If, then, one tries to interpret the time of an event analogously, one needs a means for the measurement of the difference in time (in itself determined periodic process realized by a system of sufficiently small spatial extension). A clock at rest relative to the system of inertia defines a local time. The local times of all space points taken together are the “time,” which belongs to the selected system of inertia, if a means is given to “set” these clocks relative to each other. One sees that a priori it is not at all necessary that the “times” thus defined in different inertial systems agree with one another. One would have noticed this long ago, if, for the practical experience of everyday life light did not appear (because of the high value of c), as the means for the statement of absolute simultaneity.
The presupposition of the existence (in principle) of (ideal, viz., perfect) measuring rods and clocks is not independent of each other; since a lightsignal, which is reflected back and forth between the ends of a rigid rod, constitutes an ideal clock, provided that the postulate of the constancy of the light-velocity in vacuum does not lead to contradictions.
The above paradox may then be formulated as follows. According to the rules of connection, used in classical physics, of the spatial coordinates and of the time of events in the transition from one inertial system to another the two assumptions of
(1) the constancy of the light velocity
(2) the independence of the laws (thus specially also of the law of the constancy of the light velocity) of the choice of the inertial system (principle of special relativity)
are mutually incompatible (despite the fact that both taken separately are based on experience).
The insight which is fundamental for the special theory of relativity is this: The assumptions (1) and (2) are compatible if relations of a new type (“Lorentz-transformation”) are postulated for the conversion of co-ordinates and the times of events. With the given physical interpretation of co-ordinates and time, this is by no means merely a conventional step, but implies certain hypotheses concerning the actual behavior of moving measuring-rods and clocks, which can be experimentally validated or disproved.
The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to the Lorentz-transformations (for the transition from one inertial system to any other arbitrarily chosen system of inertia). This is a restricting principle for natural laws, comparable to the restricting principle of the non-existence of the perpetuum mobile which underlies thermodynamics.
First a remark concerning the relation of the theory to “four-dimensional space.” It is a wide-spread error that the special theory of relativity is supposed to have, to a certain extent, first discovered, or at any rate, newly introduced, the four-dimensionality of the physical continuum. This, of course, is not the case. Classical mechanics, too, is based on the four-dimensional continuum of space and time. But in the four-dimensional continuum of classical physics the subspaces with constant time value have an absolute reality, independent of the choice of the reference system. Because of this [fact], the four-dimensional continuum falls naturally into a three-dimensional and a one-dimensional (time), so that the four-dimensional point of view does not force itself upon one as necessary. The special theory of relativity, on the other hand, creates a formal dependence between the way in which the spatial co-ordinates, on the one hand, and the temporal coordinates, on the other, have to enter into the natural laws.
Minkowski’s important contribution to the theory lies in the following: Before Minkowski’s investigation it was necessary to carry out a Lorentz-transformation on a law in order to test its invariance under such transformations; he, on the other hand, succeeded in introducing a formalism such that the mathematical form of the law itself guarantees its invariance under Lorentz-transformations. By creating a four-dimensional tensor-calculus he achieved the same thing for the four-dimensional space which the ordinary vector-calculus achieves for the three spatial dimensions. He also showed that the Lorentz-transformation (apart from a different algebraic sign due to the special character of time) is nothing but a rotation of the coordinate system in the four-dimensional space.
First, a remark concerning the theory as it is characterized above. One is struck [by the fact] that the theory (except for the four-dimensional space) introduces two kinds of physical things, i.e., (1) measuring rods and clocks, (2) all other things, e.g., the electro-magnetic field, the material point, etc. This, in a certain sense, is inconsistent; strictly speaking measuring rods and clocks would have to be represented as solutions of the basic equations (objects consisting of moving atomic configurations), not, as it were, as theoretically self-sufficient entities. However, the procedure justifies itself because it was clear from the very beginning that the postulates of the theory are not strong enough to deduce from them sufficiently complete equations for physical events sufficiently free from arbitrariness, in order to base upon such a foundation a theory of measuring rods and clocks. If one did not wish to forego a physical interpretation of the co-ordinates in general (something which, in itself, would be possible), it was better to permit such inconsistency—with the obligation, however, of eliminating it at a later stage of the theory. But one must not legalize the mentioned sin so far as to imagine that intervals are physical entities of a special type, intrinsically different from other physical variables (“reducing physics to geometry,” etc.).
We now shall inquire into the insights of definite nature which physics owes to the special theory of relativity.
