Causal holographic information does not satisfy the linearized quantum focusing condition

Zicao Fu,Donald Marolf,Marvin Qi

doi:10.1007/jhep04(2018)086

Zicao Fu, Donald Marolf + Show 1 more

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https://doi.org/10.1007/jhep04(2018)086

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Abstract

The Hubeny-Rangamani causal holographic information (CHI) defined by a region R of a holographic quantum field theory (QFT) is a modern version of the idea that the area of event horizons might be related to an entropy. Here the event horizon lives in a dual gravitational bulk theory with Newton’s constant Gbulk, and the relation involves a factor of 4Gbulk. The fact that CHI is bounded below by the von Neumann entropy S suggests that CHI is coarse-grained. Its properties could thus differ markedly from those of S. In particular, recent results imply that when d ≤ 4 holographic QFTs are perturbatively coupled to d-dimensional gravity, the combined system satisfies the so-called quantum focusing condition (QFC) at leading order in the new gravitational coupling Gd when the QFT entropy is taken to be that of von Neumann. However, by studying states dual to spherical bulk (anti-de Sitter) Schwarschild black holes in the conformal frame for which the boundary is a (2 + 1)-dimensional de Sitter space, we find the QFC defined by CHI is violated even when perturbing about a Killing horizon and using a single null congruence. Since it is known that a generalized second law (GSL) holds in this context, our work demonstrates that the QFC is not required in order for an entropy, or an entropy-like quantity, to satisfy such a GSL.

Highlights

JHEP04(2018)086 of black hole entropy (taken to be SBH = A/(4Gd)) and the entropy of quantum fields outside
Our work above found that the linearized single-flow quantum focusing condition (QFC) defined by causal holographic information (CHI) is violated for d = 3 holographic CFTs states on a dS background that are dual to global AdS4Schwarzschild black hole spacetimes
The adjective “linearized” refers to the term of order zero in the coupling G3 of our d = 3 holographic CFT to d = 3 Einstein-Hilbert gravity. (The term of order G−3 1 vanishes because we evaluate the generalized second law (GSL) on a Killing horizon of the dS3 background geometry.) The violation occurs near the point at which the bulk causal surface changes topology and is associated with the formation of bulk caustics

Summary

Setting the stage

The right-hand side of eq (1.3) is finite, and may be computed by separately renormalizing each term in this way This property makes the dS conformal frame useful conceptually. By time-reflection symmetry, the causal surface C(R) occurs at the same time t It is just the intersection of the bulk past light cone ∂I−(p+) and with the surface of constant time coordinate t. Since the stress tensor is defined as Tαβ := [−2/( |g|)](δS)/(δgαβ), a Weyl rescaling gαβ = Ω2gαβ of the boundary metric yields Tαβ = Ω−d+2Tαβ. Combining this with the fact that the dS affine parameters are related by d/(dλ) = Ω−2[d/(dt)], we obtain. To test our first-order single-flow QFC (eq (1.3)), we need only check positivity of the quantity

Testing the CHI QFC

Discussion