Preface 

The beginning of everything is boundless, and from whence things arise, there they must return, for things give satisfaction and reparation to one another for their injustice, as is appointed according to the ordering of time. 
Anaximander, c 600 BC 

The earliest concept of relativity was based on the simple reciprocity of spatial relations and motions. This intuitive idea led to some counterintuitive theories about the natural world. For example, Heraclides suggested that the observed rotation of the stars around the Earth might just as well be interpreted as a daily rotation of the Earth while the stars remain stationary. Aristarchus went even further, proposing that the seemingly stationary Earth not only rotates on its axis, but moves in an immense circular orbit around the Sun. However, most philosophers from antiquity through the middle ages discounted the idea of a moving Earth, regarding it as selfevidently incompatible with observation as well as common sense. Ptolemy rejected the hypothesis of a moving Earth because it requires us to believe the lightest essence (the aether) is stationary and the heaviest essence (Earth) is in motion, precisely contrary to their respective natures. He also argued that the surface of a rotating Earth would necessarily be moving faster than the clouds floating in the air above it, so we should never see clouds moving to the east. Likewise the absence of any discernable parallax in the observed positions of the stars apparently ruled out the possibility that the Earth revolves around the Sun – unless the distance to the stars is literally thousands of times the distance to the Sun, which seemed implausible. 

The same objections were raised eighteen centuries later when Copernicus revived the heliocentric model based on essentially the same primitive kinematic concept of relativity, and yet within a century of Copernicus' death the heliocentric model had been fully accepted by the scientific community − despite the fact that stellar parallax had still never been detected. (The first actual measurement of stellar parallax was not achieved until 1839.) The old conceptual objections to relativity that had once seemed irrefutable could by then be answered, but only because of a profound reinterpretation of the relativity principle brought about by the successors of Copernicus, including Kepler, Galileo, Descartes, Huygens, and Newton. The new theory of relativity was based not on purely kinematic relations, but on the dynamical concept of inertia, according to which there exists an infinite class of relatively moving coordinate systems that are all equivalent from the standpoint of mechanical dynamics. The relativity of inertia formed the conceptual basis of the Scientific Revolution. Admittedly, Newton chose to formulate laws of mechanics in terms of absolute space and time, but he explicitly recognized that those laws were nevertheless fully consistent with the principle of inertial relativity, i.e., that every system of inertial coordinates is equivalent for the description of physical laws − at least insofar as those laws pertain to the motions of material entities. Indeed the complete operational equivalence of uniformly moving inertial reference frames remained an unchallenged principle of physics for centuries. 

In addition to his theory of mechanics, Newton also tentatively suggested that light too might consist of a material entity, such as a stream of tiny corpuscles, consistent with the same overall relativistic framework of physical laws. However, due mainly to the inability of the corpuscular model of light to account for interference effects, the wave model of light (and later of electromagnetism) came to the forefront during the 19th century. This was accompanied by the hypothesis of a hitherto undetected substance, termed the luminiferous ether, to serve as the medium for the conveyance of light waves. According to this view, light does not propagate as a stream of ballistic corpuscles through empty space, with a characteristic speed relative to the source, but rather as some kind of wave in an allpervasive medium, with a characteristic speed relative to the medium and independent of the motion of the source. This too would be perfectly consistent with the principle of relativity (just as is the propagation of sound waves in air) if the hypothetical medium was itself a mechanical entity with its own identifiable state of motion, but all efforts to detect the putative ether or its state of motion failed. Indeed all experiments showed that the speed of light in vacuum is the same in all directions in the rest frame of the source, regardless of how the source is moving (as one would expect for a particle theory of light). This created a puzzle for physicists, because their theories were based on the idea that light propagates as a wave at a characteristic speed relative to a medium, but they were unable to detect the presence of that medium, let alone to determine its state of motion. 

Attempts were made to resolve this puzzle by attributing to all palpable entities and the hypothetical ether whatever properties they must possess in order to account for the ether's undetectability. This culminated in the work of Lorentz and Poincare, who, by about 1905, had arrived at the conditions that must be satisfied by all elementary entities and forces (including inertial forces) if the principle of relativity is to be satisfied. These conditions can be summarized by what Poincare called the Lorentz transformations, which were interpreted as defining the relations between the “true” space and time coordinates (associated with the ether's rest frame) and other relatively moving coordinate systems in terms of which all the laws of physics (including both mechanics and optics) take the same form that they do when expressed in terms of the “true” coordinates. 

The coordinate systems given by the Lorentz transformation for reference frames in motion relative to the ether were regarded by Lorentz and Poincare as merely apparent (or “effective”) measures of space and time, rather than “true” measures, which they continued to believe were related to the ether’s rest frame coordinates by Galilean transformations (although Poincare viewed this as at least partly a matter of convention). Lorentz had originally arrived at the nonGalilean transformations that now bear his name by determining the coordinate systems in terms of which Maxwell's equations of electromagnetism maintain the same form, which implies that the constants appearing in those equations (including the constant c representing the speed of light) maintain the same values as in the ether’s rest frame. At first Lorentz assumed this transformation applied only to electromagnetism (from which it had been derived), but in order to match all the available experimental results he found it necessary to assume more and more entities and phenomena – including the then unknown forces governing the structure of elementary particles of matter, and ultimately including mechanical inertia itself – are invariant under those same transformations. In retrospect we can see that this amounted to the assumption that the speed of light is c in terms of every system of inertial coordinates, but neither Lorentz nor Poincare explicitly identified the auxiliary coordinate systems as inertial coordinate systems. 

