RELATIVITY, THEORY OF.
### I. PHYSICAL SPACE AND TIME
This theory, like the nearly contemporary theory of "quanta" (v.), is one of the singular theories to which physics was compelled to resort in order to account, in deference to the Galilean experimental principle, for two experimental results that were in stark contrast with some of its fundamental conceptions. The singularity of these two modern theories lies in the fact that the alterations they impose are not simple modifications of some preexisting theory, but rather modifications that reverberate not merely upon some particular branch of it, but upon the whole of physics. The relativity to which the theory owes its name refers to absolute motion, in the sense that it is declared not only unobservable, but indeed a concept devoid of meaning, with the consequence that physical experiments can reveal nothing other than relative motions between different reference systems and different bodies. As for the other theory, see QUANTA.
### II. THE FIRST LIMITED RELATIVITY: OPTICS
Dynamics, codified by Newton in his great work *Philosophiae naturalis principia mathematica*, based on the discoveries of Galileo and his own, postulates as its theoretical foundation the existence of absolute time and space and thus of a fixed reference system, even though Newton himself expressed doubts about the actual existence of the latter. Amid the fervor of the development of all mechanics that followed the dissemination of the *Principia*, throughout the 18th century these fundamental difficulties were overlooked; but at the beginning of the 19th century a revolution occurred that compelled them to be taken seriously for the first time. Until then, two different theories had succeeded in interpreting optical phenomena as they were experimentally observed. The first of these was Newton’s emission theory, which interpreted light as consisting of corpuscles emitted by luminous sources and was thus predominantly mechanical in nature. The second, a development of the theory initiated by Huyghens, interpreted light through a wave hypothesis analogous to that by which sound had always been interpreted; the vibrating medium, however, was not air, but a hypothetical, fluid, all-pervading ether. The periodic vibrations were assumed to occur, like those of sound, longitudinally—that is, in the same direction as propagation.
But the discovery of polarization phenomena threw the situation into disarray; the wave theory could not explain them, whereas the corpuscular theory, albeit laboriously, could. The ascendancy of the emission theory was, however, short-lived, for A. Fresnel succeeded in demonstrating that it was not the wave hypothesis per se but the longitudinality hypothesis that impeded the explanation of the polarization of light: it sufficed to suppose that the vibrations of the hypothetical ether occurred transversely—that is, perpendicularly to the direction of propagation—for not only did that difficulty vanish, but an interpretation of all optical phenomena was thereby achieved that was vastly superior to the previous ones.
Yet the ether of the new theory, though by hypothesis permeable to any material body, had to possess characteristics of extreme rigidity in order to vibrate as was supposed, and thus could serve as the foundation of a spatial reference system and perhaps even be that absolute system which Newton and his followers had vainly sought. Moreover, optical phenomena observed in different systems in relative motion within this ether ought to present different characteristics. This last conclusion was so significant for the foundations of physics that rigorous experiments were immediately instituted to attempt to confirm it and thereby finally arrive at the determination of a motion with respect to the ether—that is, of absolute motion.
The differences that Fresnel’s theory predicted were of two distinct orders, namely of the order of v/c and (v/c)², where v denotes the velocity of the system in which the experiment is conducted relative to the ether, and c the velocity of light in the ether. Given the high value of c = 300,000 km/sec., the value of v/c is always extremely small. Even in the case where one might avail oneself in experiments of the greatest velocity at all readily available—that is, the velocity of the Earth in its orbit, namely v = 30 km/sec.—one obtains v/c = 1/10,000 and hence (v/c)² = 1/100,000,000.
The capabilities of optics at the beginning of the 19th century were already such as to allow the certain ascertainment of whether phenomena of the first order existed or not, but in no way permitted attempts at experiments of the second order. And since the former confirmed the absence of phenomena of the first order, the relativity of optical phenomena was proclaimed, at least to that already high degree of approximation, leaving to the future the decision as to whether it might extend also to the second order.
