8.8  Who Invented Relativity?

All beginnings are obscure.

                              H. Weyl

There have been many theories of relativity throughout history, from the astronomical speculations of Heraclides to the geometry of Euclid to the classical theory of space, time, and dynamics developed by Galileo, Newton and others. Each of these was based on one or more principle of relativity. However, when we refer to the “theory of relativity” today, we usually mean one particular theory of relativity, namely, the body of ideas developed near the beginning of the 20th century and closely identified with the work of Albert Einstein. These ideas are distinguished from previous theories not by the idea of relativity itself, but by the way in which relativistically equivalent coordinate systems are related to each other.

One of the interesting historical aspects of the modern relativity theory is that, although often regarded as the highly original and even revolutionary contribution of a single individual, many of the ideas and formulas of the theory had been anticipated by others. For example, the Lorentz covariance of electromagnetism and the inertia of electromagnetic energy were both (arguably) implicit in Maxwell’s equations. Also, Voigt formally derived transformations of the same form as the Lorentz transformations in 1887 based on general considerations of the wave equation. In the context of electro-dynamics, Fitzgerald, Larmor, and Lorentz had all, by the 1890s, arrived at the actual Lorentz transformations – at least for electromagnetism, although, significantly, they did not clearly understand that these represent the relationships between inertial coordinates. By 1905, Poincare had re-asserted, at least provisionally, Galileo’s principle of relativity, had pointed out the lack of empirical basis for absolute simultaneity, had challenged the ontological significance of the ether, and had even noted that the Lorentz transformations constitute a group in the same sense as do Galilean transformations. The formal synthesis of space and time into spacetime was arguably the contribution of Minkowski in 1907, and the dynamics of special relativity were first given in modern form by Planck in 1906 and Lewis and Tolman in 1909. Likewise, the Riemann curvature and Ricci tensors for n-dimensional manifolds, the tensor formalism itself, and even the crucial Bianchi identities, were all known prior to Einstein’s development of general relativity in 1915. In view of this, is it correct to regard Einstein as the sole originator of modern relativity?

The question is complicated by the fact that relativity is traditionally split into two separate theories, the special and general theories, corresponding to the two phases of Einstein's historical development, and the interplay between the ideas of Einstein and those of his predecessors and contemporaries are different in the two cases. In addition, the title of Einstein’s 1905 paper (“On the Electrodynamics of Moving Bodies”) encouraged the idea that it was just an interpretation of Lorentz's theory of electrodynamics. Indeed, Wilhelm Wein proposed that the Nobel prize of 1912 be awarded jointly to Lorentz and Einstein, saying

The principle of relativity has eliminated the difficulties which existed in electrodynamics and has made it possible to predict for a moving system all electrodynamic phenomena which are known for a system at rest... From a purely logical point of view the relativity principle must be considered as one of the most significant accomplishments ever achieved in theoretical physics... While Lorentz must be considered as the first to have found the mathematical content of relativity, Einstein succeeded in reducing it to a simple principle. One should therefore assess the merits of both investigators as being comparable.

As it happens, the physics prize for 1912 was awarded to Nils Gustaf Dalen (for the "invention of automatic regulators for lighting coastal beacons and light buoys during darkness or other periods of reduced visibility"), and neither Einstein, Lorentz, nor anyone else was ever awarded a Nobel prize for either the special or general theories of relativity. This is sometimes considered to have been an injustice to Einstein, although in retrospect it's conceivable that a joint prize for Lorentz and Einstein in 1912, as Wein proposed, assessing "the merits of both investigators as being comparable", might actually have diminished Einstein's subsequent popular image as the sole originator of special relativity.

On the other hand, despite the somewhat misleading title of Einstein’s paper, which suggested that it was primarily focused on electrodynamics, the paper really did represent a significant advance over all the previous work, precisely because it showed that the significance of the relativity principle, combined with the indifference of “empty space” to the state of motion, implies consequences that extend far beyond Lorentz's electrodynamics. As Einstein later recalled,

The new feature was the realization that the bearing of the Lorentz transformation transcended its connection with Maxwell's equations and was concerned with the nature of space and time in general.

To give just one example, note that prior to the advent of special relativity the experimental results of Kaufmann and others involving the variation of an electron’s inertial mass with velocity were thought to imply that all of the electron’s inertial mass must be electromagnetic in origin, whereas Einstein’s reasoning revealed that the inertia of all mass – even electrically neutral “ponderable” mass – would necessarily be affected by velocity in the same way. The inertia was not due to the electric charge, it was due to the mass-energy. Thus an entire research program, based on the belief that the high-speed behavior of objects represented contingent dynamics within Galilean space and time, was decisively undermined when Einstein showed that the phenomena in question could be interpreted in a much more unified and natural way in the context of relativistic spacetime.

