Affine gravity, Palatini formalism and charges

Abstract

Affine gravity and the Palatini formalism contribute both to produce a simple and unique formula for calculating charges at spatial and null infinity for Lovelock type Lagrangians whose variational derivatives do not depend on second-order derivatives of the field components. The method is based on the covariant generalization due to Julia and Silva of the Regge-Teitelboim procedure that was used to define properly the mass in the classical formulation of Einstein's theory of gravity. Numerous applications reproduce standard results obtained by other secure but mostly specialized method like in ADM energy for asymptotically flat spacetimes and in Abbot and Deser for asymptotically de Sitter and anti-de Sitter spacetimes, both at spatial infinity. As a novel application we calculate the Bondi energy loss in five dimensional gravity, based on the asymptotic solution given by Tanabe et al. and obtain, as expected, the same result. We also give the superpotential for Einstein-Gauss-Bonnet gravity and find the superpotential for Lovelock theories of gravity when the number of dimensions tends to infinity with maximally symmetrical boundaries. The paper is written in standard component formalism.

Mass and free energy of Lovelock black holes

“…al. [48] and using the Palatini formalism by Katz and Livshits [49] The mass of asymptotically AdS black holes in general higher curvature theories has also been addressed in [50].…”

Breaking generalized covariance, classical renormalization, and boundary conditions from superpotentials

“…In this section we present the main principles behind and the equations for the variation of the superpotential for the metric-affine theories. A more involved and detailed derivation was given in previous works 14,[24][25][26]28 .…”

“…U ρσ 2 possesses several remarkable properties 14 . In particular, it vanishes identically in N = 4 dimensions, which is consistent with the metric formulation, in which caseL P 2 is a pure divergence in 4 dimensions.…”

“…The problem of non-uniqueness is partially solved simply by the fact that for any known geometry of isolated sources of gravity, whether at spatial or null infinity, whether in N = 4 or more dimensions, the KBL superpotential provides the correct mass, angular momentum or radiative energy loss. In the original paper 6 , it was shown to give the correct Bondi mass loss 13 in four dimensions, and more recently, in five dimensions as well 14 . It is the only superpotential which possesses all the following properties.…”

“…In our previous studies 14,28 we solved the equations in both formulations for the first and second-order Lovelock Lagrangians. In the former case, both solutions reproduced the KBL superpotential exactly.…”