The Field Theoretical Formulation of General Relativity and Gravity with Non-zero Masses of Gravitons

It is a review paper related to the following topics. General relativity (GR) is presented in the field theoretical form, where gravitational field (metric perturbations) together with other physical fields are propagated in an auxiliary either curved, or flat background spacetime. A such reformulation of GR is exact (without approximations), is equivalent to GR in the standard geometrical description, is actively used for study of theoretical problems, and is directed to applications in cosmology and relativistic astrophysics. On the basis of a symmetrical (with respect to a background metric) energy-momentum tensor for all the fields, including gravitational one, conserved currents are constructed. Then they are expressed through divergences of antisymmetrical tensor densities (superpotentials). This form permits to connect a necessity to consider local properties of perturbations, which are analyzed in application tasks, with the academic imagination on the quasi-local nature of the conserved quantities in GR. The gauge invariance is studied, and its properties allow to consider the problem of non-localization in exact mathematical expressions. The M/string considerations point out to possible modification of GR, for example, by adding “massive terms” including masses of spin-2 and spin-0 gravitons. A such original modification on the basis of the field formulation of GR is given by Babak and Grishchuk, and we present and discuss it here. They have shown that all the local weak-field predictions of the massive theory are in agreement with experimental data. Otherwise, the exact non-linear equations of the new theory eliminate the black hole event horizons and replace a permanent power-law expansion of the homogeneous isotropic universe with an oscillator behaviour. One variant of the massive theory allows “an accelerated expansion” of the universe.