On the Green's Function in an Expanding Universe and Its Role in the Problem of Mach's Principle

In view of an equal qualification of Brans and Dicke's gravitational theory to Einstein's general theory of relativity on an experimental level, the problem of Mach's principle, fun damental in the former theory, is studied by the use of the retarded bi-~calar Green's func tion for their inertial field ifJ in a homogeneous and isotropic universe. After deriving con cretely the relevant Green's function, we show that Brans and Dicke's cosmology is compatible with their stipulation of Mach's principle, with some proviso about the radiation universe in which the trace of the energy-momentum tensor is vanishing. Four dimensional Green's functions play an essential role in the treatment of the propagation of action, not only in the Minkowski space-time, but also in a curved space-time. As regards a massless scalar field and. the electromagnetic field, a formal procedure to derive the Green's functions of these fields in any Riemannian space-time has been given by De Witt and Brehme/) but it is in general difficult to find their exact expressions by means of that procedure. Several years ago, however, the author and Kimura 2 ),*) derived concretely the Green's functions in a homogeneous and isotropic universe .to deal with the quantum nature of gravitons and photons in the universe. On the other hand, the recent discovery3) of the cosmic background radia tion of T= 3°K provides us with a renewal of inter~st in the worl9. models with the big-bang origin. Among various cosmological theories permitting the big bang models, the most interesting will be the a-/1-r-theory4) based on general relativity and Brans and Dicke's theory5) based on their gravitational theory with variable gravitation "constant". The reason is that, according to' Dicke,6) the two gravitational theories alone can reasonably explain the so-called three crucial tests and give results which are in accord with astrophysical observations ex amined in the light of modern stellar evolutionary theory. The underlying philosophy of Brans and Dicke's gravitational theory IS, however, considerably different from that of Einstein's, because they have con sidered that the E6tv6s experiment gives an experimental support of the" weak