of force as an axiom not further reducible. But the ether hypothesis was bound always to playsome part in physical science, even if at first only a latent part.When in the first half of the nineteenth century the far-reaching similarity was revealed which subsists between theproperties of light and those of elastic waves in ponderable bodies, the ether hypothesis found fresh support. Itappeared beyond question that light must be interpreted as a vibratory process in an elastic, inert medium filling upuniversal space. It also seemed to be a necessary consequence of the fact that light is capable of polarisation that thismedium, the ether, must be of the nature of a solid body, because transverse waves are not possible in a fluid, butonly in a solid. Thus the physicists were bound to arrive at the theory of the "quasi-rigid " luminiferous ether, theparts of which can carry out no movements relatively to one another except the small movements of deformationwhich correspond to light-waves.This theory — also called the theory of the stationary luminiferous ether — moreover found a strong support in anexperiment which is also of fundamental importance in the special theory of relativity, the experiment of Fizeau,from which one was obliged to infer that the luminiferous ether does not take part in the movements of bodies. Thephenomenon of aberration also favoured the theory of the quasi-rigid ether.The development of the theory of electricity along the path opened up by Maxwell and Lorentz gave thedevelopment of our ideas concerning the ether quite a peculiar and unexpected turn. For Maxwell himself the etherindeed still had properties which were purely mechanical, although of a much more complicated kind than themechanical properties of tangible solid bodies. But neither Maxwell nor his followers succeeded in elaborating amechanical model for the ether which might furnish a satisfactory mechanical interpretation of Maxwell's laws of theelectro-magnetic field. The laws were clear and simple, the mechanical interpretations clumsy and contradictory.Almost imperceptibly the theoretical physicists adapted themselves to a situation which, from the standpoint of theirmechanical programme, was very depressing. They were particularly influenced by the electro-dynamicalinvestigations of Heinrich Hertz. For whereas they previously had required of a conclusive theory that it shouldcontent itself with the fundamental concepts which belong exclusively to mechanics (e.g. densities, velocities,deformations, stresses) they gradually accustomed themselves to admitting electric and magnetic force asfundamental concepts side by side with those of mechanics, without requiring a mechanical interpretation for them.Thus the purely mechanical view of nature was gradually abandoned. But this change led to a fundamental dualismwhich in the long-run was insupportable. A way of escape was now sought in the reverse direction, by reducing theprinciples of mechanics to those of electricity, and this especially as confidence in the strict validity of the equationsof Newton's mechanics was shaken by the experiments with β-rays and rapid kathode rays.This dualism still confronts us in unextenuated form in the theory of Hertz, where matter appears not only as thebearer of velocities, kinetic energy, and mechanical pressures, but also as the bearer of electromagnetic fields. Sincesuch fields also occur in vacuo — i.e. in free ether the ether — also appears as bearer of electromagnetic fields. Theether appears indistinguishable in its functions from ordinary matter. Within matter it takes part in the motion ofmatter and in empty space it has everywhere a velocity; so that the ether has a definitely assigned velocitythroughout the whole of space. There is no fundamental difference between Hertz's ether and ponderable matter(which in part subsists in the ether).


The Hertz theory suffered not only from the defect of ascribing to matter and ether, on the one hand mechanicalstates, and on the other hand electrical states, which do not stand in any conceivable relation to each other; it wasalso at variance with the result of Fizeau's important experiment on the velocity of the propagation of light in movingfluids, and with other established experimental results.Such was the state of things when H. A. Lorentz entered upon the scene. He brought theory into harmony withexperience by means of a wonderful simplification of theoretical principles. He achieved this, the most importantadvance in the theory of electricity since Maxwell, by taking from ether its mechanical, and from matter itselectromagnetic qualities. As in empty space, so too in the interior of material bodies, the ether, and not matterviewed atomistically, was exclusively the seat of electromagnetic fields. According to Lorentz the elementaryparticles of matter alone are capable of carrying out movements; their electromagnetic activity is entirely confined tothe carrying of electric charges. Thus Lorentz succeeded in reducing all electromagnetic happenings to Maxwell'sequations for free space.As to the mechanical nature of the Lorentzian ether, it may be said of it, in a somewhat playful spirit, that immobilityis the only mechanical property of which it has not been deprived by H. A. Lorentz. It may be added that the wholechange in the conception of the ether which the special theory of relativity brought about, consisted in taking awayfrom the ether its last mechanical quality, namely, its immobility. How this is to be understood will forthwith beexpounded.The space-time theory and the kinematics of the special theory of relativity were modelled on the Maxwell-Lorentztheory of the electromagnetic field. This theory therefore satisfies the conditions of the special theory of relativity,but when viewed from the latter it acquires a novel aspect. For if K be a system of co-ordinates relatively to whichthe Lorentzian ether is at rest, the Maxwell-Lorentz equations are valid primarily with reference to K. But by thespecial theory of relativity the same equations without any change of meaning also hold in relation to any newsystem of co-ordinates K' which is moving in uniform translation relatively to K. Now comes the anxious question:— Why must I in the theory distinguish the K system above all K' systems, which are physically equivalent to it inall respects, by assuming that the ether is at rest relatively to the K system? For the theoretician such an asymmetryin the theoretical structure, with no corresponding asymmetry in the system of experience, is intolerable. If weassume the ether to be at rest relatively to K, but in motion relatively to K', the physical equivalence of K and K'seems to me from the logical standpoint, not indeed downright incorrect, but nevertheless inacceptable.The next position which it was possible to take up in face of this state of things appeared to be the following. Theether does not exist at all. The electromagnetic fields are not states of a medium, and are not bound down to anybearer, but they are independent realities which are not reducible to anything else, exactly like the atoms ofponderable matter. This conception suggests itself the more readily as, according to Lorentz's theory,electromagnetic radiation, like ponderable matter, brings impulse and energy with it, and as, according to the specialtheory of relativity, both matter and radiation are but special forms of distributed energy, ponderable mass losing itsisolation and appearing as a special form of energy.More careful reflection teaches us, however, that the special theory of relativity does not compel us to deny ether.We may assume the existence of an ether; only we must give up ascribing a definite state of motion to it, i.e. wemust by abstraction take from it the last mechanical characteristic which Lorentz had still left it. We shall see laterthat this point of view, the conceivability of which I shall at once endeavour to make more intelligible by asomewhat halting comparison, is justified by the results of the general theory of relativity.Think of waves on the surface of water. Here we can describe two entirely different things. Either we may observe how the undulatory surface forming the boundary between water and air alters in the course of time; or else — with the help of small floats, for instance — we can observe how the position of the separate particles of water alters in the course of time. If the existence of such floats for tracking the motion of the particles of a fluid were a fundamental impossibility in physics — if, in fact, nothing else whatever were observable than the shape of the space occupied by the water as it varies in time, we should have no ground for the assumption that water consists of


movable particles. But all the same we could characterise it as a medium.We have something like this in the electromagnetic field. For we may picture the field to ourselves as consisting oflines of force. If we wish to interpret these lines of force to ourselves as something material in the ordinary sense, weare tempted to interpret the dynamic processes as motions of these lines of force, such that each separate line of forceis tracked through the course of time. It is well known, however, that this way of regarding the electromagnetic fieldleads to contradictions.Generalising we must say this: — There may be supposed to be extended physical objects to which the idea ofmotion cannot be applied. They may not be thought of as consisting of particles which allow themselves to beseparately tracked through time. In Minkowski's idiom this is expressed as follows: — Not every extendedconformation in the four-dimensional world can be regarded as composed of world-threads. The special theory ofrelativity forbids us to assume the ether to consist of particles observable through time, but the hypothesis of ether initself is not in conflict with the special theory of relativity. Only we must be on our guard against ascribing a state ofmotion to the ether.Certainly, from the standpoint of the special theory of relativity, the ether hypothesis appears at first to be an emptyhypothesis. In the equations of the electromagnetic field there occur, in addition to the densities of the electriccharge, only the intensities of the field. The career of electromagnetic processes in vacua appears to be completelydetermined by these equations, uninfluenced by other physical quantities. The electromagnetic fields appear asultimate, irreducible realities, and at first it seems superfluous to postulate a homogeneous, isotropic ether-medium,and to envisage electromagnetic fields as states of this medium.But on the other hand there is a weighty argument to be adduced in favour of the ether hypothesis. To deny the etheris ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics donot harmonize with this view. For the mechanical behaviour of a corporeal system hovering freely in empty spacedepends not only on relative positions (distances) and relative velocities, but also on its state of rotation, whichphysically may be taken as a characteristic not appertaining to the system in itself. In order to be able to look uponthe rotation of the system, at least formally, as something real, Newton objectivises space. Since he classes hisabsolute space together with real things, for him rotation relative to an absolute space is also something real. Newtonmight no less well have called his absolute space "Ether"; what is essential is merely that besides observable objects,another thing, which is not perceptible, must be looked upon as real, to enable acceleration or rotation to be lookedupon as something real.It is true that Mach tried to avoid having to accept as real something which is not observable by endeavouring tosubstitute in mechanics a mean acceleration with reference to the totality of the masses in the universe in place of anacceleration with reference to absolute space. But inertial resistance opposed to relative acceleration of distantmasses presupposes action at a distance; and as the modern physicist does not believe that he may accept this actionat a distance, he comes back once more, if he follows Mach, to the ether, which has to serve as medium for theeffects of inertia. But this conception of the ether to which we are led by Mach's way of thinking differs essentiallyfrom the ether as conceived by Newton, by Fresnel, and by Lorentz. Mach's ether not only conditions the behaviourof inert masses, but is also conditioned in its state by them.Mach's idea finds its full development in the ether of the general theory of relativity. According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that "empty space" in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν), has, I think, finally disposed of the view that space is physically empty. But therewith the conception of the ether has again acquired an intelligible content, although this content differs widely from that of the ether of the mechanical undulatory theory of light. The ether of the general theory of relativity is a medium which is itself devoid of all mechanical and kinematical qualities, but helps to determine mechanical (and


