lorentz contraction: explained at the microscopic level

8/14/2019 Lorentz Contraction: Explained at the Microscopic Level

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Lorentz Contraction:

Explained at the Microscopic Level

Aeneas Wiener

Imperial College London, [email protected]

26 April 2009

Abstract: We investigate possibilities to give a microscopic explanation of theLorentz contraction. This constructivist approach is in contrast to the orthodoxopinion that the explanation of the phenomena described by special relativity is tobe sought in the geometry of space-time. We present the related philosophical debateand, after ourselves adopting the constructivist view, proceed by showing how classicalphysics and quantum mechanics can be utilised to explain the Lorentz contractionatomistically. Our motivation for this work is twofold. Firstly, we acknowledge thephilosophical importance of an explanation of Length contraction which is not basedon postulated relativity. Secondly, we agree with J. Bell on the count that an atomisticexplanation of special relativity holds clear didactic potential over a postulated theory.

Supervisor: Prof A Sutton. Assessor: Prof M Finnis.

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Contents

1 Introduction 3

2 Motivation 3

3 Philosophical interlude 4

3.1 Lorentz covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 What is special relativity? . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 Renaissance of the constructive theory 5

4.1 W. F. G. Swann: introducing quantum mechanics . . . . . . . . . . . . 64.2 J. S. Bell: model of a simplified atom . . . . . . . . . . . . . . . . . . . 64.3 General structure of the constructive approach . . . . . . . . . . . . . 8

5 Extending on the constructive approach 9

5.1 Theory of ionic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2 Stresses in ionic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.2.1 Result 1: Lorentz covariance . . . . . . . . . . . . . . . . . . . 115.2.2 Result 2: Length contraction . . . . . . . . . . . . . . . . . . . 11

6 Search for a quantum explanation of length contraction 12

6.1 Relativistic model of a finite potential well . . . . . . . . . . . . . . . 136.1.1 Stationary 1D potential well . . . . . . . . . . . . . . . . . . . . 136.1.2 Moving 1D potential well . . . . . . . . . . . . . . . . . . . . . 17

7 Conclusion 20

8 Further reading 21

Appendix A Electromagnetic fields of a moving point charge 22

A.1 Retarded potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22A.2 Retarded potentials for a point charge . . . . . . . . . . . . . . . . . . 23A.3 Fields of a point charge . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Appendix B Electromagnetic force in a dipole 26

B.1 Case A: Dipole at rest . . . . . . . . . . . . . . . . . . . . . . . . . . . 26B.2 Case B: Dipole in motion . . . . . . . . . . . . . . . . . . . . . . . . . 26

App endix C Free particle solution of the Dirac equation 27

References 30

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1 Introduction

Early versions of the equations which have now come to be known as the Lorentz trans-formations go back as early as 1887, when German physicist W. Voigt proposed a setof equations which, up to a factor ofc, reflected the Lorentz transformations developedby George Francis FitzGerald, Joseph Larmor, Hendrik Lorentz and Henri Poincar(Kittel, 1974). Their work led up to Einsteins 1905 paper, where he proposed a

radical new derivation of the Lorentz transformations based on his two postulates ofspecial relativity (Einstein, 1905).

Einsteins special relativity theory did away with the concept of the Ether andintroduced instead the principle of relativity (stating that all laws of physics are thesame in all inertial frames) and the principle of the invariance of the speed of light.This approach distinguished Einsteins principle theory from earlier constructivetheories, whose aim it was to explain the Lorentz transformations using existingphysics. Einstein often recognised this as a clear shortcoming of his theory but healso pointed out that, given the physical framework at the time, he did not see anyother way.(Einstein, 1919)

In the same article, published in The London Times, Einstein gave a detaileddefinition of the constructive versus principle theory distinction where he likened theprinciple nature of special relativity to classical thermodynamics, which was also aprinciple theory.1For thermodynamics, the advent of statistical mechanics ultimatelyprovided the constructive underpinning for the theory. In the case of special relativity,there exists a fundamental dispute about the requirement for a constructive theory,which aims to explain phenomena such as length contraction based on a considerationof the forces between particles rather than on overarching principles of space and time.

In section 3.2 we will give a more in depth overview of this philosophical debate.Having done this, we will adopt the constructivist view and, by presenting concreteexamples, show what is meant by a constructive theory of the Lorentz contractionthat is based on the dynamics of a body.

2 Motivation

We believe that the constructive theory debate is of fundamental importance forthe philosophical foundations of physics. However, being physicists, our selectiveinterest in a constructive explanation of Length contraction is principally motivatedby didactical considerations.

Students of physics are used to receiving microscopic explanations for macroscopicphenomena. Temperature is explained on the microscopic level by considering themovement of individual atoms; viscosity is understood on the basis of the forcesbetween individual molecules in a fluid and pressure is fundamentally a measure ofthe spacing between molecules related to their weight. These are only three exampleswhereby a macroscopic property of matter is accorded an intuitive explanation onthe microscopic level. This pattern is abruptly broken by most courses in special

relativity. In special relativity, changes in macroscopic variables, such as length andtime, are explained on the basis of two postulates.

1We can distinguish various kinds of theories in physics. Most of them are constructive. Theyattempt to build up a picture of the more complex phenomena out of the materials of a relativelysimple formal scheme from which they start out. Thus the kinetic theory of gases seeks to reducemechanical, thermal, and diffusional processes to movements of molecules i.e., to build them upout of the hypothesis of molecular motion. When we say that we have succeeded in understandinga group of natural processes, we invariably mean that a constructive theory has been found whichcovers the processes in question.

Along with this most important class of theories there exists a second, which I will call "principle-theories." These employ the analytic, not the synthetic, method. The elements which form theirbasis and starting-point are not hypothetically constructed but empirically discovered ones, generalcharacteristics of natural processes, principles that give rise to mathematically formulated criteriawhich the separate processes or the theoretical representations of them have to satisfy. Thus the

science of thermodynamics seeks by analytical means to deduce necessary conditions, which separateevents have to satisfy, from the universally experienced fact that perpetual motion is impossible.The advantages of the constructive theory are completeness, adaptability, and clearness, those of theprinciple theory are logical perfection and security of the foundations.

The theory of relativity belongs to the latter class. (Einstein, 1919)

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Many students accept this presentation without question, no doubt impressed bythe mathematical beauty of the theory of special relativity and the relative ease withwhich all the results of the theory follow. Historically aware students will no doubtappreciate that special relativity grew out of the efforts of people like Lorentz andLarmor, who tried to understand length contraction on an atomistic level. However,their explanations were based on the later abolished Ether hypothesis and are todayonly of historical interest.

In this work we will show that it is indeed possible to give a valid microscopicexplanation of Length contraction which is entirely based on modern physics. Wehave found that such a microscopic explanation leaves students with a much improvedand more intuitive understanding of the workings of special relativity.

3 Philosophical interlude

3.1 Lorentz covariance

The Lorentz covariance of the fundamental equations governing all non-gravitationalinteractions, as a mathematical property exhibited by physical laws which scale toarbitrary velocities, is truly the central idea that lies behind special relativity.2 Within

special relativity, the Lorentz group and the group associated with the relativityprinciple coincide. This means that if a given physical law can be shown to becovariant under a Lorentz transformation then the relativity principle is automaticallysatisfied. In Einsteins (1940) own words:

The content of the restricted relativity theory can accordingly be summar-ised in one sentence: all natural laws must be so conditioned that theyare covariant with respect to Lorentz transformations. 3

In his 1905 paper Einstein showed the Lorentz covariance of Maxwells equations. Asdiscussed by Norton (1993), Einstein then proceeded by extending this approach ofshowing the Lorentz covariance of physical laws to parts of thermodynamics. However,both electrodynamics and thermodynamics are macroscopic theories, which do notcontain any information about the microscopic properties of matter.

At this point it is important to note that we are not attempting to give a con-structive explanation of special relativity. Special relativity is a theory that makespredictions about the observations of observers moving at different constant velocitiesrelative to each other. Such a theory is inevitably based on the relativity principle,without the use of which we would never be able to make any predictions about theobservations of observers situated in an inertial frame different from our own. In thissense, imposing the relativity postulate (and ultimately Lorentz covariance) is an in-evitable step in all derivations of special relativity (at least it is if we are to replace thenihilist belief of unknowability with the realist program, which asserts that physicaltruth is something that exists outside of sense perception).