### III. THE EXTENSION FOLLOWING MICHELSON’S EXPERIMENTS
Indeed, it was necessary to wait more than 60 years before A. Michelson could attempt the final step with a prospect of success. The result of his first experiments of 1881 and of many others repeated in the following years by him and his collaborators, with means of exceptional precision specially prepared for the purpose, fully confirmed the persistence of the relativity of optical phenomena with respect to effects of the second order. It now fell to theoretical physics to account for this in deference to the experimental principle. But this precisely presented the difficulty, to which allusion was made at the outset, arising from the singularity of some of the hypotheses that had to be introduced.
To understand this essential point of the scientific situation that had arisen, it is useful to consider it with reference to some simple example. Let us suppose a railway line and a highway running parallel to each other, and on the former a train proceeding at a speed of 50 km/hr. An observer stationary on the highway will attribute to the train that same speed relative to himself; but a cyclist who instead travels on the highway at a speed of 20 km/hr will attribute to the train, relative to himself, either a speed of 30 km/hr when proceeding in the same direction as the train, or 70 km/hr when proceeding in the opposite direction; similarly, a motorist traveling at 50 km/hr will register the train’s speed relative to himself as either zero or 100 km/hr, depending on whether he is traveling in the same direction as the train or in the opposite direction; if he were instead traveling at 80 km/hr in the same direction as the train, he would register the train’s speed relative to himself as 30 km/hr, though in the opposite direction to his own motion. All this seems self-evident. Classical physics has, however, always tacitly and explicitly admitted that analogous statements could, indeed should, be made also for the speed of light relative to an observer moving in the direction of its propagation, whether in the same or in the opposite direction; thus, an observer moving with the Earth’s speed of 30 km/sec. against a ray of light should register its speed as 300,000 + 30 km/sec., whereas he should register it as 300,000 − 30 km/sec. when moving instead in the opposite direction.
These remembered experiences instead require us to hold that the observer must always measure the speed of 300,000 km/sec, regardless of his own speed and its direction. Even more drastically, this requirement of experience can be envisaged by considering two observers in uniform relative motion at any speed, even a very great one, who, at the instant they pass each other, emit a brief flash of light that propagates spherically in every direction. The requirement of experience translates in this case into the assertion that both observers, despite their relative motion and the consequent ever-increasing distance between them, must each consider themselves constantly at the center of the spherical wave that is propagating, just as, according to classical physics, one would unhesitatingly consider the observer who is at rest with respect to the ether. This requirement of a physical experience, whose reliability can no longer be doubted, determines the collapse of the notions of physical phenomena referred to absolute space and absolute time, to which one had always remained faithful despite their always unsatisfactory definitions: physical phenomena unfold in all systems in uniform relative translational motion (rectilinear and at constant speed) according to the same laws.
IV. CHARACTERISTIC FORMULATION
To formulate mathematically this singular result of experience, it is necessary to find the formula, that is, the law, capable of transforming the expression of the phenomenon found to be exact when referred to the system in which the observer is at rest, into an expression that remains equivalent to it even when referred to a system in uniform translational motion of any speed v (constant) with respect to the aforementioned system.For simplicity, let us assume—without in any way restricting the generality of the solution to the problem posed—that at time t = 0, taken as the initial moment, the given Cartesian system (x, y, z) and the analogous systems (x', y', z') in uniform translation with respect to it have their origins and axes coincident, and that the relative motion of speed v takes place along the axes x, x', which thus remain always coincident. Now physics has admitted as intuitive, since the time of Galileo, that under such conditions, to carry out the transformation of the expression of the law of any physical phenomenon from its general form (x, y, z, t) = 0, valid in the reference frame (x, y, z), to the corresponding form valid in the reference frame (x', y', z'), it was sufficient to apply the coordinate transformation:
(1) \[ x' = x - vt, \quad y' = y, \quad z' = z. \]
But it is precisely the application of this classical, intuitive transformation that raises the serious difficulty noted at the outset. It is useful to see it in action.