Now, if this interpretation applied only to electrodynamics, it’s significance might be debatable, but already by 1905 it was clear that, as Einstein put it, “the Lorentz transformation transcended its connection with Maxwell’s equations”, and must apply to all physical phenomena in order to account for the complete inability to detect absolute motion. Once this is recognized, it is clear that we are dealing not just with properties of electricity and magnetism, or any other specific entities, but with the fundamental measures of space and time themselves. This is the aspect of Einstein's 1905 theory that prompted Witkowski, after reading vol. 17 of Annalen der Physik, to exclaim: "A new Copernicus is born! Read Einstein's paper!" The comparison is apt, because the contribution of Copernicus could, after all, have been seen initially as essentially nothing but an interpretation of Ptolemy’s astronomy, just as Einstein's theory was initially seen by some as just an interpretation of Lorentz's electrodynamics. Only subsequently did men like Kepler, Galileo, and Newton, taking the Copernican insight even more seriously than Copernicus himself had done, develop a substantially new physical theory. It's clear that Copernicus was only one of several people who jointly created the "Copernican revolution" in science, and we can argue similarly that Einstein was only one of several individuals (including Maxwell, Lorentz, Poincare, Planck, and Minkowski) responsible for the "relativity revolution".

The historical parallel between special relativity and the Copernican model of the solar system is not merely superficial. In both cases the pre-existing theoretical structure was based on the naive use of a particular system of coordinates lacking any compelling physical justification. Physical expectations were shaped by these traditional coordinate systems. For example, in the Ptolemaic context it was natural to imagine that both the Sun and the planet Venus revolve around a stationary Earth in separate orbits. However, with the newly-invented telescope Galileo was able to observe the phases of Venus, clearly showing that Venus revolves around the Sun. In this way the intrinsic relations between the celestial bodies became better understood, but it was still possible (and still is possible) to regard the Earth as stationary in an absolute extrinsic sense. In fact, for many purposes we continue to do just that, but from an astronomical standpoint we now almost invariably regard the Sun rather than the Earth as the (approximate) center of the solar system. Why?

The answer is that the Sun is more nearly the inertial center. In other words, the Copernican revolution (as carried to its conclusion by the successors of Copernicus) can be summarized as the adoption of inertia as the prime organizing principle for the understanding and description of nature. The concept of physical inertia was clearly identified, and the realization of its significance evolved and matured through the works of Kepler, Galileo, Newton, and others. Nature is most easily and most perspicuously described in terms of inertial coordinates. Of course, it remains possible to adopt some non-inertial system of coordinates with respect to which the Earth can be regarded as the “center”, but it is definitely not the inertial center.

Likewise the pre-existing theoretical structure in 1905 described events in terms of coordinate systems that were not clearly understood and were lacking in physical justification. It was natural within this framework to imagine certain consequences, such as anisotropy in the speed of light, i.e., directional dependence of light speed resulting from the Earth's motion through the (assumed stationary) ether. This was largely motivated by the idea that light consists of a wave in the ether, and therefore is not an inertial phenomenon. However, experimental physicists in the late 1800's began to discover facts analogous to the phases of Venus, e.g., the symmetry of electromagnetic induction, the "partial convection" of light in moving media, the isotropy of light speed with respect to relatively moving frames of reference, and so on. Einstein accounted for all these results by showing that they were perfectly natural if things are described in terms of inertia-based coordinates − provided we apply a more profound understanding of the definition and physical significance of such coordinate systems and the relationships between them.

As a result of the first inertial revolution (initiated by Copernicus), physicists had long been aware of the existence of a preferred class of coordinate systems − the inertia-based systems − with respect to which inertial phenomena are isotropic. These systems are equivalent up to orientation and uniform motion in a straight line, and it had always been tacitly assumed that the transformation from one system in this class to another was given by a Galilean transformation. The discovery of the principle of energy conservation in the mid-1800s exposed anomalies in the conceptual foundations of Newtonian mechanics, which could have led to the discovery of the inertia of energy, and hence to the realization that inertial coordinate systems are related by Lorentz transformations. However, historically, the conceptual anomalies in mechanics went unnoticed, and the first recognized Lorentz invariant phenomena were in the field of electromagnetism. After Maxwell introduced the "displacement current" into the equations governing the electromagnetic field in vacuum, it was found that the resulting system of equations is Lorentz invariant (rather than Galilean invariant). At first this was not thought to have implications for mechanics, but the inability to discern any difference between the invariance groups of mechanical and optical phenomena (e.g., failure to detect the “ether wind”) soon led to the realization that mechanics, too, must be Lorentz invariant.