What we are attempting to show in this work, then, is that it is possible to gainvaluable insights into the workings of Length contraction by studying the physics as

viewed from a single frame. This pedagogy is very much in the spirit of J. Bell (seesec. 4.2). The advantage of such a course of investigation is that it avoids the use ofthe relativity postulate. It is this exclusion of the relativity postulate that will allowus to give a fully constructive and microscopic explanation of Length contraction.

3.2 What is special relativity?

As previously mentioned in the introduction, the theory of special relativity as pro-posed by Einstein is a principle theory. There exists a fundamental debate within thephilosophical community about what a constructive theory of special relativity wouldlook like, which results in two major opposite views.

2In general relativity (GR), the story is more difficult and a source of much debate amongst

researchers in the field. For the complications which arise in general relativity, see the work by JohnD Norton(1993; 1995). The upshot of the matter is that in GR the Lorentz group and the groupassociated with the relativity principle do no longer coincide. However, this does not need to concernus here since we are looking at the Lorentz contraction from the perspective of special relativity.

3As quoted in Norton (1995)

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Figure 1: Explanatory arrow between space-time and dynamics.

The orthodox view, recently defended by Yuri Balashov and Michel Janssen, sug-gests that the phenomena of special relativity are purely geometrical and that it issufficient to show how Minkowski space-time leads to these phenomena (Balashov &Janssen, 2003).

In contrast, the proponents of a dynamical version of a constructive theory, suchas Harvey Brown and Oliver Pooley, suggest that a constructive theory of specialrelativity is to be sought in an analysis of the dynamics of a body moving at re-lativistic velocity (Brown & Pooley, 2006). The work of Brown and Pooley and their

constructivist viewpoint was most recently criticised by Norton (2007) and Janssen(2008).

In summary, the fundamental argument is about the direction of the explanatoryarrow between the structure of space-time on one side and the dynamics of physicalobjects on the other. Constructivists argue that this arrow should start at the dynam-ics of physical objects and point towards the mathematical construct of space-time.In other words, they argue that the dynamics between particles is the thing that givesrise to the structure of space-time as we see it. The orthodox opinion, however, isthat the direction of this arrow is reversed, so that it originates at the mathematicalconstruct of Minkowski space-time and points towards the dynamics of particles (asshown in Figure1).

It should be noted that Einstein himself would have preferred a constructive ex-

planation based on the dynamics of the body over a kinematical approach. In hisautobiographical notes, he clearly states that the only reason why he decided to putforward his principle theory was that he despaired at the limited understanding ofthe laws governing the cohesion of matter in his time (Brown, 2005, chapter 5).

In the remaining sections we will ourselves adopt the constructivist view and tryto show how, using initially only classical physics, one might attempt to explain theLorentz contraction on a microscopic level.

4 Renaissance of the constructive theory

Einsteins spectacular success, following his 1905 annus mirabilis, was iconic and hasremained unparallelled to this day. The test of the general theory of relativity by Lord

Eddington in 1919 and such world changing consequences like the development of theatomic bomb by the Manhattan Project constitute a powerful legacy that, at least inthe minds of the general public, was hard to argue with. These circumstances mayhelp to explain the widespread disregard of the true achievements of trailblazers suchas Lorentz, FitzGerald and Poincar, whose work lead up to Einsteins 1905 paperon special relativity. Their utilisation of the Ether hypothesis, which Einstein hadexposed to be incorrect, has led to a stigmatisation of their efforts as being superflu-ous and backward. This point was aptly expressed by Brown, who was referring toPoincars pre 1905 work on special relativity when he wrote:

It must have been galling for him (Poincar) to see Einsteins specialrelativity given such prominence, when a good part of what the youngman seemed to be doing was merely to postulate what he and Lorentz

had been trying to prove the hard way. (2005, sec. 8.4.1)

It is against the backdrop of this revolutionary zeitgeist, filled with contentment aboutfinally having done away with the hideous Ether hypothesis, that we should view the

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attempts of people who insisted on the need of a constructive explanation of specialrelativity, even after 1905. In this chapter, we will present the work of two suchneo-constructivists.

4.1 W. F. G. Swann: introducing quantum mechanics

It was the British physicist W. F. G. Swann who was the first to publicly stress the

need for a relativistic quantum theory for the explanation of the Lorentz contraction.Picking up where Lorentz and his kinsmen had left off, Swann reiterated their pointthat the Lorentz covariance of Maxwells equations by itself was far from sufficient toaccount for the phenomenon of length contraction. (Brown, 2005, sec. 7.4)

In a 1941 paper, Swann stressed the explanatory deficiency of the property ofLorentz covariance when it comes to observable physical phenomena such as lengthcontraction. Beginning his argument by stating the well known Lorentz transforma-tions for the three spacial components and time,

x = xy = yz = (z ut)t = (t uc2 z)

, (1)

Swann proceeded by showing that there are two aspects to the meaning of invarianceof equations under these transformations. The first, A, is the mathematical fact thatif an equation that was originally expressed in terms of x, y, z, t is transformed to x,y, z, t then the equation will revert to the original form. Swann underlined that thisfact by itself is purely mathematical and, taken by itself, tells us nothing about themeaning of the primed variables. He then stressed that it was only a second aspect, B,that gives physical meaning to the primed coordinates. By postulating that if x, y, z,t are coordinates of an event in a system S as measured by an observer O, then x, y,z, t will be the coordinates of the same event when measured by an observer O in asystem S which moves at a velocity v with respect to S. It is aspect B that is usuallyassociated with the relativity postulate. Swann has to be congratulated for puttingthis point so distinctly by stressing the need for more than the simple mathematical

property of Lorentz covariance of an equation when attempting to explain lengthcontraction.In the same 1941 paper, Swann proceeded by stressing the need for a relativist-

ically covariant quantum theory in order to give physical meaning to the propertyof Lorentz covariance. It was clear to him that only a quantum picture, accountingfor all the forces involved in the cohesion of matter, will ultimately be able to givea satisfactory justification for the meaning of the primed state associated with theLorentz covariance of equations.

Nevertheless, we shall see in the next section how one can develop a model thatis akin to the one demanded above, provides a physical explanation for the primedvariables in accordance with aspect B and, at the expense of generality, does thiswithout referring to quantum mechanics.

4.2 J. S. Bell: model of a simplified atom

It is our aim to find a constructive theory of the Lorentz contraction, i.e. the shorten-ing of a piece of matter moving at velocity u by a factor of

1 u2/c2 in the direction

of u.We begin our investigations by considering a point charge with a view of extending

the theory to a simplified atom. Starting from Maxwells equations, we have derivedthe electromagnetic field of a point charge moving at uniform velocity u in the zdirection:

E = q40

x2 + y2 + z2

3/2 xyz

B = 1

c2v

E

, (2)

with = 1/

1 u2/c2 and z = (z ut). Figure 2 shows how the electric field ofthe moving charge is flattened in the direction of motion. For a detailed derivationof this result see Appendix A.

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Figure 2: The electric field of a point charge at rest (left) and moving at uniformvelocity in the z-direction (right). The field lines of the moving charge are flattenedin the direction of motion.(Bell, 2004)

Figure 3: Orbit of an electron orbiting a proton at rest (left) and moving in the zdirection (right) based on numerical integration of eqs. 4 and 5 (Bell, 2004). We seehow the orbit is flattened in the direction of motion and maintains its extent in thedirection perpendicular to it.

In his 1976 article How to teach special relativity, J. S. Bell (2004, chapter 9)points out how the Lorentz factor in eqs. 2 arises naturally from Maxwells equations,without the need to use special relativity. In order to check for consistency, one canindeed apply a Lorentz transformation to the electromagnetic field tensor (F ) of astationary point charge by writing

F

=

F

(3)

and will end up with the same fields as derived above for the moving charge4.Bell proceeds by pointing out that if we now consider an electron that is orbiting

a proton and if we ignore the fields produced by the electron the equation of motionof the electron may be written:

dp

dt= q (E+ re B) (4)

He argues that the substitution re = p/m is inadequate because it does not agree withexperimental observations. Instead, he proposes the following formula by Lorentz thatwas found to match experimental results with electrons:

re = p

m2 p2c2 (5)

We note that the inclusion of eq. 5 as a purely experimental result does not seem tobe entirely satisfying with regard to the constructive nature of the argument that weare trying to make.