electromagnetic) events.What is fundamentally new in the ether of the general theory of relativity as opposed to the ether of Lorentz consistsin this, that the state of the former is at every place determined by connections with the matter and the state of theether in neighbouring places, which are amenable to law in the form of differential equations; whereas the state ofthe Lorentzian ether in the absence of electromagnetic fields is conditioned by nothing outside itself, and iseverywhere the same. The ether of the general theory of relativity is transmuted conceptually into the ether ofLorentz if we substitute constants for the functions of space which describe the former, disregarding the causeswhich condition its state. Thus we may also say, I think, that the ether of the general theory of relativity is theoutcome of the Lorentzian ether, through relativation.As to the part which the new ether is to play in the physics of the future we are not yet clear. We know that itdetermines the metrical relations in the space-time continuum, e.g. the configurative possibilities of solid bodies aswell as the gravitational fields; but we do not know whether it has an essential share in the structure of the electricalelementary particles constituting matter. Nor do we know whether it is only in the proximity of ponderable massesthat its structure differs essentially from that of the Lorentzian ether; whether the geometry of spaces of cosmicextent is approximately Euclidean. But we can assert by reason of the relativistic equations of gravitation that theremust be a departure from Euclidean relations, with spaces of cosmic order of magnitude, if there exists a positivemean density, no matter how small, of the matter in the universe. In this case the universe must of necessity bespatially unbounded and of finite magnitude, its magnitude being determined by the value of that mean density.If we consider the gravitational field and the electromagnetic field from the standpoint of the ether hypothesis, wefind a remarkable difference between the two. There can be no space nor any part of space without gravitationalpotentials; for these confer upon space its metrical qualities, without which it cannot be imagined at all. Theexistence of the gravitational field is inseparably bound up with the existence of space. On the other hand a part ofspace may very well be imagined without an electromagnetic field; thus in contrast with the gravitational field, theelectromagnetic field seems to be only secondarily linked to the ether, the formal nature of the electromagnetic fieldbeing as yet in no way determined by that of gravitational ether. From the present state of theory it looks as if theelectromagnetic field, as opposed to the gravitational field, rests upon an entirely new formal motif, as though naturemight just as well have endowed the gravitational ether with fields of quite another type, for example, with fields ofa scalar potential, instead of fields of the electromagnetic type.Since according to our present conceptions the elementary particles of matter are also, in their essence, nothing elsethan condensations of the electromagnetic field, our present view of the universe presents two realities which arecompletely separated from each other conceptually, although connected causally, namely, gravitational ether andelectromagnetic field, or — as they might also be called — space and matter.Of course it would be a great advance if we could succeed in comprehending the gravitational field and theelectromagnetic field together as one unified conformation. Then for the first time the epoch of theoretical physicsfounded by Faraday and Maxwell would reach a satisfactory conclusion. The contrast between ether and matterwould fade away, and, through the general theory of relativity, the whole of physics would become a completesystem of thought, like geometry, kinematics, and the theory of gravitation. An exceedingly ingenious attempt in thisdirection has been made by the mathematician H. Weyl; but I do not believe that his theory will hold its ground inrelation to reality. Further, in contemplating the immediate future of theoretical physics we ought not unconditionallyto reject the possibility that the facts comprised in the quantum theory may set bounds to the field theory beyondwhich it cannot pass.Recapitulating, we may say that according to the general theory of relativity space is endowed with physicalqualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space withoutether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility ofexistence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in thephysical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media,


as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.






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The Bad Nauheim Debate

The Bad Nauheim Debate (1920) Albert Einstein, Philipp Lenard, Hermann Weyl, Ernst Gehrcke, translated by Wikisource


*Allgemeine Diskussion über die Relativitätstheorie (1920), Physikalische Zeitschrift, 21, pp. 666-668 Online [1]

• Hermann Weyl, Die Diskussion über die Relativitätstheorie (1920), Die Umschau, 24, pp. 609-611 Online [2]

• Ernst Gehrcke, (1921), Die Umschau, 25, p. 99 Online [3]

• Hermann Weyl, Antwort auf Prof. Dr. Gehrcke (1921), Die Umschau, 25, 123-124 Online [3]

• Ernst Gehrcke, Zur Relativitätsfrage (1921), Die Umschau, 25, p. 227. Online [3]

• K. Körner, Die 86. Versammlung der Deutscher Naturforscher und Ärzte in Bad Nauheim (1921), Zeitschrift fürMathematischen und Naturwissenschaftlichen Unterricht, 52, pp. 79-84 (translated 81-82) Online [4]

• Hermann Weyl, Die Relativitätstheorie auf der Naturforscherversammlung in Bad Nauheim (1922), Jahresberichtder Deutschen Mathematiker-Vereinigung, 31, pp. 51-63. (translated 51, 58-63) Online [5]

General Discussion on the Theory of RelativityLenard: I was very pleased to have heard, that someone has spoken about the aether today. Yet I have to say, that

the simple mind of a natural scientist is in conflict with a theory, in which the theory of gravitation is extended from mass-proportional forces to other forces. I refer to the example of the decelerating train. To fulfill the relativity

principle, gravitational fields are imagined when non-mass-proportional forces are in use. I also like to say, that one can use two images for physical thought, that I denoted as images of first and second kind. For example, Mr. Weyl

spoke in terms of images of the first kind, as he expressed all processes by equations. By images of the second kind, the equations can be interpreted as processes in space. I would like to prefer the images of second kind, while Mr. Einstein remains still at images of first kind. Concerning images of the second kind the aether is indispensable. It











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The Bad Nauheim Debate 74

was always one of the most important auxiliaries for progress in natural science, and its rejection means the rejectionof the thinking of all natural scientists by images of the second kind. At first I would like to pose the question: How

does it happens, that it shall be indistinguishable according to the theory of relativity, whether (in the case of thedecelerating train) the train is decelerating or whether the world around is decelerating?

Einstein: It is for sure, that we observe effects in relation to the train, and if we want we can interpret them asinertial effects. The theory of relativity can interpret them as effects of a gravitational field as well. Now, where doesthis field come from? You think, that it is the invention of Mr. Relativity-Theoretician. But it is not a free invention,as it fulfills the same differential laws as those fields, which we are accustomed to interpret as the effects of masses.It is true, that something from this solution remains arbitrary, when one looks upon a limited part of the world. Thegravitational field that dominates relative to the decelerating train, corresponds to an induction effect that is causedby the distant masses. So I would like to summarize in short: The field is not arbitrarily invented, because it fulfills

the general differential equations and because it can reduced to the effect of all distant masses.Lenard: Mr. Einstein's explanations have told me nothing new; they also have not bridged the gap between the

images of first kind and the illustrative images of the second kind. I'm of the opinion, that the imagined gravitationalfields must correspond to processes, but those processes were not noticed by experience.

Einstein: I would like to say, that what humans consider as illustrative or not, has changed. The view aboutillustrativeness is so to speak a function of time. I'm of the opinion, that physics is conceptual, not illustrative. As an

example for the changing view about illustrativeness, I remind you of the view on illustrativeness of Galileanmechanics in different periods.

