Asymptotically anti-de Sitter spacetimes and conserved quantities in higher curvature gravitational theories

TL;DR

This paper develops a new definition for asymptotically anti-de Sitter spacetimes in higher curvature gravity theories, derives conserved quantities, and demonstrates their properties and consistency with Einstein gravity results.

Contribution

It introduces an alternative definition for asymptotically anti-de Sitter spacetimes in higher curvature theories and constructs conserved quantities satisfying a balance law.

Findings

Conserved quantities are expressed as integrals of the electric part of the Weyl tensor.

These quantities vanish in pure anti-de Sitter spacetime.

The conserved quantities satisfy a balance equation similar to Einstein gravity.

Abstract

We consider $n$-dimensional asymptotically anti-de Sitter spacetimes in higher curvature gravitational theories with $n \geq 4$, by employing the conformal completion technique. We first argue that a condition on the Ricci tensor should be supplemented to define an asymptotically anti-de Sitter spacetime in higher curvature gravitational theories and propose an alternative definition of an asymptotically anti-de Sitter spacetime. Based on that definition, we then derive a conservation law of the gravitational field and construct conserved quantities in two classes of higher curvature gravitational theories. We also show that these conserved quantities satisfy a balance equation in the same sense as in Einstein gravity and that they reproduce the results derived elsewhere. These conserved quantities are shown to be expressed as an integral of the electric part of the Weyl tensor alone and hence they vanish identically in the pure anti-de Sitter spacetime as in the case of Einstein gravity.