Case Study 7

Relativity - Special and General

• How we teach relativity

• What actually happened

• A brief overview of Einstein’s route to General Relativity

1

The Usual Story

• Bradley’s observations of stellar aberration of 1727-8 – the Earth moves through astationary aether.

• The null result of Michelson-Morley experiment of 1887 – no detectable motion ofthe Earth through the aether.

• Einstein’s Principles of Relativity, in particular, the second postulate that the speedof light is the same for observers in any inertial frame of reference.

• Derivation of the Lorentz transformations and relativistic kinematics.

• Invariance of the laws of physics under Lorentz transformation.

• Relativistic dynamics.

• E = mc2, Maxwell’s equations, etc.

2

In fact, . . .

• 1887 Null result of the Michelson-Morley experiment.

• 1887 Voigt derived primitive form of the Lorentz transformation.

• 1889 Fitzgerald proposed length contraction as the solution to the null result of theMichelson-Morley experiment

• 1895 Lorentz wrote Maxwell’s equations in a moving medium and derived a versionof the Lorentz transformation.

• 1898 Poincare wrestles with the problem of the aether. A limiting speed of light?

• 1904 Lorentz’s definitive version of the Lorentz transformations.

3

Voigt (1887)

Remarkably, in 1887, Woldmar Voigt noted that Maxwell’s wave equation forelectromagnetic waves

∇2H −

1

c2∂2H

∂t2= 0

is form-invariant under the transformation

t′ = t −V x

c2,

x′ = x − V t,

y′ = y/γ,

z′ = z/γ

where γ = (1 − V 2/c2)−1/2. Except for the fact that the transformations on theright-hand side have been divided by the Lorentz factor γ, this set of equations is theLorentz transformation. Voigt derived this expression using the invariance of the phaseof a propagating electromagnetic wave, the easiest way of deriving the Lorentztransformations. This work was unknown to Lorentz when he derived what we nowknow as the Lorentz transformations.

4

G.F. Fitzgerald, Science, 13, 390, 1889.

‘I have read with much interest Messrs. Michelson and Morley’s wonderfullydelicate experiments attempting to decide the important question as to how farthe aether is carried along by the Earth. Their result seems opposed to otherexperiments showing that the aether in the air can only be carried along only toan inappreciable extent. I would suggest that almost the only hypothesis thatcan reconcile this opposition is that the length of material bodies changes,according as they are moving through the aether or across it, by an amountdepending on the square of the ratio of their velocity to that of light. We knowthat electric forces are affected by the motion of electrified bodies relative tothe aether, and it seems a not improbable supposition that the molecularforces are affected by the motion, and that the size of the body altersconsequently. . . . ’

5

Fitzgerald Contraction

The above quotation is more than 60% of his brief note. Remarkably, this paper, bywhich Fitzgerald is best remembered, was not included in his complete works edited byLarmor in 1902. Lorentz knew of the paper in 1894, but Fitzgerald was uncertain as towhether or not it had been published when Lorentz wrote to him. The reason was thatScience went bankrupt in 1889 and was only refounded in 1895.

Notice that Fitzgerald’s proposal was only qualitative and that he was proposing a realphysical contraction of the body in its direction of motion because of interaction with theaether.

6

Hendrik Lorentz

Hendrik Lorentz had agonised over the null result of the Michelson-Morley experimentand in 1892 came up with same suggestion as Fitzgerald, but with a quantitativeexpression for the length contraction. In his words,

‘This experiment has been puzzling me for a long time, and in the end I havebeen able to think of only one means of reconciling it with Fresnel’s theory. Itconsists in the supposition that the line joining two points of a solid body, if atfirst parallel to the direction of the Earth’s motion, does not keep the samelength when subsequently turned through 90◦.’

Lorentz worked out that the length contraction had to amount to

l = l0

(

1 −V 2

2c2

)

,

which is just the low velocity limit of the expression

l =l0γ

, where γ =

(

1 −V 2

c2

)−1/2

.

Subsequently, this phenomenon has been referred to as Fitzgerald-Lorentz contraction.

