Three types of superpotentials for perturbations in the Einstein-Gauss-Bonnet gravity

Abstract

Superpotentials (antisymmetric tensor densities) in Einstein-Gauss-Bonnet (EGB) gravity for arbitrary types of perturbations on arbitrary curved backgrounds are constructed. As a basis, the generalized conservation laws in the framework of an arbitrary D-dimensional metric theory, where conserved currents are expressed through divergences of superpotentials, are used. Such a derivation is exact (perturbations are not infinitesimal) and is approached, when a one solution (dynamical) is considered as a perturbed system with respect to another solution (background). Three known prescriptions are elaborated: these are the canonical N{\oe}ther theorem, the Belinfante symmetrization rule and the field-theoretical derivation. All the three approaches are presented in an unique way convenient for comparisons and a development. Exact expressions for the 01-component of the three types of the superpotentials are derived in the case, when an arbitrary static Schwarzschild-like solution in the EGB gravity is considered as a perturbed system with respect to a background of the same type. These formulae are used for calculating the mass of the Schwarzschild-anti-de Sitter black hole in the EGB gravity. As a background both the anti-de Sitter spacetime in arbitrary dimensions and a not maximally symmetric "mass gap" vacuum in 5 dimensions are considered. Problems and perspectives for a future development, including the Lovelock gravity, are discussed.