Been Wrestling With Exactly These Problems Since

Case Study 7 Relativity - Special and General • How we teach relativity • What actually happened • A brief overview of Einstein’s route to General Relativity

The Usual Story • Bradley’s observations of stellar aberration of 1727-8 – the Earth moves through a stationary aether. • The null result of Michelson-Morley experiment of 1887 – no detectable motion of the Earth through the aether. • Einstein’s Principles of Relativity, in particular, the second postulate that the speed of 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.

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 the Michelson-Morley experiment • 1895 Lorentz wrote Maxwell’s equations in a moving medium and derived a version of 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.

Voigt (1887) Remarkably, in 1887, Woldmar Voigt noted that Maxwell’s wave equation for electromagnetic waves 1 ∂ 2H =0 ∇ H− 2 2 c ∂t is form-invariant under the transformation Vx ′ t = t− 2, c x′ = x − V t, 2

y′

= y/γ,

z′

= z/γ

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

G.F. Fitzgerald, Science, 13, 390, 1889. ‘I have read with much interest Messrs. Michelson and Morley’s wonderfully delicate experiments attempting to decide the important question as to how far the aether is carried along by the Earth. Their result seems opposed to other experiments showing that the aether in the air can only be carried along only to an inappreciable extent. I would suggest that almost the only hypothesis that can reconcile this opposition is that the length of material bodies changes, according as they are moving through the aether or across it, by an amount depending on the square of the ratio of their velocity to that of light. We know that electric forces are affected by the motion of electrified bodies relative to the aether, and it seems a not improbable supposition that the molecular forces are affected by the motion, and that the size of the body alters consequently. . . . ’

Fitzgerald Contraction The above quotation is more than 60% of his brief note. Remarkably, this paper, by which Fitzgerald is best remembered, was not included in his complete works edited by Larmor in 1902. Lorentz knew of the paper in 1894, but Fitzgerald was uncertain as to whether or not it had been published when Lorentz wrote to him. The reason was that Science went bankrupt in 1889 and was only refounded in 1895. Notice that Fitzgerald’s proposal was only qualitative and that he was proposing a real physical contraction of the body in its direction of motion because of interaction with the aether.

Hendrik Lorentz Hendrik Lorentz had agonised over the null result of the Michelson-Morley experiment and in 1892 came up with same suggestion as Fitzgerald, but with a quantitative expression for the length contraction. In his words, ‘This experiment has been puzzling me for a long time, and in the end I have been able to think of only one means of reconciling it with Fresnel’s theory. It consists in the supposition that the line joining two points of a solid body, if at first parallel to the direction of the Earth’s motion, does not keep the same length when subsequently turned through 90◦.’ Lorentz worked out that the length contraction had to amount to l = l0 1 −

V2 2c2

!

,

which is just the low velocity limit of the expression l l = 0, γ

where

!−1/2 2 V

γ = 1− 2 c

.

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

Lorentz (1895) There was support for the contraction conjecture from the orbit of an electron in a moving body according to Maxwell’s equations. The diameter of its orbit in the direction of motion is flattened by a factor γ. This was an integral part of Lorentz’s theory of the electron. In 1895, Lorentz tackled the problem of the transformations which would result in form invariance of Maxwell’s equations and derived the following relations, which in SI notation are:  ′ ′ = y, z ′ = z,  = x − V t, y x          t′    

′ E 

     ′   B      P ′

Vx = t− 2 , c = E + V × B,

V ×E =B− , 2 c =P

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

Lorentz (1895) – continued Notice that time is no longer absolute. Lorentz apparently considered this simply to be a 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 to include an additional second-order compensation factor, the Fitzgerald-Lorentz contraction (1 − V 2/c2)−1/2, into the theory. One important innovation of this paper was the assumption that the force on an electron 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 electric and magnetic fields on a charged particle.

Lorentz (1899) Einstein knew of Lorentz’s paper of 1895, but was unaware of his subsequent work. In 1899, Lorentz established the invariance of the equations of electromagnetism to all orders in V /c through a new set of transformations:  ′   x   t′

= ǫγ(x − V t),

y ′ = ǫy,

z ′ = ǫz,

Vx = ǫγ t − 2 . c 



These are the Lorentz transformations, including the scale factor ǫ. By this means, he was able to incorporate length contraction into the transformations. Almost coincidentally, in 1898, Joseph Larmor wrote his prize winning essay Aether and Matter, in which he derived the standard form of the Lorentz transformations and showed that they included the Fitzgerald-Lorentz contaction. In his major paper of 1904, entitled Electromagnetic Phenomena in a System Moving with Any Velocity Smaller than Light, Lorentz presented the transformations with ǫ = 1.

Henri Poincare´ In 1898, Poincare´ wrote: ‘The simultaneity of two events or the order of their succession, as well as the equality of two time intervals, must be defined in such a way that the statement of the natural laws be as simple as possible. In other words, all rules and definitions 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 physical phenomena should be the same whether for an observer fixed or for an observer 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 an impassible limit.’

Albert Einstein (1905) Albert Einstein had been wrestling with exactly these problems since 1898. In 1905, he was working in the Patent Office in Bern and among the patents he had to review were those concerning the synchronisation of clocks for the Swiss railway system (see Einstein’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 something quite independent of the relativity postulate and I weighted which was the more probable, the principle of the constancy of c, as required by Maxwell’s equations, or the constancy of c exclusively for an observer located at the light source. I decided in favour of the former.’

