Relativity, Theory of

RELATIVITY, Theory of. - I. PHYSICAL SPACE AND TIMES. - This theory, like the almost contemporary theory of "quanta" (v.), is one of the singular theories to which physics had to come in order to account, in homage to the Galilean experimental principle, for two experimental results that were in clear contrast with some of its fundamental conceptions. The singularity of these two modern theories lies in the fact that the changes they impose are not simple modifications of some pre-existing theory, but rather modifications that have repercussions not on some particular branch of it, but on the whole of physics. The r. that gave the theory now under consideration its name refers to absolute motion, in the sense that it is declared not only undetectable, but outright a concept devoid of meaning, with the consequence that physical experiments cannot bring to light anything other than relative motions between the different frames of reference and the different bodies. As regards the other theory, v. QUANTI.

II. THE FIRST LIMITED OPTICAL RELATIVITY

Dynamics, codified by Newton in his great work Philosophiae naturalis principia mathematica, founded on the discoveries of Galileo and his own, postulates as its theoretical basis the existence of an absolute time and an absolute space and therefore of a fixed system of reference, although Newton himself expressed doubts about the actual existence of the latter. In the fervor of the development of all mechanics that followed the diffusion of the Principia, throughout the 18th century those basic difficulties were glossed over; but at the beginning of the 19th century a revolution occurred that forced 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 one by one. The first of these theories was the Newtonian theory of emission, which interpreted light as consisting of corpuscles emitted by luminous sources and was therefore predominantly mechanical in nature. The second, a development of the theory initiated by Huyghens, interpreted light through a wave hypothesis analogous to the one with which sound had always been interpreted; the vibrating medium, however, was not air, but a hypothetical ether, fluid and diffused everywhere. The periodic vibrations were supposed to occur, like those of sound, longitudinally, that is, in the same direction as the propagation.

But the discovery of the phenomena of polarization threw the situation into confusion; the wave theory could not interpret them, whereas the corpuscular theory, albeit laboriously, could. The predominance of the emission theory was short-lived, however, because A. Fresnel succeeded in demonstrating that it was not the hypothesis of wave nature but that of longitudinality that prevented the interpretation of the polarization of light: it was enough to suppose that the vibrations of the hypothetical ether occurred transversally, that is, perpendicularly to the direction of propagation, for not only did that difficulty disappear, but one also arrived at an interpretation of all optical phenomena far better than the previous ones.

But the ether of the new theory, although by hypothesis permeable to any material body, had to possess characteristics of extreme rigidity in order to vibrate as was supposed, and therefore could serve as a fundamental system of spatial reference and perhaps be that absolute system that Newton and his followers had sought in vain. Moreover, the optical phenomena observed in different systems in relative motion with respect to this ether had to present themselves with different characteristics. This last conclusion was so interesting for the foundations of physics that rigorous experiments were immediately instituted to try to confirm it and thus finally arrive at the determination of a motion with respect to the ether, that is, of an absolute motion.

The differences that Fresnel's theory predicted were of two different orders, namely of the order of v/c and of (v/c)², denoting by v the velocity with respect to the ether of the system in which one experiments, and by 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 can use in experiments the maximum velocity that is in any way available, that is, the velocity of the earth in its orbit, or v = 30 km/sec., one will have v/c = 1/10,000 and therefore (v/c)² = 1/100,000,000.

The possibilities of optics at the beginning of the 19th century were already such as to permit ascertaining with certainty whether or not phenomena of the first order existed, but did not allow in any way attempting experiments of the second order. And since the former all confirmed the non-existence of phenomena of the first order, the relativity of optical phenomena was in the meantime proclaimed, at least in that already high order of approximation, leaving to the future the decision as to whether it could also be extended to the second order.

III. THE EXTENSION AS A CONSEQUENCE OF MICHELSON'S EXPERIMENTS

Indeed, more than 60 years had to pass before A. Michelson could attempt the final step with any probability 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 prepared for the purpose, fully confirmed the existence of the relativity of optical phenomena, in relation to second-order effects. It was now up to theoretical physics to account for it in deference to the experimental principle. But this precisely presented the difficulty, mentioned at the beginning, 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, and on the former a train proceeding at a speed of 30 km/h. An observer stationary on that same highway will attribute that same speed to the train, even relative to himself: but a cyclist who instead proceeds along the highway at a speed of 20 km/h will attribute to the train, relative to himself, either the speed of 30 km/h when it proceeds in his same direction, or 70 km/h when it proceeds in the opposite direction; similarly a motorist traveling at a speed of 50 km/h will find the speed relative to himself to be either zero or 100 km/h depending on whether he proceeds in the direction of the train or in the opposite direction; if then he proceeded at a speed of 80 km/h in the direction of the train, he would attribute to it, relative to himself, the speed of 30 km/h, but in the direction opposite to his own. All this seems evident. Classical physics, however, has always, tacitly and explicitly, assumed that analogous statements could, indeed should, also be made for the speed of light relative to an observer moving in the direction of its propagation, whether in the direction opposite to it or in the same direction as it; therefore, 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 its speed to be 300,000 - 30 km/sec when instead he moved in the opposite direction.

