Relativity, Theory of

RELATIVITÀ, TEORIA della. -

I. PHYSICAL SPACE AND TIME

This theory, like the nearly contemporaneous theory of «quanta» (v.), is one of the singular theories to which physics was compelled to resort in order, in homage to the Galilean experimental principle, to account for two experimental results that were in clear conflict with some of its fundamental conceptions. The singularity of these two modern theories lies in the fact that the modifications they impose are not simple alterations of some pre-existing theory, but rather modifications that affect not merely one particular branch of it, but physics as a whole. The r. that has given its name to the theory now under consideration refers to absolute motion, in the sense that the latter is declared not only unobservable, but outright a concept devoid of meaning, with the consequence that physical experiments can bring to light nothing but relative motions between the different frames of reference and the different bodies. As regards the other theory, V. QUANTITÀ.

II. THE FIRST LIMITED OPTICAL R

Dynamics, codified by Newton in his great work Philosophiae naturalis principia mathematica, founded on the discoveries of Galilei and on his own, postulates as its theoretical basis the existence of an absolute time and space, and therefore of a fixed frame of reference, although Newton himself expressed doubts about the actual existence of the latter. In the fervor of the development of all the mechanics that followed the dissemination of the Principia, those fundamental difficulties were passed over throughout the eighteenth century; but at the beginning of the nineteenth century such a revolution took place that it compelled them to be taken seriously for the first time. Until then, two different theories had succeeded in interpreting optical phenomena as they were observed experimentally. The first of these theories was the Newtonian emission theory, which interpreted light as consisting of corpuscles emitted by luminous sources and was therefore predominantly mechanical in character. 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 ether diffused everywhere. The periodic vibrations were assumed to occur, as in the case of sound, longitudinally, that is, in the same direction as propagation.

But the discovery of polarization phenomena threw the situation into confusion; wave theory was unable to interpret them, whereas the corpuscular theory, albeit laboriously, succeeded in doing so. The predominance of emission theory was short-lived, however, because A. Fresnel succeeded in demonstrating that it was not the hypothesis of wave motion but that of longitudinal motion that prevented the interpretation 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 that difficulty to disappear, but also for an interpretation of all optical phenomena to be reached that was far superior to those previously available.

But the ether of the new theory, although by hypothesis permeable to every material body, had to possess an extreme rigidity in order to vibrate as supposed, and could therefore serve as a fundamental spatial frame of reference and perhaps be the absolute system that Newton and his followers had vainly sought. Moreover, optical phenomena observed in different systems moving relatively to one another in this ether ought to appear with different characteristics. This latter conclusion was so significant for the foundations of physics that rigorous experiments were immediately instituted in an attempt to confirm it and thus finally determine a motion with respect to the ether, that is, an absolute motion.

The differences predicted by Fresnel’s theory were of two different orders, namely, of the order of v/c and of (v/c)², where v denotes the velocity relative to the ether of the system in which the experiment is conducted, 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 event that the greatest velocity in some way available for experiments could be used, namely, the velocity of the earth in its orbit, or v = 30 km/sec., one would have v/c = 1/10,000 and therefore (v/c)² = 1/100,000,000.

The possibilities of optics at the beginning of the nineteenth century were already such as to permit the certain establishment of whether first-order phenomena existed or not, but they in no way permitted attempts at second-order experiments. And since the former all confirmed the absence of first-order phenomena, the r. of optical phenomena was provisionally proclaimed, at least to that already high degree of approximation, leaving it to the future to decide whether it could also be extended to the second order.

III. EXTENSION AS A CONSEQUENCE OF THE MICHELSON EXPERIMENTS

Indeed, it was necessary to wait more than 60 years before A. Michelson could attempt the final step with a reasonable prospect of success. The result of his first experiments in 1881, and of many others repeated in the following years by him and his collaborators, using means of exceptional precision specially devised for the purpose, fully confirmed the validity of the r. of optical phenomena with regard to second-order effects. It now fell to theoretical physics to account for this, in deference to the experimental principle. But this was precisely where the difficulty arose, mentioned at the outset, deriving from the singularity of certain hypotheses that had to be introduced.

