Holographic entropy increases in quadratic curvature gravity

Abstract

Standard methods for calculating the black hole entropy beyond general relativity are ambiguous when the horizon is nonstationary. We fix these ambiguities in all quadratic curvature gravity theories, by demanding that the entropy be increasing at every time, for linear perturbations to a stationary black hole. Our result matches with the entropy formula found previously in holographic entanglement entropy calculations. We explicitly calculate the entropy increase for Vaidya-like solutions in Ricci-tensor gravity to show that (unlike the Wald entropy) the holographic entropy obeys a Second Law.

Deriving covariant holographic entanglement

“…As discussed in [39,40] the holographic entanglement entropy functionals serve as a good starting point to examine the second law for higher derivative black hole entropy. The discussion thus far has been confined to the linear response regime of small amplitude fluctuations away from equilibrium.…”

A Second Law for higher curvature gravity

“…We shall see that there is indeed a Second Law in all such theories of higher curvature gravity theories, provided that you only consider linearized perturbations δg ab , δφ of the gravitational fields (possibly sourced by a first order perturbation to δT ab ) evaluated on a stationary black hole background (or more precisely, a bifurcate Killing horizon 1 ). This has previously been done for f (Lovelock) gravity [5], and quadratic curvature gravity [6].…”

“…We shall see that there is indeed a Second Law in all such theories of higher curvature gravity theories, provided that you only consider linearized perturbations δg ab , δφ of the gravitational fields (possibly sourced by a first order perturbation to δT ab ) evaluated on a stationary black hole background (or more precisely, a bifurcate Killing horizon 1 ). This has previously been done for f (Lovelock) gravity [5], and quadratic curvature gravity [6].Pick a gauge so that u and v are null coordinates which increase as one moves spacelike away from the horizon, so that u = 0 is the future horizon H, v is an affine parameter along the null generators of the horizon, v = 0 is the past horizon, i, j indices point in the D − 2 transverse directions, the metric obeysand the Killing symmetry acts like a standard Lorentz boost on the null coordinates:The Killing weight n of a tensor (with all indices lowered) is given by the number of v-indices minus the number of u-indices, and will sometimes be indicated by an (n) superscript. A key feature of this gauge choice is that on the horizon, any tensor with positive weight n always has at least n v-derivatives acting on it.…”

“…But in adiabatic (i.e. thermodynamically reversible) quantum processes for which δ(S + S matter ) = 0, the linearized Second Law could still be a necessary consistency condition, even in a perturbative treatment [6]. …”

“…black hole mergers), even Lovelock gravity violates the Second Law [23][24][25]! Perhaps this indicates that it is only consistent to treat the non-GR couplings perturbatively, in a consistent truncation of a UV-complete theory of quantum gravity 6 By replacing v with a general affine coordinate λ = a + bv, b > 0, where a, b can depend on the generator of the horizon, and by performing the same series of steps, it follows that S is also increasing from any initial slice on the horizon to any final slice. (Note that, in the step which uses the physical process First Law, the bifurcation surface is not generally at a constant value of λ, but this does not invalidate the proof.)…”