The experiments recalled, on the other hand, compel us to hold that the observer must always find the speed of 300,000 km/sec, whatever his own speed and its direction may be. This imposition of experiment can be envisaged even more drastically by thinking of two observers in uniform relative motion at any speed, even a very great one, who, at the instant when they pass close to each other, emit a brief flash of light that can propagate spherically in every direction. The imposition of experiment translates in this case into the assertion that the two observers, despite their relative motion and their consequent ever greater separation, must each consider themselves constantly at the center of the spherical wave that propagates, precisely as, according to classical physics, that observer who was stationary with respect to the ether would be considered without any difficulty. This imposition of a physical experiment, of whose reliability one can no longer doubt, brings about the collapse of the notions of physical phenomenon referred to absolute space and absolute time, to which one had always remained faithful despite their ever unsatisfactory definitions: physical phenomena take place in all systems in relative uniform translational motion (rectilinear and at constant speed) with 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 it was referred to the system in which the observer was fixed, into an expression that turns out to be equivalent to that one even when it is referred to a system in uniform relative translation at any velocity v (constant) with respect to the aforesaid system.

Let us suppose for simplicity (which in no way restricts the generality of the solution of the problem posed) that at time t = 0, taken as initial, the given Cartesian system (x, y, z), and the analogous systems (x', y', z') in uniform translation with respect to it, have coincident origins and axes, and that the relative motion of velocity v takes place in the direction of the axes x, x', which will thus remain always coincident. Now physics has admitted as intuitive, since the time of Galileo, that under such conditions, in order to carry out the transformation of the expression of the law of any physical phenomenon from its generic form f (x, y, z, t) = 0, valid in the reference to the system (x, y, z), to the corresponding form valid in the reference to the system (x', y', z'), it should suffice to apply the transformation of coordinates:

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

But it is precisely the application of this classical intuitive transformation that raised the grave difficulty pointed out at the beginning. It is useful to see it in operation.

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

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

This, besides having clarified how the said transformations must be made, says one more very important thing: that the expression of the dynamic phenomenon in the reference (x', y', z') thus deduced is identical to the one that the observer O' on the helicopter would have written outright by applying law (2), because the initial velocity of the falling body is, relative to him, precisely v₀ + v. In other words: the Galilean transformation (1) leaves the dynamic law to which it is applied unchanged. More physically: the unfolding of any dynamic phenomenon is not altered by a uniform translation of the system in which it occurs.

This is the so-called Galilean r., which has no mysterious origin: the phenomena of dynamics are governed by a fundamental law that does not depend on velocity, but only on the accelerations of the system considered: as long as no new accelerations are introduced, the unfolding of the 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), while not implying 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 assuming as transformation equations, instead of (2), the following, now generally called Lorentz's: 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. THE FIRST THEORY OF RELATIVITY

Equations (3) are the essential basis of the theory proposed in 1906 by Einstein, which was then called simply the theory of r. and which now, to distinguish it from later, more general theories, is usually called the theory of restricted or special r., or, better, the first theory of r. One glance at (3) is enough to note its singularity. The last equation is the most disconcerting: for the first time in physics a character relative to the system in which it is considered is attributed to time, or rather to its measurement. Its numerator confers on the time t' an essential alteration that is particularly noticeable, being of the first order in v/c. The second singularity is constituted by the radical common to the expressions for x' and t', which determines an alteration of the second order in all measurements of lengths and times when they are supposed to occur from one to the other of two systems in relative uniform transformation at velocity v. It is essential that for v/c negligible compared with unity, equations (3) reduce identically to (1). Naturally the two transformation laws (1) and (3) could not be admitted simultaneously. But here assistance came from the fact that, whereas equations (1) applied to optical phenomena determined divergences intolerable in principle whatever their magnitude, equations (3), applied to dynamical phenomena, determined no appreciable gaps from classical predictions except in the exceptional cases of bodies moving with translational velocities of the order of those of light, as occurs, for example, in the flow of electric corpuscles (electrons, mesons, β and σ rays); but above all assistance came from the fact that these gaps occur in the useful sense, which precisely experience had long since imposed as something to be admitted in the aforementioned cases, but which the classical theories were powerless to interpret. It was therefore decided to regard Lorentz's transformations as valid in all fields of physics, reducing them for simplicity to the Galilean ones when circumstances permit.

