# Relativity

## Special Relativity

Special relativity grew out of problems in electrodynamics. In 1862 James Clerk Maxwell (1831–1879) first published the set of equations named after him, providing a comprehensive description of all electric and magnetic phenomena that had been studied up to that point. He also made a new prediction. The equations allow electromagnetic waves propagating at the speed of light. Maxwell famously concluded: "light consists of the transverse undulations of the same medium which is the cause of electric and magnetic phenomena" (Niven, vol. 1, p. 500; italics original).

### The luminiferous ether.

As the quotation above illustrates, it was taken for granted in the nineteenth century that both light waves and electric and magnetic fields need a medium to support them. This medium, which was thought to fill the entire universe, was called the luminiferous (light-carrying) ether. Most physicists believed that it was completely immobile (for discussion of the reasons for this belief, see Janssen and Stachel). Ordinary matter, they thought, would move through the ether without disturbing it in the least. Earth, for instance, would zip through the ether at a velocity in the order of 30 km/s, the velocity of Earth's motion around the sun. On Earth there should therefore be a brisk ether wind blowing in the opposite direction. This ether drift, as it was called, could not be felt directly, but ever since the resurgence of the wave theory of light at the beginning of the nineteenth century, attempts had been made to detect its influence on light from terrestrial and celestial sources. To be sure, such effects were expected to be small. The velocity of light, after all, is ten thousand times greater than Earth's velocity in its orbit around the Sun. Yet optical experiments were accurate enough to detect such effects. All attempts to detect the elusive ether drift failed, however, and optical theory had to be adjusted to account for these failures.

The combination of Maxwell's theory and the concept of an immobile ether likewise faced the problem of how to explain the absence of any detectable ether drift. When Maxwell found that his equations predict electromagnetic waves propagating with the velocity of light, he quite naturally assumed that this would be their velocity with respect to the ether. As long as one accepts classical ideas about space and time as everyone before Einstein tacitly did, it trivially follows that their velocity with respect to an observer on Earth is the vector sum of the velocity of propagation in the ether and the velocity of the ether drift on Earth. This, in turn, meant that Maxwell's equations could only hold in a frame of reference at rest in the ether: in a moving frame, electromagnetic waves would have different velocities in different directions. The frame of reference of Earth is such a moving frame. One is thus driven to the conclusion that Maxwell's equations do not hold in the frame in and for which they were discovered. Experiments with electricity and magnetism were not accurate enough to detect any possible deviations from Maxwell's equations, but experiments in optics were. The failure of such experiments to detect ether drift thus posed a problem for the theory.

### Lorentz invariance.

In the 1890s Lorentz set out to explain the absence of any signs of ether drift on the basis of Maxwell's theory. Tacitly sing classical notions of space and time, Lorentz first determined the laws that electric and magnetic fields obey in a frame moving through the ether, given that they obey Maxwell's equations in a frame at rest in the ether. He then replaced the components (Ex, …, Bx, …) of the real electric and magnetic fields E and B by the components (Etx, …, Btx, ) of the auxiliary and, as far as Lorentz was concerned, purely fictitious fields Et and Bt. The components of these auxiliary fields mix components of the real electric and magnetic fields (e.g., in a frame moving with velocity v in the x-direction, Ety Ey vBz). He did the same with the space and time coordinates. In particular, he replaced the real time t by a fictitious variable tt, which he gave the suggestive name "local time," because it depends on position (in a frame moving with velocity v in the x-direction, tt t (v/c2x). Lorentz chose these quantities in such a way that in any frame moving through the ether with some constant velocity v, the fictitious fields Et and Bt as functions of the fictitious variables (xt, tt) satisfy Maxwell's equations, just as the real fields as functions of the real space and time variables in a frame at rest in the ether. Maxwell's equations, in other words, are invariant under the transformation from the real fields E and B as functions of the space and time coordinates in a frame at rest in the ether to the fictitious fields Et and Bt as functions of the fictitious space and time coordinates of a moving frame. This is the essence of what Lorentz called the theorem of corresponding states. The transformation is an example of what are now called Lorentz transformations. Maxwell's equations are invariant under Lorentz transformations—or Lorentz invariant for short.

In 1895 Lorentz proved the first version of this theorem, valid to the first order in the small quantity v/c 104. In 1899 he developed an exact version of the theorem, which he continued to improve over the following years.

With the help of his theorem was able to show in 1895 that, to first order in v/c, many phenomena on Earth or in any other frame moving through the ether will be indistinguishable from the corresponding phenomena in a frame at rest in the ether. In particular, he could show that, at least to this degree of accuracy, motion through the ether would not affect the pattern of light and shadow obtained in any optical experiment. Since the vast majority of optical experiments eventually boil down to the observation of such patterns, this was a very powerful result.