Lenard: I have expressed my opinion in the paper "On the Principle of Relativity, Aether, Gravitation", that theaether has failed in certain relations because it was not treated in the right way. The principle of relativity works witha non-euclidean space, that assumes different properties from place to place and in successive times; so there mightbe something in space whose conditions are the cause of those different properties, and this something is exactly theaether. I understand the usefulness of the relativity principle, as long as it is applied only to gravitational forces. For

forces that are not proportional to the mass, I consider it as invalid.Einstein: It lies in the nature of things, that one can only speak of the validity of the relativity principle, when it is

valid for all laws of nature.Lenard: Only if one imagines suitable fields. I think the relativity principle can only make new statements on

gravitation[6], since the gravitational fields that were included in the case of non-mass-proportional forces don't addany new point at all, except the one that gives the principle an apparent validation. Also the equality of all references

systems creates difficulties for the principle.Einstein: There is no coordinate system that is principally preferred by its simplicity; thus there is no method to

distinguish between "real" and "non-real" gravitational fields.Lenard:[7] My second question is: What says the relativity principle about the prohibited thought experiment, inwhich the earth is for example stationary, and the rest of the world rotates around the earth's axis, in which case

superluminal velocities occur?The first statement is not a claim, but a new definition of the term "aether".[8]

Einstein:[9] A thought experiment is an experiment, that is principally executable (although not factically). It servesto vividly summarize experiences, form which theoretical consequences can be drawn. A thought experiment is only

prohibited, when a realization is principally impossible.Lenard: I think it can be summarized: 1. That it's better if one refuses to proclaim the "rejection of the aether". 2.

That I still consider it necessary to restrict the relativity principle to a gravitational principle, and 3., thatsuperluminal velocities are apparently still problematic for the relativity principle; because they occur in the

relation[10] of any arbitrary body, as soon as one wants to ascribe it not to the body, but to the entire world, which isequally permitted according to the relativity principle in its most simple form that was used until now.


Rudolph: That the general theory of relativity is brilliantly confirmed, does not constitute a prove against the aether.The Einsteinian theory is correct, only its view on the aether is not correct. In addition, it will be only acceptable

together with Weyl's supplementation. But in this case, Einstein's theory even emerges from the aether hypothesis,when the gaps (that occur between the aether walls when they are moving) are forced to remain empty by the

centrifugal force due to direction changes.Palagyi: The discussion between Einstein and Lenard has made a deep impression upon me. One encounters here

again the old historic opposition between experimental and mathematical physics, as it already existed between, forexample, Faraday and Maxwell. Mr. Einstein says that there is no preferred coordinate system. But there is one. Let

me think biologically, then any man carries his coordinate system within himself. In the course of this thought, arefutation of the theory of relativity is contained.

Einstein alludes to the fact, that no opposition between theory and experiment exists.Born: The theory of relativity even prefers images of the second kind. I consider the example of the earth and the

sun. If the attraction wouldn't exist, the earth would flee in a straight line etc.Mie: I never understood, that the view in which the aether was essentially the same as seizable matter, should have

been only discovered by the theory of relativity. This was already done long before by Lorentz in his book "electricaland optical phenomena in moving bodies". Also Abraham in his textbook, in the time when he was still in opposition

to the theory of relativity, said: "The aether is empty space". I'm of the opinion, that even by approving Einstein'sgravitational theory, one has to distinguish between mere fictitious gravitational fields, that one introduces into theworld by the choice of coordinate system, and the real gravitational fields that are given by objective facts. I have

recently shown a way, how one arrives at a "preferred" coordinate system, in which all fields that are only fictitiousare excluded from the outset.

Einstein: I cannot see, why such an preferred coordinate system should exist. At most one could think to prefer suchcoordinate systems, in relation to which the Minkowskian expression for ds² is approximately true. But neglecting

the fact, that such systems do not exist for extended spaces, such coordinate system are surely not exact, but can onlybe defined approximately.

Kraus alludes to the epistemological difference between the images of first and second kind, by arguing that theimages of first kind are of higher value than the images of second kind.

Lenard: The principle of the center of gravity was introduced now; yet I think that this can be of no influence onprincipal questions.

The Discussion concerning the theory of relativityat the Meeting of Natural Scientists.

By Prof. Dr. H. Weyl (Zürich)At the Meeting of Natural Scientists in Bad Nauheim a session of the united mathematical and physical section

(September 23) was dedicated to the theory of relativity. Lectures by Weyl, Mie, von Laue and Grebe bear witnessof the current scientific work in the field of relativity theory. Besides the debate over these lectures, however, also a

general discussion concerning the principles took place, the became nearly exclusively a confrontation betweenEinstein and Lenard. One simply must conclude, that Lenard doesn't grasp the meaning of Einstein's theory;

therefore the opponents didn't find each other at all, the confrontation remained a sham fight and without result.The theory of relativity is based on two fundamental principles, that normally have nothing to do with each other;

these are 1. the relativity of simultaneity, 2. the relativity of motion. The first forces the fusion of space and time to a unified four-dimensional continuum, that is denoted by Minkowski as the "world". A world-point is a "here-now",

marked for example by a starting signal, a flashing and instantly dimming spark or some other event of minute spatial and temporal extension. My body is in any instant of my life at a certain world-(= space-time-)location; thus it traverses a one-dimensional succession of world points, a world line, as well as any other body. Who once saw a

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graphical driving plan, knows how to graphically illustrate such a world line (the world lines of trains that aredriving on a path). According to the old understanding, the spoken word "now" intersects not only the course of myinner life into past and future, but it brings this cut by a single stroke into the entire world: it intersects the world in asimilar way into two parts, that are without space between each other: the past and the future, like a horizontal plane

bisects the space into a lower and upper part. According to the relativity principle, however, the bisection ofpast-future is of a different kind when it is seen from world point O, and it corresponds to the one that in

three-dimensional space is caused by a complete circular cone (it is sketched in the vertical projection in the figure;the curved line is the world line of my body, that is of course bisected through O into two parts, the past part and thefuture part of my life). In the front cone all those world points are located that have an influence on my actions in O,

out of it are all the events, that lie finished behind, where "nothing can be changed": the mantle of the front coneseparates my active future from my active past. The border is formed by the fastest possible propagation of action atall: that of light. However, all those events are located in the back cone, for which I have either lively experiences or

by which I was informed somehow; only these events maybe have influenced me until now. In the exterior of it,however, all of that lies which I will experience or would experience, if my live lasts infinitely or my view might

penetrate everything; the mantle of the back-cone separates my passive past from my passive future. Between activefuture and passive past an empty world-area lies, with which I'm neither actively nor passively connected in the

instant O. — The stage of reality is not a stationary three-dimensional space in which things are engaged in temporalprogression, but a four-dimensional world in which time and space are inseparably connected to each other. This

objective world does not happen, but it exists; a four-dimensional continuum, but neither space nor time. Only in theview of the consciousness that crawls upon within the world-lines of the bodies, a section of this world "lives up"

and passes by as an image that undergoes spatial and temporal changes.The fusion of space and time that is taught by the theory of relativity, was not attacked at the Nauheim discussion. Only the second point was concerned, the relativity of motion. It lies in the essence of the expression "motion", that

one can only speak about motion in relation to a solid reference body (such a role is played by the "lasting and well-founded earth" in ordinary live). By itself, none of the possible states of motion of a body is preferred over

other ones, i.e. so preferred that it would deserve the name "stationary". This evident principle is seemingly contradicted in a clear way by experience. Experience shows, that centrifugal forces arise at a "rotating" flywheel which stresses the flywheel - maybe until breaking; but in the "stationary" case nothing can be experienced. In an immediately stopped train everything falls apart; why, asks Lenard, is the church steeple not falling into pieces as

well, as it experiences an equally strong motion pressure relative to the train, as the train relative to the church steeple. The old solution of the dilemma as defended by Lenard in Nauheim is as follows: If we attribute to space, independent of all matter by which it is filled, a certain geometrical structure by which (in particular) straight lines can be distinguished form curved ones, then the same is true for the world. The Galilean principle of inertia says,

that a body that is not influenced from outside, executes a motion with a straight line as its world line. While according to Einstein: It can not be denied of course, and it comes clearly from the Galilean principle of Inertia, that there must exist some sort of "guidance" in the world which forces a very certain "natural" motion upon the body,

when one brings it into motion with a certain direction and a certain velocity. But this "guidance" is a physical force-field, exactly like the electrical field from which electrical forces do arise. Together with inertial forces,

something comes into light which was until now be interpreted as the "geometrical structure": the "guidance field", as something real, as an acting power of shaking force in some cases. Therefore it impossibly can be a formal or

given condition of the world independent of matter and its states; on the contrary, the guidance field must interact with matter and must change when the state of matter is changing. In the development of this thought it became clear

- and in this lies the success of Einstein's theory - that what was known as gravitation until now, is only a partial phenomenon of the guidance field. The planetary motion follows the path that is given by the guidance; while in the old mechanics the motion doesn't comply with the Galilean principle, but is diverted from its path by a special force that was ad-hoc invented by Newton, the "force of gravity". However, according to Einstein, inertia and gravitation constitute a inseparable unity; therefore inertial and gravitational masses must necessarily be the same; earlier it was


impossible to understand the meaning of this connection.To Einstein's principles of relativity of simultaneity and relativity of motion, the principle of relativity of magnitude

was added by me. It makes it possible to derive the electromagnetic phenomena (besides gravitation), withoutintroducing a specific electric or magnetic force whose laws have to be taken from experience. I think, only by this

extension the theory of relativity has found its natural completion; but at the moment it is disputed by an even higherextent than Einstein's theory.

