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LECTURES on GRAVITATION and RELATIVITY Goddard Institute For LECTURES ON GRAVITATION AND RELATIVITY Goddard Institute for Space Studies New York, New York Volume II w THE SIGNIFICANCE FOR THE SOLAR SYSTEM OF TIME VARYING GRAVITATION N88- 14305 Lecture VIII R. H. Dicke Princeton University Seminar on Gravitation and Relativity, NASA Goddard Space \Ylight Center, Institute for Space Studies, New York, N.Y.; \edited by H. Y. Chiu and W. F. Hoffmann. I ! l Preface These notes were taken from a series of seminars on Gravitation and Relativity presented at the Institute for Space Studies, NASA Goddard Space Flight Center during the academic year 1961-1962. Professor R. H. Dicke of Princeton University organized the series as an introduction to the subject for non-experts, emphasizing the observable implications of the theory and the potential contribution the space sciences may make towards a better understanding of general relativity. The approach has been conceptual rather than formal. For this reason, this record does not include a complete mathematical development of the subject, but, we hope, does contain sufficient mathematics to elaborate on the conceptual discussions. The notes were prepared with a minimum amount of editing from a transcript made from recordings of the lectures. The speakers have not had the opportunity to read and correct the final manuscript. Hence, we accept responsi- bility for errors and omissions. H. Y. Chiu W. F. Hoffmann I In my last lecture, I discussed a number of general features of gravitation theory. Two of these features furnish the starting point for this lecture. i) The requirements that a theory of gravitation be expressed in generally covariant equations and that inertial and gravitational forces both be obtained from a single invariant lead us naturally to represent gravitational effects by a tensor field. Einstein's theory is a particular example of a tensor theory in which the tensor field is the only field exhibiting gravitational effects and the geometry is so defined that this tensor is the metric tensor of the geometry. 2) Einstein's general theory of relativity is not rela- tivistic in the Machian sense. That is, this theory is not limited to a description of the relations between positions of matter. Rather, properties such as fixed directidns are ascribed to empty space in the complete absence of matte_ and motion is referred to a preferred, or absolute, geometry. In this lecture, I will discuss how Einstein's theory can be modified to overcome in part its absolute space-time character by introducing a second field quantity into the equations. Then} I will go on to relate this modification to some of the problems connected with geophysics and astrophysics. I anticipated some of the characteristics of this modi- fication in the last lecture with the discussion of Sciama's model. Sciama and others have Pointed out that it is possible to incorporate Mach's principle more completely into general relativity by introducing a gravitational constant which is not strictly a constant but is a function of some field variable. This possibility has also been raised in connection with a time varying gravitational constant. The suggestion of a time varying gravitational constant may have first appeared in physics in connection with Milne's ideas of cosmology. Later Dirac (I) suggested that certain numerical coincidences of large cosmologi- cal numbers might imply time-varying gravitation, that is, a gravitational constant which is not truly a constant, but a function of time. Later Jordan (2) attempted to put Dirac's ideas into a proper field-theoretic form by introducing a gravi- tational "constant," a variable "constant" as function of a scalar. I shall approach this matter in a slightly different way, by pointing out explicitly that in addition to the gravitational field which is connected with the geometry of space, one can (1) P. A. M. Dirac, Proceedings Royal Society (London) A 165, 199 (1938). (2) P. Jordan, Schwerkraft und Weltall (Vieweg und Sohn, Braunschweig, 1955). --2-- have long-range matter-type fields. We are already familiar with one of these, the electromagnetic field. Two charged particles can interact with each other over a long distance through electro- magnetic fields. In the previous lecture I showed that this kind of vector field appears to be ruled out for producing long range effects of cosmological importance. Large sections of the universe cannot interact with each other through vector interactions obeying Maxwell-type (gauge invariant) equations in a uniformly isotropic universe. Within the framework of relativity there are two other fields that might play important roles in cosmology. One of these is a scalar long-range field, and the other is a tensor long-range field, that is, a second tensor field, in addition to the metric tensor