Eqs. 4 and 5 can be integrated numerically using a computer. If one starts witha proton at rest and then gradually increases the velocity one finds that the electronorbit is flattened in the direction of motion as shown in Figure 3. In fact, what Bellfound is that the orbit of the moving electron is flattened by a factor of precisely

1 u2/c2 (6)4Compare for example J. D. Jacksons book Classical Electrodynamics (Jackson, 1998)

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and that the period of the moving electron is dilated by a factor of

1/

1 u2/c2 (7)

These are of course the famous Lorentz contraction and time dilation results of specialrelativity. Bell uses them to derive the moving system in terms of new (primed)variables and hence shows the Lorentz invariance of Maxwells equations and of eq. 4(the equation of motion of the electron).

Bell made two provisos regarding his approach and in particular to its extensionto multi-atom systems. (1) The classical Maxwell model he used fails to reproducethe empirical stability of actual matter. (2) If the acceleration is not gentle enoughfor the given system, we might damage the system rather than moving it from theunprimed into the primed state.

The ideas presented in Bells article are important for our project because theyshow how it is possible to obtain the Lorentz contraction result for an idealised singleatom by using classical physics. This treatment could be seen as a constructive theoryof the Lorentz transformation for a simplified atom. Naturally, it would be desirableto extend the model to include real atoms and also collections of atoms, representinga real piece of matter.

We believe that the problems posed by Bells approach may be addressed by

modelling the interactions using quantum mechanics and by using an appropriatemodel from modern solid state physics.

4.3 General structure of the constructive approach

In the preceding sections we have presented the work by two early constructivists.In this section, we attempt to use their work to illustrate the general structure ofthe constructive approach. Due to its simplicity, Bells idealised atom lends itselfparticularly well to such an analysis. The scheme which we deduce in this section willserve as a guideline for our own constructive models of length contraction.

In broad terms, what Bell has done in his article can be broken down into thefollowing five steps:

1. Determine the stationary system forces involved in the cohesion ofthe material under consideration. In Bells case, this is the attractiveCoulomb force and the repulsive force, resulting from the orbit of the electron.

2. Show the rest frame length of the system. Bell does this by solving theequation of motion numerically.

3. Determine forces in the moving frame. Bell does this by analysing Max-wells equations in the case of the attractive Coulomb force. The repulsiveforce, due to the orbit of the electron, is modelled by an empirical relativisticequation that happens to be valid both in the stationary and in the movingframe.

4. Show the length of the system in the moving frame. Bell does this bynumerical integration of the equation of motion of the atom, slowly acceleratingthe atom to its final relativistic speed. He observes that the system is contractedby the Lorentz factor and that it can be expressed in terms of new variables:

x = xy = yz = (z vt)t = (t uc2 z)

(8)

5. Show that step 4 implies Lorentz covariance of the physical laws underconsideration. For the atom this is done by taking for example the movingframe force and showing that, upon substitution of eq. 8, it takes the same form

as the stationary force.

We can see from steps 4 and 5 that the acceleration of a system to relativistic velo-cities uncovers the Lorentz covariance of the physical laws under consideration. In

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section 3.1 we have already expanded on the importance of Lorentz covariance inspecial relativity. In the principle formulation of special relativity, Lorentz covari-ance is a mathematical construct. It is precisely this point in which the constructiveapproach goes a step further. By showing Lorentz covariance without presuming it,the constructive approach assigns meaning to the fact that all natural laws have theproperty of Lorentz covariance.

5 Extending on the constructive approach

Based on the reasoning presented in section 4.1 it is clear that any attempt at amicroscopic explanation of the Lorentz contraction will be based on a considerationof the forces involved in the cohesion of matter.

Traditional solid state models of ionic crystals are based on a long range electro-magnetic force and a short range repulsive force, which is due to the Pauli exclusionprinciple. In this chapter, we will show that it is possible to use classical electro-dynamics to show how the electromagnetic force transforms in the relativistic limit.The inherent Lorentz covariance of Maxwells equations is the underlying reason whysuch a program can succeed without the use of special relativity.

The repulsive Pauli force, on the other hand, is based on quantum mechanics.

Schrdinger quantum mechanics, however, is not Lorentz covariant and as a result itonly holds for non-relativistic velocities. The use of relativistic quantum mechanicshas to be considered very carefully, as it is based on special relativity. Any reasoningthat is based on principle theories, such as special relativity, would lead to a circularargument in our attempt to explain the Lorentz contraction constructively. However,we will show in this section that there are ways to explain the Lorentz contractionqualitatively without having to refer to the repulsive Pauli force.

5.1 Theory of ionic crystals

The theory of ionic crystals involving the Madelung energy as described in Kittel(2004, chapter 3) approximates the forces in a crystal by a repulsive potential of the

form er

, whose exact form is determined empirically, and a 1/r Coulomb potential.According to this theory, the cohesive energy of the crystal is

Utot = N Ui = N

z exp(R/) q

2

R

(9)

where N is the number of ions, z is the number of nearest neighbours of any ion, is the strength of repulsive potential, is a measure of its range, q is the charge ofthe ions and R is the smallest distance between two ions. is the Madelung constantand is defined as

R=

j

()rij

(10)

where rij is the distance between the i-th and the j-th ion.In section 4.2 on page 6 we have shown how the electromagnetic field of a point

charge transforms when the charge is set in motion. This result could be used totransform the Coulomb term in eq. 9. If we were to use the above theory of ioniccrystals to demonstrate the Lorentz contraction we would need to consider how therepulsive er-force transforms. We would also need to modify the representation ofR, the smallest inter ionic distance, and the Madelung constant in a way thatwould allow us to observe distance changes in the direction of motion, which wouldof course be expected to be in accordance with the Lorentz contraction.

Once this is achieved, we could then differentiate eq. 9 (the total energy of thecrystal) with respect to a homogeneous strain in the direction of motion of the crystaland compare this with the derivative with respect to the directions perpendicularto the direction of motion. We would expect these two quantities to be different,reflecting the fact that the crystal has undergone Lorentz contraction in the directionof motion.5 This would constitute a classical treatment of the Lorentz contraction. A

5This approach was suggested to us by Prof Adrian Sutton.

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clear account of the theory of elastic strains in crystals can be found in Kittel (2004,73ff).

Unfortunately, it is at the point where we need to model the repulsive Pauliforce that this approach has to be truncated because we do not (yet) have a fullyconstructive and relativistically covariant expression for the repulsive potential.

5.2 Stresses in ionic crystals

Noting the problems associated with relativistically valid formulations of the repulsiveforce between ions, we decided to focus our attention on electromagnetic interactions,for which we have a constructive formulation that scales to arbitrary velocities. Notingthat electromagnetic forces play a central role in the cohesion of ionic crystals, we willnow investigate the stresses in ionic crystals that are due to electromagnetic effects.

The basic premise of this part of our investigation will be to show how the stressesin an ionic crystal change when it is set in motion. Intuition tells us that the stresswill increase in the direction of motion of the crystal, leading to the contraction resultof special relativity investigated here. We will now show how the stress componentsin the direction of motion as well as in the direction perpendicular to the direction ofmotion change as functions of velocity. In conclusion, we will show that this approachcan be used to show length contraction constructively by requiring Lorentz covariance

(see section 3.1).Consider three orthogonal unit vectors x, y, z that are embedded in an ionic

crystal. If the solid undergoes a small uniform deformation these vectors will changeto

x (1 + xx) x + xyy + xzzy yxx + (1 + yy) y + yzzz zxx + zyy + (1 + zz) z

. (11)

Thus, if a small strain is applied to the crystal, a given point r(j) in the crystal

is displaced according to the following infinitesimal displacement vector:

r(j) = r(j) . (12)

As a result of the applied strain, the energy of a lattice point in a lattice on whichindividual sites are labelled by j will change by the amount:

E(i) =

j

f(j) r(j) , (13)

where f(j) is the force between sites i and j. Upon substitution of eq. 12 this becomes

E(i) =

j

f(j) r(j) . (14)

The stress on site i in a perfect crystal is defined as

(i) =1V

j

E(j)

= 1V

N E(i)

= 1V(i)

E(i)

. (15)

We note that E(i)/ does not depend on the position in the lattice as the appliedstrain is uniform and we are dealing with a perfect crystal where all sites are equival-ent. V is the total volume of the lattice and V(i) is the volume associated with thei-th cell.

