New perspective on thermodynamics of spacetime: The emergence of unimodular gravity and the equivalence of entropies

Abstract

We present a novel derivation of Einstein equations from the balance between Clausius entropy crossing the boundary of a local causal diamond and entanglement entropy associated with its horizon. Comparing this derivation with the entanglement equilibrium approach developed by Jacobson, we are able to argue for the equivalence of matter entanglement and Clausius entropy in the semiclassical regime. We also provide a direct comparison of both entropies for conformal matter, showing their equivalence without appealing to gravitational dynamics. Furthermore, we determine that gravitational dynamics implied by thermodynamics of spacetime, in fact, corresponds to unimodular gravity rather than general relativity.

Quantum phenomenological gravitational dynamics: a general view from thermodynamics of spacetime

“…The second derivation utilises as the starting point an expression for the Clausius entropy flux across any null bifurcate surface [15] applied to the specific case of causal diamonds. Equations of motion are then found by comparing the decrease of Clausius entropy of the matter fields with the corresponding change of entanglement entropy of the horizon [20]. As it was shown there, both derivations give rise to the same gravitational equations of motion in the semiclassical case.…”

“…Then, the introduction of quantum gravity effects could provide a preferred direction towards one or another theory of gravity. Likewise, the entropy equivalence reported in [20] might break due to quantum gravity corrections. These questions provide further motivation for extending thermodynamics of spacetime into the realm of phenomenological quantum gravity.…”

“…Here, we will focus on two recent derivations, extensively reviewed in a previous paper of the authors [20] (see this reference also for a more detailed historical review of thermodynamics of spacetime). The first one obtains the dynamics from thermodynamic equilibrium of geodesic local causal diamonds (GLCD), by performing a simultaneous variation of the entanglement entropy associated with the horizon of GLCD and the entanglement entropy of the matter present inside it [10].…”

“…The interest of considering both derivations lies, on one side, in checking the consistency of results, and, on the other side, in studying the correspondence among the different concepts of entropy. The most relevant results of the analysis of these derivations were the recovery of unimodular gravity (UG) instead of GR and the semiclassical equivalence of Clausius and entanglement entropy [20]. Note that UG implies the same classical dynamics as GR, but the quantisation of both theories might differ.…”

“…In section 3, we first introduce a geometric tool we will use in the derivation, that is, geodesic local causal diamonds. Then, we use both above described thermodynamic methods [20] to derive the quantum phenomenological gravitational equations of motion. Section 4 discusses diffeomorphism and local Lorentz invariance of the dynamics we obtained.…”