Thermodynamics of local causal horizons

We propose an expression for the entropy density associated with the local causal horizons in any diffeomorphism-invariant theory of gravity. If the black-hole entropy of the theory satisfies the physical process version of the first law of thermodynamics, then our proposed entropy satisfies the Clausius relation. Thus, our study shows that the thermodynamic nature of the spacetime horizons is not restricted to the black holes; it also applies to the local causal horizons in the neighborhood of any point in the spacetime.

Figure 1

LCH associated with the approximate boost Killing vector field ${\xi }^a$. $p_0$ is the bifurcation point where ${\xi }^a$ vanishes. $k^a$ is the affinely parametrized tangent to the null generators of the LCH with the affine parameter $\lambda$. $p$ is the final equilibrium point where the expansion $\theta$ and shear $\sigma$ of $k^a$ vanishes. The affine parameter vanishes at $p_0$ and takes a value ${\lambda }_f$ at $p$. On the generator connecting $p_0$ to $p$ we have ${\xi }^a=\lambda k^a$.