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Einstein’s Electrodynamical

Pathwayto Special Relativity

John D. Norton

Department of History

and Philosophy of Science

University of Pittsburgh

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“…the seven and more years that the development ofthe Special Theory of Relativity had been my entire life.”

Einstein’s tribute to A. A. Michelson, Dec. 19, 1952.

“On the electrodynamics...”received by Annalen der Physik, Jun. 30, 1905

“The step” 5-6 weeks before

7 and more years= 364 and more weeks

new considerations of space and time enter

What was Einstein doing for the remaining 364 - 5.5 = 358.5 and

more weeks?

…reflecting on electrodynamics in the context of Newtonian

space and time.

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We know...

The Michelson-Morley experiment had no decisive presence in his deliberations. (Holton) Its importance was to affirm the principle of relativity, not the light postulate (Stachel).

Einstein’s early deliberations were driven by the magnet and conductor thought experiment, which brought him the device of field transformations well before the Lorentz transformation of space and time. (Rynasiewicz et al.)

Einstein seriously investigated an emission theory of light (= Galilean covariant) akin to Ritz’s approach. (Stachel)

…and very little more can be said if we demand unequivocal foundation in documentary evidence.

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This talk will try nonetheless to answer…

What could Einstein recover from the device of field transformations?

Two partial theories of electrodynamics, jointly not adequate to the principle of relativity.

What was Ritz’s emission “theory”?

Part polemic against Einstein and part program for finding Galilean covariant force laws.

How might Einstein have used a Ritz type approach?

To construct a promising, Galilean covariant electrodynamics.

What thought experiment shows its failure most cogently?

How did Einstein make “the step”? Several possibilities; it may not have been by reflecting on light signals and clock synchronization.

Einstein’s chasing-a-light-beam thought experiment.

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Read all about it in:

"Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905," Archive for History of Exact Sciences, forthcoming.

"Einstein's Special Theory of Relativity and the Problems in the Electrodynamics of Moving Bodies that Led him to it." in Cambridge Companion to Einstein, M. Janssen and C. Lehner, eds., Cambridge University Press.

Links at www.pitt.edu/~jdnorton

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The Magnet and Conductor Thought

Experiment

and the Device of Field Transformations

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A device that reveals motion through the ether?

A magnet at rest in the ether is surrounded by a pure magnetic field H, but no electric field E.

H

H

E

A magnet moving through the ether creates in addition an induced magnetic field.

Is this induced magnetic field the detectible mark of motion through the ether?

IMPROVEDVERSION

conductor moves with magnet

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What is observable?

conductor

no current

no current

current due to electric

field

exactly balanced

by

current due to motion of conductor in magnetic field

Observables obey principle of relativity. Theoretical account should as well.

…but how?Einstein 1905

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“The idea, however, that these were two, in principle different cases was unbearable for me.

The difference between the two, I was convinced, could only be a difference in choice of viewpoint and not a real difference.

Judged from the magnet, there was certainly no electric field present.Judged from the [resting observer], there certainly was one present.

Thus the existence of the electric field was a relative one, according to the state of motion of the coordinate system used, and only the electric and magnetic field together could be ascribed a kind of objective reality, apart from the state of motion of the observer or the coordinate system.

The phenomenon of magneto-electric induction compelled me to postulate

the (special) principle of relativity.”

Einstein’s 1920 recollection of his reaction:

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The device of field transformations.

H’

E’ = - (1/c)vxH

“Judged from the [resting observer], there certainly was one present.”

“Judged from the magnet, there was certainly no electric field present.”

H

Pure magnetic field H

Magnetic field H’ = H andElectric field E’ = - (1/c)vxH

change observer’s velocity by v

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Which transformation for the general case?

Lorentz transformation (First order)

E = E’ + (1/c) uxH’H = H’ - (1/c) uxE’

Unusable without Lorentz’s local time. Einstein is still years from “the step.”

E = E’ + (1/c) uxH’H = H’

Simplified transformation Unique, linear field transformation law under which Lorentz force lawf/e = E + (1/c) uxHis covariant

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Maxwell’s electrodynamics splits into two partial theories

(“magnet and conductor partial theory”)

is covariant under

t=t’ r=r’-utE = E’ + (1/c) uxH’ H = H’

Full theory of the magnet and conductor thought experiment.

BUTA moving charge does not induce a magnetic field

(“two charges partial theory”)

is covariant under

t=t’ r=r’-utE = E’ H = H’ - (1/c) uxE’

Full theory of Föppl’s (1894) two charges thought experiment, which violates the principle of relativity.

BUTA moving magnet does not induce an electric field.Lorentz force law not included.