Response by Prof. Dr. E. Gehrcke.The report that was given by Weyl (Umschau, October 23, 1920; p. 610) on the relativity session in Nauheim

requires an extension in various ways.A not completely unimportant point that arose in the Nauheim meeting with remarkable clarity, has to be

supplemented to Mr. Weyl's report: In the discussion Einstein has in fact unequivocally expressed his disapproval ofWeyl's theory and has given an explanation, that a theory that is based on the pure mathematical demands of

symmetry, like that of Weyl, has to be rejected. — When Mr. Weyl tries to bring his considerations more closer tothe public, then such an interesting point like the statement of Einstein concerning Weyl's theory should not remain

unmentioned by him, so that in the public from the outset no false opinion can occur, how the creator of the theory ofrelativity stands to the specific relativism of Weyl.

Mr. Weyl thinks that he is allowed to conclude, that Lenard hasn't grasped the meaning of the theory of relativity.That is only a rebuttal to a statement of Lenard in the meeting in Nauheim, that the relativists had not shown anyunderstanding for the requirements of reality search in physics, and that they undertake no attempt to bridge this

"gap". Weyl should consider, that even when someone possesses as a mathematician a virtuous skill in usingmathematical symbols, he might show a lack of understanding for other abstractions than the quantitative relations ofmathematics, of which more universally gifted natures are free. By using Weyl's papers it might be easy to create a

list of epistemological errors and conceptual confusions; in this connection it shall be alluded to the recentlypublished paper by Rikpe-Kühn: Kant contra Einstein.

The point that was more closely examined by Mr. Weyl concerning the discussion between Einstein and Lenard,namely the example of the decelerated train, lacks an essential objection that was more closely explained by Lenard,

that for the creation of a gravitational field according to our current knowledge, some masses must be present thatgenerate the gravitational field. In the case of the train accident, where according to the relativists not the train, but

the entire surrounding should have been decelerated, no formation of masses or nothing that might produce thegravitational field that decelerates the surrounding can be seen. The relativist was therefore forced in Nauheim toexplicitly assume gravitational fields without gravitating masses that generate them. Yet besides other things it

remained unanswered, where this energy of this gravitational field is coming from. From all of those things nothingwas mentioned by Weyl.

Finally the discussion in Nauheim has brought Einstein to the explanation, that according to the general theory ofrelativity, bodies can have any arbitrary velocity faster than that of light. Also this case, with its conclusions that will

not treated further at this place, was not mentioned by Weyl. So the debate in Nauheim was in no way "withoutresult".


Response to Prof. Dr. Gehrckeby Prof. Dr. H. Weyl.

I cannot leave unchallenged the remarks by Mr. Gehrcke concerning my "Umschau"-article on the theory ofrelativity. It do not care of the first purely demagogic part of his statements, in which he brings the authority of

Einstein's name against me into this field (of all people Mr. Gehrcke, according to whom this authority is only basedon advertising and suggestion!) and who presented me to the public as an "Only-Virtuoso" of the mathematical

technique and (with reference to Ripke-Kühn) as a philosophical scatterbrain. I only want to conclude, that Einsteinand I are indeed not of the same opinion concerning my extension of the theory of relativity, and regarding the

relations to philosophy, I wish to allude the reader of "Umschau" to the paper "Zur Einsteinschen Relativitätstheorie"by E. Cassirer, the member of Neo-Kantianism who is most competent for this confrontation.

It followed from my article (which was the result of the invitation of the editor of Umschau) that I didn't wanted togive a report of the Nauheim discussion, but following this discussion to highlight the two points of relativity theory,which I consider as decisive. But also, if I had that other intention, I wouldn't been able to report on the "results" that

were mentioned by Mr. Gehrcke. It cannot be spoken about, that Einstein's theory would be forced to assumegravitational fields without masses that generate them. The "fictitious gravitational fields" that were alway

mentioned by Lenard as well, are only necessary when the law by which the masses generate the guidance field, ismisinterpreted so as if everything happens the same way in any reference system as in Newton's "absolute space".

Such a formulation is impossible, when one really has understood the unity of inertia and gravitation that was calledby me the "guidance field". Also inertia, the "Galilean guidance field" that brings a body to fly in a straight line with

constant velocity, and which according to Newton-Lenard exists once and for all time without material cause, hasaccording to Einstein its cause in the generating masses. Namely, this neutral basis of gravitation is caused by the

collective action of all masses in the universe; analogous to, for example, the charges on the plates of a capacitor thatcause the homogeneous electrical field between the plates, from which the field in the immediate surrounding of the

excited electrons are lifted out as small steep conical mountains from a plane. Here as well as there it is of courseimpossible to separate in a strict manner the homogeneous field that is caused by the collective action of all particles

(the "Galilean guidance field"), from the conical mountains the belong to any single particle ("gravitation"). InNauheim also Einstein alluded to this solution, which I can only sketch at this place, and which removes the great

cosmological difficulties with which Newton's theory of gravitation has to struggle.Also the second "result" that was constructed by Mr. Gehrcke, is only a repetition of old misunderstandings; I take

on any guarantee, that in Nauheim no "relativist" has admitted the possibility of body motions by superluminalvelocity. Lenard, as his closing words in the discussion show, heard something like that in the words of Einstein; but

completely erroneously.Zürich.

On the Theory of Relativity.by Prof. Dr. E. Gehrcke

I want to express at this place that Einstein, at the Nauheim meeting of natural scientists, has admitted the possibilityof superluminal velocities from the standpoint of his general theory of relativity. When Mr. Weyl thinks that he can

deny that, then only another contradiction between him and Einstein - at least at the time of the Nauheim meeting - isto be stated. The explanation of Einstein concerning superluminal velocities, as unsatisfactory as it may by, has

actually been given, and it would have been better for Mr. Weyl to verify the evidence before assuming an error ofLenard.


The 86. Meeting of the German Natural ScientistsBy K. Körner

The debate didn't deviate from the objective and factual stream, into which it was directed by the readers and thevigorous chairman. At first a dialog between Lenard and Einstein evolved, the first lively presenting his objections,

the second responding calmly and with extraordinary clarity.While Lenard accepts the special theory of relativity, he rejects the general theory. He denies, that the aether wasabolished by the theory of relativity; it cannot vanish at all, as it is the only thing that makes it possible for us to

understand optical and electrical phenomena in an illustrative way. Other reasons for contention were for him thefictitious gravitational fields introduced by Einstein, and the termination of the "common sense" which was

necessary for both the physicists and the (in the scientific sense) naive man. When e.g. in an immediately stoppedtrain everything is falling around, then by the common sense one can decide, whether the change of the train'smotion or that of the earth is the cause for it. If one additionally thinks, that one cannot say whether we are in

rotational motion, or the environment - then one only has to assume the latter, to see that one has to admitsuperluminal velocities. That was a result, to which the theory of relativity leads by itself, and which contradicts its

own assumption, that the speed of light cannot be superseded. The theory of relativity kills itself!On the other hand, Einstein stresses that illustrativeness is a changing concept. The Galilean mechanics is for us thehighest point of illustrativeness, while it was very non-illustrative for Galileo's contemporaries. And in the present

we find electricians, for whom nothing is more illustrative than the electric field, and for whom the electricalphenomena even become images for mechanical ones. Thus one cannot use such a changing concept for or against

the theory. To the example of the decelerating train he remarks, that this is without any doubt an interaction betweenmasses, and for the success it is irrelevant, which mass is moved against the other. To let decide the "common sense"in this question, is no less problematic as it was before in respect to illustrativeness. To the example of the rotationalmotion it has to be said, that the role of the speed of light in the general theory of relativity is completely different as

in the special theory, and that the first requires no constant speed of light at all. — Nearly all other speakers in thedebate agreed with Einstein in the essential points - for example von Laue, Mie (who responded to Lenard that theaether was abolished not only by the theory of relativity, but already three decades earlier by H. A. Lorentz) and

particularly inspired by Born, who feels attracted to Einstein's theory just because of its illustrativeness.

The Theory of Relativity at the Meeting of Natural Scientists in Bad NauheimBy H. Weyl in Zürich.

On occasion of the German Mathematical Society, at the meeting of natural scientists in Bad Nauheim last year in acombined session of the mathematical and physical section, the theory of relativity was the center of a series of

lectures and a general discussion; about that a report should be given at this place - after a considerable long timewhich, however, is maybe beneficial for clarification and calm judgment of the facts.

