Comment on "A comment on metric vs metric-affine gravity"

TL;DR

This paper clarifies that adding a boundary term to the Einstein-Palatini action does not alter the field equations, reaffirming that Lovelock gravity uniquely ensures equivalence between metric and metric-affine formulations.

Contribution

It refutes a recent claim by showing that the proposed counterexample is invalid because the added term is a boundary divergence that does not affect the equations of motion.

Findings

The Pontryagin term is a total divergence and does not change field equations.

The claimed counterexample does not invalidate the Lovelock gravity uniqueness.

Boundary terms only affect boundary conditions, not the bulk dynamics.

Abstract

It has been recently claimed in [arXiv:2211.12327] that an action of the Einstein-Palatini form plus a torsionless Pontryagin term (multiplied by a constant) represents a counterexample to the conclusions of [arXiv:0705.1879], namely, that Lovelock gravity is the only case in which the metric and metric-affine formulations of gravity are equivalent. However, given that the Pontryagin term (multiplied by a constant) can be written as a total D-divergence, it is a textbook matter to realise that the addition of such (or any other) D-divergence only affects at the boundary, leaving invariant the field equations and its solutions, which are those of GR \`{a} la Palatini. We thus conclude that the example provided in [arXiv:2211.12327] is not a valid counterexample of [arXiv:0705.1879].