(M1) .E = 4

(M3) xH = (4/c)j + (1/c)(∂E/∂t)

(M2) .H = 4

(M4) xE = - (1/c)(∂H/∂t)

(L) f/e = E + (1/c) uxH

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August Föppl’s Two Charges Thought Experimentfrom Einführung in die Maxwell’sche Theorie der Elektricität. Leipzig:

B. G. Tuebner, 1894, Part 5, Ch.1.

To what extent is the principle of relativity respected in electrodynamics?

It holds in the case of the magnet and conductor thought experiment.

It fails in the case of the two charges thought experiment.

+

-

Force between charges at rest in the ether is f.

+

-

Force between same charges when they share a common motion in the ether isf’ = (1-v2/c2)f.

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What now?

Einstein concluded his 1920 recollections:“The difficulty to be overcome lay in the constancy of the velocity of light in a vacuum, which I first believed had to be given up. Only after years of groping did I notice that the difficulty lay in the arbitrariness of basic kinematical concepts.”

(“magnet and conductor partial theory”)

Supports an account of the magnet and conductor thought experiment fully in accord with the principle of relativity.

(“two charges partial theory”)

.E = 4

(M3) xH = (4/c)j + (1/c)(∂E/∂t)

(M2) .H = 4

(M4) xE = - (1/c)(∂H/∂t)

(L) f/e = E + (1/c) uxH

How can it be extended to cover all electro-dynamics?

The two partial theories jointly entail the constancy of the velocity of light.

keep this?

modify this?

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Ritz’s Emission “Theory” of Light

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Ritz’s Emission “Theory”

A modified electrodynamics in which the velocity of the light emitter is added vectorially to the velocity of light.

All Galilean covariant theories of light must be emission theories (but not conversely).

Synonym for a Galilean covariant theory?

Einstein to Ehrenfest,June, 1912 and elsewhere

“…Ritz’s conception, which incidentally was also mine before rel. theory.”

Reasons to take Einstein’s remark seriously.

Closeness of 1912 to actual events.

Einstein was defending his relativity theory from Ritz’s theory. Ehrenfest was proposing experimental tests.

Einstein knew Ritz and had co-authored a note with him.

Einstein won his first job in Zurich in 1909 only after the first choice, Ritz--Einstein’s critic, fell ill.

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Ritz’s Negative Program

“Recherches Critique sur l’Électrodynamique Génénerale,” Annales des Chimie et de Physique,” 13 (1908), pp. 145-275. (and other works)

Died July 1909, aged 31,from tuberculosis.

We should be skeptical about many quantities used in electromagnetic theory, especially electric and magnetic fields.

Fields laws should be eliminated in favor of action at a distance laws such as due to Weber and others.

The ether should be eliminated from electrodynamics and the principle of relativity restored.

The principle of relativity should not be restored by means of Einstein’s strange kinematical notions.

Presentations of electrodynamics should be given in terms of retarded potentials. The advanced potentials admitted by Maxwell’s equations are unphysical.

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Ritz’s Positive Program

To reconfigure and reformulate all of electrodynamics in terms of Galilean covariant, Weber-like action at a distance force laws. The program was not completed. Ritz developed many examples of such laws for special cases.

The force F between two charges e, e’, moving with velocity u and acceleration w is given by:

€

Fx =e ′ e

r2 1−r ′ w rc2

⎛ ⎝ ⎜

⎞ ⎠ ⎟cos(r,x)ϕ

ur

c,u2

c2

⎛

⎝ ⎜

⎞

⎠ ⎟−

uxur

c2ψ

ur

c,u2

c2

⎛

⎝ ⎜

⎞

⎠ ⎟−

r ′ w xc2

χur

c,u2

c2

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Fy = …

€

ϕ =+3−k

4

u2

c2+

3(1− k)

4

ur2

c2+ a1

u4

c4+L

€

ψ =k +1

2+ b1

u2

c2+ b2

ur2

c2+L

€

χ =+ c1u2

c2+ c2

ur2

c2+L

Note the many undetermined constants!

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This time delay encodes the propagation of electromagnetic action at speed c in the ether.

Ritz’s Theory as reported in Pauli’s 1921 Relativitätstheorie

Formulate electrodynamics in terms of scalar and vector potentials , A

E = - (1/c)∂A/∂t

H = xA

€

ϕ (x,y,z, t) =[ ′ ρ ]

r∫∫∫ d ′ τ

€

A (x,y,z, t) = 1c

[ ′ ρ ′ v ]

r∫∫∫ d ′ τ

“[…]” means a quantity is evaluated at event (x’,y’z’,t’) where the retardation time t’ist’ = t - r/c

Replace the propagating retardation time

t’=t-r/c

by a Galilean covariant, projected retardation time

t’=t-r/(c+vr)

vr=speed of source in direction of point at

r=(x,y,z).