### Contraction hypothesis.

In 1899—and more systematically in 1904—Lorentz extended his analysis to higher powers of v/c. He now found that there is a tiny difference between the pattern of light and shadow obtained in an experiment performed while moving through the ether and the pattern of light and shadow obtained in the corresponding experiment performed at rest in the ether. Compared to the latter, the former pattern is contracted by a factor 1 v2/ c2 in the direction of motion. Lorentz had come across this contraction factor before. In 1887, the American scientists Albert A. Michelson and Edward W. Morley had tried to detect ether drift in an experiment accurate to order (v / c) 2. They had not found any. Independently of each other (in 1889 and 1892, respectively), the Irish physicist George Francis FitzGerald and Lorentz had suggested that this negative result could be accounted for by assuming that material bodies, such as the optical components in the experiment, contract by a factor 1 v2/ c2 in the direction of motion. Lorentz's analysis of Einstein's magnet-conductor experiment 1899 and 1904 showed that this contraction hypothesis, as it came to be known, could be used not only to explain why the Michelson-Morley experiment had not detected any ether drift but also to explain much more generally why no observation of patterns of light and shadow ever would.

In modern terms, the hypothesis that Lorentz added to his theory in 1899 is that the laws governing matter, like Maxwell's equations, are Lorentz invariant. To this end, Lorentz had to amend the Newtonian laws that had jurisdiction over matter in his theory. As he had shown in the case of Maxwell's equations, it is a direct consequence of the Lorentz invariance of the laws governing a physical system that the system will undergo the Lorentz-FitzGerald contraction when moving with respect to the ether. From a purely mathematical point of view, Lorentz had thereby arrived at special relativity. To meet the demands of special relativity, all that needs to be done is to make sure that any proposed physical law is Lorentz invariant.

### Concepts of space-time from Lorentz to Einstein.

Conceptually, however, Lorentz's theory is very different from Einstein's. In Einstein's theory, the Lorentz invariance of all physical laws reflects a new space-time structure. Lorentz retained Newton's conception of space and time, the structure of which is reflected in the invariance of the laws of Newtonian physics under what are now called Galilean transformations. It is an unexplained coincidence in Lorentz's theory that all laws are invariant under Lorentz transformations, which have nothing to do with the Newtonian space-time structure posited by the theory.

This mismatch between the Newtonian concepts of space and time (and the invariance under the classical so-called Galilean transformations associated with it) and the Lorentz invariance of the laws governing matter and fields in space-time manifests itself in many other ways. Einstein hit upon a particularly telling example of this kind and used it in the very first paragraph of his 1905 paper. The example is illustrated in Figure 1.

Consider a bar magnet and a conductor—a piece of wire hooked up to an ammeter—moving with respect to one another at relative velocity v. In Lorentz's theory it is necessary to distinguish two cases, (a) with the conductor and (b) with the magnet at rest in the ether. In case (a) the approaching magnet causes the magnetic field at the location of the wire to grow. According to Faraday's law of induction, this change in magnetic field induces an electric field, producing a current in the wire, which is registered by the ammeter. In case (b) the magnetic field is not changing and there is no induced electric field. The ammeter, however, still registers a current. This is because the electrons in the wire are moving in the magnetic field and experience a Lorentz force that makes them go around the wire.

It turns out that the currents in cases (a) and (b) are exactly the same. Yet, Lorentz's theoretical account of what produces these currents is very different for the two cases. This is, in Einstein's words, an example of theoretical "asymmetries that do not appear to be inherent in the phenomena" (Lorentz et al., 1952, p. 37). Einstein proposed to remove the asymmetry by insisting that cases (a) and (b) are just one and the same situation looked at from different perspectives. Even though this meant that he had to jettison the ether, Einstein took the relativity principle for uniform motion from mechanics and applied it to this situation in electrodynamics. He then proposed to extend the principle to all of physics.

In 1919, in an article intended for Nature but never actually submitted, Einstein explained the importance of the example of the magnet and the conductor for the genesis of special relativity:

The idea that we would be dealing here with two fundamentally different situations was unbearable to me … The existence of the electric field was therefore a relative one, dependent on the coordinate system used, and only the electric and magnetic field taken together could be ascribed some kind of objective reality. This phenomenon of electromagnetic induction forced me to postulate the … relativity principle. (Stachel et al., Vol. 7, pp. 264–265)

The lack of documentary evidence for the period leading up to his creative outburst in his miracle year 1905 makes it very hard to reconstruct Einstein's path to special relativity (for a valiant attempt see Rynasiewicz). It seems clear, however, that Einstein hit upon the idea of "the relativity of electric and magnetic fields" expressed in the quotation above before he hit upon the new ideas about space and time for which special relativity is most famous. His reading of works of Lorentz and Poincaré probably helped him connect the dots from one to the other.

