Contributions of Riemann invariants to the Entropy of Extremal Black Holes

TL;DR

This paper investigates how Riemann invariants influence the entropy of extremal black holes in four and five dimensions, extending the entropy function formalism to include higher derivative terms and deviations from the Bekenstein-Hawking law.

Contribution

It introduces a comprehensive analysis of scalar invariants from Riemann tensors in the entropy calculation of extremal black holes with higher derivatives.

Findings

Derived entropy expressions including higher derivative corrections.

Identified deviations from the Bekenstein-Hawking area law.

Analyzed the impact of Riemann invariants on black hole entropy.

Abstract

We use the entropy function formalism introduced by A. Sen to obtain the entropy of $AdS_{2}\times S^{d-2}$ extremal and static black holes in four and five dimensions, with higher derivative terms of a general type. Starting from a generalized Einstein--Maxwell action with nonzero cosmological constant, we examine all possible scalar invariants that can be formed from the complete set of Riemann invariants (up to order 10 in derivatives). The resulting entropies show the deviation from the well known Bekenstein--Hawking area law $S=A/4G$ for Einstein's gravity up to second order derivatives.