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Propagation versus Projection

Ritz imagined that charges emit fictitious particles that are projected by ordinary rules of Galilean kinematics.

PropagationElectromagnetic action propagates from fixed point in space that is left behind by a moving source.

ProjectionThe apparent source

of electromagnetic action is boosted,

and moves with uniformly moving

source.

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Propagation versus Projection

PropagationElectromagnetic action propagates from fixed point in space that is left behind by a moving source.

ProjectionThe apparent source

of electromagnetic action is boosted,

and moves with uniformly moving

source.

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My conjecture:This is also the theory Einstein hit upon and associated with Ritz’s name.

“Ritz’s ideas on electrodynamics and optics are not so far developed that one can call them a ‘theory.’ What is special in them is that there does not exist a definite speed for light propagation at a position and in a given direction, but that this [speed] depends on the state of motion of the light source. Then one cannot trace light propagation back to differential equations, but one must introduce “retarded potentials,” which is a kind of action at a distance. Before setting up the special theory of rel., I had myself thought of investigating such a possibility.”

Draft of a response written on the back of a letter dated 1 February 1952 to Einstein from C. O. Hines.(Einstein Archive 12 250, 12 251.)

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Why Einstein might think of “investigating such a possibility”:

Galilean covariant, magnet and conductor partial theory

Might we render the theory Galilean covariant by replacing propagating retardation times by projected retardation times?

Retarded integrals capture content of remaining Maxwell equations (M1) and (M3).

€

ϕ (x,y,z, t) =[ ′ ρ ]

r∫∫∫ d ′ τ

€

A (x,y,z, t) = 1c

[ ′ ρ ′ v ]

r∫∫∫ d ′ τ

Necessary and sufficient for existence of scalar and vector potentialsE = - (1/c)∂A/∂t H = xA

(M2) .H = 4

(M4) xE = - (1/c)(∂H/∂t)

(L) f/e = E + (1/c) uxH

Promising, BUT…

H = H’ still precludes magnetic fields induced by charge currents.

Might some other variant of this theory escape these troubles? Is any Galilean covariant electrodynamics admissible?

€

A (x,y,z, t) = 1c

[ ′ ρ ′ v ]

r∫∫∫ d ′ τ

Transformations for A must conform, so A=A’ under which

is not covariant.

t=t’ r=r’-utE = E’ + (1/c) uxH’ H = H’

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Einstein’s Objections to All Emission Theories of Light.

The physical state of a light ray is determined completely by its intensity and color [and polarization].

“ I decided [against an emission theory], since I was convinced that each light [ray] should be defined by frequency and intensity alone, quite independently of whether it comes from a moving or a resting light source.” Einstein to Ehrenfest, mid June 1912

Problems with shadow formation by a moving screen.

Collected from remarks in many places.

e.g. To Mario Viscardini, April 1922

The theory cannot be formulated in terms of differential equations.e.g. Einstein to Shankland, 1950sDifferent velocities entail that

light can back up on itself (later parts overtakes earlier).e.g. Einstein to Shankland, 1950s; to Hines Feb. 1952

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Einstein Chases a Beam of Light

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Einstein, Autobiographical Notes, 1946

“After ten years of reflection such a principle resulted from a paradox upon which I had already hit at the age of sixteen:

If I pursue a beam of light with the velocity c (velocity of light in a vacuum),

I should observe such a beam of light as an electromagnetic field at rest though spatially oscillating. There seems to be no such thing, however, neither on the basis of experience nor according to Maxwell’s equations.

From the very beginning it appeared to me intuitively clear that, judged from the standpoint of such an observer, everything would have to happen according to the same laws as for an observer who, relative to the earth, was at rest. For how should the first observer know or be able to determine, that he is in a state of fast uniform motion?

One sees in this paradox the germ of the special relativity theory is already contained.”

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Albert Einstein, “Autobiographical Sketch”

As recounted to Max Wertheimer in 1916

published 1956

“During this year in Aarau the following question came to me: if one chases a light wave with the speed of light, then one would have before one a time independent wave field. But such a thing appears not to exist! This was the first child-like thought experiment related to the special theory of relativity. Discovery is not a work of logical thought, even if the final product is bound in logical form.”

“The problem began when Einstein was sixteen years old, a pupil in the Gymnasium (Aarau, Kantonschule)…

The process started in a way that was not very clear, and is therefore difficult to describe—in a certain state of being puzzled. First came such questions as: What if one were to run after a ray of light? What if one were riding on the beam? If one were to run after a ray of light as it travels, would its velocity thereby be decreased? If one were to run fast enough, would it no longer move at all?…[W’s ellipses] To young Einstein this seemed strange.