The discovery of Lorentz invariance was similar to the discovery of the phases of Venus, in the sense that it irrevocably altered our awareness of the intrinsic relations between events. We can still go on using coordinate systems related by Galilean transformations, but we now realize that only one of those systems (at most) is a truly inertial system of coordinates. The electrodynamic theory of Lorentz was in some sense analogous to Tycho Brahe's model of the solar system, in which the planets revolve around the Sun but the Sun revolves around a stationary Earth. Tycho's model was kinematically equivalent to Copernicus' Sun-centered model, but expressed – awkwardly – in terms of a coordinate system with respect to which the Earth is stationary, i.e., a non-inertial coordinate system.

It's worth emphasizing that we still define inertia-based coordinates just as Galileo did, i.e., systems of coordinates with respect to which inertial phenomena are homogeneous and isotropic, so our definition hasn't changed. All that has changed is our understanding of the relations between such coordinate systems. Einstein's famous "synchronization procedure" (which was actually first proposed by Poincare) was expressed in terms of light rays, but the physical significance of this procedure is due to the empirical fact that it yields exactly the same synchronization as does Galileo's synchronization procedure based on mechanical inertia. To establish simultaneity between spatially separate events while floating freely in empty space, throw two identical objects in opposite directions with equal force, so that the thrower remains stationary in his original frame of reference. These objects then pass equal distances in equal times, i.e., they serve to assign inertially simultaneous times to separate events as they move away from each other. In this way we can theoretically establish complete slices of inertial simultaneity in spacetime, based solely on the inertial behavior of material objects. Someone moving uniformly relative to us can carry out this same procedure with respect to his own inertial frame of reference and establish his own slices of inertial simultaneity throughout spacetime. The intrinsic relations that were discovered at the end of the 19th century show that these two sets of simultaneity slices are not identical. The two main approaches to the interpretation of these facts were discussed in Sections 1.5 and 1.6. The approach advocated by Einstein was to adhere to the principle of inertia as the basis for organizing our understanding and descriptions of physical phenomena − which in itself was certainly not a novel idea. The novelty was in recognizing that the inertial coordinate systems are related by Lorentz transformations, and all that this implies, including the inertia of energy.

In his later years Einstein observed "there is no doubt that the Special Theory of Relativity, if we regard its development in retrospect, was ripe for discovery in 1905". The person (along with Lorentz) who most nearly anticipated Einstein's special relativity was undoubtedly Poincare, who had already in 1900 discussed an operational definition of clock synchronization using light signals, and in 1904 had suggested that the ether was in principle undetectable to all orders of v/c. However, he did not achieve the crucial realization that light-speed synchronization gives the same result as synchronization by mechanical inertia. Also, as late as 1909 Poincare was not prepared to say that the equivalence of all inertial frames (for the expression of physical laws) combined with the invariance of (two-way) light speed were sufficient to infer Einstein's model. He maintained that one must also stipulate a particular contraction of physical objects in their direction of motion, as if this were an independent hypothesis. This is typically cited as evidence that Poincare still failed to understand the situation, although he was strictly correct that the two famous principles of Einstein's 1905 paper are not sufficient to uniquely identify the inertial coordinates, as Einstein himself later acknowledged. One must also stipulate, at the very least, homogeneity, memorylessness, and isotropy. Of these, the first two are rather innocuous, and one could be forgiven for failing to explicitly mention them, but not so the assumption of isotropy, which serves precisely to single out the inertia-based simultaneity convention from all the other − equally viable − conventions. (See Section 4.5). This is also precisely the aspect that is fixed by Poincare's postulate of contraction as a function of velocity.

In a sense, the failure of Poincare to found the modern theory of relativity can be attributed to his philosophical sophistication, preventing him from subscribing to the young patent examiner's more pragmatic focus on the inertia-based convention. Poincare recognized too well the extent to which our physical models are both conventional and provisional. In retrospect, Poincare's scruples have the appearance of someone arguing that we could just as well regard the Earth rather than the Sun as the center of the solar system, i.e., his reservations were (and are) technically valid, but in some sense misguided. Also, as Max Born remarked, to the end of Poincare’s life his expositions of relativity “definitely give you the impression that he is recording Lorentz’s work”, and yet “Lorentz never claimed to be the author of the principle of relativity”, but invariably attributed it to Einstein. Indeed Lorentz himself often expressed reservations about the relativistic interpretation.