To understand this essential point in the scientific situation that had arisen, it is useful to consider it with reference to a few simple examples. Let us suppose that a railway line and a motorway run parallel to one another, and that on the former a train is proceeding at a speed of 30 km per hour. An observer stationary on the same motorway will attribute to the train that same speed also relative to himself; but a cyclist proceeding along the motorway at a speed of 20 km per hour will attribute to the train, relative to himself, either a speed of 30 km per hour when proceeding in the same direction, or 70 km per hour when proceeding in the opposite direction. Similarly, a motorist travelling at a speed of 50 km per hour will find the speed relative to himself to be either zero or 100 km per hour, depending on whether he is proceeding in the train’s direction or in the opposite direction; if, on the other hand, he were proceeding at a speed of 80 km per hour in the train’s direction, he would attribute to it, relative to himself, a speed of 30 km per hour, but in the direction opposite to his own. All this seems evident. Classical physics, however, has always, tacitly and explicitly, admitted that analogous statements could, indeed ought to, be made also concerning the speed of light relative to an observer moving in the direction of its propagation, either in the opposite or in the same direction; hence, an observer moving at the speed of the earth, 30 km/sec., toward a ray of light should find its speed to be 300,000 + 30 km/sec., whereas he should find it to be 300,000 − 30 km/sec. when he moved in the opposite direction.

The experiments recalled above instead compel us to hold that the observer must always find the speed to be 300,000 km/sec., whatever his own speed and its direction may be. This demand of experience can be formulated even more drastically by considering two observers in uniform relative motion, at any speed, however great, who, at the instant when they pass close to one another, emit a brief flash of light capable of propagating itself spherically in every direction. In this case the demand of experience translates into the assertion that the two observers, despite their relative motion and their consequent ever-increasing separation, must each constantly regard himself as being at the centre of the spherical wave that is propagating, precisely as, on the basis of classical physics, one would have no difficulty in regarding as such an observer who was at rest relative to the ether. This demand imposed by a physical experience, whose reliability can by now no longer be doubted, brings about the collapse of the notions of a physical phenomenon referred to absolute space and absolute time, to which adherence had always been maintained despite their increasingly unsatisfactory definitions: physical phenomena take place in all systems in relative uniform translational motion (rectilinear and at constant speed) according to the same laws.

IV. CHARACTERISTIC FORMULATION

In order 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 was at rest into an expression equivalent to it even when referred to a system in relative uniform translation at any velocity v (constant) with respect to the aforementioned system.

Let us suppose for simplicity (which in no way restricts the generality of the solution to the problem posed) that at time t = 0, taken as the initial time, 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 at velocity v takes place in the direction of the axes x, x', which will thus always remain coincident. Now, since the time of Galileo, physics has admitted as intuitive that, under such conditions, in order to transform the expression of the law of any physical phenomenon from its general form f (x, y, z, t) = 0, valid when referred to the system (x, y, z), into the corresponding form valid when referred to the system (x', y', z'), it should suffice to apply the transformation of the coordinates:

(1) x' = x - vt , y' = y , z' = z.

But it was precisely the application of this classical intuitive transformation that raised the serious difficulty mentioned at the outset. It is useful to see it in operation.

Let us consider the expression of any dynamic phenomenon, for example, the free fall of a body from the altitude x = x₀ on the vertically oriented x-axis. It is notoriously given by the general (Galilean) formula:

(2) x = initial altitude x₀ + initial velocity v₀ · t - ½ · gt², assuming the x-axis to be vertical, with due regard for the sign to be assigned to v₀ (positive if v₀ is directed upward, negative otherwise), taking t = 0 at the instant when the body passes through the initial altitude x₀ and denoting by g the acceleration of gravity in the vicinity of the Earth's surface. Let us now suppose that at the same time a helicopter descending at the constant velocity - v likewise passes through the altitude x₀ at time t = 0. How will the phenomenon of the fall appear to an observer O' who is on the helicopter when it is referred to the system x', y', z' fixed to the latter? The transformation equations lead to the entirely intuitive expression:

x' = x₀ + (v₀ + v) t - ½ · gt².