7

Lorentz (1895)

There was support for the contraction conjecture from the orbit of an electron in amoving body according to Maxwell’s equations. The diameter of its orbit in the directionof motion is flattened by a factor γ. This was an integral part of Lorentz’s theory of theelectron.

In 1895, Lorentz tackled the problem of the transformations which would result in forminvariance of Maxwell’s equations and derived the following relations, which in SInotation are:

x′ = x − V t, y′ = y, z′ = z,

t′ = t −V x

c2,

E′ = E + V × B,

B′ = B −V × E

c2,

P ′ = P

where P is the polarisation. Under this set of transformations, Maxwell’s equations areform-invariant to first order in V/c.

8

Lorentz (1895) – continued

Notice that time is no longer absolute. Lorentz apparently considered this simply to bea convenient mathematical tool in order to ensure form-invariance to first order in V/c.He called t the general time and t′ the local time.

In order to account for the null result of the Michelson-Morley experiment, he had toinclude an additional second-order compensation factor, the Fitzgerald-Lorentzcontraction (1 − V 2/c2)−1/2, into the theory.

One important innovation of this paper was the assumption that the force on anelectron should be given by the first order expression

f = e(E + V × B)

This is the origin of the expression for the Lorentz force for the joint action of electricand magnetic fields on a charged particle.

9

Lorentz (1899)

Einstein knew of Lorentz’s paper of 1895, but was unaware of his subsequent work. In1899, Lorentz established the invariance of the equations of electromagnetism to allorders in V/c through a new set of transformations:

x′ = ǫγ(x − V t), y′ = ǫy, z′ = ǫz,

t′ = ǫγ

(

t −V x

c2

)

.

These are the Lorentz transformations, including the scale factor ǫ. By this means, hewas able to incorporate length contraction into the transformations. Almostcoincidentally, in 1898, Joseph Larmor wrote his prize winning essay Aether andMatter, in which he derived the standard form of the Lorentz transformations andshowed that they included the Fitzgerald-Lorentz contaction.

In his major paper of 1904, entitled Electromagnetic Phenomena in a System Movingwith Any Velocity Smaller than Light, Lorentz presented the transformations with ǫ = 1.

10

Henri PoincareIn 1898, Poincare wrote:

‘The simultaneity of two events or the order of their succession, as well as theequality of two time intervals, must be defined in such a way that the statementof the natural laws be as simple as possible. In other words, all rules anddefinitions are but the result of unconscious opportunism.’

In 1904, Poincare surveyed the current problems in physics and included the statement:

‘. . . the principle of relativity, according to which the laws of physicalphenomena should be the same whether for an observer fixed or for anobserver carried along by uniform movement or translation.’

He concluded by remarking

‘Perhaps likewise, we should construct a whole new mechanics, . . . where,inertia increasing with the velocity, the velocity of light would become animpassible limit.’

11

Albert Einstein (1905)

Albert Einstein had been wrestling with exactly these problems since 1898. In 1905, hewas working in the Patent Office in Bern and among the patents he had to review werethose concerning the synchronisation of clocks for the Swiss railway system (seeEinstein’s Clocks, Poincare’s Maps: Empires of Time by Peter Galison).

According to Einstein in a letter of 25 April 1912 to Paul Ehrenfest,

‘I knew that the principle of the constancy of the velocity of light was somethingquite independent of the relativity postulate and I weighted which was the moreprobable, the principle of the constancy of c, as required by Maxwell’sequations, or the constancy of c exclusively for an observer located at the lightsource. I decided in favour of the former.’

12

Einstein (1905)

In 1924, he stated:

‘After seven years of reflection in vain (1898-1905), the solution came to mesuddenly with the thought that our concepts of space and time can only claimvalidity insofar as they stand in a clear relation to our experiences; and thatexperience could very well lead to the alteration of these concepts and laws.By a revision of the concept of simultaneity into a more malleable form, I thusarrived at the special theory of relativity.’

Once he had discovered the concept of the relativity of simultaneity, it took him only fiveweeks to complete his great paper, On the Electrodynamics of Moving Bodies.

