Jacobson's thermodynamic approach to classical gravity applied to non-Riemmanian geometries: remarks on the simplicity of Nature

Abstract

Three decades ago, Ted Jacobson surprised us with a very appealing approach to classical gravity. According to his proposal, the gravitational field equations are the consequence of the first law of thermodynamics applied, locally, to a Rindler observer. Together with the dynamical laws of black holes, Jacobson's approach has become a very strong piece of evidence supporting the intimate connection between gravity, thermodynamics, and quantum theory. Jacobson's approach being formulated for Riemannian geometries, we have wondered what its consequences would be for non-Riemannian geometries, i.e., those that involve torsion, non metricity, or both. The results of our quest have been particularly appealing: we have found that the gravitational theory that derives from the Einstein-Hilbert action, arguably ``the simplest one'', does not belong to the pool of gravitational theories available for Nature's selection (except in the Riemannian case). In the search of a unique alternative, we have also considered the hypotheses employed in the formulation of the Lanczos-Lovelock theories of gravity. Together, the two approaches point towards the gravitational theory that derives from the Einstein-Hilbert action plus a term quadratic in the torsion vector as the one that would be selected by Nature in the non-Riemannian case without non metricity. The same strategy cannot be followed in the full non-Riemannian case as the two approaches are mutually inconsistent.