Einstein (1905) In 1924, he stated:

‘After seven years of reflection in vain (1898-1905), the solution came to me suddenly with the thought that our concepts of space and time can only claim validity insofar as they stand in a clear relation to our experiences; and that experience 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 thus arrived at the special theory of relativity.’

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

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 a revolutionary act, but a natural development of a line which can be pursued through the centuries.’

Lorentz published his final form of the transforms in 1904 and Einstein was not aware of them when he published his paper in 1905. Further, Lorentz had to assume the transformations, rather than deriving them from Einstein’s two postulates of Special Relativity. It is interesting to contrast Lorentz’s paper of 1904 with Einstein’s of 1905. Besides the two postulates, Einstein made only four assumptions, one concerning the isotropy and homogeneity of space, the others concerning three logical properties of the definition of synchronisation of clocks.

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 intimately bound up with his theory of the electron.

Pedagogical Note: Lorentz Force (1) We can use Lorentz contraction to illustrate the equivalence of electric and magnetic forces. Suppose we have a current carrying wire in which, in the frame S, the electrons drift at velocity v while the ions are stationary. The current is I = ρev , where ρe is the number density of electrons per unit length and is equal to ρi, the number density of ions, which ` are stationary. Applying Ampere’s law, the magnetic flux density at radial distance r from the wire is I µ ρe v B · ds = µ0I B= 0 2πr If a charge q is moving at speed u, parallel to the wire, the Lorentz force is µ ρeuv f L = q(u × B ) = q 0 2πr away from the wire, if q > 0.

Pedagogical Note: Lorentz Force (2) Now repeat in the frame of the moving charge. The ions are moving in the negative x-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 uv 1 − 1 2 ′ ′ c γi = γe = !1/2 !1/2 !1/2 2 2 2 u u v 1− 2 1− 2 1− 2 c c c and the corresponding charge densities are ρ′i = ρiγi′

ρ′e = ρeγe′

Therefore, there is a net positive charge on the wire which results in an electrostatic repulsive force. The field at distance r from the wire is ρ′i − ρ′e ρeuv µ0ρeuv fE = qE = q ≈q γ = q 2πǫ0r 2πǫ0c2r 2πr Notice that we had to add together the speeds relativistically.

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 of special relativity except for the law of gravitation. I felt a deep desire to understand the reason behind this . . . It was most unsatisfactory to me that, although the relation between inertia and energy is so beautifully derived [in special relativity], there is no relation between inertia and weight. I suspected that 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 a thought 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. It impelled me towards a theory of gravitation.’

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

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

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

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

The Deflection of Light by Massive Bodies Applying Maxwell’s equation to the propagation of light in a gravitational potential, he found that the equations are form-invariant, provided the speed of light varies in the radial direction as ! Φ(r) , c(r) = c 1 + 2 c recalling that Φ is always negative. Einstein realised that, as a result of Huygens’ principle, or equivalently Fermat’s principle of least time, light rays are bent in a non-uniform gravitational field. He was disappointed to find that the effect was too small to be detected in any terrestrial experiment.

The Deflection of Light by Massive Bodies (1911) Einstein published nothing on gravity and relativity until 1911. He reviewed his earlier ideas, but noted that the gravitational dependence of the speed of light would result in the deflection of the light of background stars by the Sun. Applying Huygens’ principle to 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 2GM , ∆θ = pc2 where p is the collision, or impact, parameter. For the Sun, this deflection amounts to 0.87 arcsec, although Einstein estimated 0.83 arcsec. Einstein urged astronomers to attempt to measure this deflection.

Einstein (1912-1915) Following the Solvay conference of 1911, Einstein returned to the problem of incorporating gravity into the theory of relativity and, from 1912 to 1915, his efforts were principally devoted to formulating the relativistic theory of gravity. It was to prove to be a titanic struggle. During 1912, he realised that he needed more general space-time transformations than those of special relatively. Two quotations illustrate the evolution of his thought.

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

‘If all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them.’

Einstein and Grossmann Towards the end of 1912, he realised that what was needed was non-Euclidean geometry. From his student days, he vaguely remembered Gauss’s theory of surfaces. Einstein consulted his old school friend, the mathematician Marcel Grossmann, about the most general forms of transformation between frames of reference for metrics of the form ds2 = gµν dxµ dxν .

(1)

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

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

‘Finally, grateful thoughts go at this place to my friend the mathematician Grossmann, who by his help not only saved me the study of the relevant mathematical literature but also supported me in the search for the field equations of gravitation.’

The Final Form of General Relativity The Einstein-Grossmann paper of 1913 was the first exposition of the role of Riemannian geometry in the search for a relativistic theory of gravity. The details of Einstein’s struggles over the next three years are fully recounted by Pais. It was a huge and exhausting intellectual endeavour which culminated in the presentation of the theory in its full glory in November 1915. In that month, Einstein discovered that he could account precisely for the perihelion shift of Mercury, discovered by Le Verrier in 1859, as a natural consequence of his General Relativity of Relativity. He knew he must be right.

The Eclipse Expeditions of 1919 Einstein and Eddington In 1919, the famous eclipse expeditions to Principe off the coast of Spanish Guinea in West Africa and to Sobral in Brazil led by Eddington and Crommelin measured the deflection of the positions of stars grazing the Sun and found results consistent with the predictions of General Relativity, ∆θ = 1.75 arcsec. Sobral ∆θ = 1.98 ± 0.12 arcsec Principe ∆θ = 1.61 ± 0.3 arcsec An example of the results from the Sobral expedition.

Einstein’s Achievement