Regarding Born’s impression that Poincare was just “recording Lorentz’s work”, it should be noted that Poincare habitually wrote in a self-effacing manner. He named many of his discoveries after other people, and expounded many important and original ideas in writings that were ostensibly just reviewing the works of others, with “minor amplifications and corrections”. Poincare’s style of writing, especially on topics in physics, tend to give the impression that he was just reviewing someone else’s work (which he was) – in contrast with Einstein, whose style of writing, as Born said, “gives you the impression of quite a new venture”. Of course, Born went on to say, when recalling his first reading of Einstein’s paper in 1907, “Although I was quite familiar with the relativistic idea and the Lorentz transformations, Einstein’s reasoning was a revelation to me… which had a stronger influence on my thinking than any other scientific experience”.

Lorentz’s reluctance to fully embrace the relativity principle (that he himself did so much to uncover) is partly explained by his belief that "Einstein simply postulates what we have deduced... from the equations of the electromagnetic field". If this were true, it would be a valid reason for preferring Lorentz's approach. However, if we closely examine Lorentz's electron theory we find that full agreement with experiment required not only the invocation of Fitzgerald's contraction hypothesis, but also the assumption that mechanical inertia is Lorentz covariant. It's true that, after Poincare complained about the proliferation of hypotheses, Lorentz realized that the contraction could be deduced from more fundamental principles (as discussed in Section 1.5), but this was based on yet another hypothesis, the so-called molecular force hypothesis, which simply asserts that all physical forces and configurations (including the unknown forces that maintain the shape of the electron) transform according to the same laws as do electromagnetic forces. Needless to say, it obviously cannot follow deductively "from the equations of the electromagnetic field" that the necessarily non-electromagnetic forces which hold the electron together must transform according to the same laws. (Both Poincare and Einstein had already realized by 1905 that the mass of the electron cannot be entirely electromagnetic in origin, although Poincare seems not to have grasped the full significance of this fact.) Even less can the Lorentz covariance of mechanical inertia be deduced from electromagnetic theory. To this day we still do not know the origin of inertia, so no one can claim to have deduced Lorentz covariance in any constructive sense, let alone from the laws of electromagnetism.

Hence Lorentz's molecular force hypothesis and his hypothesis of covariant mechanical inertia together are simply a disguised and piece-meal way of postulating universal Lorentz invariance − which is precisely what Lorentz claims to have deduced rather than postulated. The whole task was to reconcile the Lorentzian covariance of electromagnetism with the Galilean covariance of mechanical dynamics, and Lorentz simply recognized that one way of doing this is to assume that mechanical dynamics (i.e., inertia) is actually Lorentz covariant. This is presented as an explicit postulate (not a deduction) in the final edition of his book on the Electron Theory. In essence, Lorentz’s program consisted of performing a great deal of deductive labor, at the end of which it was still necessary, in order to arrive at results that agreed with experiment, to simply postulate the same principle that forms the basis of special relativity. (To his credit, Lorentz candidly acknowledged that his deductions were "not altogether satisfactory", but this is actually an understatement, because in the end he simply postulated what he claimed to have deduced.)

In contrast, Einstein recognized the necessity of invoking the principle of relativity and Lorentz invariance at the start, and then demonstrated that all the other "constructive" labor involved in Lorentz's approach was superfluous, because once we have adopted these premises, all the experimental results follow unavoidably, with no need for molecular force hypotheses or any other exotic and dubious conjectures regarding the ultimate constituency of matter. On some level Lorentz grasped the superiority of the purely relativistic approach, as is evident from the words he included in the second edition of his "Theory of Electrons" in 1916. After noting that Einstein’s theory of relativity gives a simpler account of the phenomena of electromagnetism than his own (Lorentz’s) theory, he added

The chief cause of my failure was my clinging to the idea that the variable t only can be considered as the true time, and that my local time t' must be regarded as no more than an auxiliary mathematical quantity.

Still, neither Lorentz nor Poincare ever whole-heartedly embraced special relativity, for reasons that may best be summed up by Lorentz when he wrote

Yet, I think, something may also be claimed in favor of the form in which I have presented the theory. I cannot but regard the aether, which can be the seat of an electromagnetic field with its energy and its vibrations, as endowed with a certain degree of substantiality, however different it may be from all ordinary matter. In this line of thought it seems natural not to assume at starting that it can never make any difference whether a body moves through the aether or not, and to measure distances and lengths of time by means of rods and clocks having a fixed position relatively to the aether.