Besides clarifying how the said transformations are to be carried out, this also states something extremely important: that the expression of the dynamic phenomenon in the reference system (x', y', z') thus deduced is identical to what the observer O' on the helicopter would certainly have written by applying law (2), because the initial velocity of the falling body relative to him is precisely v₀ + V. In other words: the Galilean transformation (1) leaves unchanged the dynamic law to which it is applied. In more physical terms: the course of any dynamic 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; so long as no new accelerations are introduced, the course 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 implying no accelerations, predict alterations in the course of optical phenomena, alterations that experience decisively rules out.

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 assuming as the transformation equations, instead of (2), the following, now generally called Lorentz equations:

x' = frac{x - vt}{√{1 - (v/x)²}}, y' = y, z' = z, t' = frac{t - frac{vx}{c²}}{√{1 - (v/x)²}} tag{3}

V. LA PRIMA TEORIA DELLA R

Equations (3) constitute the essential basis of the theory proposed in 1906 by Einstein, which was then simply called the theory of relativity and which is now, to distinguish it from later, more general theories, usually called the theory of restricted or special relativity, or, better, the first theory of relativity. A glance at (3) suffices to reveal its 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 time t' an essential alteration, particularly perceptible because it is of the first order in v/c. The second singularity consists in the radical common to the expressions for x' and t', which determines a second-order alteration in all measurements of length and time when they are supposed to take place from one to another of two systems undergoing relative uniform motion at velocity V. The essential fact is that, when v/c is negligible in comparison with unity, equations (3) reduce identically to (1). Naturally, the two transformation laws (1) and (3) could not be admitted simultaneously. Here, however, the fact came to our aid that, whereas (1), when applied to optical phenomena, produced divergences that were intolerable in principle, whatever their magnitude, (3), when applied to dynamical phenomena, produced no appreciable deviations from classical predictions except in the exceptional cases of bodies moving with translational velocities of the order of that of light, as occurs, for example, in the flow of electric corpuscles (electrons, mesons, β and σ rays); above all, we were helped by the fact that these deviations occur in the useful sense, which experience had already long compelled us to admit in the cases in question, but which classical theories were powerless to interpret. It was therefore decided to regard the Lorentz transformations as valid in all fields of physics, reducing them, for the sake of simplicity, to the Galilean transformations whenever circumstances permit.

The most conspicuous and important effects are, besides the radical elimination of the difficulties identified at the outset: 1° the prediction of an increase in the dynamical mass m of every body with its velocity, expressed by the formula m = m₀ / √{1 - (v/c)²}, in which m₀ denotes the mass of that body evaluated in a system in which it is at rest, that is, its rest mass; 2° the prediction of the equivalence between the increases in the mass of every body multiplied by the square of the velocity of light c and the increases in its energy E, exactly expressed by Δ mc² = Δ E and often extended by extrapolation and formulated as mc² = E, not without reservations concerning the scope of its validity.

This first theory of relativity, applied in all fields of physics, consistently proved markedly superior to the non-relativistic theories that preceded it, in the sense that, wherever it differed from them, this always occurred not only without conflict with experience, but in many cases with appreciable advantages.

VI. THE THEORY OF GENERAL R

Just over a year after presenting his first theory of r., while most scientists admired its simplicity and generality, Einstein noted some of its shortcomings and expressed his intention of perfecting II. He thus began work lasting almost a decade which

finally, after a series of attempts that he himself subsequently demonstrated to be unsatisfactory, led him at the end of 1915 to the theory he called the general theory of r.

