Gravity from equilibrium thermodynamics of stretched light cones

TL;DR

This work shows that Einstein's equations can be recovered from equilibrium thermodynamics of stretched light cones (SLCs) formed by uniformly accelerating observers, with the Unruh temperature $T_U=\hbar a/(2\pi)$. It analyzes two complementary methods to determine the SLC's expansion, shear, and vorticity—direct velocity-based computation and Raychaudhuri's equation—and proves their equivalence while ensuring shear vanishes to leading order, enabling a fully equilibrium thermodynamic description. By combining Clausius' relation with an entropy-area law, and accounting for a work term due to observer acceleration, the authors derive $G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu}$, with a freedom in the timelike-surface entropy normalization that reduces to a known fixed value for null horizons. The approach unifies prior work by Jacobson, Chirco-Liberati, and Parikh-Svesko, clarifying the roles of energy flux, geometry, and observer-dependent entropy in gravitational dynamics, and raises questions about the interpretation of entropy and work in curved spacetime.

Abstract

This work digs into the connection between gravity and thermodynamics of stretched light cones (SLC). They are associated with uniformly accelerating observers, who endow the SLC with a physical notion of temperature via the Unruh effect. We compute the expansion, shear, and vorticity of the SLC to fully study its dynamics and account for the possibility of previously predicted non-equilibrium entropy production. For consistency, we prove the equivalence of the two different geometrical methods available for studying the SLCs' properties. Then, we apply the energy balance and use Clausius' relation to relate the geometrical properties of the SLC with energy fluxes crossing its surface, showing that it encodes the equations governing the gravitational dynamics. We show here how this analysis can be fully carried in terms of equilibrium thermodynamics due to the vanishing of shear, and how one can identify a work term related to the acceleration of the observer.

Figures (2)

• Figure 1: Graphical depiction of a SLC $\Sigma$, the velocity field $u^\mu$ defined on it, its acceleration $a^\mu$, and the heat flux crossing its surface $\delta Q$. The darker blue lines denote the null surface which the SLC approaches in the limit of infinite acceleration, i.e. the light cone.
• Figure 2: Flow diagram of the derivation of Einstein equations from the Clausius equilibrium relation. Blue boxes in the diagram correspond to matter contributions, yellow boxes to geometry contributions, and green boxes to their combination.