that is associated with gravitation in Einstein's theory. I shall take these in inverse order, discussing the tensor field first. It is possible to have a tensor interaction in addition to the metric tensor of space. But it would be very difficult to incorporate into the theory another such long-range tensor field. A second tensor field would be expected to lead to some queer results, results that would show up in experiments of the kind that Hughes and his students performed. -3- Professor Hughes, and independently Drever, was able to demonstrate the isotropy of space to a very high precision. This experiment is discussed in detail in Lecture 6 • The reason the tensor field is expected to run into difficulty with the Hughes experiment was first discussed quantitatively by J. (2a) Peebles. It is this: Assume there is some second tensor field. We may choose a coordinate system for which the metric tensor is locally Minkowskian. Then, generally speaking, the second tensor will not be locally Minkowskian but willl have some form for which the spatial parts of this tensor exhibit an anisotropy. If there are forces associated with this second tensor field, it would be expected that this spatial anisotropy would appear in the results of the Hughes experiment. If we should happen to find isotropy simultaneously for both tensors in a particular coordinate system, then we could simply transform to a moving coordinate system. If the tensors are not identical, we can always, by moving, obtain a lack of spatial isotropy in one of the tensors. I believe that, because of the tremendous precision and sensitivity of the experiment, this is a compelling argument against the second tensor interaction. I see no very obvious way of getting a second long-range tensor (2a) J. Peebles & R. H. Dicke, "The Significance of Spatial Isotropy" (to be published). --4-- field into physics. The case for the scalar field is more promising. I shall summarize its properties which were mentioned in my last lecture. It is remarkable that for the little we know about this inter- action in an observational way (in fact,even its existence is in doubt) we can delineate its properties so well. This is because the interaction is so simple that a couple of observa- tions and the requirement of Lorentz invariance are sufficient to specify fairly completely its properties. The properties of a long-range scalar interaction (a neutral, scalar, massless field) are the following: i) The scalar field can only be weak. Its strength must be of the order of the gravitational interaction. 2) The interaction of a scalar field with a particle cannot occur unless the mass of the particle is a function of the scalar. This means that work must be done on the particle to move it in a non-constant scalar field. This implies an extra force which acts on matter by virtue of its interaction with the scalar. We express this in the form of an equation by J ' ,i = (i) -5- where , (2) and (3) The great accuracy of the E_tv_s experiment imposes the requirement that if there is such a scalar interaction, all particles must suffer essentially the same type of scalar interaction. Otherwise, there would be anomalous accelerations. In other words, if the mass of the proton plus electron varied with the scalar with some different functional dependence than the mass of the neutron, then, generally speaking, a neutron would fall with a different acceleration from that of an ordinary hydrogen atom. So we face the requirement that the functional dependence of the mass on the scalar field variable should be equal to some constant times some standard function which is the same for all particles. -6- The assumption that the mass of a particle is a function of a scalar leads to some rather strange effects. In particu- lar, the gravitational coupling "constant," then, is not really constant. The gravitational constant can be given as a dimensionless number, a coupling constant, in terms of atomic constants as G .z - _0 P _ I0 (5) {c where _p is the mass of the proton. But if the mass of the proton varies from one place to the other, this ratio also will vary so that the gravitational interaction expressed in atomic units is not constant. Normally the dimensionless coupling constants of physics, such as oL = e -_ _ (6) and _ C_ (7) -7- / l / / are regarded as fai___t accompli of nature, numbers preordained and unrelated to other physical dimensionless numbers. However, if the gravitational coupling "constant" is variable, and is determined as a function of some scalar field variable, in turn determined by the matter distribution in the universe, it becomes possible to understand the extraordinary value of this number. In my opinion, the number 10 -40, contrary to what Eddington thought, would not be expected to appear in a formalism as a pure number of simple mathematical origin.
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