The sign of the gravitational force

Abstract

The theoretical assumptions determining the sign of the static long-range (“Newtonian”) force between particles in general relativity, and in a number of other theories, are identified. It is first shown that in a Lorentz-covariant theory of a massless field, the sign of the Newtonian force is determined by the spin of the field and by the requirement, essential for quantization, that the field energy be positive. Without this requirement, the sign may be fixed arbitrarily, as is the case in direct action theories where no corresponding criterion exists. Despite unavoidable complications, which are discussed in some detail, the situation in general relativity theory is similar to that in Lorentzcovariant theories. However, the interaction between gravitational geons (self-maintaining purely gravitational excitations) is exceptional, in that its sign is completely determinate, without ancillary assumption. The geon-geon and geon-particle interactions are discussed at length; in the latter, the effect of the geon on the particle is also completely determinate, while the effect of the particle on the geon may be chosen arbitrarily. These results are related to the fact that a geon has no independently defined inertial mass, since its interactions are purely gravitational. The arguments which show that the gravitational interactions of matter and of anti-matter must be the same are also reviewed briefly. Some alternatives to Einstein theory, in which the positive energy requirement is inapplicable, and the sign of the Newtonian force is fixed by enlarging the invariance group of the theory, are discussed.