Equivalence of Noether charge and Hilbert action boundary term formulas for the black hole entropy in $F(R_{abcd})$ gravity theory

TL;DR

This paper proves the equivalence of Noether charge and Hilbert action boundary term formulas for black hole entropy in $F(R_{abcd})$ gravity using covariant phase space formalism, supported by explicit calculations.

Contribution

It demonstrates the equivalence of two different entropy formulas in $F(R_{abcd})$ gravity, providing a unified understanding of black hole entropy in this theory.

Findings

Established the equivalence of Noether charge and Hilbert boundary term formulas.

Expressed the Hamiltonian as a contraction with an $\xi$-independent tensor.

Validated the equivalence through explicit computations.

Abstract

By working with the covariant phase space formalism, we have shown that not only can the Hamiltonian conjugate to a Killing vector field $\xi$ be expressed as the sum of the associated Noether charge and $\xi$ contracted with the Hilbert action boundary term for $F(R_{abcd})$ gravity, but also be written as its contraction with another $\xi$ independent tensor field. With this, we have proven the equivalence of Noether charge and Hilbert action boundary term formulae for the stationary black hole entropy in $F(R_{abcd})$ gravity, which is further substantiated by our explicit computation using both formulae.