This passage implies that Lorentz's rationale for retaining a substantial aether and attempting to refer all measurements to the rest frame of this aether (without, of course, specifying how that is to be done) was the belief that it might, after all, make some difference whether a body moves through the aether or not. In other words, we should continue to look for physical effects that violate Lorentz invariance (by which we now mean local Lorentz invariance), both in new physical forces and at higher orders of v/c for the known forces. A century later, our present knowledge of the weak and strong nuclear forces and the precise behavior of particles at 0.99999c has vindicated Einstein's judgment that Lorentz invariance is a fundamental principle whose significance and applicability extends far beyond Maxwell's equations, and apparently expresses a general attribute of all physical phenomena, rather than a specific attribute of particular physical entities. Einstein cited both Ernst Mach and David Hume as inspirations for the willingness to challenge deeply held notions (e.g., Hume's view that our sense of causality is merely a habit of association), which emboldened Einstein to critically examine the basis for the traditional measures of space and time.

In addition to the formulas expressing the Lorentz transformations, we can also find precedents for other results commonly associated with special relativity, such as the equivalence of mass and energy. Isaac Newton famously asked "Are not gross bodies and light convertible into one another...?", although it’s debatable whether this actually represents a suggestion of mass-energy equivalence. In a more modern context, the general idea of associating mass with energy in some way had been around for about 25 years prior to Einstein's 1905 papers. Indeed, as Thomson and even Einstein himself noted, this association is already implicit in Maxwell's theory. With electric and magnetic fields e and b, the energy density is (e² + b²)/(8π) and the momentum density is (e x b)/(4πc), so in the case of radiation (when e and b are equal and orthogonal) the energy density is E = e²/(4π) and the momentum density is p = e²/(4πc), which implies p = E/c. Given two initially resting particles at a distance D from each other, a pulse of light with arbitrarily small energy E transmitted from one to the other will cause the first to have recoiled a distance Dv/c when the pulse is absorbed by the second. To conserve the center of mass immediately before and after the exchange we must have (D/2 + Dv/c)(m−Δm) = (D/2)(m+Δm), and conservation of momentum implies E/c = (m−Δm)v, so we have E = Δmc². This should not be surprising, because Maxwell’s equations are Lorentz invariant. Indeed, in the 1905 paper containing his original deduction of mass-energy equivalence, Einstein acknowledges that it was explicitly based on "Maxwell's expression for the electromagnetic energy of space". We can also mention the pre-1905 writings on the “apparent mass” of charged particles due to electromagnetic self-induction, and the work of Hasenohrl on how the mass of a cavity increases when it is filled with radiation. However, Hasenohrl proposed the wrong constant of proportionality, and the studies of apparent mass were interpreted as evidence that all mass is electromagnetic in origin, which we now know is not true. Moreover, these early suggestions were all very restricted in their applicability, and didn't amount to the assertion of a fundamental equivalence applicable to all forms of energy (not just electromagnetic) such as emerges so clearly from Einstein's relativistic interpretation. Although few of the formulas in Einstein's two 1905 papers on relativity were new, the papers provided a single conceptual framework within which all those formulas flow quite naturally from a simple set of general principles. In contrast, Poincare was still writing in 1908 (Science and Method) that Lorentz’s theory violates conservation of momentum (the reaction principle), since he still discounted the idea that massless electromagnetic energy (between its emission and absorption by material bodies) could have actual momentum.

Occasionally one hears of other individuals who are said to have discovered one or more aspect of relativity prior to Einstein. To take just one example, in November of 1999 there appeared in newspapers around the world a story announcing that "The mathematical equation that ushered in the atomic age was discovered by an unknown Italian dilettante two years before Albert Einstein used it in developing the theory of relativity...". The "dilettante" in question was an Italian business man named Olinto De Pretto, whose 1903 paper claimed that every particle of matter is agitated by its exposure to an ultra-mundane flux of hypothetical ether particles in a "shadow theory” of gravity. De Pretto reasoned that the mean vibrational speed of the particles of matter must approach the speed of the ether particles, which he supposed might be the speed of light. He then asserted (erroneously) that the kinetic energy of a mass m moving at speed v is mv². On this basis, De Pretto asserted that the mean kinetic energy in a quantity of mass m would be mc². In addition to being erroneous and counter-factual, this line of reasoning was not original to De Pretto. The shadow theory of gravity was first conceived by Newton’s friend Nicholas Fatio in the 1690’s, and subsequently re-discovered by many individuals, notably George Louis Lesage in the late 18th century. The fact that the bombardment of such an intense ultramundane flux would necessarily elevate the temperature of ordinary matter to incredible temperatures was noted by both Kelvin and Maxwell in the late 19th century. Poincare and Lorentz both realized the same thing, and used this fact to conclude that the shadow model of gravity is not viable (since it entails the vaporization of the Earth in a fraction of a second). Hence, De Pretto said nothing new, and simply repeated what was then already a long since discredited idea, which of course bears no resemblance to the concept of mass-energy equivalence that emerges from special relativity.

