Jacobson's thermodynamic approach to classical gravity applied to non-Riemannian geometries: remarks on the simplicity of Nature
Three decades ago, Ted Jacobson surprised us with a very appealing approach to classical gravity. According to him, the gravitational field equations are the consequence of the first law of thermodynamics applied to a Rindler observer. Jacobson's approach being formulated for Riemannian geometries, we have wondered what its consequences would be for non-Riemannian geometries. The results of our quest have been particularly appealing: we have found that the 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 considered the hypotheses employed in the formulation of the Lanczos-Lovelock theories of gravity. Together, the two approaches point towards the 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 (when the energy-momentum tensor is identified as its metric version). The same strategy cannot be followed in the full non-Riemannian case (and in the previous case when the energy-momentum tensor is identified as its canonical version) as the two approaches are mutually inconsistent.