…When I asked him whether, during this period, he had already had some idea of the constancy of light velocity, independent of the movement of the reference system, Einstein answered decidedly: ‘No, it was just curiosity. That the velocity of light could differ depending upon the movement of the observer was somehow characterized by doubt. Later developments increased that doubt.’”

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The Thought

A frozen waveform!

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The thought experiment generates no trouble for an ether based Maxwellelectrodynamics.

…but only because we have no experience of moving at the speed light in the ether.

…but it is allowed by Maxwell’s equations through the simplest transformation.

…but the observer would know he is moving rapidly because the light would appear frozen.

“…I should observe such a beam of light as an electromagnetic field at rest though spatially oscillating.

There seems to be no such thing, however, neither on the basis of experience

nor according to Maxwell’s equations.From the very beginning it appeared to me intuitively clear that, judged from the standpoint of such an observer, everything would have to happen according to the same laws as for an observer who,

relative to the earth, was at rest. For how should the first observer know or be able to determine, that he is in a state of fast uniform motion?”

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Why does the thought experiment merit pride of place in Einstein’s defining autobiography?

Or does it have a cogency that extends beyond Einstein’s final high school year?

Einstein (16yrs) in 1896 in the cantonal school of Aarau

Is it merely the recording of the visceral hunches of a precocious sixteen year old, who did not study Maxwell’s theory until two years later?

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The Thought Experiment succeeds against an emission theory of light.i.e. a theory that conforms to the principle of relativity

using Newtonian notions of space and time.

Frozen lightwaves? “…There seems to be no such thing, however, neither on the basis of experience...”

A light source receding at c leaves a frozen wave behind.

We should expect to experience these frozen waves if there are rapidly receding light sources. There is

no need for us to move at c.

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“…or according to Maxwell’s equations…”

Frozen electromagnetic waves are possible in any inertial frame of reference.

Frozen electromagnetic waves must be admissible in electrostatics and magnetostatics.

Electrostatics and magnetostatics of an emission theory should agree with the electrostatics and magnetostatics of Maxwell’s theory. (Oldest and most secure part of theory.)

BUT Maxwell’s equations prohibit frozen waves.

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“…For how should the first observer know or be able to determine, that he is in a state of fast uniform motion?”

A light wave of definite color, amplitude, polarization.

Is it a propagating wave?

Or a frozen wave?

An extra property is needed to separated the two cases.(Equivalent to:Is the observer moving rapidly with respect to the source?)

But color, amplitude and polarization are the only properties light has.

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Present state of field

Rate of change of field

Future time development of field.

An emission theory of light cannot be formulated in terms of differential field equations.

xH = (1/c)(∂E/∂t)

xE = - (1/c)(∂H/∂t)

Example: Maxwell’s theory

Field theory formulated with differential equations: present, local state of the field determines its future time development.

Precluded in an emission theory of light. An extra property is needed to distinguish frozen from propagating waves.

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“But the strongest argument [against an emission theory] seemed to me: If there is no fixed velocity for light at all, then why should it be that all light emitted by “stationary” bodies has a velocity completely independent of the color? This seemed absurd to me. Therefore I rejected this possibility as a priori improbable.” Einstein to Hines, Feb. 1952,

The obvious escape…

A field theory in which the color of a wave fixes its velocity of propagation.

Example: The differential field equation (∂2/∂t2 - ∂2/∂x2 -m2) (x,t)= 0admits waves (x,t)= exp i (t-kx) where m2=k2-2

Color (wave number k) fixes velocityv = /k = (1- m2/ k2) 1/2

k = m --> v=0

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Pathways to “the Step”

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Or was the celebrated analysis of clock synchronization a convenient way to present a result already found by other means?

Did Einstein actually discover the relativity of simultaneity by reflecting on clocks and their synchronization by light signals?

Light enters virtually everywhere as an electromagnetic waveform and not a spatially localized signal.

e.g. aberration, Einstein unaware that Fresnel drag coefficient can be derived by velocity addition on light pulses.

Other pathways are possible:

Simple thought experiments show how field transformations can force the relativity of simultaneity.

The aberration of starlight and the Fresnel drag in Maxwell-Lorentz electrodynamics is a manifestation of Lorentz’s local time. Reverse the derivation to read relativity of simultaneity from observations.

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Conclusion

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What could Einstein recover from the device of field transformations?

Two partial theories of electrodynamics, jointly not adequate to the principle of relativity.

What was Ritz’s emission “theory”?

Part polemic against Einstein and part program for finding Galilean covariant force laws.

How might Einstein have used a Ritz type approach?

To construct a promising, Galilean covariant electrodynamics.

What thought experiment shows its failure most cogently?

How did Einstein make “the step”? Several possibilities; it may not have been by reflecting on light signals and clock synchronization.

Einstein’s chasing-a-light-beam thought experiment.