Preface |
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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. |
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Anaximander, c 600 BC |
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The earliest conceptions of relativity were based on the apparent reciprocity of spatial relations and motions. 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 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 regarded the idea of a moving Earth as self-evidently 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 ether) 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 rules 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. |
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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 still had 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 now be answered, but only because of a profound re-interpretation 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 became 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., 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. |
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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 aether, to serve as the medium for the conveyance of light waves (similar to air as the medium of sound 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 all-pervasive 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 (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 aether or its state of motion failed. This created a puzzle for physicists, because their theories were based on the idea that light propagates at a characteristic speed relative to the medium, but they were unable to detect the presence of that medium, let alone to determine its state of motion. |
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Attempts were made to resolve this puzzle by attributing to the hypothetical aether (and all other entities) whatever properties they must possess in order to account for the aether'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 aether'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 in the “true” coordinates. |
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The coordinate systems given by the Lorentz transformation for reference frames in motion relative to the aether 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 aether’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 non-Galilean 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 aether’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 – 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. |
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At about the same time, Einstein presented a simplified 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 principles – which were just a small subset of the assumptions made by Lorentz and Poincare – were sufficient to derive all the known phenomena of electrodynamics, as well as new relativistic formulas for aberration, Doppler shift, and time dilation. But 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 empirical meanings of space and time, he also derived the consequence that all energy must possess inertia, and that the inertial mass of an object is reduced by E/c2 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. |
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Thus 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 had been necessary to rescue relativity from the objections of Ptolemy, it had been necessary to once again re-interpret the principle to assimilate the phenomena of electromagnetic wave propagation, and this in turn led to a deeper understanding of a multitude of other phenomena. Minkowski followed in 1907 with a geometrical interpretation, emphasizing the fact that the invariant dt2 - dx2 - dy2 - dz2 is simply a generalization of the Pythagorean theorem, giving the invariant measure of intervals in a unified spacetime manifold, which he called the “absolute world”. Minkowski also made extensive use of four-vectors, which had been introduced by Poincare. Important contributions and clarifications to the new relativity theory were also made by Planck, Lewis and Tolman, and others. |
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However, soon after the classical relativity principle was reconciled with electro-magnetism, 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 equivalence of inertial and gravitational mass, and the equivalence of inertia and energy. It seemed once again that relativity would have to be abandoned. Then, in 1915, Einstein extended the principle of relativity yet again, with a still more profound re-interpretation of space and time, building on the mathematical insights of Gauss, Riemann, Minkowski and others. The general theory of relativity established equivalence between the members of an even larger class of reference 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 |
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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 ... |
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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, once again making it appear that the principle of relativity would have to be abandoned. Not surprisingly, Einstein was reluctant to concede the issue, having rescued relativity 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 agreed that this was possible, but continued to believe in (or at least hope for) one more re-interpretation of space, time, and motion that would allow the phenomena of quantum mechanics to fit naturally within the relativistic framework. To this day the beauty and elegance of general relativity challenges the imaginations of scientists seeking to reconcile it with the latest theories of physics. |
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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. |
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