In Newton’s theory one can write the field-law of gravitation thus:
(φ = gravitation-potential) at points, where the density of matter, , vanishes. In general one may write (Poisson equation)
In the case of the relativistic theory of the gravitational field Rik takes the place of Δφ. On the right side we shall then have to place a tensor also in place of . Since we know from the special theory of relativity that the (inert) mass equals energy, we shall have to put on the right side the tensor of energy-density—more precisely the entire energy-density, insofar as it does not belong to the pure gravitational field. In this way one gets the field-equations
The second member on the left side is added because of formal reasons; for the left side is written in such a way that its divergence disappears identically in the sense of the absolute differential calculus. The right side is a formal condensation of all things whose comprehension in the sense of a field-theory is still problematic. Not for a moment, of course, did I doubt that this formulation was merely a makeshift in order to give the general principle of relativity a preliminary closed expression. For it was essentially not anything more than a theory of the gravitational field, which was somewhat artificially isolated from a total field of as yet unknown structure.
If anything in the theory as sketched—apart from the demand of the invariance of the equations under the group of the continuous coordinate-transformations—can possibly make the claim to final significance, then it is the theory of the limiting case of the pure gravitational field and its relation to the metric structure of space. For this reason, in what immediately follows we shall speak only of the equations of the pure gravitational field.
The peculiarity of these equations lies, on the one hand, in their complicated construction, especially their non-linear character, as regards the field-variables and their derivatives, and, on the other hand, in the almost compelling necessity with which the transformation-group determines this complicated field-law. If one had stopped with the special theory of relativity, i.e., with the invariance under the Lorentz-group, then the field-law Rik = 0 would remain invariant also within the frame of this narrower group. But, from the point of view of the narrower group there would at first exist no reason for representing gravitation by so complicated a structure as is represented by the symmetric tensor gik. If, nonetheless, one would find sufficient reasons for it, there would then arise an immense number of field-laws out of quantities gik, all of which are co-variant under Lorentz-transformations (not, however, under the general group). However, even if, of all the conceivable Lorentz-invariant laws, one had accidentally guessed precisely yet the law which belongs to the wider group, one would still not yet be on the plane of insight achieved by the general principle of relativity. For, from the standpoint of the Lorentz-group two solutions would incorrectly have to be viewed as physically different from each other, if they can be transformed into each other by a non-linear transformation of co-ordinates, i.e., if they are, from the point of view of the wider field, only different representations of the same field.
One more general remark concerning field-structure and the group. It is clear that in general one will judge a theory to be the more nearly perfect the simpler a “structure” it postulates and the broader the group is concerning which the field-equations are invariant. One sees now that these two demands get in each other’s way. For example: according to the special theory of relativity (Lorentz-Group) one can set up a covariant law for simplest structure imaginable (a scalar field), whereas in the general theory of relativity (wider group of the continuous transformations of co-ordinates) there is an invariant field-law only for the more complicated structure of the symmetric tensor. We have already given physical reasons for the fact that in physics invariance under the wider group has to be demanded:1 from a purely mathematical standpoint I can see no necessity for sacrificing the simpler structure to the generality of the group.
The group of the general relativity is the first one which demands that the simplest invariant law be no longer linear or homogeneous in the field-variables and in their differential quotients. This is of fundamental importance for the following reason. If the field-law is linear (and homogeneous), then the sum of two solutions is again a solution; as, for example: in Maxwell’s field-equations for the vacuum. In such a theory it is impossible to deduce from the field equations alone an interaction between bodies, which can be described separately by means of solutions of the system. For this reason all theories up to now required, in addition to the field equations, special equations for the motion of material bodies under the influence of the fields. In the relativistic theory of gravitation, it is true, the law of motion (geodetic line) was originally postulated independently in addition to the field-law equations. Afterwards, however, it became apparent that the law of motion need not (and must not) be assumed independently, but that it is already implicitly contained within the law of the gravitational field.
The essence of this genuinely complicated situation can be visualized as follows: A single material point at rest will be represented by a gravitational field which is everywhere finite and regular, except at the position where the material point is located: there the field has a singularity. If, however, one computes by means of the integration of the field-equations the field which belongs to two material points at rest, then this field has, in addition to the singularities at the positions of the material points, a line consisting of singular points, which connects the two points. However, it is possible to stipulate a motion of the material points in such a way that the gravitational field which is determined by them does not become singular anywhere at all except at the material points. These are precisely those motions which are described in first approximation by Newton’s laws. One may say, therefore: The masses move in such fashion that the solution of the field-equation is nowhere singular except in the mass points. This attribute of the gravitational equations is intimately connected with their non-linearity, and this is a consequence of the wider group of transformations.
