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

Abstract

In view of an equal qualification of Brans and Dicke's gravitational theory to Einstein's general theory of relativity on an experimental level, problem of Mach's principle, fun­ damental in former theory, is studied by use of retarded bi-~calar Green's func­ tion for their inertial field ifJ in a homogeneous and isotropic universe. After deriving con­ cretely 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 radiation universe in which trace of energy-momentum tensor is vanishing. Four dimensional Green's functions play an essential role in treatment of propagation of action, not only in Minkowski space-time, but also in a curved space-time. As regards a massless scalar field and. electromagnetic field, a formal procedure to derive 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, author and Kimura 2 ),*) derived concretely Green's functions in a homogeneous and isotropic universe .to deal with quantum nature of gravitons and photons in universe. On other hand, recent discovery3) of cosmic background radia­ tion of T= 3°K provides us with a renewal of inter~st in worl9. models with big-bang origin. Among various cosmological theories permitting big­ bang models, most interesting will be 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) two gravitational theories alone can reasonably explain so-called three crucial tests and give results which are in accord with astrophysical observations ex­ amined in 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 E6tv6s experiment gives an experimental support of the weak