The first part of the session was constituted by four lectures in the field of the theory of relativity: 1. H. Weyl,Elektrizität und Gravitation; 2. G. Mie, Das elektrische Feld eines um ein Gravitationszentrum rotierenden geladenenPartikelchens; 3. M. v. Laue, Theoretisches über neuere optische Beobachtungen zur Relativitätstheorie; 4. L. Grebe,

Über die Gravitationsverschiebungen der Fraunhoferschen Linien. The four lectures were followed by "special"discussions that were related to the content of the lecture. The last and most dramatic part, the general discussion

concerning the theory of relativity, was essentially a duel between Einstein and Lenard. Planck fulfilled his duty aschairman with great skill, strength and impartiality; it was mostly thanks to him, that this "Nauheim Relativity

Discussion", in which opposing epistemological fundamental-views of science were confronted with each other, hastaken a worthy development.

...[11]


The two last points that were discussed before were also mentioned in the general discussion (that was mostly usedby Lenard) between Lenard and Einstein. For clarities sake it may be allowed, to separate two additional

controversies from this dialog, that are of minor importance in respect to the main difference which will be discussedat the end.

At first it's about the existence of the aether. Lenard thinks that Einstein in the course of formulating the specialtheory of relativity, was much to rush to proclaim the rejection of the aether. Indeed he can refer to the fact, that

Einstein again speaks of an aether in the general theory of relativity.[12] However, one shall not be deceived by theequal terminology about the factual difference! The old aether of the light theory was a substantial medium, athree-dimensional continuum, from which any place P in any time t is located at a certain space point p (or at a

specific world location); the recognizability of the same aether location at different times is essentially for it. By thisaether the four-dimensional world disintegrates into a three times infinite continuum of one-dimensional world lines;

consequently it allows to absolutely distinguish between at rest and in motion. Only in this sense (somethingdifferent was never claimed by Einstein) the aether was rejected by the theory of special relativity; it was replaced bythe affine geometric structure of the world, that doesn't determine the difference between "at rest" and "in motion",

but separates uniform motion from all other motions. The substantial aether was considered by its inventors assomething real, that is comparable to ponderable bodies. In Lorentzian electrodynamics it was transformed into apure geometrical, i.e. a forever solid structure that won't be influenced by matter. In Einstein's special theory of

relativity another one took its place, the affine geometric structure. In the general theory of relativity eventually thelatter was, as an "affine connection" or "guidance field", transformed back into a state-field of physical reality, that is

in effective connection with matter. And therefore Einstein considered it convenient, to reintroduce the old wordaether for the completely changed concept; whether this was useful or not, is less a physical than a philological

question.Second: the superluminal velocity. Lenard thinks, that the general theory of relativity reintroduces superluminal

velocities, since it allows e.g. the rotating earth as reference frame; so that superluminal velocities occur insufficiently great distances. This is an obvious misunderstanding. If are the space coordinates that are

measured in respect to the rotating earth, is the corresponding "time" (its precise definition isn't relevant now),then the coordinate lines upon which at constant only varies, won't all have a time-like direction,i.e. in those coordinates there isn't everywhere . Einstein claims indeed, that such a coordinate system is

valid; also in those coordinate systems his general invariant gravitational laws are valid. However, he insists that theworld line of a material body always has a time-like direction, so that concerning a material body (and its "signal

velocity") no superluminal velocity can occur. A coordinate system of the kind mentioned before, cannot berepresented in its entire extension by a "reference mollusk", i.e. one cannot think about a material medium, whose

single elements follow the coordinate lines of that coordinate system as world lines. —But it's time now, that I come to speak about the decisive difference between Lenard and Einstein. Lenard claims, that Einstein's theory operates with fictitious gravitational fields, to which no generating mass can be found and

which was only introduced for the sake of the relativity principle. The illustrative example by Lenard, i.e. the train that is immediately stopped by an oncoming train, should serve as the basis of the discussion. Why, asks Lenard,

falls the train into pieces but not the church steeple next to the train, although according to Einstein it can be said of the church steeple with the same justification as of the train, that it became decelerated? It seems to me that the

answer to this is simple. In Einstein's theory like in the older interpretation there is a guidance field that is followed by a body according to the Galilean principle, as long as no forces act on it. The catastrophe happens at the train and not at the church steeple, since the first is thrown out of its path by the molecular forces of the oncoming train, but

the church steeple is not. This answer is also completely in agreement with "commons sense", which whole-heartedly agrees that the inertial-tendency of the guidance field (that opposes to the forces) can be seen as a

physical reality. But the question is now: is the guidance field a single unity, or can the two components "inertia" and "gravitation" principally be separated, so that only the first exists once and forever as the affine linear structure of the

four-dimensional world, and only the second is produced by matter? Here, concerning the equality of all types of


motion, the state of affairs is quite analogues as for the equality of all directions in space. According to Democrit,there exists an absolute "above-below" per se; the real falling direction of a body is composed of its absolute

direction and the deviation from it that stems from a physical cause. For example, Democrit could argue againstNewton (who sees the fall direction as an unity) in the same way as Lenard against Einstein: If another direction asthe true one is installed as the normal direction, then one has to introduce in addition to it and the real deviation, a

third fictitious deviation that is everywhere the same and is not connected with matter; and this is only to satisfy theprinciple of the equality of all directions in space. As soon as one admits an absolute direction above-below, one can

distinguish between real and fictitious deviations; as soon as one admits a preferred, "rational" coordinate system,then one has (with Mie and Lenard) to distinguish between real and fictitious gravitational fields. However, from the

relativity standpoint such a separation is impossible. But when we, with Newton and against Democrit, claim theindecomposability of the real falling direction into an absolute above-below and a deviation from it, then we must

provide a physical cause not only for the deviation, but also for the falling direction as a whole; in the same way alsoEinstein has the duty to show, how and by which law the guidance field as a whole is produced by matter. This was

required by Lenard with full justification from him, and that is the deepest and essentially decisive point of hisobjections. It must openly be admitted, that there still exist serious difficulties for the theory of relativity in its

current formulation. To answer, Einstein alludes to his cosmology of the spatial closed world; and he responds toLenard: The field is not arbitrarily invented, since it fulfills the general differential equations and because it can bereduced to the effect of all distant masses. As long as one adheres to the difference of matter and field at all (only

then the requirement, that matter generates the field, is meaningful and valid), then Einstein's cosmology means, thatbesides the inner seams of the field upon which the single matter particles act so that they dominate the field, noother infinitely distant seam (that act as an agent that determines the field in the infinite) is added; the entirety ofdistant masses took its place. The co-rotation of the plane of Foucault's pendulum with the fixed stars makes that

very clear. However, the difficulty is not solved yet. At first it is to say, that by Einstein only the laws are given, thatbind the inner differential connection of the field, but there is no clear formulation of the laws, by which matter

determines the field (by the way, that is not essentially different in the case of the electromagnetic field). But secondand especially it is completely excluded that matter can uniquely determine the field, if one considers mass, charge,

and state of motion as the characteristics of matter. Because one can introduce such an coordinate system into theworld, that by the following projection of the world on a four-dimensional Cartesian image space, not only the worldchannel of one particle, but of all particles assume a simultaneously given form, e.g. that all those channels become

vertical straight lines. Compared to Mach, whose reference body is always a rigid body, Einstein's coordinate systembecame so "softened", that it can simultaneously cling to the motion of all bodies, so that one can transform all

particles to rest at the same time; thus it has no meaning anymore, to even speak about the relative state of motion ofdifferent bodies against each other. This problem was recently highlighted by Reichenbächer more clearly.[13] The

principle, that matter generates the field, can afterwards only be upheld, when within the concept of motion adynamical factor is integrated; so the analysis of the concept of motion is not about the contrast between absolute or

relative, but between kinematics or dynamics. —In the second meeting at the other day, F. P. Liesegang (Düsseldorf) showed some excellent graphs for the

illustration of the space-time relations in special relativity, and H. Dingler (Munich) read, as it appeared only as aformal protest against the relativity theory and without noticing the public, his critical remarks concerning the

fundamentals of the theory; it is strange, that in Dingler the Poincaré-oriented conventionalism is connected with adogmatical stubbornness of the born aprioristic philosopher. That the tragedy is not missing a the end of the satyrplay, Mr. Rudolph developed a fantastic aether theory with "gaps" between streaming aether walls, star lines, etc.

and with their aid he (out of nothing) determined the mass of the sun up to an arbitrary number of decimals...I have tried in a free way to allude to the questions, that came to speak at the Nauheim discussion, but not to give an

objective report about the development of the session; for a shortened, but equivalent reproduction of the lecturesand the discussion, the reader is referred to the issue of December 1920 of the Physikalische Zeitschrift.