When substituting eq. 14 into eq. 15 we find that cancels and we end upwith the following definition for the stress

(i)

=1

V(i) jf(j) r

(j) , (16)

which we will now use to compute the stresses.In Appendix B we have shown in detail how the force between two point charges

in constant and uniform motion can be derived. This result, which we reproduce here

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for clarity, can be used to evaluate the force term in eq. 16. According to eq. 121 theforce between two sites i and j may be written as:

f(j) =qiqj40

1x2j + y

2j +

2z2j3/2 1

xjyj

2 zj

. (17)

We observe that this reduces to the simple Coulomb force in the static case ( = 1).We also note that the off diagonal stress components, which represent shear stresses,all vanish by symmetry.

Using the above results, we find the stresses in the x, y and z directions on anysite i in an infinite ionic stationary crystal:

(i)xx =

q2

401

V(i)

j

x2j

(x2j+y2j+z2j )3/2

(i)yy =

q2

401

V(i)

j

y2j

(x2j+y2j+z2j )3/2

(i)zz =

q2

401

V(i)

j

z2j

(x2j+y2j+z2j )3/2

. (18)

Further we find that if the crystal is moving uniformly along the z axis, these stresses

become:(i)xx =

q2

401

V(i)1

j

x2j

(x2j+y2j+2z2j)3/2

(i)

yy =q2

401

V(i)1

j

y2j

(x2j+y2j+2z2j)3/2

(i)

zz =q2

401

V(i)1

j

2z2j

(x2j+y2j+2z2j)3/2

, (19)

where we have primed to signify that it refers to the moving configuration. Thelast two results are of central importance to our argument. Comparing eqs. 19 and18 we note that the latter takes the form of the former if the z coordinate undergoesthe following, as yet unidentified, substitutions:eq:

z = z

V

= V . (20)

5.2.1 Result 1: Lorentz covariance

What this means mathematically is that eq. 16 (the stress tensor) is Lorentz co-variant. As argued in section 3.1, Lorentz covariance is a natural property of allnon-gravitational interactions and that we uncovered it here in the case of the stressin an ionic crystal should come as no surprise.

At this point it is important to keep in mind that we are trying to give a fullyconstructive explanation of the Lorentz contraction. For this reason, we cannot referto any preconceived notions that we might have from the study of special relativity,tempting us to identify the variable z in eq. 20 as the rest length. Rather, we notethat z is at this point just a substitution that allows us to write the equation for the

moving crystal stress in the same form as that of the stationary crystal.

5.2.2 Result 2: Length contraction

In order to interpret the meaning of the above coordinate transformation it is usefulto consider the stresses in the moving direction with those that are acting in thedirections perpendicular to the direction of motion. These values are given in eq. 19.They show that

>

=

, (21)

i.e. when the crystal is set in motion the stress in the direction of motion is greatercompared to the stresses acting in the directions perpendicular to it. This is in obvious

contrast to the results for a stationary crystal, where we observe the stresses to be thesame in all directions. In other words, we observe that the stress is no longer isotropicwhen the crystal is set in motion. We can conclude qualitatively that the change inthe electrodynamic forces in the moving crystal has led to a shift in the distribution

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of the stresses in the crystal, acting in a way that will lead to a contraction in thedirection of motion.

Finally, we can give further physical meaning to the substitution in eq. 20 bycomparing it to Bells contracting orbit result (see eq. 6). As Bell has shown (seesection 4.2), it is possible to make a fully constructive case for the contraction of theelectron orbit in a simplified atom by a factor of 1/. Comparing this (Lorentz)-contraction result with eq. 20 leads us to the conclusion that the transformations

we have uncovered actually refer to a spacial transformation that relates distancemeasurements for the resting crystal (z) to the distance measurements for the movingcrystal (z). In other words, we conclude that if an ionic crystal is set in motion, apoint that has coordinates x, y, z when at rest will have have coordinates

x = x

y = y

z = z

(22)

when the crystal is moving in the z direction. In other words: the crystal has under-gone Lorentz contraction.

6 Search for a quantum explanation of length con-traction

We will now give a summary of the work presented in the preceding sections. Thepurpose of this review is to explain our motivation for the quantum mechanical in-vestigations which form the final part of this paper.

Up to this point, the only system for which we were able to provide a fully con-structive and quantitative explanation of the Lorentz contraction was the idealisedsingle electron atom in section 4.2. This was achieved by considering the forces inthe system at an arbitrary velocity and then finding the steady state solution of theelectron at a given velocity by integrating the equations of motion. This approachwas first introduced by J. Bell, who found the orbit of the electron to be contracted

by a factor of 1/. This contraction of the electron orbit by a factor of 1/ was usedto show the Lorentz covariance of the equations governing the electron, includingMaxwells equations.

Noting that it is not satisfactory to show the Lorentz contraction for a single atomalone we concluded that a more extensive model was required. Thus, we extended ourapproach by looking at an infinite crystal lattice made up of charges. In such a crystal,there are two forces to consider - the electromagnetic Coulomb force and the repulsivePauli force, which is quantum mechanical in nature. Ideally, we would have liked togive representations of both forces at arbitrary velocities, as this would have allowedus to proceed in a manner analogous to the single atom. We could have consideredthe crystal at an arbitrarily high velocity smaller than the speed of light and, notingthat the steady state solution will occur when all the forces are balanced, determinedthe length of the moving crystal. Naturally, we would have expected the contraction

to be by a factor of 1/ and, just like in the case of the single atom, we would haveconcluded our analysis by using this result to show the Lorentz covariance of theequations governing the dynamics of the crystal. In summary, we expected that amulti-atom model of length contraction would be based on both quantum mechanicaland electromagnetic elements and that only a combination of the two would ultimatelybe sufficient to show length contraction in such a model.

Noting that the exact form of the repulsive Pauli force can only be determinedexperimentally, we were not able to proceed with our argument in this way andinstead focused solely on the electromagnetic interactions in an infinite crystal. Thus,we were able to conclude that an infinite crystal will undergo a shape deformation atrelativistic velocities due to an increase in stress in the direction of motion. This wasthe central result presented in section 5.2.

Ultimately, we would like to return to the above argument and, by adding to itthe missing component of a model of the repulsive Pauli force at arbitrary velocity,conclude that a moving crystal will not only change shape but in fact undergo a lengthcontraction by a factor of 1/.

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Figure 4: Stationary potential well of width l and depth V0

The attempt to combine the repulsive Pauli force and the electromagnetic Lorentzforce to this end will be the focus of future work. It will require a detailed invest-igation into the quantum mechanical nature of the repulsive Pauli force, which is afundamentally quantum mechanical endeavour.

In this section we will present our initial attempts to employ relativistic quantummechanics in a constructive theory of length contraction. Although it is still ourultimate aim to give a constructive explanation of length contraction based on acombination of the electromagnetic and quantum mechanical interactions in a crystal,this section will solely focus on quantum mechanics.

The investigations detailed here were carried out to gain valuable insights into theways relativistic quantum mechanics behaves in a constructive model. We will presentour attempt to find a purely quantum mechanical model of length contraction basedon the study of a potential well. The aim is to show that a potential well, when setin motion, will show length contraction. We will detail the current limitations of ourmodel and how we plan to address them in our future work.

6.1 Relativistic model of a finite potential well

In undergraduate courses the potential well problem receives ample attention. How-ever, the treatments are usually based on the Schrdinger equation, which is knownto break down at high velocities because it is not Lorentz covariant. For this reason,we will now show how the Dirac equation can be solved to model a potential well.The use of the Dirac equation (which is a Lorentz covariant equation) is an importantingredient if our model of the potential well is to hold at arbitrarily high velocitiessmaller than the speed of light.

Despite the somewhat tedious mathematics involved in solving the relativisticpotential well we present it here in full detail, breaking with our habit of givinglengthy derivations in the Appendix. The reason for this is that we were unable tofind the derivation for the moving potential well anywhere else in the literature. Also,the questions posed by this part of our work remain mostly unanswered at this pointand could form the focus of future work.