Consider the expression of any dynamical phenomenon, for example, the free fall of a body from the height x = x₀ along the x-axis oriented vertically. It is well known to be given by the general (Galilean) formula:
(2) \[ x = \text{initial height } x₀ + \text{initial velocity } v₀ \cdot t - \frac{1}{2} \cdot g \cdot t² \]
assuming the x-axis is vertical, taking proper account of the sign to be assigned to v₀ (positive if v₀ is directed upward, negative otherwise), setting t = 0 at the instant the body passes through the initial height x₀, and denoting by g the acceleration due to gravity near the Earth’s surface. Now suppose that at the same time a descending helicopter moving at constant speed v also passes through the height x₀. How will the phenomenon of the fall appear to an observer O' located on the helicopter when referred to the system x', y', z' fixed to it? The transformation equations lead to the expression, quite intuitive:
\[ x' = x₀ + (v₀ + v) \cdot t - \frac{1}{2} \cdot g \cdot t². \]
This not only clarifies how such transformations must be made, but also reveals something of great importance: that the expression of the dynamical phenomenon in the reference frame (x', y', z') thus derived is identical to the one that observer O' on the helicopter would have immediately written by applying law (2), since the initial velocity of the falling body relative to him is precisely v₀ + V. In other words: the Galilean transformation (1) leaves the dynamical law to which it is applied unchanged. More physically: the unfolding of any dynamical phenomenon is not altered by a uniform translation of the system in which it occurs.
This is the so-called Galilean relativity, which has no mysterious origin: the phenomena of dynamics are governed by a fundamental law that depends not on velocity but only on the accelerations of the system under consideration. As long as no new accelerations are introduced, the unfolding of phenomena is not altered, even if uniform velocities are attributed to the systems in which they occur. But, as has already been said, the same transformation equations (1), although they do not involve accelerations, predict alterations in the unfolding of optical phenomena—alterations that experience decisively excludes.
As H. A. Lorentz and H. Poincaré demonstrated, and independently of them A. Einstein, the invariance of optical phenomena under generic uniform translations is obtained only by adopting as transformation equations, instead of (1), the following, now generally called Lorentz transformations:
\[
(3) \quad x' = \frac{x - vt}{\sqrt{1 - (v/c)^2}}, \quad y' = y, \quad z' = z, \quad t' = \frac{t - \frac{vx}{c^2}}{\sqrt{1 - (v/c)^2}}
\]
V. THE FIRST THEORY OF RELATIVITY
Equations (3) are the essential basis of the theory proposed in 1906 by Einstein, which was then simply called the theory of relativity and which, to distinguish it from later, more general theories, is now referred to as the special or restricted theory of relativity, or more precisely, the first theory of relativity. A glance at (3) is enough to reveal their singularity. The last equation is the most disconcerting: for the first time in physics, time—or rather its measurement—is attributed a character relative to the system in which it is considered. Its numerator gives t' an essential alteration that is particularly noticeable, being of first order in v/c. The second singularity consists in the common radical in the expressions for x' and t', which produces an alteration of second order in all measurements of lengths and times when they are made from one to another of two systems related by uniform transformation at speed V. It is essential that when v/c is negligible compared to unity, equations (3) reduce identically to (1). Naturally, the two transformation laws (1) and (3) could not be admitted simultaneously. But here the facts came to the rescue: while equations (1) applied to optical phenomena produce divergences in principle intolerable no matter how small, equations (3), applied to dynamical phenomena, produced no appreciable deviations from classical predictions except in exceptional cases of bodies moving with translational speeds of the order of that of light, as occurs, for example, in the flux of electric corpuscles (electrons, mesons, β and α rays); but above all, the facts came to the rescue because these deviations occur in the direction required by experience, which had long since imposed their acceptance in the aforementioned cases, but which classical theories were forced to interpret differently. It was therefore decided to consider the Lorentz transformations valid in all fields of physics, reducing them to the Galilean ones whenever circumstances permit.The most striking and important effects, in addition to the radical elimination of the difficulties noted at the outset, are: 1° the prediction of the increase in the dynamic mass of every body with its velocity, expressed by the formula \( m = m_0 / \sqrt{1 - (v/c)^2} \), in which \( m_0 \) indicates the mass of that body as measured in a system in which it is at rest, i.e., the rest mass; 2° the prediction of the equivalence between the increases in the mass of every body multiplied by the square of the speed of light and the increases in its energy \( E \), exactly expressed by \( \Delta m c^2 = \Delta E \) and often extended by extrapolation and formulated as \( m c^2 = E \), though not without reservations regarding the scope of its validity.