[1] http:/ / hdl. handle. net/ 2027/ mdp. 39015086723080[2] http:/ / hdl. handle. net/ 2027/ mdp. 39015023538898[3] http:/ / hdl. handle. net/ 2027/ mdp. 39015039295228[4] http:/ / hdl. handle. net/ 2027/ mdp. 39015035226052[5] http:/ / resolver. sub. uni-goettingen. de/ purl?GDZPPN002126729[6][6] "gravitation" presumably means in this context "mass-proportional" gravitational fields generated by real masses, as opposed to the

"non-mass proportional" forces that follow after the comma.[7][7] The name of is not present in the original version at this place, but it seems to be clear that the following question was posed by him.[8][8] It's unclear to whom this line belongs. However, the "new definition of the term aether" evidently refers to 's new definition in his Leyden

speech, as it was discussed by below.[9][9] The name of Einstein is not present in the original version at this location, but as this is the answer to the question above, it seems to belong to

.[10][10] The word "Relation" presumably means "relative motion" in this context. See also the discussion on superluminal velocity given by below.[11][11] p. 51-58 are omitted in this translation, as they contain the discussion of the lectures by , , , and . It follows 's description of the "general

discussion" between and , in which two critical points presented by (preferred reference frame, relativity of acceleration) are concludinglydiscussed as well.

[12] See especially the Leiden inauguration speech of concerning Ether and the Theory of Relativity, Springer 1920[13][13] Schwere und Trägheit, Physik. Zeitschr. 22 (1921), p. 234-243

Notes by Wikisource


The author died in 1955, so this work is also in the public domain in countries and areas where the copyright term is the author's lifeplus 50 years or less. This work may also be in the public domain in countries and areas with longer native copyright terms that apply

the rule of the shorter term to foreign works.

Geometry and Experience

Geometry and Experience (1922) by Albert Einstein, translated by George Barker Jeffery and Wilfrid Perrett


From: Sidelights on Relativity (1922), pp. 25-56, London: Methuen, Online [1]

German original: Geometrie und Erfahrung (1921), Berlin: Springer

GEOMETRY AND EXPERIENCEAn expanded form of an Address tothe Prussian Academy of Sciencesin Berlin on January 27th, 1921.

One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certainand indisputable, while those of all other sciences are to some extent debatable and in constant danger of beingoverthrown by newly discovered facts. In spite of this, the investigator in another department of science would notneed to envy the mathematician if the laws of mathematics referred to objects of our mere imagination, and not toobjects of reality. For it cannot occasion surprise that different persons should arrive at the same logical conclusionswhen they have already agreed upon the fundamental laws (axioms), as well as the methods by which other laws areto be deduced therefrom. But there is another reason for the high repute of mathematics, in that it is mathematicswhich affords the exact natural sciences a certain measure of security, to which without mathematics they could notattain.





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Geometry and Experience 83

At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be thatmathematics, being after all a product of human thought which is independent of experience, is so admirablyappropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able tofathom the properties of real things.In my opinion the answer to this question is, briefly, this: — As far as the laws of mathematics refer to reality, theyare not certain; and as far as they are certain, they do not refer to reality. It seems to me that complete clearness as tothis state of things first became common property through that new departure in mathematics which is known by thename of mathematical logic or "Axiomatics." The progress achieved by axiomatics consists in its having neatlyseparated the logical formal from its objective or intuitive content; according to axiomatics the logical-formal aloneforms the subject-matter of mathematics, which is not concerned with the intuitive or other content associated withthe logical-formal.Let us for a moment consider from this point of view any axiom of geometry, for instance, the following: —Through two points in space there always passes one and only one straight line. How is this axiom to be interpretedin the older sense and in the more modern sense?The older interpretation: — Every one knows what a straight line is, and what a point is. Whether this knowledgesprings from an ability of the human mind or from experience, from some collaboration of the two or from someother source, is not for the mathematician to decide. He leaves the question to the philosopher. Being based upon thisknowledge, which precedes all mathematics, the axiom stated above is, like all other axioms, self-evident, that is, itis the expression of a part of this à priori knowledge.The more modern interpretation: — Geometry treats of entities which are denoted by the words straight line, point,etc. These entities do not take for granted any knowledge or intuition whatever, but they presuppose only the validityof the axioms, such as the one stated above, which are to be taken in a purely formal sense, i.e. as void of all contentof intuition or experience. These axioms are free creations of the human mind. All other propositions of geometryare logical inferences from the axioms (which are to be taken in the nominalistic sense only). The matter of whichgeometry treats is first defined by the axioms. Schlick in his book on epistemology has therefore characterisedaxioms very aptly as "implicit definitions."This view of axioms, advocated by modern axiomatics, purges mathematics of all extraneous elements, and thusdispels the mystic obscurity which formerly surrounded the principles of mathematics. But a presentation of itsprinciples thus clarified makes it also evident that mathematics as such cannot predicate anything about perceptualobjects or real objects. In axiomatic geometry the words "point," "straight line," etc., stand only for empty conceptualschemata. That which gives them substance is not relevant to mathematics.Yet on the other hand it is certain that mathematics generally, and particularly geometry, owes its existence to theneed which was felt of learning something about the relations of real things to one another. The very word geometry,which, of course, means earth-measuring, proves this. For earth-measuring has to do with the possibilities of thedisposition of certain natural objects with respect to one another, namely, with parts of the earth, measuring-lines,measuring-wands, etc. It is clear that the system of concepts of axiomatic geometry alone cannot make any assertionsas to the relations of real objects of this kind, which we will call practically-rigid bodies. To be able to make suchassertions, geometry must be stripped of its merely logical-formal character by the co-ordination of real objects ofexperience with the empty conceptual frame-work of axiomatic geometry. To accomplish this, we need only add theproposition: — Solid bodies are related, with respect to their possible dispositions, as are bodies in Euclideangeometry of three dimensions. Then the propositions of Euclid contain affirmations as to the relations ofpractically-rigid bodies.Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch ofphysics. Its affirmations rest essentially on induction from experience, but not on logical inferences only. We willcall this completed geometry "practical geometry," and shall distinguish it in what follows from "purely axiomaticgeometry." The question whether the practical geometry of the universe is Euclidean or not has a clear meaning, and


its answer can only be furnished by experience. All linear measurement in physics is practical geometry in this sense,so too is geodetic and astronomical linear measurement, if we call to our help the law of experience that light ispropagated in a straight line, and indeed in a straight line in the sense of practical geometry.I attach special importance to the view of geometry which I have just set forth, because without it I should have beenunable to formulate the theory of relativity. Without it the following reflection would have been impossible: — In asystem of reference rotating relatively to an inert system, the laws of disposition of rigid bodies do not correspond tothe rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inert systems we mustabandon Euclidean geometry. The decisive step in the transition to general co-variant equations would certainly nothave been taken if the above interpretation had not served as a stepping-stone. If we deny the relation between thebody of axiomatic Euclidean geometry and the practically-rigid body of reality, we readily arrive at the followingview, which was entertained by that acute and profound thinker, H. Poincaré: — Euclidean geometry isdistinguished above all other imaginable axiomatic geometries by its simplicity. Now since axiomatic geometry byitself contains no assertions as to the reality which can be experienced, but can do so only in combination withphysical laws, it should be possible and reasonable — whatever may be the nature of reality — to retain Euclideangeometry. For if contradictions between theory and experience manifest themselves, we should rather decide tochange physical laws than to change axiomatic Euclidean geometry. If we deny the relation between thepractically-rigid body and geometry, we shall indeed not easily free ourselves from the convention that Euclideangeometry is to be retained as the simplest. Why is the equivalence of the practically-rigid body and the body ofgeometry — which suggests itself so readily — denied by Poincare and other investigators? Simply because undercloser inspection the real solid bodies in nature are not rigid, because their geometrical behaviour, that is, theirpossibilities of relative disposition, depend upon temperature, external forces, etc. Thus the original, immediaterelation between geometry and physical reality appears destroyed, and we feel impelled toward the following moregeneral view, which characterizes Poincaré's standpoint. Geometry (G) predicates nothing about the relations of realthings, but only geometry together with the purport (P) of physical laws can do so. Using symbols, we may say thatonly the sum of (G) + (P) is subject to the control of experience. Thus (G) may be chosen arbitrarily, and also partsof (P); all these laws are conventions. All that is necessary to avoid contradictions is to choose the remainder of (P)so that (G) and the whole of (P) are together in accord with experience. Envisaged in this way, axiomatic geometryand the part of natural law which has been given a conventional status appear as epistemologically equivalent.Sub specie aeterni Poincaré, in my opinion, is right. The idea of the measuring-rod and the idea of the clockco-ordinated with it in the theory of relativity do not find their exact correspondence in the real world. It is also clearthat the solid body and the clock do not in the conceptual edifice of physics play the part of irreducible elements, butthat of composite structures, which may not play any independent part in theoretical physics. But it is my convictionthat in the present stage of development of theoretical physics these ideas must still be employed as independentideas; for we are still far from possessing such certain knowledge of theoretical principles as to be able to give exacttheoretical constructions of solid bodies and clocks.Further, as to the objection that there are no really rigid bodies in nature, and that therefore the properties predicatedof rigid bodies do not apply to physical reality, this objection is by no means so radical as might appear from a hastyexamination. For it is not a difficult task to determine the physical state of a measuring-rod so accurately that itsbehaviour relatively to other measuring-bodies shall be sufficiently free from ambiguity to allow it to be substitutedfor the "rigid" body. It is to measuring-bodies of this kind that statements as to rigid bodies must be referred.All practical geometry is based upon a principle which is accessible to experience, and which we will now try torealise. We will call that which is enclosed between two boundaries, marked upon a practically-rigid body, a tract.We imagine two practically-rigid bodies, each with a tract marked out on it. These two tracts are said to be "equal toone another" if the boundaries of the one tract can be brought to coincide permanently with the boundaries of theother. We now assume that:If two tracts are found to be equal once and anywhere, they are equal always and everywhere.