The profound physical characteristics of this new theory, even more singular than the preceding one, are: 1) that it included in its treatment the age-old problem of gravitation, which had never been satisfactorily resolved, so much so that it would be far more logical simply to regard it as a relativistic theory of gravitation, as many in fact did from the outset and as Einstein himself now often does; 2) that it interpreted the gravitational field, and therefore the phenomenon itself, through the no longer Euclidean but curved structure of the physical space in which phenomena occur, making use for the first time of the possibility, foreseen by B. Riemann as early as 1834, of representing mysterious physical fields by means of the clear geometrical characteristics of curved spaces, far more general than Euclidean space, which he discovered and which are today precisely called Riemannian. Another notable characteristic of the new theory is constituted by the extensive, or rather exclusive, use, for its formal development, of the methods of the absolute differential calculus of G. Ricci and T. Levi-Civita, without which it would not have been possible, as the numerous fruitless earlier attempts demonstrate. With this premise, one may conclude that this Einsteinian theory constitutes a complete geometrical synthesis of all phenomena of a mechanical nature, including gravitational phenomena, concerning whose reliability only experience could pronounce the definitive judgment, as will be seen below.

It is not possible to expound even the broadest outlines of the theory accurately without resorting to mathematical assistance. We shall therefore limit ourselves to considerations that are only sufficiently approximate. The existence of gravitating masses determines in the space surrounding them a modification, intense in the immediate vicinity of very large masses, which diminishes with distance. Since our Euclidean mentality, perhaps through lack of sufficient preparation, recoils from an intuitive conception of the curvature of three-dimensional space as an extension of the intuitive curvature of surfaces, let us limit ourselves to a consideration involving only curvatures of two-dimensional geometrical entities, which are easy to conceive, because our mind spontaneously regards them as immersed in our customary three-dimensional Euclidean space, precisely as simple surfaces. In the absence of any mass, the geometrical entity in question must evidently be regarded as without curvature, that is, as a plane; but if a concentrated material mass, or a so-called material point, is placed at one point of it, the plane curves as though a small conical protuberance rose upon it, very sharply curved at the vertex and progressively less so toward the plane, with which it tends to merge again at a distance. This curvature may be interpreted as a geometrical representation of the gravitational field of the material mass, or even, without further qualification, of the mass itself. Indeed, if a second material mass is thought of as placed near that first deformation of the plane, thereby producing a consequent protuberance analogous to the first, it is easy to demonstrate that the two protuberances must tend toward one another until they form a single protuberance equivalent to the two, once the masses that produced them have come together; and this is strikingly analogous to the behavior of two nearby bubbles on the surface of soapy water, which tend to approach one another until they unite as though they attracted each other.

The essential fact is that calculation demonstrates that the attraction of two masses immersed in Riemannian space, as predicted by the theory of general gravity, differs extremely little from Newtonian attraction, which it can consequently replace while improving upon it, because, besides resolving all the cases resolved by the latter, it also accounts for a series of slight phenomena, especially astronomical, which the latter cannot in any way explain. These include, among others, the slight motion around the sun of the perihelion of the planets, particularly perceptible in Mercury and never interpretable on the basis of Newtonian law; and the curvature of light rays passing in the immediate vicinity of large masses, observed for the mass of the sun on the occasion of certain total eclipses.

VII. SPACE AND TIME IN PRESENT-DAY PHYSICS

The preceding intimate association of the important physical phenomenon of gravitation with the nature of space and time poses for science a problem perhaps without precedent. It is generally admitted that the abstract formulation of geometric postulates and the corresponding constitution of geometry must have been preceded by a period of approximate physical geometry. But as soon as the familiar abstract geometry had been established by Euclid, it was immediately adopted by the natural sciences and by philosophy itself, not only as the geometry suitable also to our physical space, but as the only logically possible and therefore true geometry; an absolute philosophical value was attributed to its theorems even after Kant, despite the fact that toward the end of the eighteenth century G. Saccheri had already set forth considerations capable of prompting doubt about, or even categorically denying, the absoluteness of that value. But precisely when, in the first half of the nineteenth century, the well-known work of N. Lobachewski and G. Bolyai definitively demonstrated, through their own construction, the possibility of geometries different from the Euclidean but equally logical, the question arose whether the space of physics could still be regarded as Euclidean. Triangulations of large terrestrial and astronomical triangles promoted by Gauss (which, in a certain sense, may be regarded as genuine experiments for testing physical space) demonstrated that, at least within the high limits of precision already 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 consequent geometries, and proposed the aforementioned question, which in a certain sense implied their physical reality: whether they might not serve to represent physical fields. For sixty years physics took no account of this possibility and referred all its fields to a cosmic ether; but after the advent of general relativity, the gravitational field was generally interpreted through the curvature of Riemannian space, a development from which astronomy benefited greatly. Since, moreover, a space of positive curvature, in vague analogy with what occurs on a surface, tends to close in upon itself, the preceding considerations led to the replacement of the ancient hypothesis of an infinitely extended space with that of a space still enormously large, but no longer infinite, though unbounded—that is, not bounded by insurmountable obstacles; a space which, in two dimensions, can be represented, for example, as a great spherical surface on which it is possible to move without encountering limitations, although it is finite.