6.1.1 Stationary 1D potential well

The approach we used for the stationary one dimensional relativistic finite potentialwell follows closely the one given by Strange (1998). A more involved treatment ofthe problem can be found in Greiner (2000).

The problem statement is detailed in Figure 4. We are trying to find the allowedeigenvalues in the stationary potential well of depth V0 and length l. Just like inthe familiar Schrdinger case, we begin by writing down the wave functions and thegoverning equations for the three regions of the potential well (as defined in Figure4).

Region 1: In this region of zero potential, the Dirac equation is given by

c p1 + mc2

1 = W 1, (23)where is the wave function of the particle trapped by the potential well, c is thespeed of light, and are the usual four by four matrices and have been definedin Appendix C, m is the mass of the particle and W is its total energy. The wave

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function is just that of a free particle (as derived in Appendix C) and is given by

1 = AI

100

c p1(W+mc2)

exp

ip1x

+ AII

100

c p1(W+mc2)

exp

ip1x

, (24)

with p21c2

= W2

m2

c4

. Note that the continuity requirement of the wave functionat the boundaries implies that there are no spin flips at the boundary.

Region 2: In this region, where the potential is reduced to V0, the Diracequation is given by

id2dt

= H2 =

c p2 + mc2

2 = W 2 (25)

and the wave function is thus given by

2 = BI

100

c p2

(W+V0+mc2)

exp

ip2x

+ BII

100

c p2

(W+V0+mc2)

exp

ip2x

,

with p22c2 = (W + V0)

2 m2c4.Region 3: This is similar to Region 1, with the Dirac equation reading

c p3 + mc2

3 = W 3 (26)

and the wave function

3 = CI

100

c p3(W+mc2)

exp

ip3x

+ CII

100

c p3(W+mc2)

exp

ip3x

, (27)

with p23c2 = W2m2c4 = p21c2. We note that in Region 3 the momentum is equal to

the momentum in Region 1. From now on we will thus replace p3 with p1.It is now our aim to solve for the allowed spectrum of the eigenenergies W in the

system described above. The final equation which we obtained and used to computethe eigenenergies is given in eq. 45 and a sample of the eigenenergies is plotted inFigure 6. The reader may wish to take a look at these results before attempting tofollow the somewhat lengthy derivation.

Again in close analogy to the Schrdinger treatment of the potential well, webegin by exploiting the fact that 1, 2 and 3 are subject to the following boundaryconditions:

1(0) = 2(0)2(l) = 3(l) . (28)

Applying the boundary condition at x = 0 and writing out the first and the fourthcomponent of on separate lines gives:

AI + AII = BI + BIIc p1

(W+mc2)(AIAII) = c p2(W+V0+mc2) (BIBII)

. (29)

We will now write these two equations in matrix form, the reason for which will soonbecome apparent. Defining the dimensionless quantity as

=p1

W + V0 + mc2

p2 (W + mc2)=

(BI BII)(AI AII) (30)

and omitting a few lines of algebra we can can write the set of equations defined ineq. 29 as

AIAII

=

1

2

+ 1 1 1 + 1

BIBII

. (31)

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Figure 5: Plot of eq. 46, which was used to determine the eigenenergies W of astationary 1D finite relativistic potential well.. All constants, including l and V0,equal to unity. The energy levels determined from this plot are shown in Figure 6.

Similarly, applying the boundary condition at x = l and writing out the first and thefourth component of on separate lines gives:

BIeip2l/ + BIIeip2l/ = CIe

ip1l/ + CIIeip1l/c p2

(W+V0+mc2)

BIe

ip2l/ BIIeip2l/

= c p1(W+mc2)

CIeip1l/ CIIeip1l/

.(32)

After again defining as

=p1

W + V0 + mc2

p2 (W + mc2)=

BIe

ip2l/ BIIeip2l/

CIeip1l/ CIIeip1l/ (33)

and working through the associated straightforward algebra the matrix form of eq.

32 reads BIBII

=

1

2

(1 + )ei(p1p2)l/ (1 )ei(p1+p2)l/(1 )ei(p1+p2)l/ (1 + )ei(p1p2l/

CICII

. (34)

Eliminating BI and BII by substituting eq. 34 into eq. 31 we findAIAII

=

1

4

+ 1 1 1 + 1

(1 + )ei(p1p2)l/ (1 )ei(p1+p2)l/(1 )ei(p1+p2)l/ (1 + )ei(p1p2l/

CICII

.

(35)Carrying out the matrix multiplication we find

AIAII

=

1

2

4 1+1 e il(p1p2)

11+

eil(p1+p2)

eli(p1+p2)

e

il(p1p2)

eil(p1+p2)

e il(p1p2) 1+1 eil(p1p2)

11+ eli(p1+p2)

CICII

.

(36)Now recall that p21c

2 = W2 m2c4. From this we see that |W| < mc2 results in amomentum that is imaginary outside the well. We will now investigate these boundstates, where |W| < mc2.

If the momentum in the well is imaginary, i.e. ip1 = p0 with p0 positive, werequire AI = 0 and CII = 0 in order for the wave function to remain finite in allspace. Hence the first line of eq. 36 yields the relationship:

0 =1 2

4

1 +

1 eil(p1p2)

1 1 +

eil(p1+p2)

CI, (37)

which has the non-trivial (not CI = 0) solution1 +

1

eilp2 =

1 1 +

eilp2 . (38)

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Figure 6: Eigenenergy spectrum of a stationary, 1D, finite, relativistic potential well.Determined numerically using Newton-Raphson method. All constants, including land V0, are equal to unity. Compare Figure 5 and eq. 46.

We see that the left hand side of eq. 38 is equal to the complex conjugate of the righthand side (note that is purely imaginary because we restricted p1 to be imaginary).If an expression is equal to its complex conjugate it is real. Hence, setting the ima-

ginary part of the left hand side of 38 to zero and writing the exponential part intrigonometric form we obtain:

Im

1 + i

1 i

cos

p2l

i sin

p2l

= 0, (39)

with = i and R. Taking the imaginary part, this can be written

2cos

lp2

+

2 1 sin lp2

1 + 2

= 0. (40)

Or more simply2

1 2 = tan lp

2

. (41)

Remembering the definition of and simplifying the resulting expression thisbecomes:

tan

lp2

=

2p0p2c2

W + mc2

W + V0 + mc2

p22c2 (W + mc2)2 p20c2 (W + V0 + mc2) 2

. (42)

Using the definition of the momenta,

p22c2 = (W + V0)

2 m2c4 = W + V0 mc2 W + V0 + mc2 (43)p21c

2 = p20c2 = W2 m2c4 = Wmc2

W + mc2

, (44)

eq. 42 simplifies to1

p2ctan

lp2

= p0c

p20c2 V0W. (45)

which we have found to be identical to the result found by Strange (1998).In order to find the eigenenergies W we used the definitions of p0 (p0 = ip1), p1

and p2 (eqs. 43-44) to write eq. 45 as a function of W only:

1(W + V0) 2 m2c4

tan

l

(W + V0) 2 m2c4c

=

i

W2 m2c4V0W (W2 m2c4) . (46)

A plot of this equation is given in Figure 5. The intersection points of the two curves

plotted represent the eigenenergies W. The value of the first five interception pointswere determined using the Newton-Raphson method and are shown in Figure 6.

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6.1.2 Moving 1D potential well

We now proceed by re-deriving the eigenvalue spectrum for a potential well moving inthe x direction (see Figure 7). To our knowledge, such an attempt is unprecedented inthe literature. The ultimate aim of our derivation is to show that a moving potentialwell is contracted by a factor of 1/ when it is compared to the same well at rest.

The derivation follows the same structure as in the stationary case, except for thefact that the wave functions are now solutions of the time dependent Dirac equationand that the sides of the potential well are moving in the x direction with speed u.Let us begin by again writing down the wave functions in the three regions:

Region 1: In this region of zero potential, the Dirac equation is given by

id1dt

=

c p1 + mc2

1 = W1, (47)

where all symbols take the same meaning as in the stationary case, with primesindicating that they are pertaining to a moving well. The wave function is a solutionof the time dependent Dirac equation for a free particle (as derived in Appendix C)and is given by

1 = AI

1

00c p1

(W+mc2)

e ip1xiWt + AII1

00c p1

(W+mc2)

eip1xiWt , (48)

with p21c2 = W2 m2c4.