(1) There is no such thing as simultaneity of distant events; consequently there is also no such thing as immediate action at a distance in the sense of Newtonian mechanics. Although the introduction of actions at a distance, which propogate with the speed of light, remains thinkable, according to this theory, it appears unnatural; for in such a theory there could be no such thing as a reasonable statement of the principle of conservation of energy. It therefore appears unavoidable that physical reality must be described in terms of continuous functions in space. The material point, therefore, can hardly be conceived any more as the basic concept of the theory.
(2) The principles of the conservation of momentum and of the conservation of energy are fused into one single principle. The inert mass of a closed system is identical with its energy, thus eliminating mass as an independent concept.
Remark. The speed of light c is one of the quantities which occurs as “universal constant” in physical equations. If, however, one introduces as unit of time instead of the second the time in which light travels 1 cm, c no longer occurs in the equations. In this sense one could say that the constant c is only an apparently universal constant.
It is obvious and generally accepted that one could eliminate two more universal constants from physics by introducing, instead of the gram and the centimeter, properly chosen “natural” units (for example, mass and radius of the electron).
If one considers this done, then only “dimension-less” constants could occur in the basic equations of physics. Concerning such I would like to state a theorem which at present can not be based upon anything more than upon a faith in the simplicity, i.e., intelligibility, of nature: there are no arbitrary constants of this kind; that is to say, nature is so constituted that it is possible logically to lay down such strongly determined laws that within these laws only rationally completely determined constants occur (not constants, therefore, whose numerical value could be changed without destroying the theory). - - -
The special theory of relativity owes its origin to Maxwell’s equations of the electromagnetic field. Inversely the latter can be grasped formally in satisfactory fashion only by way of the special theory of relativity. Maxwell’s equations are the simplest Lorentz-invariant field equations which can be postulated for an anti-symmetric tensor derived from a vector field. This in itself would be satisfactory, if we did not know from quantum phenomena that Maxwell’s theory does not do ju
stice to the energetic properties of radiation. But how Maxwell’s theory would have to be modified in a natural fashion, for this even the special theory of relativity offers no adequate foothold. Also to Mach’s question: “how does it come about that inertial systems are physically distinguished above all other co-ordinate systems?” this theory offers no answer.
That the special theory of relativity is only the first step of a necessary development became completely clear to me only in my efforts to represent gravitation in the framework of this theory. In classical mechanics, interpreted in terms of the field, the potential of gravitation appears as a scalar field (the simplest theoretical possibility of a field with a single component). Such a scalar theory of the gravitational field can easily be made invariant under the group of Lorentz-transformations. The following program appears natural, therefore: The total physical field consists of a scalar field (gravitation) and a vector field (electromagnetic field); later insights may eventually make necessary the introduction of still more complicated types of fields; but to begin with one did not need to bother about this.
The possibility of the realization of this program was, however, dubious from the very first, because the theory had to combine the following things:
(1) From the general considerations of special relativity theory it was clear that the inert mass of a physical system increases with the total energy (therefore, e.g., with the kinetic energy).
(2) From very accurate experiments (specially from the torsion balance experiments of Eötvös) it was empirically known with very high accuracy that the gravitational mass of a body is exactly equal to its inert mass.
It followed from (1) and (2) that the weight of a system depends in a precisely known manner on its total energy. If the theory did not accomplish this or could not do it naturally, it was to be rejected. The condition is most naturally expressed as follows: the acceleration of a system falling freely in a given gravitational field is independent of the nature of the falling system (specially therefore also of its energy content).
It then appeared that, in the framework of the program sketched, this elementary state of affairs could not at all or at any rate not in any natural fashion, be represented in a satisfactory way. This convinced me that, within the frame of the special theory of relativity, there is no room for a satisfactory theory of gravitation.
Now it came to me: The fact of the equality of inert and heavy mass, i.e., the fact of the independence of the gravitational acceleration of the nature of the falling substance, may be expressed as follows: In a gravitational field (of small spatial extension) things behave as they do in a space free of gravitation, if one introduces in it, in place of an “inertial system,” a reference system which is accelerated relative to an inertial system.
If then one conceives of the behavior of a body, in reference to the latter reference system, as caused by a “real” (not merely apparent) gravitational field, it is possible to regard this reference system as an “inertial system” with as much justification as the original reference system.
So, if one regards as possible, gravitational fields of arbitrary extension which are not initially restricted by spatial limitations, the concept of the “inertial system” becomes completely empty. The concept, “acceleration relative to space,” then loses every meaning and with it the principle of inertia together with the entire paradox of Mach.
The fact of the equality of inert and heavy mass thus leads quite naturally to the recognition that the basic demand of the special theory of relativity (invariance of the laws under Lorentz-transformations) is too narrow, i.e., that an invariance of the laws must be postulated also relative to non-linear transformations of the co-ordinates in the four-dimensional continuum.