At about the same time, influenced by his discovery of the particlelike behavior of light responsible for the photoelectric effect, Einstein presented a more direct derivation – and a much broader interpretation – of the Lorentz transformation, based on the principle of relativity combined with the principle that the speed of light is c in terms of every system of inertial coordinates. He showed that these two empiricallybased principles – a small subset of the assumptions made by Lorentz and Poincare – were sufficient to account for all the known phenomena of electrodynamics, and to derive new relativistic formulas for aberration, Doppler shift, and time dilation. More significantly, Einstein explicitly recognized that the Lorentz transformations describe the relationships between inertial coordinate systems (i.e., coordinate systems in terms of which “the laws of mechanics hold good”), which are by definition the "true" coordinate systems of Newtonian physics. He highlighted the reciprocity of those transformations, emphasizing the symmetry between relatively moving systems of inertial coordinates, and pointed out the crucial relativity of simultaneity exhibited by these systems. From these simple considerations of the operational measures of space and time, he also deduced that all energy must possess inertia, and that the inertial mass of an object is reduced by E/c^{2} when the object emits energy E. From this he inferred the fundamental equivalence of mass and energy, thereby unifying the previously separate conservations laws of mass and energy. 

In this way relativity was restored by reinterpreting the measures of time and space on a more profound level. Just as a deepening of the principle and the associated concepts of space, time, and motion were needed to rescue relativity from the objections of Ptolemy, it had been necessary to once again reinterpret the principle to assimilate the phenomena of electromagnetism, and this in turn led to a deeper understanding of a multitude of other phenomena, including mechanics. Minkowski followed in 1907 with a profound and elegant geometrical interpretation, emphasizing that the invariance of the spacetime interval is simply a generalization of the ancient Pythagorean theorem for spatial intervals, characterizing a unified spacetime manifold which he called the “absolute world”. Important contributions and clarifications to the new relativity theory were also made by Planck, Laue, Lewis and Tolman, and many others. 

However, soon after the classical relativity principle was reconciled with electromagnetism, a new challenge appeared. Einstein himself was among the first to realize that the special theory of relativity which he had described in 1905 was fundamentally incompatible with gravitation and the two principles of equivalence, i.e., the wellestablished equivalence of inertial and gravitational mass, and the equivalence of inertia and energy. The latter is the most fundamental and important feature of the special theory. Taken together, these principles imply that every quantity of energy, including (for example) rays of light, must interact gravitationally on the same footing as the energy of material objects. It follows that a plane wave of light moving through a gravitational field must undergo deflection in terms of any static and global system of coordinates, which signifies that the propagation speed differs at different points of the wave front. This is incompatible with the fundamental premise of the special theory of 1905, according to which there exist static and global systems of coordinates (the global inertial coordinates) in terms of which the speed of light is invariant. Thus it seemed once again that the theory of relativity would have to be abandoned. 

This led Einstein, in the years between 1907 and 1915, to extend the principle of relativity yet again, with a still more profound reinterpretation of space and time, building on the mathematical insights of Gauss, Riemann, Minkowski and others. The general theory of relativity established an equivalence between the members of an even larger class of coordinate systems, and in so doing achieved a conceptual unification of inertia and gravity, while retaining the structure of special relativity locally at every point of spacetime. Instead of conceiving of gravitation as something that takes place within Newton’s immutable space and time, Einstein found that it could be more accurately described as an attribute of a dynamical spacetime. One of Einstein's contemporaries, the physicist Max Born, later said 

The theory appeared to me then, and it still does, the greatest feat of human thinking about nature, the most amazing combination of philosophical penetration, physical intuition, and mathematical skill… It appealed to me like a great work of art ... 

Nevertheless, during the same years in which Einstein was developing and extending the modern theory of relativity, another class of phenomena came under study, leading to the theory of quantum mechanics. Einstein himself made important contributions to the early development of quantum theory, and later the union of special relativity and quantum mechanics led directly to the Dirac equation and modern quantum field theory. However, the representations of the fundamental interactions in quantum field theory seem to be completely different in character from the representation of the gravitational interaction in general relativity. This has led many to suspect that general relativity, and perhaps even the underlying concept of continuous fields in space and time, will have to be abandoned. However, Einstein was reluctant to take this step. He had rescued the relativistic model of spacetime twice from seemingly intractable problems, both times showing that in fact relativity was the key to a deeper understanding of the very phenomena that were thought to be incompatible with it. Could those apparent successes have been illusory? He acknowledged that this was possible, but continued to search for one more reinterpretation of space, time, and motion that would unify his theory of gravitation with the phenomena of quantum mechanics, in such a way that the latter emerge naturally within the relativistic framework. 

To this day the beautiful and elegant theory of general relativity continues to challenge the imaginations of scientists seeking to reconcile it with the latest theories of physics. This book examines the evolution of the principle of relativity in its classical, special, and general incarnations, both from a technical and a historical perspective, with the aim of showing how it has repeatedly inspired advances in our understanding of the physical world. 
