Revisiting Conserved Charges in Higher Curvature Gravitational Theories

TL;DR

This paper applies the solution phase space method to higher curvature gravitational theories, specifically $f(R)$ models with quadratic curvature terms, to compute conserved charges and verify the first law of thermodynamics for various black hole solutions.

Contribution

It extends the solution phase space method to $f(R)$ gravity with quadratic curvature terms, enabling unambiguous conserved charge calculations for diverse black hole solutions.

Findings

Conserved charges can be computed on arbitrary surfaces, not just horizons or asymptotics.

The first law of thermodynamics holds as a local identity in these theories.

Explicit calculations for warped AdS$_3$, charged BTZ, and Lifshitz black holes confirm the method's effectiveness.

Abstract

Restricting the covariant gravitational phase spaces to the manifold of parametrized families of solutions, the mass, angular momenta, entropies, and electric charges can be calculated by a single and simple method. In this method, which has been called "solution phase space method," conserved charges are unambiguous and regular. Moreover, assuming the generators of the charges to be exact symmetries, entropies and other conserved charges can be calculated on almost arbitrary surfaces, not necessarily horizons or asymptotics. Hence, the first law of thermodynamics would be a local identity relating the exact symmetries to which the mass, angular momentum, electric charge, and entropy are attributed. In this paper, we apply this powerful method to the $f(R)$ gravitational theories accompanied by the terms quadratic in the Riemann and Ricci tensors. Furthermore, conserved charges and the first law of thermodynamics for some of their black hole solutions are exemplified. The examples include warped AdS$_3$, charged static BTZ, and 3-dimensional $z=3$ Lifshitz black holes.