On the Connections between Thermodynamics and General Relativity

TL;DR

This thesis investigates the deep links between thermodynamics and general relativity, analyzing temperature gradients, equilibrium conditions, and black hole thermodynamics, with new insights into non-Killing flows and the trans-Planckian problem.

Contribution

It generalizes the concept of thermal equilibrium for non-Killing flows and explores conditions under which exact thermodynamic equilibrium can be achieved in curved spacetimes.

Findings

Exact thermodynamic equilibrium occurs only along Killing flows.

Non-Killing flows can exhibit some equilibrium-like properties under certain conditions.

A model shows the trans-Planckian problem can be avoided if Hawking radiation is emitted close to the horizon.

Abstract

In this thesis, the connections between thermodynamics and general relativity are explored. We review the concept of gravitationally induced temperature gradients in equilibrium states, first introduced by Richard Tolman. We explore these Tolman-like temperature gradients, understanding their physical origin and whether they can be generated by other forces or not. We then generalize this concept for fluids following generic four-velocities, which are not necessarily generated by Killing vectors, in general stationary space-times. We then move to a more fundamental question: can we still define thermal equilibrium for non-Killing flows? To answer this we review two of the main theories of relativistic non-perfect fluids: Classical Irreversible Thermodynamics and Extended Irreversible Thermodynamics. We also take a tour through the interesting concept of Born-rigid motion, showing some explicit examples of non-Killing rigid flows for Bianchi Type I space-times. These results are important since they show that the Herglotz--Noether theorem cannot be extended for general curved space-times. We then connect the Born-rigid concept with the results obtained by the relativistic fluid's equilibrium conditions and show that the exact thermodynamic equilibrium can only be achieved along a Killing flow. We do, however, introduce some interesting possibilities which are allowed for non-Killing flows. Following, we launch into black hole thermodynamics, specifically studying the trans-Planckian problem for Hawking radiation. We construct a kinematical model consisting of matching two Vaidya spacetimes along a thin shell and show that, as long as the Hawking radiation is emitted only a few Planck lengths (in proper distance) away from the horizon, the trans-Plackian problem can be avoided.