An Attempt to Derive the Gravitational Potential

Abstract

A model theory is proposed to introduce both the mass of the matter field and the repulsive gravitational potential between matters within the framework of Minkowski's space-time. In consideration of the expansion of universe, the conformal invariant equations for two coup led scalar fields are taken as starting equations. By two procedures breaking conformal invariancy, it is shown that one field is the negative energy field and the self­ consistent mass equations, while they allow zero for the negative energy field, allow a non­ zero value for the mass of the other field. The negative energy field, in the energy density, is replaced by the repulsive adiabatic potential with a retardation effect. The transformed energy density docs not satisfy the usual. equal-time commutator of energy density which is valid for the ordinary system of fields with spin;:--::;:1. The additive potential is introduced so that the total energy density satisfies the usual energy density commutator. The total Hamiltonian is shown to be nothing but the one which gives the post Newtonian equation of Einstein, Infelcl and Hoffmann for the two-body problem in the general relativity. § L Introduction In spite of many approaches to the quantization of the gravitational field in the theory of the general relativity, there are neither necessary reasons for the quantization of the general relativity nor experimental evidence for the ex­ istence of the gravitational wave. On the other hand, since the general rela­ tivity has little to do with the ordinary theory of fields in Minkowski's space­ time, there are attempts to yield the same result as that of the general relativi­ ty standing on the usual theory of fields in Minkowski's space-time. As an example, there has been the approach of Gupta.l) l--Ie introduced a graviton with spin 2 and mass 0 and obtained the Lagrangian density for the graviton coupled with the matter field by the successive approximation method so as to bring about the energy-momentum density of the system on the right-hand side of the equation for the graviton as the source term. However, unless the experi­ mental evidence of the graviton exists, it may be no more than the expansion of the general relativity on Minkowski's space-time, though the equivalence prin­ ciple may be derived from the existence of the spin 2 particle with mass 0 and the Lorentz invariancy of S-matrix as shown by Weinberg. 2 l