The Field-Theoretic Approach in General Relativity and Other Metric Theories. A Review

Abstract: GR and other metric theories of gravity are formulated with an arbitrary auxiliary curved background in a Lagrangian formalism. A new sketch of how to include spinor fields is included. Conserved quantities are obtained using Noether's theorem and expressed as divergences of antisymmetric densities, connecting local perturbations with quasi-local conserved quantities. The background's arbitrariness matches the so-called non-localizability of gravitational energy (infinity of localizations). The formalism has two partly overlapping uses: practical applications of pure GR (with fictitious background) and foundational considerations in which background causality facilitates quantization. The Schwarzschild solution is a primary application. Various possibilities for calculating the mass using surface integration are given. A field-theoretic curved spacetime is given from spatial infinity to the horizon and even to the true singularity. Trajectories of test particles in the Schwarzschild geometry are gauge-dependent in that even breakdowns at the horizon can be suppressed (or generated) by naive gauge transformations. This fact illustrates the auxiliary nature of the background metric and the need for some notion of maximal extension---much as with coordinate transformations in geometric GR. A continuous collapse to a point mass in the field-theoretic framework is given. The field-theoretic method is generalized to arbitrary metric theories in $D$ dimensions. The results are developed in the framework of Lovelock gravity and applied to calculate masses of Schwarzschild-like black holes. The bimetric formalism makes it natural to consider a graviton mass. Babak and Grishchuk's numerical and hence nonperturbative work sheds light on questions of a (dis)continuous massless limit for massive pure spin-2 and the classical (in)stability of spin-2/spin-0 theory.