Mechanics of apparent horizons

An equation for the variation in the surface area of an apparent horizon is derived which has the same form as the thermodynamic relation $TdS=dQ$. For a stationary vacuum black hole, the expression corresponding to a temperature equals the temperature of the event horizon. Also, if the black hole is perturbed infinitesimally by weak matter and gravitational fields, the area variation of the apparent horizon asymptotically approaches the Hartle-Hawking result for the event horizon. These results support the idea that a local version of black-hole thermodynamics in nonstationary systems can be constructed for apparent horizons.