It’s also historically inaccurate to say that Einstein used the idea of the equivalence of mass and energy to develop the theory of relativity. Mass-energy equivalence wasn’t even mentioned in his foundational paper of June 1905. Only a few months later did he recognize this implication of the theory, prompting him to write in a letter to his close friend Conrad Habicht:

One more consequence of the paper on electrodynamics has also occurred to me. The principle of relativity, in conjunction with Maxwell's equations, requires that mass be a direct measure of the energy contained in a body; light carries mass with it. A noticeable decrease of mass should occur in the case of radium [as it emits radiation]. The argument [which he intends to present in the paper] is amusing and seductive, but for all I know the Lord might be laughing over it and leading me around by the nose.

The most obvious proof of the originality of Einstein’s path to special relativity is the wonderfully lucid sequence of thoughts presented in his 1905 paper, beginning from first principles and a careful examination of the physical significance of the inertial measures of time and space, from which the inertia of energy emerges naturally. This is not to deny that the idea of a relation between mass and some forms of energy had occurred in the work of Einstein’s predecessors, but it’s clear that Einstein arrived at the idea of complete mass-energy equivalence as a consequence of his ideas on relativistic dynamics, and not the other way around.

As mentioned above, special relativity could have been derived purely from energy considerations as soon as the conservation of energy was established as a fundamental principle in the mid 1800’s. According to Newton’s laws the center of mass of an isolated system moves uniformly in a straight line, but the center of energy does not. Modifying Newton’s laws to correct this problem leads directly to special relativity. However, even in the early 1900s, confidence in the concept  of energy as an actual entity that can be unambiguously tracked through time was not sufficient to support this line of reasoning. (For example, Lorentz wrote “The flow of energy can, in my opinion, never have quite the same distinct meaning as the flow of material particles... It might even be questioned whether the transfer of electromagnetic energy takes place in the way indicated by Poynting’s law...”.)

Although Einstein’s 1905 papers on special relativity were remarkably thorough and mature, there were many important contributions to the foundations of special relativity made by others in the years that followed. For example, in 1907 Max Planck greatly clarified relativistic mechanics, basing it on the conservation of momentum with his "more advantageous" definitions, as did Tolman and Lewis. Planck also critiqued Einstein's original deduction of mass-energy equivalence, and gave a more general and comprehensive argument. (This led Johannes Stark in 1907 to cite Planck as the originator of mass-energy equivalence, prompting an angry letter from Einstein saying that he "was rather disturbed that you do not acknowledge my priority with regard to the connection between mass and energy". In later years Stark became an outspoken critic of Einstein's work.)

Another crucially important contribution was made by Hermann Minkowski (one of Einstein's former professors), who recognized that what Einstein had described was simply ordinary dynamics in a four-dimensional spacetime manifold with the pseudo-metric (dτ)² = (dt)² – (dx)² – (dy)² – (dz)². (It’s interesting to compare this with how Max Born recognized that Heisenberg’s seemingly unintuitive quantum operations were simply matrix multiplication.) Poincare had noted the invariance of this quadratic form under Lorentz transformations as early as 1905. This was vital for the generalization of relativity which Einstein – with the help of his old friend Marcel Grossmann – constructed on the basis on the theory of curved manifolds developed in the 19th century by Gauss and Riemann.

The tensor calculus and generally covariant formalism employed by Einstein in his general theory had been developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita around 1900 at the University of Padua, building on the earlier work of Gauss, Riemann, Beltrami, and Christoffel. In fact, the main technical challenge that occupied Einstein in his efforts to find a suitable field law for gravity, which was to construct from the metric tensor another tensor whose covariant derivative automatically vanishes, had already been solved in the form of the Bianchi identities, which lead directly to the Einstein tensor as discussed in Section 5.8.

Several other individuals are often cited as having anticipated some aspect of general relativity, although not in any sense of contributing seriously to the formulation of the theory. John Michell wrote in 1783 about the possibility of "dark stars" that were so massive light could not escape from them, and Laplace contemplated the same possibility in 1796. Around 1801 Johann von Soldner predicted that light rays passing near the Sun would be deflected by the Sun’s gravity, just like a small corpuscle of matter moving at the speed of light (at a particular point on its trajectory). This gives a deflection of just half the relativistic value. Ironically, in accord with the German literature of the time, the parameter Soldner used to represent the “acceleration of gravity” was half the modern definition of that term, so his formulas included a factor of 2, which some people subsequently took as an indication that he had predicted the relativistic value. However, the Newtonian derivation he presented is unambiguous, and leads to the numerical value of 0.84 arc seconds, which was explicitly stated by Soldner, so there is no doubt that his prediction was half of the relativistic value.