Now it would of course be possible to object: If singularities are permitted at the positions of the material points, what justification is there for forbidding the occurrence of singularities in the rest of space? This objection would be justified if the equations of gravitation were to be considered as equations of the total field. [Since this is not the case], however, one will have to say that the field of a material particle may the less be viewed as a pure gravitational field the closer one comes to the position of the particle. If one had the field-equation of the total field, one would be compelled to demand that the particles themselves would everywhere be describable as singularity-free solutions of the completed field-equations. Only then would the general theory of relativity be a complete theory.
Before I enter upon the question of the completion of the general theory of relativity, I must take a stand with reference to the most successful physical theory of our period, viz., the statistical quantum theory which, about twenty-five years ago, took on a consistent logical form (Schrödinger, Heisenberg, Dirac, Born). This is the only theory at present which permits a unitary grasp of experiences concerning the quantum character of micro-mechanical events. This theory, on the one hand, and the theory of relativity on the other, are both considered correct in a certain sense, although their combination has resisted all efforts up to now. This is probably the reason why among contemporary theoretical physicists there exist entirely differing opinions concerning the question as to how the theoretical foundation of the physics of the future will appear. Will it be a field theory; will it be in essence a statistical theory? I shall briefly indicate my own thoughts on this point.
Physics is an attempt conceptually to grasp reality as it is thought independently of its being observed. In this sense one speaks of “physical reality.” In pre-quantum physics there was no doubt as to how this was to be understood. In Newton’s theory reality was determined by a material point in space and time; in
Maxwell’s theory, by the field in space and time. In quantum mechanics it is not so easily seen. If one asks: does a ψ-function of the quantum theory represent a real factual situation in the same sense in which this is the case of a material system of points or of an electromagnetic field, one hesitates to reply with a simple “yes” or “no”; why? What the ψ-function (at a definite time) asserts, is this: What is the probability for finding a definite physical magnitude q (or p) in a definitely given interval, if I measure it at time t? The probability is here to be viewed as an empirically determinable, and therefore certainly as a “real” quantity which I may determine if I create the same ψ-function very often and perform a q-measurement each time. But what about the single measured value of q? Did the respective individual system have this q-value even before the measurement? To this question there is no definite answer within the framework of the [existing] theory, since the measurement is a process which implies a finite disturbance of the system from the outside; it would therefore be thinkable that the system obtains a definite numerical value for q (or p), i.e., the measured numerical value, only through the measurement itself. For the further discussion I shall assume two physicists, A and B, who represent a different conception with reference to the real situation as described by the ψ-function.
A. The individual system (before the measurement) has a definite value of q (i.e., p) for all variables of the system, and more specifically, that value which is determined by a measurement of this variable. Proceeding from this conception, he will state: The ψ-function is no exhaustive description of the real situation of the system but an incomplete description; it expresses only what we know on the basis of former measurements concerning the system.
B. The individual system (before the measurement) has no definite value of q (i.e., p). The value of the measurement only arises in cooperation with the unique probability which is given to it in view of the ψ-function only through the act of measurement itself. Proceeding from this conception, he will (or, at least, he may) state: the ψ-function is an exhaustive description of the real situation of the system.
We now present to these two physicists the following instance: There is to be a system which at the time t of our observation consists of two partial systems S1 and S2, which at this time are spatially separated and (in the sense of the classical physics) are without significant reciprocity. The total system is to be completely described through a known ψ-function ψ12 in the sense of quantum mechanics. All quantum theoreticians now agree upon the following: If I make a complete measurement of S1, I get from the results of the measurement and from ψ12 an entirely definite ψ-function ψ2 of the system S2. The character of ψ2 then depends upon what: kind of measurement I undertake on S1.
Now it appears to me that one may speak of the real factual situation of the partial system S2. Of this real factual situation, we know to begin with, before the measurement of S1, even less than we know of a system described by the ψ-function. But on one supposition we should, in my opinion, absolutely hold fast: the real factual situation of the system S2 is independent of what is done with the system S1, which is spatially separated from the former. According to the type of measurement which I make of S1, I get, however, a very different ψ2 for the second partial system Now, however, the real situation of S2 must be independent of what happens to S2. For the same real situation of S2 it is possible therefore to find, according to one’s choice, different types of ψ-function. (One can escape from this conclusion only by either assuming that the measurement of S2 ((telepathically)) changes the real situation of S2 or by denying independent real situations as such to things which are spatially separated from each other. Both alternatives appear to me entirely unacceptable.)