Not only the practical geometry of Euclid, but also its nearest generalisation, the practical geometry of Riemann, andtherewith the general theory of relativity, rest upon this assumption. Of the experimental reasons which warrant thisassumption I will mention only one. The phenomenon of the propagation of light in empty space assigns a tract,namely, the appropriate path of light, to each interval of local time, and conversely. Thence it follows that the aboveassumption for tracts must also hold good for intervals of clock-time in the theory of relativity. Consequently it maybe formulated as follows: — If two ideal clocks are going at the same rate at any, time and at any place (being thenin immediate proximity to each other), they will always go at the same rate, no matter where and when they are againcompared with each other at one place. — If this law were not valid for real clocks, the proper frequencies for theseparate atoms of the same chemical element would not be in such exact agreement as experience demonstrates. Theexistence of sharp spectral lines is a convincing experimental proof of the above-mentioned principle of practicalgeometry. This is the ultimate foundation in fact which enables us to speak with meaning of the mensuration, inRiemann's sense of the word, of the four-dimensional continuum of space-time.The question whether the structure of this continuum is Euclidean, or in accordance with Riemann's general scheme,or otherwise, is, according to the view which is here being advocated, properly speaking a physical question whichmust be answered by experience, and not a question of a mere convention to be selected on practical grounds.Riemann's geometry will be the right thing if the laws of disposition of practically-rigid bodies are transformable intothose of the bodies of Euclid's geometry with an exactitude which increases in proportion as the dimensions of thepart of space-time under consideration are diminished.It is true that this proposed physical interpretation of geometry breaks down when applied immediately to spaces ofsub-molecular order of magnitude. But nevertheless, even in questions as to the constitution of elementary particles,it retains part of its importance. For even when it is a question of describing the electrical elementary particlesconstituting matter, the attempt may still be made to ascribe physical importance to those ideas of fields which havebeen physically defined for the purpose of describing the geometrical behaviour of bodies which are large ascompared with the molecule. Success alone can decide as to the justification of such an attempt, which postulatesphysical reality for the fundamental principles of Riemann's geometry outside of the domain of their physicaldefinitions. It might possibly turn out that this extrapolation has no better warrant than the extrapolation of the ideaof temperature to parts of a body of molecular order of magnitude.It appears less problematical to extend the ideas of practical geometry to spaces of cosmic order of magnitude. Itmight, of course, be objected that a construction composed of solid rods departs more and more from ideal rigidity inproportion as its spatial extent becomes greater. But it will hardly be possible, I think, to assign fundamentalsignificance to this objection. Therefore the question whether the universe is spatially finite or not seems to medecidedly a pregnant question in the sense of practical geometry. I do not even consider it impossible that thisquestion will be answered before long by astronomy. Let us call to mind what the general theory of relativity teachesin this respect. It offers two possibilities: —1. The universe is spatially infinite. This can be so only if the average spatial density of the matter in universal space,concentrated in the stars, vanishes, i.e. if the ratio of the total mass of the stars to the magnitude of the space throughwhich they are scattered approximates indefinitely to the value zero when the spaces taken into consideration areconstantly greater and greater.2. The universe is spatially finite. This must be so, if there is a mean density of the ponderable matter in universalspace differing from zero. The smaller that mean density, the greater is the volume of universal space.I must not fail to mention that a theoretical argument can be adduced in favour of the hypothesis of a finite universe.The general theory of relativity teaches that the inertia of a given body is greater as there are more ponderable masses in proximity to it; thus it seems very natural to reduce the total effect of inertia of a body to action and reaction between it and the other bodies in the universe, as indeed, ever since Newton's time, gravity has been completely reduced to action and reaction between bodies. From the equations of the general theory of relativity it can be deduced that this total reduction of inertia to reciprocal action between masses — as required by E. Mach, for


example — is possible only if the universe is spatially finite.On many physicists and astronomers this argument makes no impression. Experience alone can finally decide whichof the two possibilities is realised in nature. How can experience furnish an answer? At first it might seem possible todetermine the mean density of matter by observation of that part of the universe which is accessible to ourperception. This hope is illusory. The distribution of the visible stars is extremely irregular, so that we on no accountmay venture to set down the mean density of star-matter in the universe as equal, let us say, to the mean density inthe Milky Way. In any case, however great the space examined may be, we could not feel convinced that there wereno more stars beyond that space. So it seems impossible to estimate the mean density.But there is another road, which seems to me more practicable, although it also presents great difficulties. For if weinquire into the deviations shown by the consequences of the general theory of relativity which are accessible toexperience, when these are compared with the consequences of the Newtonian theory, we first of all find a deviationwhich shows itself in close proximity to gravitating mass, and has been confirmed in the case of the planet Mercury.But if the universe is spatially finite there is a second deviation from the Newtonian theory, which, in the language ofthe Newtonian theory, may be expressed thus: — The gravitational field is in its nature such as if it were produced,not only by the ponderable masses, but also by a mass-density of negative sign, distributed uniformly throughoutspace. Since this factitious mass-density would have to be enormously small, it could make its presence felt only ingravitating systems of very great extent.Assuming that we know, let us say, the statistical distribution of the stars in the Milky Way, as well as their masses,then by Newton's law we can calculate the gravitational field and the mean velocities which the stars must have, sothat the Milky Way should not collapse under the mutual attraction of its stars, but should maintain its actual extent.Now if the actual velocities of the stars, which can, of course, be measured, were smaller than the calculatedvelocities, we should have a proof that the actual attractions at great distances are smaller than by Newton's law.From such a deviation it could be proved indirectly that the universe is finite. It would even be possible to estimateits spatial magnitude.Can we picture to ourselves a three-dimensional universe which is finite, yet unbounded?The usual answer to this question is "No," but that is not the right answer. The purpose of the following remarks is toshow that the answer should be "Yes." I want to show that without any extraordinary difficulty we can illustrate thetheory of a finite universe by means of a mental image to which, with some practice, we shall soon growaccustomed.First of all, an obervation of epistemological nature. A geometrical-physical theory as such is incapable of beingdirectly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicityof real or imaginary sensory experiences into connection in the mind. To "visualise" a theory, or bring it home toone's mind, therefore means to give a representation to that abundance of experiences for which the theory suppliesthe schematic arrangement. In the present case we have to ask ourselves how we can represent that relation of solidbodies with respect to their reciprocal disposition (contact) which corresponds to the theory of a finite universe.There is really nothing new in what I have to say about this; but innumerable questions addressed to me prove thatthe requirements of those who thirst for knowledge of these matters have not yet been completely satisfied. So, willthe initiated please pardon me, if part of what I shall bring forward has long been known?What do we wish to express when we say that our space is infinite? Nothing more than that we might lay anynumber whatever of bodies of equal sizes side by side without ever filling space. Suppose that we are provided witha great many wooden cubes all of the same size. In accordance with Euclidean geometry we can place them above,beside, and behind one another so as to fill a part of space of any dimensions; but this construction would never befinished; we could go on adding more and more cubes without ever finding that there was no more room. That iswhat we wish to express when we say that space is infinite. It would be better to say that space is infinite in relationto practically-rigid bodies, assuming that the laws of disposition for these bodies are given by Euclidean geometry.


Another example of an infinite continuum is the plane. On a plane surface we may lay squares of cardboard so thateach side of any square has the side of another square adjacent to it. The construction is never finished; we canalways go on laying squares — if their laws of disposition correspond to those of plane figures of Euclideangeometry. The plane is therefore infinite in relation to the cardboard squares. Accordingly we say that the plane is aninfinite continuum of two dimensions, and space an infinite continuum of three dimensions. What is here meant bythe number of dimensions, I think I may assume to be known.Now we take an example of a two-dimensional continuum which is finite, but unbounded. We imagine the surface ofa large globe and a quantity of small paper discs, all of the same size. We place one of the discs anywhere on thesurface of the globe. If we move the disc about, anywhere we like, on the surface of the globe, we do not come upona limit or boundary anywhere on the journey. Therefore we say that the spherical surface of the globe is anunbounded continuum. Moreover, the spherical surface is a finite continuum. For if we stick the paper discs on theglobe, so that no disc overlaps another, the surface of the globe will finally become so full that there is no room foranother disc. This simply means that the spherical surface of the globe is finite in relation to the paper discs. Further,the spherical surface is a non-Euclidean continuum of two dimensions, that is to say, the laws of disposition for therigid figures lying in it do not agree with those of the Euclidean plane. This can be shown in the following way.Place a paper disc on the spherical surface, and around it in a circle place six more discs, each of which is to besurrounded in turn by six discs, and so on. If this construction is made on a plane surface, we have an uninterrupteddisposition in which there are six discs touching every disc except those which lie on the outside.