In the theory of general relativity, electric and magnetic fields are still referred to a special ether. The notion of time, which is generally said to have been most profoundly disturbed by relativistic conceptions, remains, from the standpoint of physics, equally mysterious, as s. Agostino so acutely observed. The only innovation that physics found it necessary to introduce concerns not the essence of time at all, but only its measurement when carried out in a system moving relative to the person making the measurement. And this innovation, connected with the fact of relativity, must be regarded as valid in every case and not only in the two theories of relativity under consideration.

It is still necessary to focus attention for a moment on an ambiguity that must absolutely be avoided, although it remains fairly widespread. Since, as was noted, relativistic physics involves a correlation, an interdependence, between measurements of space and time, the erroneous opinion arose that this must imply an identity between the time variable and the spatial coordinates or, as was often said, that time constituted a fourth dimension of space. For a certain period, only the author of this account resisted that suggestion. But finally, in 1921, Einstein himself, who had accepted it, stated in the most explicit terms that the inseparability of the time variable from the spatial coordinates in no way implies the identity of their notions, and that physical meaning requires their absolute distinction.

VIII. EINSTEIN’S RECENT AND MORE GENERAL THEORY

The brilliant result of the geometrization of the gravitational field prompted attempts at an analogous geometrization of the electric field. Less than two years after the publication of Einstein’s theory of gravitation, a first attempt was made by H. Weyl, followed shortly afterwards by a related attempt by A. Eddington. Although they did not satisfactorily resolve the extremely difficult problem, they nevertheless had the merit of bringing to light an essential fact, which proves, at the very least, how greatly the physical mentality had suddenly adapted itself to the idea of using the properties of space to interpret its own phenomena. These first attempts at geometrizing the electric field clearly showed that the extensive properties of Riemannian spaces were not yet sufficient, and that it was necessary to rise to the consideration of spaces of greater generality.

This idea, which only a few years earlier would have been judged absurd, was readily accepted and gave rise to an enormous quantity of studies, most of them purely geometric.

Among the scholars who, from around 1920, addressed the question also from the physical point of view was Einstein himself. But the theories submitted for discussion always proved not entirely satisfactory. Thus, little by little, all the others became discouraged, and only Einstein continued; finally, after approximately three decades of study and attempts, he had the satisfaction of arriving at a mathematically rigorous solution to the bold problem.

Whether this is also the definitive natural solution, only experience will have to say. Since it is impossible to outline, even summarily, the principal characteristics of this supreme synthesis of the human mind, it may merely be recalled that the necessary amplification of the nature of space was achieved through the apparently minor innovation of excluding a simplifying symmetry that had been retained in Riemannian geometric construction.

The simplification disappeared, but a new, grandiose possibility took its place.

BIBLI. A. Einstein. Über die spezielle und die allgemeine Relativitätstheorie, Italian translation by G. L. Calisse, Bologna 1921; id., Il significato fisico della r., Turin 1930; M. Born, Die Relativitätstheorie Einsteins, French translation, Paris 1923; C. Castelnuovo, Spazio e tempo, Bologna 1923; P. Straneo, Teoria della r. secondo il senso fisico, Rome 1924. Paolo Straneo