Interestingly, the work of Soldner had been virtually forgotten until being rediscovered and publicized by Philipp Lenard in 1921, along with the claim that Hasenohrl should be credited with mass-energy equivalence. Similarly in 1917 Ernst Gehrcke arranged for the re-publication of a 1898 paper by a secondary school teacher named Paul Gerber which contained a formula for the precession of elliptical orbits identical (at the lowest order of approximation) to the one Einstein had derived from the field equations of general relativity. Gerber's approach was based on the premise that the gravitational potential propagates at the speed of light, and that the effect of the potential on the motion of a body depends on the body's velocity through the potential field. His potential was similar in form to the Gauss-Weber theories. However, Gerber's "theory" was (and still is) regarded as unsatisfactory, mainly because his conclusions don’t follow from his premises, but also because the combination of Gerber's proposed gravitational potential with the rest of (nonrelativistic) physics results in predictions (such as 3/2 the relativistic prediction for the deflection of light rays near the Sun) which are inconsistent with observation. In addition, Gerber's free mixing of propagating effects with action-at-a-distance tended to undermine the theoretical coherence of his proposal.

The writings of Michell, Soldner, Gerber, and others were, at most, anticipations of some of the phenomenology later associated with general relativity, but had nothing to do with the actual theory of general relativity, i.e., a theory that conceives of gravity as a manifestation of the curvature of spacetime. Some hints of this latter conception can be found in the notional writings of William Kingdon Clifford, who wrote about a possible connection between matter and curved space in 1873, but like Gauss and Riemann he lacked the crucial idea of including time as one of the dimensions of the manifold. As noted above, the formal means of treating space and time as a single unified spacetime manifold was conceived by Poincare and Minkowski, and the tensor calculus was developed by Ricci and Levi-Civita, with whom Einstein corresponded during the development of general relativity. It’s also worth mentioning that Einstein and Grossmann, working in collaboration, came very close to discovering the correct field equations in 1913, but were diverted by an erroneous argument that led them to believe no fully covariant equations could be consistent with experience. In retrospect, this accident may have been all that prevented Grossmann from being perceived as a co-creator of general relativity. On the other hand, Grossmann had specifically distanced himself from the physical aspects of the 1913 paper, and Einstein wrote to Sommerfeld in July 1915 (i.e., prior to arriving at the final form of the field equations) that

Grossmann will never lay claim to being co-discoverer. He only helped in guiding me through the mathematical literature but contributed nothing of substance to the results.

The views of Ernst Mach on the origin of inertia had a profound effect on Einstein, who for many years believed that general relativity was actually the fulfillment of Mach's idea. However, Mach's attitude toward Einstein's theory is unclear. In the forward to a posthumous book published in 1921 Mach was quoted as stating (in 1913) that he disavowed relativity theory, although questions have been raised about the authenticity of the quote. In any case, Einstein certainly came to disavow Mach's Principle, as it became clear that general relativity does not actually fulfill that principle − at least not in a direct way. So, while general relativity has flourished, one of the main motivating ideas for it (Mach's Principle) has been almost completely discarded.

In the summer of 1915 Einstein gave a series of lectures at Gottingen on the general theory, and apparently succeeded in convincing both Hilbert and Klein that he was close to an important discovery, despite the fact that he had not yet arrived at the final form of the field equations. Hilbert took up the problem from an axiomatic standpoint, and carried on an extensive correspondence with Einstein until the 19th of November. On the 20th, Hilbert submitted a paper to the Gesellschaft der Wissenschaften in Gottingen with a derivation of the field equations. Five days later, on 25 November, Einstein submitted a paper with the correct form of the field equations to the Prussian Academy in Berlin. The exact sequence of events leading up to the submittal of these two papers – and how much Hilbert and Einstein learned from each other – is somewhat murky, especially since Hilbert’s paper was not actually published until March of 1916, and seems to have undergone some revisions from what was originally submitted. However, the question of who first wrote down the fully covariant field equations (including the trace term) is less significant than one might think, because, as Einstein wrote to Hilbert on 18 November after seeing a draft of Hilbert’s paper

The difficulty was not in finding generally covariant equations for the g_μν’s; for this is easily achieved with the aid of Riemann’s tensor. Rather, it was hard to recognize that these equations are a generalization – that is, a simple and natural generalization – of Newton’s law.