Fig. I.

On the spherical surface the construction also seems to promise success at the outset, and the smaller the radius ofthe disc in proportion to that of the sphere, the more promising it seems. But as the construction progresses itbecomes more and more patent that the disposition of the discs in the manner indicated, without interruption, is notpossible, as it should be possible by Euclidean geometry of the the plane surface. In this way creatures which cannotleave the spherical surface, and cannot even peep out from the spherical surface into three-dimensional space, mightdiscover, merely by experimenting with discs, that their two-dimensional "space" is not Euclidean, but sphericalspace.From the latest results of the theory of relativity it is probable that our three-dimensional space is also approximatelyspherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, butapproximately by spherical geometry, if only we consider parts of space which are sufficiently great. Now this is theplace where the reader's imagination boggles. "Nobody can imagine this thing," he cries indignantly. "It can be said,but cannot be thought. I can represent to myself a spherical surface well enough, but nothing analogous to it in threedimensions."

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We must try to surmount this barrier in the mind, and the patient reader will see that it is by no means a particularlydifficult task. For this purpose we will first give our attention once more to the geometry of two-dimensionalspherical surfaces. In the adjoining figure let K be the spherical surface, touched at S by a plane, E, which, for facilityof presentation, is shown in the drawing as a bounded surface. Let L be a disc on the spherical surface. Now let usimagine that at the point N of the spherical surface,

Fig. 2.

diametrically opposite to S, there is a luminous point, throwing a shadow L' of the disc L upon the plane E. Everypoint on the sphere has its shadow on the plane. If the disc on the sphere K is moved, its shadow L' on the plane Ealso moves. When the disc L is at S, it almost exactly coincides with its shadow. If it moves on the spherical surfaceaway from S upwards, the disc shadow L' on the plane also moves away from S on the plane outwards, growingbigger and bigger. As the disc L approaches the luminous point N, the shadow moves off to infinity, and becomesinfinitely great.Now we put the question. What are the laws of disposition of the disc-shadows L' on the plane E? Evidently they areexactly the same as the laws of disposition of the discs L on the spherical surface. For to each original figure on Kthere is a corresponding shadow figure on E. If two discs on K are touching, their shadows on E also touch. Theshadow-geometry on the plane agrees with the the disc-geometry on the sphere. If we call the disc-shadows rigidfigures, then spherical geometry holds good on the plane E with respect to these rigid figures. Moreover, the plane isfinite with respect to the disc-shadows, since only a finite number of the shadows can find room on the plane.At this point somebody will say, "That is nonsense. The disc-shadows are not rigid figures. We have only to move atwo-foot rule about on the plane E to convince ourselves that the shadows constantly increase in size as they moveaway from S on the plane towards infinity." But what if the two-foot rule were to behave on the plane E in the sameway as the disc-shadows L'? It would then be impossible to show that the shadows increase in size as they moveaway from S; such an assertion would then no longer have any meaning whatever. In fact the only objective assertionthat can be made about the disc-shadows is just this, that they are related in exactly the same way as are the rigiddiscs on the spherical surface in the sense of Euclidean geometry.We must carefully bear in mind that our statement as to the growth of the disc-shadows, as they move away from Stowards infinity, has in itself no objective meaning, as long as we are unable to employ Euclidean rigid bodies whichcan be moved about on the plane E for the purpose of comparing the size of the disc-shadows. In respect of the lawsof disposition of the shadows L', the point S has no special privileges on the plane any more than on the sphericalsurface.The representation given above of spherical geometry on the plane is important for us, because it readily allows itselfto be transferred to the three-dimensional case.

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Let us imagine a point S of our space, and a great number of small spheres, L', which can all be brought to coincidewith one another. But these spheres are not to be rigid in the sense of Euclidean geometry; their radius is to increase(in the sense of Euclidean geometry) when they are moved away from S towards infinity, and this increase is to takeplace in exact accordance with the same law as applies to the increase of the radii of the disc-shadows L' on theplane.After having gained a vivid mental image of the geometrical behaviour of our L' spheres, let us assume that in ourspace there are no rigid bodies at all in the sense of Euclidean geometry, but only bodies having the behaviour of ourL' spheres. Then we shall have a vivid representation of three-dimensional spherical space, or, rather ofthree-dimensional spherical geometry. Here our spheres must be called "rigid" spheres. Their increase in size as theydepart from S is not to be detected by measuring with measuring-rods, any more than in the case of the disc-shadowson E, because the standards of measurement behave in the same way as the spheres. Space is homogeneous, that is tosay, the same spherical configurations are possible in the environment of all points.[1] Our space is finite, because, inconsequence of the "growth" of the spheres, only a finite number of them can find room in space.In this way, by using as stepping-stones the practice in thinking and visualisation which Euclidean geometry givesus, we have acquired a mental picture of spherical geometry. We may without difficulty impart more depth andvigour to these ideas by carrying out special imaginary constructions. Nor would it be difficult to represent the caseof what is called elliptical geometry in an analogous manner. My only aim to-day has been to show that the humanfaculty of visualisation is by no means bound to capitulate to non-Euclidean geometry.

[1] This is intelligible without calculation — but only for the two-dimensional case — if we revert once more to the case of the disc on thesurface of the sphere.














The Meaning of Relativity 90

The Meaning of Relativity

The Meaning of Relativity (1922) by Albert Einstein

• information about this edition. • related portals: Physics.

A compilation of Stafford Little Lectures made by Albert Einstein in 1921 at Princeton University. The lectures werepublished in 1922 by the Princeton University Press and address the consequences of Einstein's special and generaltheories or relativity. (Index)

On the Relative Motion of the Earth and theLuminiferous Ether

On the Relative Motion of the Earth and the Luminiferous Ether (1887) by Albert Abraham Michelson and Edward Morley


American Journal of Science, 1887, 34 (203): 333–345, Online [1]

Notes[1] http:/ / www. aip. org/ history/ gap/ PDF/ michelson. pdf




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Article Sources and Contributors 91

Article Sources and ContributorsOn a Heuristic Point of View about the Creation and Conversion of Light Source: http://en.wikisource.org/w/index.php?oldid=4422351 Contributors: C. Trifle, Cleonis, D.H, George OrwellIII, John Vandenberg, Pathoschild, Simon Villeneuve, Stleger, Webarnett, WillowW, 7 anonymous edits

On the Electrodynamics of Moving Bodies Source: http://en.wikisource.org/w/index.php?oldid=4151389 Contributors: Billinghurst, Cantons-de-l'Est, D.H, Eroica, Jusjih, Yann, 3 anonymousedits

The Development of Our Views on the Composition and Essence of Radiation Source: http://en.wikisource.org/w/index.php?oldid=4281060 Contributors: AllanHainey, Billinghurst, D.H,Knotwork, Pathoschild, Schlafly, WillowW, 1 anonymous edits

The Field Equations of Gravitation Source: http://en.wikisource.org/w/index.php?oldid=4281130 Contributors: Billinghurst, D.H, Simon Peter Hughes, StephenDaedalus

The Foundation of the Generalised Theory of Relativity Source: http://en.wikisource.org/w/index.php?oldid=4300126 Contributors: Billinghurst, D.H, Simon Peter Hughes, 10 anonymousedits

Dialog about Objections against the Theory of Relativity Source: http://en.wikisource.org/w/index.php?oldid=4278898 Contributors: Billinghurst, BirgitteSB, Cleonis, D.H, JohnVandenberg, Prosfilaes, Yann, 6 anonymous edits

Time, Space, and Gravitation Source: http://en.wikisource.org/w/index.php?oldid=2625721 Contributors: D.H

A Brief Outline of the Development of the Theory of Relativity Source: http://en.wikisource.org/w/index.php?oldid=3578203 Contributors: Billinghurst, D.H, ResidentScholar

Ether and the Theory of Relativity Source: http://en.wikisource.org/w/index.php?oldid=2625717 Contributors: D.H

The Bad Nauheim Debate Source: http://en.wikisource.org/w/index.php?oldid=4280879 Contributors: Billinghurst, D.H, ResidentScholar

Geometry and Experience Source: http://en.wikisource.org/w/index.php?oldid=3836556 Contributors: D.H, Marc

The Meaning of Relativity Source: http://en.wikisource.org/w/index.php?oldid=4474814 Contributors: Ozhu

On the Relative Motion of the Earth and the Luminiferous Ether Source: http://en.wikisource.org/w/index.php?oldid=4289460 Contributors: Billinghurst, D.H, Giro720, Wild Wolf, 1anonymous edits

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