Abstract
According to the classical Penrose inequality, the mass at spatial infinity is bounded from below by a function of the area of certain trapped surfaces. We exhibit quantum field theory states that violate this relation at the semiclassical level. We formulate a Quantum Penrose Inequality, by replacing the area with the generalized entropy of the lightsheet of an appropriate quantum trapped surface. We perform a number of nontrivial tests of our proposal, and we consider and rule out alternative formulations. We also discuss the relation to weak cosmic censorhip.
Highlights
Semiclassical general relativity allows for quantum matter while keeping the gravitational field classical, by coupling the metric to the expectation value of the stress tensor: Gab 1⁄4 8πGhTabi: ð1:1ÞSince hTabi receives quantum contributions proportional to ħ, this approximation can be organized as a perturbative expansion in Għ and solved iteratively
In Appendix B, we present a perturbative construction of Q screens [9], which plays a role in our discussion of the quantum Penrose inequality in anti–de Sitter (AdS) spacetimes
We will construct an explicit counterexample that is based on a Boulware-like state outside a Schwarzschild black hole. It violates the classical Penrose inequality (CPI) by a substantial, classical amount. This will be a counterexample to the CPI in the same sense as black hole evaporation is a counterexample to Hawking’s area theorem: we identify a physically allowed state in which a key assumption of the classical statement, the null energy condition, does not hold, and we verify that the conclusion fails as well
Summary
Semiclassical general relativity allows for quantum matter while keeping the gravitational field classical, by coupling the metric to the expectation value of the stress tensor: Gab 1⁄4 8πGhTabi: ð1:1Þ. We will study an important conjecture in classical general relativity, the Penrose inequality [15]. This is a relation between the area of certain marginally trapped surfaces μ in the spacetime and the total mass defined at spatial infinity [16]: 2470-0010=2019=100(12)=126010(22). In Appendix A, we compute the expansion of outgoing null rays and the positions of classical and quantum marginally trapped surfaces for an evaporating Schwarzschild black hole. In Appendix B, we present a perturbative construction of Q screens [9], which plays a role in our discussion of the quantum Penrose inequality in anti–de Sitter (AdS) spacetimes. A brief summary of the main results of our investigation has appeared elsewhere [18]
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