This first theory of relativity, applied across all fields of physics, proved to be consistently superior to the non-relativistic theories that preceded it, in the sense that, wherever it differed from them, it did so not only without contradicting experience but in many cases with notable advantages.
VI. THE THEORY OF GENERAL RELATIVITY
Barely more than a year after the exposition of his first theory of relativity, while most scientists admired its simplicity and generality, Einstein noted certain shortcomings and his intention to refine II. He thus embarked on a nearly decade-long endeavor that, after a series of attempts which he himself later deemed unsatisfactory, culminated at the end of 1915 in the theory he termed general relativity.The profound physical characteristics of this new theory, even more singular than the previous one, are: 1) it incorporated the unsatisfactorily resolved problem of gravitation, dating back to 1906, to the extent that it would be more logical to consider it outright a relativistic theory of gravitation—a view adopted by many from the beginning and often held even by Einstein himself; 2) it interpreted the gravitational field, and thus the phenomenon itself, through the non-Euclidean, curved structure of physical space in which phenomena occur, for the first time utilizing the possibility, anticipated by B. Riemann as early as 1854, of representing physical fields by means of the characteristic geometry of curved spaces—spaces far more general than Euclidean space, which Riemann himself discovered and which are now known as Riemannian. Another notable feature of the new theory is the extensive, or rather exclusive, use of the methods of the absolute differential calculus of G. Ricci and T. Levi-Civita for its formal development, without which, as demonstrated by numerous failed prior attempts, it would not have been possible. With this in mind, it can be concluded that this Einsteinian theory constitutes a complete geometric synthesis of all natural phenomena, including gravitational ones, upon which only experience could ultimately pronounce the definitive word regarding its validity, as will be seen later.
It is not possible to present even the broadest outline of the theory without recourse to mathematical aids. Therefore, we shall limit ourselves to considerations that are only sufficiently approximate. The existence of gravitating masses determines in the space around them a modification, intense in the immediate vicinity of very large masses, which diminishes with distance. Our Euclidean mindset, perhaps due to insufficient preparation, may recoil from an intuitive conception of a three-dimensional space curvature as an extension of the intuitive notion of curved surfaces. Let us instead consider only the curvature of two-dimensional geometric entities, which are easily grasped because our minds spontaneously conceive of them as immersed in our familiar three-dimensional Euclidean space, much like simple surfaces. In the absence of any mass, the geometric entity in question must evidently be considered without curvature, i.e., as a plane; but if a concentrated material mass, or a so-called point mass, is placed at a point on this plane, the plane curves as if a small, cone-like protuberance were rising from it—highly curved at the apex and flattening out toward the plane, with which it tends to merge at a distance. This curvature can be interpreted as a geometric representation of the gravitational field of the material mass, or indeed of the mass itself. For if a second material mass is imagined placed near the first deformation of the plane, producing a similar protuberance, it is easy to demonstrate how these two protuberances must tend toward one another until they form a single protuberance equivalent to the two, when the masses that caused them have merged—an analogy to the behavior of two nearby bubbles on the surface of soapy water that approach one another until they unite as if attracted to each other.
The essential fact is that calculation demonstrates that the attraction between two masses immersed in Riemannian space, as predicted by the theory of general relativity, differs extremely little from Newtonian attraction, which it can therefore improve upon, as it not only resolves all cases that Newtonian theory does but also accounts for a series of subtle phenomena, particularly astronomical ones, which the latter cannot explain in any way. Among these are the slight motion of the perihelion of planets around the sun, especially noticeable in the case of Mercury and never interpretable under Newton’s law; and the bending of light rays passing in the immediate vicinity of large masses, observed for the mass of the sun during certain total eclipses.
