Einstein’s equations from the stretched future light cone

Abstract

We define the stretched future light cone, a timelike hypersurface composed of the worldlines of radially accelerating observers with constant and uniform proper acceleration. By attributing temperature and entropy to this hypersurface, we derive Einstein's equations from the Clausius theorem. Moreover, we show that the gravitational equations of motion for a broad class of diffeomorphism-invariant theories of gravity can be obtained from thermodynamics on the stretched future light cone, provided the BekensteinHawking entropy is replaced by the Wald entropy.

From entanglement to thermodynamics and to gravity

“…Recently it was shown how to generalize (2) to higher derivative theories of gravity [5]. By attributing a temperature and entropy to a stretched future light cone-a timelike hypersurface composed of the worldlines of constant and uniformly radially accelerating observers-the equations of motion for a broad class of higher derivative theories of gravity are a consequence of the Clausius relation TΔS rev ¼ Q, where ΔS rev is the reversible entropy, i.e., the entropy growth solely due to a flux of matter crossing the horizon of the stretched light cone.…”

“…For holographic CFTs the first law of entanglement entropy (5) can be understood as a geometric constraint on the dual gravity side. By substituting (4) into the left-hand side (lhs) of (5), and relating the energy-momentum tensor of the CFT to a metric perturbation in anti-de Sitter (AdS) space, one arrives at the linearized Einstein equations [21]…”

“…By considering the higher derivative gravity generalization of (4), similar arguments lead to the linearized equations of motion for higher derivative theories of gravity [22]. The nonlinear behavior of gravitational equations of motion is encoded in a generalized form of (5), where one must take into account the relative entropy of excited CFT states [23,24]. In this way, gravity emerges from spacetime entanglement.…”

“…Here we aim to extend the work of [5,28] and develop the connection between thermodynamics, entanglement, and gravity. Specifically, we will consider the geometric setup of [28] and provide a "physical process" derivation of the geometric identity known as the first law of causal diamond mechanics (FLCD), crucial in deriving the entanglement equilibrium condition.…”

“…We accomplish this as follows: First attribute thermodynamics to sections of causal diamonds in an arbitrary spacetime, and compute the Clausius relation TΔS rev ¼ Q, where ΔS rev is the reversible (gravitational) entropy change computed via a reversible thermodynamic process. Using the techniques developed in [5], we will then show that the Clausius relation is geometrically equivalent to the nonlinear gravitational equations of motion for a broad class of diffeomorphism invariant theories of gravity, thereby connecting gravity to thermodynamics. Next, we show how the FLCD relates to the Clausius condition, by explicitly showing that the leading contribution to generalized volumeW is precisely the entropy change due to the natural increase of the causal diamond, presenting an equivalence between entanglement equilibrium and (reversible) equilibrium thermodynamics in theories of gravity.…”