13

Should Einstein Get the Credit?

Einstein’s point of view:

‘With respect to the theory of relativity, it is not at all a question of arevolutionary act, but a natural development of a line which can be pursuedthrough the centuries.’

Lorentz published his final form of the transforms in 1904 and Einstein was not aware ofthem when he published his paper in 1905. Further, Lorentz had to assume thetransformations, rather than deriving them from Einstein’s two postulates of SpecialRelativity.

It is interesting to contrast Lorentz’s paper of 1904 with Einstein’s of 1905. Besides thetwo postulates, Einstein made only four assumptions, one concerning the isotropy andhomogeneity of space, the others concerning three logical properties of the definition ofsynchronisation of clocks.

14

Lorentz’s Assumptions

Lorentz’s paper contains 11 ad hoc hypotheses, for example:

• Restriction to v ≪ c

• postulation a priori of the transformations

• stationary aether

• stationary electron is round, with uniform charge.

• all its mass is electromagnetic

• one dimension is shrunk by a factor of γ

• . . .

The reason for the complexity of his approach was that the transforms were intimatelybound up with his theory of the electron.

15

Pedagogical Note: Lorentz Force (1)

We can use Lorentz contraction to illustrate the equivalence of electric and magneticforces.

Suppose we have a current carrying wire in which, in the frame S, the electrons drift atvelocity v while the ions are stationary. The current is I = ρev , where ρe is the numberdensity of electrons per unit length and is equal to ρi, the number density of ions, whichare stationary. Applying Ampere’s law, the magnetic flux density at radial distance r

from the wire is∮

B · ds = µ0I B =µ0ρev

2πr

If a charge q is moving at speed u, parallel to the wire, the Lorentz force is

fL = q(u × B) = qµ0ρeuv

2πr

away from the wire, if q > 0.

16

Pedagogical Note: Lorentz Force (2)

Now repeat in the frame of the moving charge. The ions are moving in the negativex-direction at speed u and the electrons have speed v′ which is the relativistic sum of v

and u in opposite directions, with u ≪ c, v ≪ c. The appropriate Lorentz factors are

γ′i =

1(

1 −u2

c2

)1/2γ′e =

1 −uv

c2(

1 −u2

c2

)1/2(

1 −v2

c2

)1/2

and the corresponding charge densities are

ρ′i = ρiγ′i ρ′e = ρeγ

′e

Therefore, there is a net positive charge on the wire which results in an electrostaticrepulsive force. The field at distance r from the wire is

fE = qE = qρ′i − ρ′e2πǫ0r

≈ qρeuv

2πǫ0c2rγ = q

µ0ρeuv

2πr

Notice that we had to add together the speeds relativistically.

17

Einstein 1907 – Relativistic Gravity

To quote Einstein’s own words from his Kyoto address of December 1922.

‘In 1907, while I was writing a review of the consequences of special relativity,. . . I realised that all the natural phenomena could be discussed in terms ofspecial relativity except for the law of gravitation. I felt a deep desire tounderstand the reason behind this . . . It was most unsatisfactory to me that,although the relation between inertia and energy is so beautifully derived [inspecial relativity], there is no relation between inertia and weight. I suspectedthat this relationship was inexplicable by means of special relativity.’

In the same lecture, he remarks

‘I was sitting in a chair in the patent office in Bern when all of a sudden athought occurred to me: ‘If a person falls freely he will not feel his own weight.’I was startled. This simple thought made a deep impression upon me. Itimpelled me towards a theory of gravitation.’

18

The Principle of Equivalence

In his comprehensive review of relativity published in 1907, Einstein devoted the wholeof the last section, Section V, to The Principle of Relativity and Gravitation. In the veryfirst paragraph, he raised the question,

‘Is it conceivable that the principle of relativity also applies to systems that areaccelerated relative to one another?’

He had no doubt about the answer and stated the principle of equivalence explicitly forthe first time:

‘. . . in the discussion that follows, we shall therefore assume the completephysical equivalence of a gravitational field and a corresponding accelerationof the reference system.’

