Variational principle and boundary terms in gravity \`a la Palatini

TL;DR

This paper investigates boundary terms in Palatini formalism of $f( ext{R})$ gravity, emphasizing their importance in black hole entropy calculations and showing their explicit form through variational principles.

Contribution

It explicitly derives the boundary terms in Palatini $f( ext{R})$ gravity, highlighting their role in black hole entropy, extending previous metric-based results.

Findings

Boundary terms in Palatini $f( ext{R})$ gravity are explicitly derived.

Surface terms are crucial for correct black hole entropy calculations.

The results align with metric-compatible gravity theories.

Abstract

A general $f(\mathcal{R})$ gravitational theory is considered within the Palatini formalism. By applying the variational principle and the usual conditions on the boundary, we show explicitly that a surface term remains such that as in their metric-compatible counterparts, an additional surface term has to be added in the gravitational action, which plays a fundamental role when calculating the entropy of the black hole.