If now the physicists, A and B, accept this consideration as valid, then B will have to give up his position that the ψ-function constitutes a complete description of a real factual situation. For in this case it would be impossible that two different types of ψ-functions could be co-ordinated with the identical factual situation of S2.
The statistical character of the present theory would then have to be a necessary consequence of the incompleteness of the description of the systems in quantum mechanics, and there would no longer exist any ground for the supposition that a future basis of physics must be based upon statistics. - - -
It is my opinion that the contemporary quantum theory by means of certain definitely laid down basic concepts, which on the whole have been taken over from classical mechanics, constitutes an optimum formulation of the connections. I believe, however, that this theory offers no useful point of departure for future development. This is the point at which my expectation departs most widely from that of contemporary physicists. They are convinced that it is impossible to account for the essential aspects of quantum phenomena (apparently discontinuous and temporally not determined changes of the situation of a system, and at the same time corpuscular and undulatory qualities of the elementary bodies of energy) by means of a theory which describes the real state of things [objects] by continuous functions of space for which differential equations are valid. They are also of the opinion that in this way one can not understand the atomic structure of matter and of radiation. They rather expect that systems of differential equations, which could come under consideration for such a theory, in any case would have no solutions which would be regular (free from singularity) everywhere in four-dimensional space. Above everything else, however, they believe that the apparently discontinuous character of elementary events can be described only by means of an essentially statistical theory, in which the discontinuous changes of the systems are taken into account by way of the continuous changes of the probabilities of the possible states.
All of these remarks seem to me to be quite impressive. However, the question which is really determinative appears to me to be as follows: What can be attempted with some hope of success in view of the present situation of physical theory? At this point it is the experiences with the theory of gravitation which determine my expectations. These equations give, from my point of view, more warrant for the expectation to assert something precise than all other equations of physics. One may, for example, call on Maxwell’s equations of empty space by way of comparison. These are formulations which coincide with the experiences of infinitely weak electro-magnetic fields. This empirical origin already determines their linear form; it has, however, already been emphasized above that the true laws can not be linear. Such linear laws fulfill the super-position-principle for their solutions, but contain no assertions concerning the interaction of elementary bodies. The true laws can not be linear nor can they be derived from such. I have learned something else from the theory of gravitation: No ever so inclusive collection of empirical facts can ever lead to the setting up of such complicated equations. A theory can be tested by experience, but there is no way from experience to the setting up of a theory. Equations of such complexity as are the equations of the gravitational field can be found only through the discovery of a logically simple mathematical condition which determines the equations completely or [at least] almost completely. Once one has those sufficiently strong formal conditions, one requires only little knowledge of facts for the setting up of a theory; in the case of the equations of gravitation it is the four-dimensionality and the symmetric tensor as expression for the structure of space which, together with the invariance concerning the continuous transformation-group, determine the equations almost completely.
Our problem is that of finding the field equations for the total field. The desired structure must be a generalization of the symmetric tensor. The group must not be any narrower than that of the continuous transformations of co-ordinates. If one introduces a richer structure, then the group will no longer determine the equations as strongly as in the case of the symmetrical tensor as structure. Therefore it would be most beautiful, if one were to succeed in expanding the group once more, analogous to the step which led from special relativity to general re
lativity. More specifically I have attempted to draw upon the group of the complex transformations of the co-ordinates. All such endeavors were unsuccessful. I also gave up an open or concealed raising of the number of dimensions of space, an endeavor which was originally undertaken by Kaluza and which, with its projective variant, even today has its adherents. We shall limit ourselves to the four-dimensional space and to the group of the continuous real transformations of co-ordinates. After many years of fruitless searching I consider the solution sketched in what follows as the logically most satisfactory.
In place of the symmetrical gik (gik = gki), the non-symmetrical tensor gik is introduced. This magnitude is constituted by a symmetric part sik and by a real or purely imaginary anti-symmetric aik, thus:
Viewed from the standpoint of the group the combination of s and a is arbitrary, because the tensors s and a individually have tensor-character. It turns out, however, that these gik (viewed as a whole) play a quite analogous rôle in the construction of the new theory as the symmetric gik in the theory of the pure gravitational field.
This generalization of the space structure seems natural also from the standpoint of our physical knowledge, because we know that the electro-magnetic field has to do with an anti-symmetric tensor.
For the theory of gravitation it is furthermore essential that from the symmetric gik it is possible to form the scalar density as well as the contravariant tensor gik according to the definition
These concepts can be defined in precisely corresponding manner for the non-symmetric gik, also for tensor-densities.
A Stubbornly Persistent Illusion Page 36