The most conspicuous and important effects are, besides the radical elimination of the difficulties noted at the beginning: 1st the prediction of the increase of the dynamical mass m of every body with its velocity, expressed by the formula m = m₀ / √{1 - (v/c)²}, in which m₀ indicates the mass of that body evaluated in a system in which it is at rest, that is, the rest mass; 2nd the prediction of the equivalence of the increases of the mass of every body multiplied by the square of the velocity of light c and of the increases of 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 field of its validity.

This first theory of r., applied in all fields of physics, always proved notably superior to the non-relativistic theories that preceded it, in the sense that, where it differed from them, this always happened not only without conflicts with experience, but in many cases with appreciable advantages.

VI. THE THEORY OF GENERAL RELATIVITY

Barely a little more than a year after the exposition of his first theory of r., while most scientists admired its simplicity and generality, Einstein noted certain shortcomings in it and his own intention to perfect it. He thus began a nearly ten-year labor which finally, after a series of attempts, which he himself then showed to be unsatisfactory, led him at the end of 1915 to the theory that he called that of general r.

Profound physical characteristics of this new theory, even more singular than the preceding one, are: 1) having included in its treatment the age-old and never satisfactorily resolved problem of gravitation, so much so that it would be far more logical to consider it outright a relativistic theory of gravitation, as moreover many did from the outset and as Einstein himself now often does; 2) having interpreted the field of gravitation, and therefore the phenomenon itself, through the no longer Euclidean but curved structure of the physical space in which phenomena unfold, using for the first time the possibility foreshadowed by B. Riemann, as early as 1834, of representing the mysterious physical fields by means of the clear geometric characteristics of curved spaces, by far more general than Euclidean space, discovered by him and today precisely called Riemannian. Another notable characteristic of the new theory is constituted by the ample, or rather, exclusive employment, 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 and vain previous attempts demonstrate. That said, one may conclude that this Einsteinian theory constitutes a complete geometric synthesis of all phenomena of a mechanical nature, including gravitational ones, on which only experience could pronounce the definitive word regarding its reliability, as will be seen below.

It is not possible to expound exactly, without resorting to mathematical aid, even the broadest outlines of the theory. 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 great masses, which gradually diminishes with distance. Since our Euclidean mentality recoils, perhaps for lack of sufficient preparation, from an intuitive conception of a curvature of three-dimensional space as an extension of the intuitive one of surfaces, let us limit ourselves to a consideration involving only curvatures of two-dimensional geometric entities, easily conceivable, because our mind spontaneously considers them immersed in our usual three-dimensional Euclidean space precisely as simple surfaces. In the absence of any mass the geometric entity in question must evidently be considered without curvature, that is, a plane; but if at a point of this a concentrated material mass is placed, or a so-called material point, the plane curves as if upon it there rose a small conical protuberance, much curved at the vertex and ever less so toward the plane with which it tends to merge again at a distance. The said curvature can be interpreted as a geometric representation of the gravitational field of the material mass, or even, outright, of the mass itself. Indeed, if in the vicinity of that first deformation of the plane a second material mass is thought to be placed, and therefore the consequent protuberance analogous to the first is produced, it is easy to demonstrate how those two protuberances must tend toward one another, until they constitute a single one equivalent to the two, when the masses that provoked them have come together; and this with a marked analogy to the behavior of two nearby little bubbles on the surface of soapy water that tend to approach until they unite as if they attracted each other.

The essential fact is that calculation demonstrates that the attraction of two masses immersed in Riemannian space, which the theory of general gravitation predicts, differs extremely little from Newtonian attraction, which it can therefore replace while improving upon it, because, besides solving all the cases that the latter solves, it also accounts for a series of subtle phenomena, especially astronomical, which the latter can in no way reach. These are, among others, the slight motion around the sun of the perihelion of the planets, especially noticeable for Mercury and never able to be interpreted on the basis of the Newtonian law; the curvature of light rays passing in the immediate vicinity of great masses, confirmed for the mass of the sun on the occasion of certain total eclipses.