Once again, consider the example of the magnet and the conductor in Fig. 1. From the point of view of the magnet (b), the electromagnetic field only has a magnetic component. From the point of view of the conductor (a), this same field has both a magnetic and an electric component. Lorentz's work provides the mathematics needed to describe this state of affairs. Einstein came to recognize that the fictitious fields of Lorentz's theorem of corresponding states are in fact the fields measured by a moving observer. (He also recognized that the roles of Lorentz's two observers, one at rest and one moving in the ether, are completely interchangeable.) If the observer at rest with respect to the magnet measures a magnetic field with z-component BZ, then the observer at rest with respect to the conductor will not only measure a magnetic field but also an electric field with y-component Ety. This is captured in Lorentz's formula Ety Ey vBz for one of the components of his fictitious fields.

Maxwell's theory is compatible with the relativity principle if it can be shown that the observer measuring the Lorentz-transformed electric and magnetic fields Et and Bt also measures the Lorentz-transformed space and time coordinates (xt, tt). Carefully analyzing how an observer moving through the ether would synchronize her clocks, Poincaré had already shown to first order in v / c that such clocks register Lorentz's local time. A direct consequence of this is that observers in relative motion to one another will disagree about whether two events occurring in different places happened simultaneously or not. Distant simultaneity is not absolute but depends on the state of motion of the observer making the call. This insight made everything fall into place for Einstein about six weeks before he published his famous 1905 paper.

### The relativity postulate and the light postulate.

Einstein modeled the presentation of his theory on thermodynamics (as he explained, for instance, in an article for The London Times in 1919; see Einstein 1954, p. 228). He started from two postulates, the analogues of the two laws of thermodynamics. The first is the relativity postulate, which extends the principle of the relativity of uniform motion from mechanics to all of physics; the other, known as the light postulate, is the key prediction that Einstein needed from electrodynamics to develop his theory: "light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body" (Lorentz et al., 1952 p. 37).

The combination of these two postulates appears to lead to a contradiction: two observers in relative motion will both claim that one and the same light beam has velocity c with respect to them. Einstein reassured the reader that the two postulates are "only apparently irreconcilable." Reconciling the two, however, requires giving up several commonsense notions about space and time and replacing them with unfamiliar new ones, such as the relativity of simultaneity, length contraction (moving objects are shorter than identical objects at rest), and time dilation (processes in moving systems take longer than identical processes in systems at rest). Einstein derived these effects from his two postulates and the plausible assumption that space and time are still homogeneous and isotropic in his new theory.

Following Poincaré's lead but without neglecting terms smaller than of order v / c, Einstein showed that the time and space coordinates of two observers in uniform relative motion are related to one another through a Lorentz transformation. Einstein thus introduced a new space-time structure. In the second part of his paper, he showed that this removes the incompatibility of Maxwell's equations with the relativity principle. He did this simply by proving that Maxwell's equations are Lorentz invariant. Since he was familiar with the early version of Lorentz's theorem of corresponding states (valid to order v / c), he would have had no trouble with this proof.

In 1908, Minkowski supplied the geometry of the Einstein's new space-time. The geometry of this Minkowski space-time is similar to Euclidean geometry. Frames of reference in different states of motion resemble Cartesian coordinate systems with different orientations of their axes. Lorentz transformations in Minkowski space-time are akin to rotations in Euclidean space. The demand that physical laws be Lorentz invariant thus acquired the same status as the demand that physical laws should be independent of the orientation of the axes of the coordinate system used to formulate them.

In a passage that echoes Einstein's comments about electric and magnetic fields quoted above, Minkowski wrote: "space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality" (Lorentz et al., 1952, p. 73). Different observers in Minkowski space-time will agree on the space-time distance between any two events, but disagree on how to break down that spatiotemporal distance into a spatial and a temporal component. It is this disagreement that lies behind relativity of simultaneity, length contraction, and time dilation, the effects Einstein had derived from his two postulates. In light of this analysis, Minkowski pointed out, the phrase relativity postulate seemed ill-chosen:

Since the postulate comes to mean that only the four-dimensional world in space and time is given … but that the projection in space and in time may still be undertaken with a certain degree of freedom, I prefer to call it the postulate of the absolute world. (Lorentz et al., 1952, p. 83)

Einstein agreed with Minkowski on this point but felt that it was already too late to change the theory's name. However, he initially resisted Minkowski's geometrical reformulation of the theory, dismissing it as "superfluous learnedness" (Pais, p. 152). He only came to value Minkowski's contribution in his work on general relativity.