Horizon entropy and higher curvature equations of state

The Clausius relation between entropy change and heat flux has previously been used to derive Einstein’s field equations as an equation of state. In that derivation the entropy is proportional to the area of a local causal horizon, and the heat is the energy flux across the horizon, defined relative to an approximate boost Killing vector. We examine here whether a similar derivation can be given for extensions beyond Einstein gravity to include higher-derivative and higher-curvature terms. We review previous proposals which, in our opinion, are problematic or incomplete. Refining one of these, we assume that the horizon entropy depends on an approximate local Killing vector in a way that mimics the diffeomorphism Noether charge that yields the entropy of a stationary black hole. We show how this can be made to work if various restrictions are imposed on the nature of the horizon slices and the approximate Killing vector. Also, an integrability condition on the assumed horizon entropy density must hold. This can yield field equations of a Lagrangian constructed algebraically from the metric and Riemann tensor, but appears unlikely to allow for derivatives of curvature in the Lagrangian.

Figure 1

Causal spacetime diagram of the local causal horizon (LCH) and Killing field in the old (left) and new (right) setups. Each point in the diagram represents a patch of spacelike 2-surface. The boundary of the past of a patch of 2-surface through $p$ defines the LCH. The arrows indicate the flow lines of the local Killing field ${\xi }^a$, which is of third order on the bifurcation surface $\mathcal{B}$, and which vanishes at $p$ (left) or $p_0$ (right). The central horizon generator $\Gamma$ runs from $p_0$ to $p$. The “heat” is the boost energy flux across the horizon. In the new setup the Killing vector is timelike within the horizon and spacelike beyond, while the old setup is the opposite.

Figure 2

The local causal horizon is part of the boundary of the past of ${\Sigma }_p$, a spacelike $D-2$ dimensional surface. The vertical lines are horizon generators with affine parameter $V$ and tangent vector $k^a$, and $\Gamma$ is the central generator. ${\Sigma }_p$ is geodesic at $p$, so the expansion $\theta$ and shear vanish at $p$. The horizontal lines represent constant $V$ slices. The local Killing vector vanishes at the bifurcation point $p_0$. The entropy is compared on two slices, ${\Sigma }_0$ and $\Sigma$, which differ only in a compact region.

Figure 3

A local causal horizon patch enclosed by a special pair of slices. The entropy change associated with $s_{GR}^{ab}=-(8\hslash G{)}^{-1}{\nabla }^{[a}{\xi }^{b]}$ is equal at leading order to the area change over $4\hslash G$ if (i) the horizon expansion and shear vanish at $p$, and (ii) the patch between the slices ${\Sigma }_0$ and $\Sigma$ nearly coincides with the interval $[V_0,0]$ across the entire patch.