This happened in 1908. Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that co-ordinates must have an immediate metrical meaning. The transformation took place in approximately the following fashion.
We start with an empty, field-free space, as it occurs—related to an inertial system—in the sense of the special theory of relativity, as the simplest of all imaginable physical situations. If we now think of a non-inertial system introduced by assuming that the new system is uniformly accelerated against the inertial system (in a three-dimensional description) in one direction (conveniently defined), then there exists with reference to this system a static parallel gravitational field. The reference system may thereby be chosen as rigid, of Euclidian type, in three-dimensional metric relations. But the time, in which the field appears as static, is not measured by equally constituted stationary clocks. From this special example one can already recognize that the immediate metric significance of the co-ordinates is lost if one admits non-linear transformations of co-ordinates at all. To do the latter is, however, obligatory if one wants to do justice to the equality of gravitational and inert mass by means of the basis of the theory, and if one wants to overcome Mach’s paradox as concerns the inertial systems.
If, then, one must give up the attempt to give the co-ordinates an immediate metric meaning (differences of co-ordinates = measurable lengths, viz., times), one will not be able to avoid treating as equivalent all co-ordinate systems, which can be created by the continuous transformations of the co-ordinates.
The general theory of relativity, accordingly, proceeds from the following principle: Natural laws are to be expressed by equations which are covariant under the group of continuous co-ordinate transformations. This group replaces the group of the Lorentz-transformations of the special theory of relativity, which forms a sub-group of the former.
This demand by itself is of course not sufficient to serve as point of departure for the derivation of the basic concepts of physics. In the first instance one may even contest [the idea] that the demand by itself contains a real restriction for the physical laws; for it will always be possible thus to reformulate a law, postulated at first only for certain coordinate systems, such that the new formulation becomes formally universally co-variant. Beyond this it is clear from the beginning that an infinitely large number of field-laws can be formulated which have this property of covariance. The eminent heuristic significance of the general principles of relativity lies in the fact that it leads us to the search for those systems of equations which are in their general covariant formulation the simplest ones possible; among these we shall have to look for the field equations of physical space. Fields which can be transformed into each other by such transformations describe the same real situation.
The major question for anyone doing research in this field is this: Of which mathematical type are the variables (functions of the coordinates) which permit the expression of the physical properties of space (“structure”)? Only after that: Which equations are satisfied by those variables?
The answer to these questions is today by no means certain. The path chosen by the first formulation of the general theory of relativity can be characterized as follows. Even though we do not know by what type field-variables (structure) physical space is to be characterized, we do know with certainty a special case: that of the “field-free” space in the special theory of relativity. Such a space is characterized by the fact that for a properly chosen co-ordinate system the expression
belonging to two neighboring points, represents a measurable quantity (square of distance), and thus has a real physical meaning. Referred to an arbitrary system this quantity is expressed as follows:
whereby the indices run from 1 to 4. The gik form a (real) symmetrical tensor. If, after carrying out a transformation on field (1), the first derivatives of the gik with respect to the co-ordinates do not vanish, there exists a gravitational field with reference to this system of co-ordinates in the sense of the above consideration, a gravitational field, moreover, of a very special type. Thanks to Riemann’s investigation of n-dimensional metrical spaces this special field can be inva
riantly characterized:
(1) Riemann’s curvature-tensor Riklm formed from the coefficients of the metric (2) vanishes.
(2) The orbit of a mass-point in reference to the inertial system (relative to which (1) is valid) is a straight line, therefore an extremal (geodetic). The latter, however, is already a characterization of the law of motion based on (2).
The universal law of physical space must now be a generalization of the law just characterized. I now assume that there are two steps of generalization:
(a) pure gravitational field
(b) general field (in which quantities corresponding some-how to the electromagnetic field occur, too).
The instance (a) was characterized by the fact that the field can still be represented by a Riemann-metric (a), i.e., by a symmetric tensor, whereby, however, there is no representation in the form (1) (except in infinitesimal regions). This means that in the case (a) the Riemanntensor does not vanish. It is clear, however, that in this case a field-law must be valid, which is a generalization (loosening) of this law. If this law also is to be of the second order of differentiation and linear in the second derivations, then only the equation, to be obtained by a single contraction
came under consideration as field-equation in the case of (a). It appears natural, moreover, to assume that also in the case of (a) the geodetic line is still to be taken as representing the law of motion of the material point.
It seemed hopeless to me at that time to venture the attempt of representing the total field (b) and to ascertain field-laws for it. I preferred, therefore, to set up a preliminary formal frame for the representation of the entire physical reality; this was necessary in order to be able to investigate, at least preliminarily, the usefulness of the basic idea of general relativity. This was done as follows.
A Stubbornly Persistent Illusion Page 35
