2019, no. 4

### THE FIELD-THEORETIC APPROACH IN GENERAL RELATIVITY AND OTHER METRIC THEORIES. A REVIEW

#### Petrov A.N., Pitts J.B.

The representation of General Relativity (GR) and other metric theories of gravity in field-theoretic form on a background is reviewed. The gravitational field potential (metric perturbation) and other physical fields are propagated in an auxiliary background spacetime, which may be curved and may lack symmetries. Such a reformulation of a metric theory is exact and generally equivalent to its initial formulation in the standard geometrical form. The formalism is Lagrangian-based, in that the equations for the propagating fields are obtained by varying the related Lagrangian, as are the background field equations. A new sketch of how to include spinor fields is included.
Conserved quantities are obtained by applying the Noether theorem to the Lagrangian as well. Conserved currents are expressed through divergences of antisymmetric tensor densities (superpotentials), connecting local perturbations with quasi-local conserved quantities. The gauge dependence due to the background metric is studied, reflecting the so-called non-localizability of gravitational energy in exact mathematical expressions formally, an infinity of localized energy distributions that, combined with the material energy, satisfy the continuity equation. The exact expressions can be related to pure GR pseudotensors (especially Papapetrou’s) employing the matrix (-1, 1, 1, 1), as Nester et al. consider on independent grounds.
The field-theoretic formalism admits two partially overlapping uses. The first one is practical applications of pure GR, where the background presents merely a useful fiction. The second one is foundational considerations in which a background notion of causality, $$\eta$$-causality, is useful for making sense of equal-time or space-like commutation relations, in which case the background metric via inequalities has qualitative but not strict quantitative physical meaning.
The Schwarzschild solution is the main object for demonstration of the power of the method. Various possibilities for calculating the mass of the Schwarzschild black hole using surface integration of superpotentials are given. Presenting the Schwarzschild solution as a field configuration on a Minkowski background, we describe a curved spacetime from spatial infinity to the horizon and even to the true singularity, which is represented in consistently as a point particle using the Dirac $$\delta$$-function. 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-like state modelled by the Dirac $$\delta$$-function in the framework of the field-theoretic method is presented.
The field-theoretic method is generalized to arbitrary metric theories in arbitrary D dimensions. The results are developed in the framework of Lovelock gravity and applied to calculate masses of Schwarzschild-like black holes. Future applications are discussed. The formalism also makes it natural to consider adding a graviton mass. The works of Babak and Grishchuk, which are partly numerical and hence nonperturbative, are reviewed, shedding light on the traditional questions of a (dis)continuous massless limit for massive pure spin-2 and the (in)stability of a classical theory including massive spin-2 and spin-0 gravitons.

Keywords: conservation laws, general relativity, modified metric theories.

UDC: 530.12 + 531.51

PACS: 04.20-q 04.20.Cv 04.20.Fy 04.20.Jb 04.50.Kd

DOI: 10.17238/issn2226-8812.2019.4.66-124