Reparametrization Symmetry of Local Entropy Production on a Dynamical Horizon

Abstract

Recently, it has been shown that for a dynamical black hole in any higher derivative theory of gravity, one could construct a spatial entropy current, characterizing the in/outflow of entropy at every point on the horizon, as long as the dynamics of the amplitude is small enough. However, the construction is very much dependent on how we choose the spatial slicing of the horizon along its null generators. In this note, we have shown that though both the entropy density and the spatial entropy current change nontrivially under a reparametrization of the null generator, the net entropy production, which is given by the 'time' derivative of entropy density plus the divergence of the spatial current is invariant. We have explicitly verified this claim for the particular case of dynamical black holes Einstein-Gauss-Bonnet theory.1 We are deeply saddened to lose our friend and collaborator, Pooja Jethwani. She was an active collaborator in this project and had seen its completion before the end of our time together.

The second law of black hole mechanics in effective field theory

“…Therefore it is very unclear whether the result of Wall's procedure must be gauge-invariant. (This has also been noted in [12]. )…”

“…It arises because we have chosen our coordinates such that the perturbation does not change the location of H, i.e., the horizon of the perturbed black hole remains at r = 0. For conventional GR (where (12) is simply δK = 0) this was explained in [11]. Integrating (12) over C(v) (with measure √ µ) the divergence drops out and we obtain δ ṠIWW = 0 where the Iyer-Wald-Wall entropy is 3…”

“…For conventional GR (where (12) is simply δK = 0) this was explained in [11]. Integrating (12) over C(v) (with measure √ µ) the divergence drops out and we obtain δ ṠIWW = 0 where the Iyer-Wald-Wall entropy is 3…”

“…The IWW entropy is simply given by the horizon area: s v = 1. Now consider sv = 1 + cℓ 2 K K. Recall that K = 0 on H in the background, and that, on-shell, linear perturbations satisfy (12), which here reduces to δK = 0. Hence we have δ( √ µs v ) = δ( √ µs v ) on-shell.…”

“…This is because it is written in terms of D A rather than D A . So the entropy current is not gauge-invariant in this example (see also [12] for discussion of this point).…”