Region 2: In this region, where potential is reduced to V0 , the Dirac equationis given by

id2dt

=

c p2 V0 + mc2

2 = W2 (49)

and the wave function is thus given by

2 = B

I

10

0c p2

(W+V0+mc2)

eip2xiW

t

+ BII

10

0c p2

(W+V0+mc2)

eip2xiW

t

, (50)

with p22c2 = (W + V0)

2 m2c4.Region 3: This is similar to Region 1, with the Dirac equation reading

id3dt

=

c p3 + mc2

3 = W3 (51)

and the wave function

3 = C

I

10

0c p3(W+mc2)

e

ip3xiWt

+ C

II

10

0c p3(W+mc2)

e

ip3xiWt

, (52)

with p23c2 = W2 m2c4 = p21c2. We note that in Region 3 the momentum is equal

to the momentum in Region 1. From now on we will thus replace p3 with p1.

We continue by imposing the following boundary conditions:

1(ut) = 2(ut)

2(l + ut) = 3(l

+ ut)

. (53)

Applying the boundary condition at x = ut and writing out the first and the fourthcomponent of on separate lines gives

A

Ie

i(p1uW)t

+ AIIe

i(p1u+W)t

= B

Ie

i(p2uW)t

+ BIIe

i(p2u+W)t

c p1W+mc2

AIe

i(p1uW)t

AIIei(p1u+W

)t

=

c p2W+V0+mc

2

BIe

i(p2uW)t

BIIei(p2u+W

)t

.(54)

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Figure 7: Moving potential well of width l and depth V0

We will now write these two equations in matrix form, the reason for which will soonbecome apparent. Defining the dimensionless quantity as

=p1

W + V0 + mc2

p2 (W + mc2)

=

BIe

ip2ut/ BIIeip

2ut/

AIe

ip1ut/ AIIeip

1ut/ (55)

and omitting a few lines of algebra we can can thus write the set of equations definedin eq. 54 as

AIAII

=1

2 ( + 1)ei(p2p1)ut/ ( 1)e

i(p2+p

1)ut/

( 1)ei(p2+p1)ut/ ( + 1)ei(p2p1)ut/ B

IBII

. (56)

Similarly, applying the boundary condition at x = l + ut and writing out the firstand the fourth component of on separate lines gives

BIe + BIIe

+ = CIe + CIIe

+

c p2W+V0+mc

2

BIe

BIIe+

=c p1

W+mc2

CIe

CIIe+

, (57)

where we have made the following definitions so that the equation can fit on the page:

=i (p2(l

+ ut) Wt)

, (58)

+ = i (p

2(l

+ ut) + W

t)

, (59)

=i (p1(l

+ ut) Wt)

, (60)

+ =i (p1(l

+ ut) + Wt)

. (61)

After defining as

=p1

W + V0 + mc2

p2 (W + mc2)

=

BIe

i p2(l+ut)/ BIIei p

2(l+ut)/

CIei p1(l

+ut)/ CIIei p

1(l+ut)/

(62)and working through the associated straightforward algebra the matrix form of eq.57 reads

BIBII

=

1

2

(1 + )e

i(p1p

2)(l+ut)

(1 )ei(p

1+p

2)(l+ut)

(1 )e i(p

1+p

2)(l+ut)

(1 + )ei(p1p

2)(l+ut)

CICII

. (63)

Eliminating BI and BII by substituting eq. 63 into eq. 56 we obtain

AIAII

=

1 24

e 1+

1 e+ 1

1+1

2 e 1+

1 ei(p

1+p

2)l

1

1+

CICII

,

(64)where we have made the following definitions:

= i (p1 p2) l

, (65)

+ =i (p1 +p

2) l

, (66)

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1 = e

i(l+ut)(p1+p

2)+iut(p

1p

2) + e

tui(p1+p

2)+i(l+ut)(p1p

2) , (67)

2 = ei((l+2ut)p1p

2l)+2ip2l

ei((l+2ut)p1p

2l)

. (68)

In what follows, the result will have exactly the same form as in the stationarycase. This is because the first component of our matrix has the same form as in thestationary case (in fact both diagonal components are) and only the first component

is needed to find the eigenenergy spectrum.Comparing eq. 64 with its stationary counterpart (eq. 36) we see that the reas-oning continues in the exact same way and that we obtain the following eigenvalueequation for the moving potential well:

1

p2ctan

p2l

= p

0c

p20c2 V0W

. (69)

This equation has the exact same form as the equation for the stationary potential well(45, except for the fact that all the variables are now primed). This is a manifestationof the fact that the Dirac equation is Lorentz covariant and as such, it is an expectedresult.

We now make some speculative conclusions about a possible continuation of the

above argument with a view to showing that a moving potential well in fact undergoeslength contraction by a factor of 1/. First, we note that if we make the well knownrelativistic substitutions for the momentum (p = p) and if we require form invarianceof the argument of the tan() function, the only possible value ofl is l = l/. However,this is not the kind of argument we want to make within the constructive pedagogy.

In section 4.2 we presented a constructive model of a simplified atom. A key step ofthe argument was to give a fully constructive equation for the potential of a movingpoint charge as seen from a stationary observer, the analogue of V in the aboveequation. For our model of the atom, we obtained this expression from Maxwellsequations, without ever referring to special relativity. It would be our aim to treatthe potential well in a similar way, finding constructive expressions for the potentialV, the momentum p and the eigenenergies W. Having fixed these variables for themoving system as observed by a stationary observer, we would then be able to deducethe value of l unambiguously. Naturally, we expect l (the length of the well) tocontract, taking on the value l/.

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7 Conclusion

We have presented the philosophical debate between the advocates of a kinematicalexplanation of special relativity and those who argue for a dynamical explanation.The former is based on Minkowski space-time where as the latter urges us to con-sider the inter-atomic dynamics of actual matter. We have shown what is meant bya microscopic explanation of the Lorentz contraction and how such an explanation

might constitute a constructive theory of the Lorentz contraction as envisioned byLorentz and others. Following the constructivist pedagogy, our classical treatment ofa simplified atom (inspired by Bell) led us to the conclusion that we would need toexpand our model to a multi-atom system. We then considered the electromagneticstress in an infinite, simple cubic, crystal both in a stationary and in a moving con-figuration. Comparing the stresses in the moving and the stationary crystal we wereable to show microscopically that a moving crystal will undergo shape deformation.This shape deformation is due to an increase in stress in the direction of motion ofthe crystal. To our knowledge, such a treatment is unprecedented. We are confidentthat our model could be expanded to not only explain a general shape deformationbut in fact the precise Lorentz contraction. Specifically, we concluded that in orderto be able to show Lorentz contraction of a moving crystal it would be necessary tonot only consider the stress due to electromagnetic interactions but also those stresses

due to the repulsive Pauli force. We believe that only the combination of these twoforces would give an accurate model of a crystal which would allow us to show lengthcontraction microscopically. We have documented our initial attempts to this end,where we considered a 1D, finite, potential well moving at a constant velocity smallerthan the speed of light.

Acknowledgements

The author thanks Prof Adrian Sutton (project supervisor) for proposing the projectand for his continued guidance and support. Special thanks go also to my lab partnerVignesh Venkataraman and all other members of our group, namely Christopher C.Chan, Benjamin Lok, Gino Abdul-Jabbar, Hussain Anwar and Chen Lin for manyuseful discussions. Finally, we thank Prof Harvey Brown for ongoing and always verystimulating discussions surrounding the subject.

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8 Further reading

The idea of showing Lorentz contraction atomistically is far from new. However,research interest in the field has plummeted after the success of Einsteins specialrelativity theory, which seemingly made an atomistic explanation obsolete. In thepreceding sections we have presented the reasons why we think that such an atomisticexplanation is in fact still desirable today.

In this section we present a selection of papers which may offer a starting pointfor further research in the areas of atomistic explanations of Lorentz contraction inparticular and physical relativity in general. Most of the recent work on these topicshas been purely philosophical and as far as we know, the adaptation of these ideasby material scientists is unprecedented. For this reason, the immediate relevance ofthe papers listed here may vary. We encourage interested researchers to keep an openmind and not to dismiss seemingly irrelevant avenues of investigation prematurely.