It might be argued that Einstein was underestimating the mathematical difficulty, since he hadn’t yet included the trace term in his published papers, but in fact he repeated the same comment in a letter to Sommerfeld on 28 November, this time explicitly referring to the full field equations, with the trace term. He wrote

It is naturally easy to set these generally covariant equations down; however, it is difficult to recognize that they are generalizations of Poisson’s equations, and not easy to recognize that they fulfill the conservation laws. I had considered these equations with Grossmann already 3 years ago, with the exception of the [trace term], but at that time we had come to the conclusion that it did not fulfill Newton’s approximation, which was erroneous.

Thus he regards the purely mathematical task of determining the most general fully covariant expression involving the g_μν’s and their first and second derivatives as comparatively trivial and straightforward – as indeed it is for a competent mathematician. The Bianchi identities were already known, so there was no new mathematics involved. The difficulty, as Einstein stressed, was not in writing down the solution of this mathematical problem, but in conceiving of the problem in the first place, and then showing that it represents a viable law of gravitation. In this, Einstein was undeniably the originator, not only in showing that the field equations reduce to Newton’s law in the first approximation, but also in showing that they yield Mercury’s excess precession in the second approximation. Hilbert was suitably impressed when Einstein showed this in his paper of 18 November, and it’s important to note that this was how Einstein was spending his time around the 18th of November, establishing the physical implications of the fully covariant field equations, while Hilbert was busying himself with elaborating the mathematical aspects of the problem that Einstein had outlined the previous summer.

It’s also worth noting that although they arrived at the same formulas, Hilbert and Einstein were working in fundamentally different contexts, so it would be somewhat misleading to say that they arrived at the same theoretical result. Already in 1921 Pauli commented on both the simultaneous discoveries and on the distinctions between what the two men discovered.

At the same time as Einstein, and independently, Hilbert formulated the generally covariant field equations. His presentation, though, would not seem to be acceptable to physicists, for two reasons. First, the existence of a variational principle is introduced as an axiom. Secondly, of more importance, the field equations are not derived for an arbitrary system of matter, but are specifically based on Mie’s theory of matter.

Whatever the true sequence of events and interactions, it seems that Einstein initially had some feelings of resentment toward Hilbert, perhaps thinking that Hilbert had acted ungraciously and stolen some of his glory. Already on November 20 Einstein had written to a friend

The theory is incomparably beautiful, but only one colleague understands it, and that one works skillfully at "nostrification". I have learned the deplorableness of humans more in connection with this theory than in any other personal experience. But it doesn't bother me.

(Literally the word “nostrification” refers to the process by which a country accepts foreign academic degrees as if they had been granted by one of its own universities, but the word has often been used to suggest the appropriation and re-packaging of someone else’s ideas and making them one’s own.) However, by December 20 he was able to write a conciliatory note to Hilbert, saying

There has been between us a certain unpleasantness, whose cause I do not wish to analyze. I have struggled against feelings of bitterness with complete success. I think of you again with untroubled friendliness, and ask you to do the same with me. It would be a shame if two fellows like us, who have worked themselves out from this shabby world somewhat, cannot enjoy each other.

Thereafter they remained on friendly terms, and Hilbert never publicly claimed any priority in the discovery of general relativity, and always referred to it as Einstein’s theory.

As it turned out, Einstein can hardly have been dissatisfied with the amount of popular credit he received for the theories of relativity, both special and general. Nevertheless, one senses a bit of annoyance when Max Born mentioned to Einstein in 1953 (two years before Einstein's death) that the second volume of Edmund Whittaker's book “A History of the Theories of Aether and Electricity” had just appeared, in which special relativity is attributed to Lorentz and Poincare, with barely a mention of Einstein except to say that "in the autumn of [1905] Einstein published a paper which set forth the relativity theory of Poincare and Lorentz with some amplifications, and which attracted much attention". In the same book Whittaker attributes some of the fundamental insights of general relativity to Planck and a mathematician named Harry Bateman (a former student of Whittaker’s). Einstein replied to his old friend Born

Everybody does what he considers right... If he manages to convince others, that is their own affair. I myself have certainly found satisfaction in my efforts, but I would not consider it sensible to defend the results of my work as being my own 'property', as some old miser might defend the few coppers he had laboriously scrapped together. I do not hold anything against him [Whittaker], nor of course, against you. After all, I do not need to read the thing.

On the other hand, in the same year (1953), Einstein wrote to the organizers of a celebration honoring the upcoming fiftieth anniversary of his paper on the electrodynamics of moving bodies, saying

I hope that one will also take care on that occasion to suitably honor the merits of Lorentz and Poincare.

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