Comment on “A comment on metric vs metric-affine gravity”

Abstract

It has been recently claimed in [1] that an action of the Einstein-Palatini form plus a torsionless Pontryagin term (multiplied by a constant) represents a counterexample to the conclusions of [2], 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 à la Palatini. We thus conclude that the example provided in [1] is not a valid counterexample of [2].