Physical relativity (Brown, 2005). This is an extremely comprehensive mono-graph about the philosophical arguments surrounding the constructive explan-ations of special relativity. It gives a detailed overview of the subject fromthe constructivist point of view and offers a perfect starting point for someonelooking to grasp the philosophical intricacies of the field.

Why Constructive Relativity Fails (Norton et al. , 2007). This paper is a criticalresponse to the ideas put forward by Harvey Brown and other constructivists.The merit of this paper is to show step by step what is meant by a constructiveversion of special relativity. It is interesting that the authors manage to spell outthe exact form of the argument more clearly than many constructivists. Theyconclude by giving arguments supporting the view that a constructive versionof special relativity is impossible to achieve, if not obsolete.

The above publications are sufficient to equip the reader with a working knowledge ofthe spectrum of philosophical opinions surrounding the subject. We now conclude bygiving a selection of papers which could be of interest to someone who is approachingthe subject matter from a physicists or a materials scientists point of view:

Relativistic effects on the wave function of a moving system (Friar, 1975,?) Stress Effects due to Lorentz Contraction (Dewan, 1963) Testable scenario for relativity with minimum length (Amelino-Camelia, 2001) The fields of a moving electric dipole (Mott et al. , 1965) Lorentz contraction of bound states in 1+1 dimensional gauge theory (Jrvinen,

2004)

Hydrogen atom in relativistic motion (Jrvinen, 2005) On the meaning of Lorentz covariance (Szab, 2004)

The electromagnetic field in a uniformly moving crystal (Hillion, 1997) Treating some solid state problems with the Dirac equation (Renan et al. , 2000) Cerenkov effect and the Lorentz contraction (Pardy, 1997) Solution of the generalised Dirac equation in a time-dependent linear potential:

Relativistic geometric amplitude factor (Maamache & Lakehal, 2007)

Wave functions for a Dirac particle in a time-dependent potential (Landim &Guedes, 2000)

History and physics of the Klein paradox (Calogeracos & Dombey, 1999)

Conveyance of Quantum Particles by a Moving Potential Well (Miyashita, 2007)

New theory of nuclear forces. Relativistic origin of the repulsive core (Gross,1974)

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Appendix A Electromagnetic fields of a moving point

charge

A.1 Retarded potentials

In electrodynamics, the electric and magnetic fields are defined as

E = V A

t , (70)

B = A. (71)respectively, where V is the scalar potential and A is the vector potential. Choosingthe Lorentz gauge ( A = 00(V/t) and defining the dAlembertian operatoras 2 2 00(2/t2), Maxwells equations may be written

2V =

0, (72)

2A = 0J. (73)

where is the charge density, 0 is the magnetic permeability constant, 0 is the elec-tric permeability constant and J is the current density vector. A detailed derivation

of these results can be found in Griffiths (1999, chapter 10).From electrostatics we know the potential for a time independent charge distribu-

tion (r). Choosing the reference point at infinity, it may be written:

A(r) =04

J(r)

|r r|dr, (74)

V(r) =1

4 0

(r)

|r r|dr. (75)

Things change if the charge and current densities are functions of position as well.This is because of the speed of light at which electromagnetic signals travel. It meansthat the potential at a point r is not due to the current charge distribution. What itis due to can be imagined as a sort of shell structure, the inner most shell is the one

that is closest to the present and the further out we go the older (more retarded) theshells become (compare Figure 8).

From Figure 8 we see that we need to evaluate J(r) and (r) in eqs. 74 and 75at a time that is earlier than the time at the field point r by an amount that dependson the distance of the integration point r to the field point r. The time at which weneed to evaluate the integrands is called the retarded time and denoted by

tr = t |r r|

c. (76)

So for non-static sources, the equations for V and A become

A(r, t) =04

J(r, tr)

|r

r

|dr, (77)

V(r, t) =1

4 0

(r,tr)

|r r| dr. (78)

Figure 8: Showing the concept of retarded time in two dimensions.

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For the proof see Griffiths (1999, 423). The retarded potential can be used to calculateelectric and magnetic fields of a time dependent charge distribution. These equationsare then the electrodynamic equivalent to Coulombs law and the Biot-Savart law.They are called the causal solutions of Maxwells equations or Jefimenkos Equations.

A.2 Retarded potentials for a point charge

We proceed by working out the forms of eqs. 78 and 77 for a moving point charge atposition r0(t). Note that the integrands are zero except at the position of the pointcharge. Therefore we can write

V(r, t) =1

4 0

1

|r r0|

(r,tr)dr. (79)

It would seem that the integration would simply give q. This is not true because thecharge is moving and therefore appears longer (if coming towards) or shorter (if goingaway from the field point r). This is an effect that can be understood by consideringa moving train. The result of the following example makes no reference to the size ofthe train and applies therefore equally well to a point charge. Note that this is not arelativistic effect.

Consider a train of length L as depicted in Figure 9. What is the observed lengthL of a train of rest length L moving at speed v and related to an observer by a unitvector n?

1. How much longer does the light from the back of the train have travel to get tothe observer?

t =L cos

c. (80)

2. How is this time related to the additional distance the train moves in this time?

L cos

c=

L Lv

(81)

L L v cos c

= L (82)

L =L

1 v cos /c . (83)

3. Thus, the volume is changed as

V =V

1 n v/c . (84)

Figure 9: Train example(Griffiths, 1999).

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We can now use this result to evaluate the integral over the charge density, whoseresult is not just q but q, distorted by the above factor. The same procedure appliesto the vector potential. Hence:

V(r, t) =q

4 0

1

|r r0 (tr)| (r r0 (tr)) v/c , (85)

A(r, t) =0

4

v (tr)

|r r0 (tr)| (r r0 (tr)) v (tr) c=v (tr)

c2V(r, t). (86)

These are the Lienard-Wiechert potentials for a moving point charge.In order to simplify these expressions we consider a charge q moving on the point

z(t) in the z direction at constant velocity and starting at the origin at t = 0. Westart by finding an expression for the retarded time in terms of t, v and r:

1. The magnitude of the retarded position is:

rr = |r z (tr)| = c (t tr) (87)

x2 + y2 + (z vtr)2 = c2 (t tr)2 (88)x2 + y2 + z2

2vztr + v

2t2r = c2t2

2c2ttr + c

2t2r (89)v2 c2 t2r 2 vz c2t tr + x2 + y2 + z2 c2t2 = 0. (90)

2. Solving for the retarded time:

tr =c2t vz

(vz c2t )2 (v2 c2) (x2 + y2 + z2 c2t2)

c2 v2 (91)

1 v

2

c2

tr = t vz

c2 1

c

vzc ct

2+

1 v

2

c2

(x2 + y2 + z2 c2t2)

(92)

= tvz

c2 1

cvz

c ct 2

+ (z2

c2

t2

)

1 v2

c2

+

1v2

c2

(x2

+ y2

)

(93)

= t vzc2 1

c

t2v2 2tvz + z2 +

1 v

2

c2

(x2 + y2) (94)

= t vzc2 1

c

(z vt)2 +

1 v

2

c2

(x2 + y2). (95)

3. To decide on the sign, consider the limit v 0, when the point charge is atz = 0:tr = t z/c as v 0.From 87 we can see that the retarded time should here betr = t |rz(tr)|c = t z/c.So we want the minus sign. Hence

1 v

2

c2

tr = t vz

c2 1

c

(z vt)2 +

1 v

2

c2

(x2 + y2). (96)

We can use this result to find the vector and scalar potentials. Recall the result forthe Lienard-Wiechert scalar potential

V(r, t) =q

4 0

1

|r r0 (tr)| (r r0 (tr)) v/c , (97)

where, in our case

r r0 (tr) = xy

z vtr

, (98)

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v =

00

v

, (99)

hence

V(r, t) =q

4 0

1

c (t tr) v (z vtr)/ c (100)

= q4 01ct ctr vzc + vtrc (101)

=q

4 0

1

ct vzc c

1 v2c2

tr. (102)

Substituting for c

1 v2c2

tr from 95:

V(r, t) =q

4 0

1(z vt)2 + 1 v2c2 (x2 + y2) (103)

=q

4 0

1

1 v2

c2

1

zvt1v2/c22

+ x2 + y2

.

