The Particle and Energy Cost of Entanglement of Hawking Radiation with the Final Vacuum State

Abstract

A semiclassical analysis shows that in the process of black hole formation and evaporation, an initial pure state will evolve to a mixed state, i.e., information will be lost. One way of avoiding this conclusion without invoking drastic modifications of the local laws of physics in a low curvature regime would be for the information to be restored at the very end of the evaporation process. It is normally envisioned that this would require a final burst of particles entangled with the early time Hawking radiation. This would imply the emission of an extremely large number of particles from an object of Planck size and mass and would appear to be blatantly ruled out by energy considerations. However, Hotta, Schutzhold, and Unruh have analyzed a $(1+1)$-dimensional moving mirror analog of the Hawking process and have found that, in this model, information is restored via entanglement of the early time Hawking radiation with vacuum fluctuations in the spacetime region to the future of the event where the mirror returns to inertial motion. We analyze their model here and give a precise formulation of this entanglement by introducing the notion of "Milne particles." We then analyze the inertial particle and energy cost of such an entanglement of Hawking radiation with vacuum fluctuations. We show that that, in fact, the entanglement of early time Hawking radiation with vacuum fluctuations requires the emission of at least as many late time inertial particles as Hawking particles. Although the energy cost can be made small in the $(1+1)$-dimensional mirror system, this should not be the case for the $(3+1)$-dimensional evaporating black hole system. Thus, vacuum entanglement has the same difficulties as the more usual burst scenarios for attempting to avoid information loss.

28 pages, 3 figures; typos corrected and other very minor changes made in v2

References in corpus (2)

• Entanglement of the Vacuum between Left, Right, Future, and Past: The Origin of Entanglement-Induced Quantum Radiation
• On the partner particles for moving mirror radiation and black hole evaporation

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• Vacuum-purified Hawking radiation from evaporating black holes: Lessons from moving mirrors
• Vacuum Energy in Modern Cosmology: an analysis of quantum field theory in curved spaces and its application to cosmological spacetimes