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The Deflection of Light by Massive Bodies

Applying Maxwell’s equation to the propagation of light in a gravitational potential, hefound that the equations are form-invariant, provided the speed of light varies in theradial direction as

c(r) = c

(

1 +Φ(r)

c2

)

,

recalling that Φ is always negative.

Einstein realised that, as a result of Huygens’ principle, or equivalently Fermat’sprinciple of least time, light rays are bent in a non-uniform gravitational field. He wasdisappointed to find that the effect was too small to be detected in any terrestrialexperiment.

20

The Deflection of Light by Massive Bodies (1911)

Einstein published nothing on gravity and relativity until 1911. He reviewed his earlierideas, but noted that the gravitational dependence of the speed of light would result inthe deflection of the light of background stars by the Sun. Applying Huygens’ principleto the propagation of light rays with a variable speed of light, he found the standard‘Newtonian’ result that the angular deflection of light by a mass M would amount to

∆θ =2GM

pc2,

where p is the collision, or impact, parameter. For the Sun, this deflection amounts to0.87 arcsec, although Einstein estimated 0.83 arcsec. Einstein urged astronomers toattempt to measure this deflection.

21

Einstein (1912-1915)

Following the Solvay conference of 1911, Einstein returned to the problem ofincorporating gravity into the theory of relativity and, from 1912 to 1915, his efforts wereprincipally devoted to formulating the relativistic theory of gravity. It was to prove to be atitanic struggle.

During 1912, he realised that he needed more general space-time transformations thanthose of special relatively. Two quotations illustrate the evolution of his thought.

‘The simple physical interpretation of the space-time coordinates will have tobe forfeited, and it cannot yet be grasped what form the general space-timetransformations could have.’

‘If all accelerated systems are equivalent, then Euclidean geometry cannothold in all of them.’

22

Einstein and Grossmann

Towards the end of 1912, he realised that what was needed was non-Euclideangeometry. From his student days, he vaguely remembered Gauss’s theory of surfaces.Einstein consulted his old school friend, the mathematician Marcel Grossmann, aboutthe most general forms of transformation between frames of reference for metrics of theform

ds2 = gµν dxµ dxν. (1)

Although outside Grossmann’s field of expertise, he soon came back with the answerthat the most general transformation formulae were the Riemannian geometries, butthat they had the ‘bad feature’ that they are non-linear. Einstein instantly recognisedthat, on the contrary, this was a great advantage since any satisfactory theory ofrelativisitic gravity must be non-linear.

23

Einstein and Grossmann

The collaboration between Einstein and Grossmann was crucial in elucidating thefeatures of Riemannian geometry essential for the development of the theory, Einsteinfully acknowledging the central role which Grossmann had played. At the end of theintroduction to his first monograph on General Relativity, Einstein wrote

‘Finally, grateful thoughts go at this place to my friend the mathematicianGrossmann, who by his help not only saved me the study of the relevantmathematical literature but also supported me in the search for the fieldequations of gravitation.’

24

The Final Form of General Relativity

The Einstein-Grossmann paper of 1913 was the first exposition of the role ofRiemannian geometry in the search for a relativistic theory of gravity. The details ofEinstein’s struggles over the next three years are fully recounted by Pais. It was a hugeand exhausting intellectual endeavour which culminated in the presentation of thetheory in its full glory in November 1915.

In that month, Einstein discovered that he could account precisely for the perihelionshift of Mercury, discovered by Le Verrier in 1859, as a natural consequence of hisGeneral Relativity of Relativity. He knew he must be right.

25

The Eclipse Expeditions of 1919

Einstein and Eddington

In 1919, the famous eclipse expeditionsto Principe off the coast of SpanishGuinea in West Africa and to Sobral inBrazil led by Eddington and Crommelinmeasured the deflection of the positionsof stars grazing the Sun and foundresults consistent with the predictions ofGeneral Relativity, ∆θ = 1.75 arcsec.

Sobral ∆θ = 1.98 ± 0.12 arcsecPrincipe ∆θ = 1.61 ± 0.3 arcsec

An example of the results from theSobral expedition.

26

Einstein’s Achievement

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