Spacetime thermodynamics in the presence of torsion

It was shown by Jacobson in 1995 that the Einstein equation can be derived as a local constitutive equation for an equilibrium spacetime thermodynamics. With the aim to understand if such thermodynamical description is an intrinsic property of gravitation, many attempts have been made so far to generalize this treatment to a broader class of gravitational theories. Here we consider the case of the Einstein-Cartan theory as a prototype of theories with nonpropagating torsion. In doing so, we study the properties of Killing horizons in the presence of torsion, establish the notion of local causal horizon in Riemann-Cartan spacetimes, and derive the generalized Raychaudhuri equation for these kinds of geometries. Then, starting with the entropy that can be associated to these local causal horizons, we derive the Einstein-Cartan equation by implementing the Clausius equation. We outline two ways of proceeding with the derivation depending on whether we take torsion as a geometric field or as a matter field. In both cases we need to add internal entropy production terms to the Clausius equation as the shear and twist cannot be taken to be 0 a priori for our setup. This fact implies the necessity of a nonequilibrium thermodynamics treatment for the local causal horizon. Furthermore, it implies that a nonzero twist at the horizon in general contributes to the Hartle-Hawking tidal heating for black holes with possible implications for future observations.