VII. SPACE AND TIME IN CONTEMPORARY PHYSICS

The preceding intimate bringing-together of the important physical phenomenon of gravitation with the nature of space and time poses for science a problem that perhaps has no precedent. It is generally admitted that the abstract formulation of geometrical 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 constituted by Euclid, it was forthwith adopted by the natural sciences and by philosophy itself, not only as the geometry suited also to our physical space, but as the only logically possible and therefore true geometry; its theorems were attributed an absolute philosophical value even after Kant, despite the fact that toward the end of the 18th century G. Saccheri had already set forth considerations capable of casting doubt on, or indeed of denying outright, the absoluteness of that value. But precisely when, in the first half of the 19th century, through the well-known work of N. Lobachevsky and G. Bolyai, the possibility of geometries different from the Euclidean but equally logical was definitively demonstrated by their very construction, the question arose whether the space of physics could still be considered Euclidean. Triangulations of large terrestrial and astronomical triangles promoted by Gauss (which in a certain sense can be regarded as true experimental tests on physical space) showed that, at least within the high limits of accuracy already permitted by the measuring instruments then in use, space presented itself as Euclidean.

Shortly afterward B. Riemann demonstrated the possibility of a broader generalization by means of the conception of curved spaces and their consequent geometries, and proposed the already-mentioned question that implied in a certain sense their physical reality: whether they might serve to represent physical fields. For sixty years physics paid no heed to this possibility and referred all its fields to a cosmic ether, but after the advent of general relativity the field of gravitation was generally interpreted by means of the curvature of Riemannian space, a thing from which astronomy greatly benefited. Since then a space of positive curvature, by a vague analogy with what happens for a surface, has a tendency to close in upon itself, the preceding considerations led to replacing the old hypothesis of an infinitely extended space with that of a space still exceedingly vast, but no longer infinite, though unbounded, that is, not limited by insurmountable obstacles; of a space that, in two dimensions, we can picture to ourselves, for example, as that of a large spherical surface upon which it is possible to move about without encountering limitations, although it is finite.

The electric and magnetic fields in the theory of general relativity are still always referred to a special ether. The notion of time, which is generally asserted to have been the one most disturbed by relativistic conceptions, from the point of view of physics remains always equally mysterious, as St. Augustine so acutely observed. The only innovation that physics found it necessary to introduce does not concern at all the essence of time, but only its measurement when this is carried out in a system in motion relative to the one who performs it. And this innovation, bound to the fact of relativity, must be considered valid in every case and not only in the two relativistic theories considered.

We must still fix our attention for a moment on a misunderstanding that must absolutely be avoided although it is still fairly widespread. Since, as was pointed out, in relativistic physics there exists a correlation, an interdependence between the measurements of space and of time, the erroneous opinion insinuated itself 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 of time only the author of this account resisted that suggestion. But finally, in 1921, Einstein himself, who had adhered to it, affirmed in the most explicit manner that the fact of the inseparability of the time variable from the spatial coordinates does not at all imply the identity of their notions, and that physical sense demands their absolute separation.

VIII. EINSTEIN'S RECENT AND MORE GENERAL THEORY

The brilliant result of the geometrization of the gravitational field prompted an attempt at an analogous geometrization of the electric field. Less than two years after the publication of Einstein's theory of gravitation there came a first attempt by H. Weyl, followed shortly afterward by a similar attempt by A. Eddington, which, although they did not satisfactorily solve the extremely difficult problem, nevertheless had the merit of bringing to light an essential fact, which proves, at the very least, how suddenly the physical mentality had adapted to the idea of using the properties of space to interpret its own phenomena. From these first attempts at the geometrization of the electric field it emerged clearly that the ample properties of Riemannian spaces were not yet sufficient, and that it was still necessary to rise to the consideration of spaces of greater generality.

This idea, which a few years earlier would have been judged an absurdity, was immediately accepted and provoked an enormous quantity of studies, the majority purely geometrical.

Among the scholars who from about 1920 concerned themselves with the question also from the physical side was Einstein himself. But the theories that were submitted for discussion always turned out to be not entirely satisfactory. Thus little by little all the others became discouraged, and only Einstein continued and finally had the satisfaction of arriving, after about thirty years of studies and attempts, at a mathematically rigorous solution of the bold problem.

Whether this is also the definitive natural solution, only experience will have to say. In the impossibility of tracing, even summarily, the major characteristics of this supreme synthesis of the human mind, let it be recalled only that the necessary amplification of the nature of space was achieved with the innovation, apparently of little consequence, of excluding a simplifying symmetry that had been preserved in the Riemannian geometrical construction.

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

Paolo Straneo