Notes on Spacetime Thermodynamics and the Observer-dependence of Entropy

TL;DR

This paper shows that entropy for a localized object in flat spacetime is not invariant across inertial and accelerated (Rindler) observers due to the Unruh effect. It first presents a general thermodynamic argument that the Rindler-observer entropy change for a small perturbation on a thermal bath satisfies $\Delta S = \Delta E / T$, and then provides explicit calculations for free bosonic and fermionic fields to illustrate the bound. For bosons, the results give $\Delta E = \omega/(1 - e^{-\omega/T})$ and $\Delta S = \Delta E / T$, independent of the number of internal states $n$; for fermions, $\Delta E = \omega/(1 + e^{-\omega/T})$ with $\Delta S = \Delta E / T$, also $n$-independent. These findings imply a fundamental observer-dependence of entropy and have implications for generalized second-law discussions and horizon thermodynamics, aligning with ideas from Wald and Sorkin, while noting that a full dynamical proof remains an open challenge.

Abstract

Due to the Unruh effect, accelerated and inertial observers differ in their description of a given quantum state. The implications of this effect are explored for the entropy assigned by such observers to localized objects that may cross the associated Rindler horizon. It is shown that the assigned entropies differ radically in the limit where the number of internal states $n$ becomes large. In particular, the entropy assigned by the accelerated observer is a bounded function of $n$. General arguments are given along with explicit calculations for free fields. The implications for discussions of the generalized second law and proposed entropy bounds are also discussed.