Spacetime thermodynamics with contorsion

Abstract

We prove that a conserved effective energy-momentum tensor for Einstein-Cartan theory can be identified from the Noether identities of the matter Lagrangian, using the torsion field equations relating them. More precisely, a one-parameter family labelled by the Immirzi parameter. We use this result and the contorsion description to show that Jacobson's thermodynamical derivation of the Einstein equations follows as in the metric theory, namely from the equilibrium Clausius relation and the fact that a Killing horizon is metric-geodetic. Our derivation works for an arbitrary torsion field. In the course of our discussion we review the laws of black hole mechanics and their dependence on torsion.

A note on dual gravitational charges

“…The starting point for the cohomological methods is the field equations E (e) and E (ω) obtained varying the tetrad Lagrangian (2.1). Specializing the variation to a general gauge transformation, we have [10,17,26,34] δ (λ,ξ) e I ∧ E (e) I + δ (λ,ξ) ω IJ ∧ E (ω)…”

“…But the difference of the symplectic potentials by an exact 3-form remains, therefore this dual contribution is inequivalent to the one computed using tetrad variables. For the related topic of the contribution of torsion to the first law of black hole mechanics, see [18,26].…”

Raychaudhuri and optical equations for null geodesic congruences with torsion

“…The decomposition of the various projections are reported in (A. 14), in particular T mlm ≡T mlm , and we also notice that…”

“…The torsional geodesics coincide in this case with the metric ones up to a difference in inaffinity determined by the vector (with spin 1 and spin 0 components) trace-part of torsion. Accordingly, also the Raychaudhuri equation coincides with the metric one, a result that was used in [14], and so do the optical equations for the shear and twist (up to gauge choice on the space-like dyad used). The drift equation on the other hand always differs, because the notion of drift depends on the choice of transverse vector, and its evolution is non geodetic and feels even a completely antisymmetric torsion.…”

“…It would be also interesting to further elaborate on the geometric possibility of the alternative Lie dragging (58). Finally, even though torsion-full geodesics do not arise from the conservation law of test matter in Einstein-Cartan of Poincaré gauge theory of gravity [7], it could be interesting to explore what happens with the conserved energy momentum tensor used for instance in [14].…”