Physical process first law for bifurcate Killing 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 κ 8π ∆A = ∆E − Ω∆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 ≥ 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 ≥ 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 ≥ 3. We also comment on the situation for d = 2.

Physical process first law and caustic avoidance for Rindler horizons

“…Clearly when 90 > 1, then 9 increases as we move along the past and diverges at some finite Killing time before the bifurcation surface. Therefore, if for any cross section v -v0, the horizon expansion is strong enough so as to satisfy 9 > 2k, (9) then the Raychaudhuri equation implies that a caustic will be developed at some finite v < v0. In the following section, this condition will be used to get a bound on the size of the object that crosses the horizon.…”

“…Thus for any finite sized object PPFL will not hold as the process cannot be made quasistationary. But, in [9], it was clarified that the above argum ent is not com pletely acceptable because in the analysis o f Price et al [8] the planet under consideration had started its journey from rest at an infinite distance apart from the black hole horizon, which does not have a smooth limit for the case o f the R indler horizon. W ith the help o f a m ore refined treatm ent and the consideration o f a local characterization o f the form ation o f caustics, it was shown in [9] that the PPFL for RH holds nontrivially; in fact it holds for any general bifurcate Killing horizon.…”

“…Since the derivation o f PPFL traces generators o f the horizon only back to the unperturbed past horizon, one only has to make sure no caustic forms on the future bifurcation surface. In [9], it was shown one can indeed avoid the form ation of caustics in the region o f interest and the first law holds for R indler horizon. Moreover, there is a bound on the size o f the object given by the relation r > s/ E J k for which caustic will not form and the process will rem ain quasista tionary, where Ex = mxz0 and k are the Killing energy of the incident object and the surface gravity defined by the Killing field / , respectively.…”

“…Next we have considered the case when a slowly rotating object of mass m falls across the Rindler horizon from rest. The primary goal of these investigations is to generalize the analysis depicted in [9]. For the rotating case, the horizon is now imparted by the angular momentum of the falling object on top of its mass.…”

Collisions with black holes and deconfined plasmas

“…Furthermore, the incident object need not be another black hole for this to occur; any sufficiently rapid flux of incoming energy results in the formation of a similar black hole arm that reaches out toward the incoming object. As discussed in [16] (which generalizes the arguments of [11]), in any spacetime dimension d the threshold for such nucleation to occur is 1) where E, A are the energy and transverse area of the incident object, T is the temperature of the target black hole, G d is Newton's gravitational constant, and we have set = 1. The purpose of this paper is to calculate the shape of the nucleating arm and to interpret the results in terms of deconfined plasmas; we shall be less concerned with the final relaxation to equilibrium.…”