Area law unification and the holographic event horizon

Abstract

We prove a new, large family of area laws in general relativity, which apply to certain classes of untrapped surfaces that we dub generalized holographic screens. Our family of area laws contains, as special cases, the area laws for marginally-trapped surfaces (holographic screens) and the event horizon (Hawking's area theorem). In addition to these results in general relativity, we show that in the context of holography the geometry of a generalized holographic screen is related to the outer entropy of the screen. Specifically, we show for spherically-symmetric spacetimes that the area of the largest HRT surface consistent with the outer wedge can be computed in terms of the geometry of the general (not necessarily marginally-trapped) codimension-two surface defining the wedge. This outer entropy satisfies a second law of thermodynamics, growing monotonically along the generalized holographic screen. In particular, this result provides the holographic dual for the geometry of the event horizon for spherically-symmetric spacetimes.

Outer entropy and quasilocal energy

“…(The surface at r = r + can correspond to either the past or the future horizon.) This spacetime exhibits a qualitative difference from the spherically-symmetric geometries considered by NR [28]: nonzero twist ω a . Computing the twist on σ according to Eq.…”

“…Note that, a priori, this condition does not make Y 0 a marginally-antitrapped surface: the ingoing null congruence orthogonal to Y 0 has tangent 3 As discussed in Ref. [28], we can set T kk and T k to zero along N −k (σ) consistent with our energy conditions and energy-momentum conservation via a limiting procedure, and a similar argument applies for T . Moreover, we can set ς k and ς to zero discontinuously via a shock wave in the Weyl tensor [46], which has no effect on T ab .…”

“…Given the area law for holographic screens [26,27], this implies a thermodynamic second law associated with the evolution of the entropy along the holographic screen. NR [28] generalized the concept of a holographic screen to a particular class of surfaces that are not marginally trapped or antitrapped, including the event horizon. It was shown there that these generalized holographic screens also satisfy an area law and, for spherically-symmetric surfaces, a second law for the outer entropy (despite the fact that S (outer) [σ] = A[σ]/4G for surfaces that are not marginally trapped or antitrapped).…”

“…Throughout, we assume the NEC, along with the version of the dominant energy condition that ignores the cosmological constant (dubbed the ΛDEC in Ref. [28]), which requires that −T a b t b be a future-directed, causal vector for all future-directed, causal t a , so that the energy-momentum flow (excepting the cosmological constant) is causal in any reference frame. In particular, the ΛDEC implies that T k ≥ 0, just as the NEC implies that T kk and T are nonnegative.…”

“…1). Nomura and Remmen (NR) [28] previously formulated this generalization to normal surfaces in the case of spherically-symmetric spacetimes, but in this work we will consider general normal surfaces without assuming spherical symmetry.…”