The Physical Process First Law for Bifurcate Killing Horizons

TL;DR

This work extends the physical process version of the first law from black holes to any bifurcate Killing horizon in spacetime dimensions $d \ge 3$, using a quasi-stationary perturbation framework based on the Raychaudhuri equation. It derives the horizon-area change formula $\frac{\kappa \Delta A}{8 \pi} = \Delta E_\chi$ (and $\frac{\kappa \Delta A}{8 \pi} = \Delta E - \Omega \Delta J$ with angular momentum) from linearized null focusing, and identifies a precise caustic-formation threshold $r \lesssim \left( \frac{4 \pi \sqrt{d-3}}{\Omega_{d-3}} \frac{E_\chi}{\kappa} \right)^{1/(d-2)}$ that guarantees quasi-stationarity for weak, small objects. The analysis confirms the non-trivial validity of horizon mechanics for Rindler and other bifurcate Killing horizons in $d \ge 3$, while showing that no general analogue exists for $d=2$ in stabilized or compactified theories. Together, these results solidify the parallel between black hole and horizon thermodynamics in higher dimensions and clarify the limits of the first-law analogy.

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 $\fracκ{8 π} ΔA = ΔE - ΩΔJ$, 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.