Similarly, using c = 1

00 and eq. 86, the vector potential for our charge is

A(r, t) =0q

4

11 v2c2

vzvt1v2/c2

2+ x2 + y2

. (104)

A.3 Fields of a point charge

Recall the equations for the electric and magnetic fields in terms of the vector andscalar potentials (eqs. 70 and 71):

E = V At

,

B = A.Thus, we need to compute V, At andA in order to find the fields.

1. Finding V:

V = q4 0

x2 + y2 + z2

3/2

1 v2c2

1/2x

1 v2c21/2

y1 v2c2

3/2(z vt)

. (105)

2. Finding At :

Azt

=0q

4

x2 + y2 + z2

3/21 v

2

c2

3/2(z vt)v2 (106)

=q

40

x2 + y2 + z2

3/21 v

2

c2

3/2(z vt) v

2

c2. (107)

Ax and Ay are zero because of the direction of the velocity vector.

3. Finding A

B =

A = yAz zAyzAx

xAz

xAy yAx =

yAz

xAz

0 = v

c2

EyEx0

(108)=

v Ac2

. (109)

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So the fields of a point charge q moving in the z direction at velocity v are given by

E =q

40

x2 + y2 + z2

3/2 xyz

, (110)

B =1

c2v E. (111)

with = 1/

1 v2/c2 and z = (z vt).

Appendix B Electromagnetic force in a dipole

Consider the dipole in shown in figure 10. The dipole consists of two charges, qi at theorigin and qj at position (x,y,z) with respect to qi. In the following, we determinethe force between these two charges when they are at rest (case A) and when theyare both moving with a constant velocity v in the z direction.

B.1 Case A: Dipole at rest

In the static case, the force between the two charges in the dipole is trivially given byCoulombs law:

Fij =qiqj

4 0

r

r2. (112)

B.2 Case B: Dipole in motion

If the two charges in Figure 10 are moving at velocity v in the z direction the forcebetween the charges can be obtained from the Lorentz force law

Fij = q(E+ vB). (113)

The fields E and B for a moving charge have been derived in A and are given by eqs.110 and 111 . The velocity vector is v = (0, 0, v). Thus, noting that Fij is a functionof (xij, yij, zij + vt)) we proceed by writing

F (xij , yij , zij + vt) = q

E+

v vEc2

. (114)

using the vector identity A (B C) = B(A C) C(A B) for the triple crossproduct this becomes

F (xij , yij , zij + vt) = q

E+

v(v E)Ev2c2

(115)

F (xij , yij , zij + vt) =1

4 0q

E

2+v(v E)

c2 . (116)

Figure 10: Dipole with qi at the origin and qj at position (x,y,z).

26


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Substituting for the electric field (eq. 110) into F (xij , yij , zij + vt):

Fij =qiqj

4 0

1

(x2 + y2 + z2)3/2

12

xy

z

+ 1

c2

00

v

00

v

xy

z

,(117)

with z = zvt1v2/c2

. Thus F (xij , yij , zij ) is

Fij =qiqj

4 0

1x2ij + y

2ij +

zij+vtvt1v2/c2

23/2 1

2

xijyij

zij+vtvt1v2/c2

+

00

v2

c2zij+vtvt1v2/c2

(118)

Fij =qiqj

4 0

1x2ij + y

2ij +

2z2ij3/2 12

xijyij

zij

1 + v

2/c2

1v2/c2

(119)

Fij =qiqj

4 0

1x2ij + y2ij + 2z2ij

3/2 12

xijyij

3 zij . (120)

So the force in the dipole is

Fij =qiqj

4 0

1x2ij + y

2ij +

2z2ij3/2 1

xijyij

2 zij

. (121)

Appendix C Free particle solution of the Dirac equa-

tion

Consider the Dirac Equation

i

t= H = (c

p + mc2) = W , (122)

with p = p and all other symbols have their usual meaning, especially

=

I 00 I

,

=

0 0

, (123)

=

xyz

, x =

0 11 0

,

y =

0 ii 0

,

z =

1 00 1

. (124)

For a free particle this Dirac equation has solutions of the form

= u exp(i(ppp rrr W t)/), (125)where and uuu are 4 component vectors.

Separating this into space and time components and singling out the spacial com-ponent for now we can write

= u exp(ippp rrr/). (126)In order to find the vector u we substitute this into the Dirac equation and obtain

c ppp +

mc2u = Wu. (127)

The LHS of the this eigenvalue problem can be expressed as a block diagonal matrixmc2 c ppp ppp ppp

c ppp ppp ppp mc2

=

W 00 W

(128)

27


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Wmc2 c ppp ppp pppc ppp ppp ppp W + mc2

= 0. (129)

We look for eigenvectors of the form

u =

, (130)

where and are 2 component vectors.Finding eigenvalues W:

W mc2 c ( ppp ppp ppp))) = 0 (131)c ( ppp ppp ppp))) + W + mc2 = 0 (132)

=c ( ppp ppp ppp)))

(W + mc2) (133)

=c ( ppp ppp ppp)))

(Wmc2) (134)

=c ( ppp ppp ppp)))

(W mc2)c ( ppp ppp ppp)))

(W + mc2) (135)

Wmc2 W + mc2 c2 ( ppp ppp ppp)))2 = 0. (136)

We note that

p =

i =

iU(ppp))) exp(ippp rrr/) = pppU(ppp))) exp(ippp rrr/) = ppp (137)

and that

( ppp ppp ppp))) 2 =

pz px ipypx + ipy pz

2 (138)

( ppp ppp ppp)))2 = p2z +

px ipy

2 0

0 p2z + px ipy2 = p

2 (139)

So Wmc2 W + mc2 c2p2 = 0. (140)

So we have the conditionW2 = p2c2 + m2c4 (141)

as expected.Finding basis states: First, choose two independent vectors normalised and

orthogonal to each other (i.e.: rs = rs), e.g.:

1 =

10

2 =

01

. (142)

So

u = N

(143)

=c ( ppp ppp ppp)))

(W + mc2) =

c

(W + mc2)

pz px ipy

px ipy pz

(144)

1 =c

(W + mc2)

pz

px + ipy

2 =

c

(W + mc2)

px ipypz

(145)

u1 = N1

10

c pz(W+mc2)

c (((px+ipy)(W+mc2)

u2 = N2

01

c (((pxipy)(W+mc2)c pz

(W+mc2)

. (146)

Normalise

1 = u1u1 = N2

1 1

c(ppppppppp)))(W+mc2)

1c(ppppppppp)))

(W+mc2)1

(147)

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1 = N2

11 + 1

c ( ppp ppp ppp)))(W + mc2)

c ( ppp ppp ppp)))(W + mc2)

1

(148)

1 = N2

11 +

1

c2 ( ppp ppp ppp)))2(W + mc2)

2 1

(149)

1 = N211 + 1 c2p2

(W + mc2)2

1 = N2c2p2 + W + mc

2

2

(W + mc2)2 11 (150)

1 = N2

c2p2 + W2 + 2W mc2 + m2c4

(W + mc2)2

= N2

2W2 + 2W mc2

(W + mc2)2

(151)

1 = N22W

W + mc2. (152)

And so

u1 =

W + mc2

2W

10

c pz(W+mc2)

c (((px+ipy)

(W+mc2

)

u2 =

W + mc2

2W

01


(W+mc2

)

. (153)

From this we can get the wave functions

1 = N

W + mc2

2W

10

c pz(W+mc2)

c (((px+ipy)(W+mc2)

exp(i(ppp rrr W t)/) (154)

2 = N

W + mc2

2W

01


(W+mc2

)

exp(i(ppp rrr W t)/). (155)

Normalising

dVN2 exp(i(p r W t)/) exp(i(p r W t)/) = N2V = 1 (156)

so the normalised wave functions for a free, relativistic, positive energy particle are

1 =1V

W + mc2

2W

1/2

10

c pz(W+mc2)

c (((px+ipy)(W+mc2)

exp(i(ppp rrr W t)/) (157)

2 =1V

W + mc2

2W

1/201


(W+mc2)

exp(i(ppp rrr W t)/). (158)

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lorentz contraction: explained at the microscopic level

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