The Physical Process First Law for Bifurcate Killing Horizons

TL;DR

This paper generalizes the physical process first law of black hole mechanics to any bifurcate Killing horizon in higher dimensions, establishing conditions for its applicability and completing the analogy with black hole thermodynamics.

Contribution

It extends the physical process first law to arbitrary bifurcate Killing horizons in dimensions d ≥ 3, including Rindler horizons, and clarifies the conditions for its validity.

Findings

Derived a condition for quasi-stationary processes to satisfy the first law.

Extended the first law to Rindler horizons in higher dimensions.

Discussed the special case of d=2 horizons.

Abstract

The physical process version of the first law for black holes states that the passage of energy and angular momentum through the horizon results in a change in area $\frac{\kappa}{8 \pi} \Delta A = \Delta E - \Omega \Delta J$, so long as this passage is quasi-stationary. A similar physical process first law can be derived for any bifurcate Killing horizon in any spacetime dimension $d \ge 3$ using much the same argument. However, to make this law non-trivial, one must show that sufficiently quasi-stationary processes do in fact occur. In particular, one must show that processes exist for which the shear and expansion remain small, and in which no new generators are added to the horizon. Thorne, MacDonald, and Price considered related issues when an object falls across a d=4 black hole horizon. By generalizing their argument to arbitrary $d \ge 3$ and to any bifurcate Killing horizon, we derive a condition under which these effects are controlled and the first law applies. In particular, by providing a non-trivial first law for Rindler horizons, our work completes the parallel between the mechanics of such horizons and those of black holes for $d \ge 3$. We also comment on the situation for d=2.