VII. SPACE AND TIME IN MODERN PHYSICS
The intimate connection between the important physical phenomenon of gravitation and the nature of space and time presents science with a problem perhaps without precedent. It is generally accepted that the abstract formulation of geometric postulates and the subsequent construction of geometry must have been preceded by a period of approximate physical geometry. Yet, once Euclidean geometry was established by Euclid, it was immediately adopted by the natural sciences and even philosophy, not only as the geometry suitable for physical space but as the only logically possible and thus true geometry; its theorems were attributed absolute philosophical value even after Kant, despite the fact that by the late 18th century G. Saccheri had already presented considerations casting doubt on, or outright denying, the absoluteness of that value. However, precisely when, in the first half of the 19th century, the seminal work of N. Lobachevsky and J. Bolyai definitively demonstrated—through their very constructions—the possibility of geometries different from Euclid’s but equally logical, the question arose as to whether physical space could still be considered Euclidean. Triangulations of large terrestrial and astronomical triangles, undertaken by Gauss (which can in a sense be regarded as genuine experimental tests of physical space), demonstrated that at least within the high limits of accuracy permitted by the measuring instruments then in use, space appeared Euclidean.Shortly afterward, B. Riemann demonstrated the possibility of a broader generalization through the conception of curved spaces and their resulting geometries, and he posed the most remembered question that, in a certain sense, implied their physical reality: whether they might serve to represent physical fields. For sixty years, physics took no heed of this possibility and referred all its fields to a cosmic ether, but after the advent of the general theory, the gravitational field was generally interpreted through the curvature of Riemannian space, a development that greatly benefited astronomy. Since, moreover, a space of positive curvature—with a vague analogy to what occurs on a surface—tends to close in upon itself, the foregoing considerations led to the replacement of the ancient hypothesis of an infinitely extended space with that of a space still vast beyond measure, yet no longer infinite, though unbounded—that is, not limited by insurmountable obstacles. A space that, in two dimensions, we may represent, for example, as that of a vast spherical surface upon which one may move without encountering limitations, even though it is finite.
Electric and magnetic fields in the general theory are still referred to a special ether. The notion of time, which in general is affirmed to have been most disturbed by relativistic conceptions, from the standpoint of physics remains equally mysterious, as St. Augustine so acutely observed. The only innovation that physics found necessary to introduce does not concern the essence of time at all, but solely its measurement when performed in a system in motion relative to the observer. And this innovation, linked to the fact of relativity, must be considered valid in every case and not merely in the two relativistic theories considered.
It is necessary, moreover, to fix attention for a moment on a misunderstanding that must absolutely be avoided, though it is still fairly widespread. Since, as has been noted, in relativistic physics there exists a correlation, an interdependence between the measurements of space and time, the erroneous opinion is insinuated that this must imply an identity between the variable time and spatial coordinates, as is often said—that time constitutes a fourth dimension of space. For a certain period, only the author of this account resisted this suggestion. But finally, in 1921, Einstein himself, who had adhered to it, stated in the most explicit manner that the fact of the inseparability of the time variable from spatial coordinates does not in any way imply the identity of their notions, and that physical meaning demands their absolute distinction.
VIII. THE RECENT AND MORE GENERAL THEORY OF EINSTEIN
The brilliant result of the geometrization of the gravitational field encouraged an attempt at an analogous geometrization of the electric field. Within less than two years of the publication of the kinematic theory of gravitation, H. Weyl made a first attempt, soon followed by a similar one by A. Eddington, which, though they did not satisfactorily resolve the exceedingly difficult problem, nonetheless had the merit of highlighting an essential fact: namely, that the physical mentality had suddenly adapted itself to the idea of utilizing the properties of space to interpret its own phenomena. From these early attempts at geometrization of the electric field, it clearly emerged that the ample properties of Riemannian spaces were still insufficient, and that it was necessary to ascend to the consideration of spaces of greater generality.This idea, which only a few years earlier would have been judged absurd, was immediately embraced and provoked an enormous quantity of studies, most purely geometric.
Among the scholars who, from about 1920, addressed the question even from the physical side was Einstein himself. But the theories subjected to discussion were always less than fully satisfactory. Thus, little by little, all the others became discouraged, and only Einstein persisted and was finally rewarded with the satisfaction of arriving, after about thirty years of study and effort, at a mathematically rigorous solution to the bold problem.
Whether this is also the definitive and natural solution, only experience can tell. In the impossibility of outlining, even summarily, the major features of this greatest synthesis of the human mind, we recall only that the necessary amplification of the nature of space was achieved through the innovation—apparently of little significance—of excluding a simplifying symmetry that had been preserved in the Riemannian geometric construction.
The simplification vanished, but a new and grand possibility took its place.