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arXiv:hep-th/0405183 · International Journal of Modern Physics D

NO BLACK HOLE INFORMATION PUZZLE IN A RELATIONAL UNIVERSE

Rodolfo Gambini, Rafael A. Porto, Jorge A. Pullin

Published Dec 1, 2004• 4 citations
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Abstract

The introduction of a relational time in quantum gravity naturally implies that pure quantum states evolve into mixed quantum states. We show, using a recently proposed concrete implementation, that the rate at which pure states naturally evolve into mixed ones is faster than that due to collapsing into a black hole that later evaporates. This is rather remarkable since the fundamental mechanism for decoherence is usually very weak. Therefore the "black hole information puzzle" is rendered de-facto unobservable.

DOI
10.1142/s0218271804006383

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The introduction of a relational time in quantum gravity naturally implies that pure quantum states evolve into mixed quantum states. We show, using a recently proposed concrete implementation, that the rate at which pure states naturally evolve into mixed ones is faster than that due to collapsing into a black hole that later evaporates. This is rather remarkable since the fundamental mechanism for decoherence is usually very weak. Therefore the "black hole information puzzle" is rendered de-facto unobservable.
Suppose a black hole forms from a pure quantum state ∣ψ⟩\ketψ∣ψ⟩. The black hole information loss paradox arises from semiclassical arguments suggesting that, even in a closed system, the process of black hole formation and evaporation evolves a pure state into a mixed state. Resolution to the paradox typically demands violation of quantum mechanics or relativity in domains where they should hold. Instead, I propose that in a complete theory of quantum gravity, any region U\mathcal{U}U that could collapse into a black hole should already be described by a mixed state, thus bypassing the paradox entirely. To that end, I present a model in which the universe is in a quantum error-corrected state, such that any local black hole appears mixed and encodes no information locally.
In semiclassical gravity, the final state of black-hole evaporation cannot be described by a pure state. Nevertheless, we point out that the system can be described by a generalized pure state, which is not defined on a three-dimensional hypersurface but on the four-dimensional spacetime describing the whole Universe at all times. Unlike the conventional quantum state, such a generalized state treats time on an equal footing with space, which makes it well suited for systems that are both quantum and relativistic. In particular, such a generalized state contains a novel type of information encoded in the correlations between future and past, which avoids the black-hole information paradox.
Stephen Hawking has argued that universes containing evaporating black holes can evolve from pure initial states to mixed final ones. Such evolution is non-unitary and so contravenes fundamental quantum principles on which Hawking's analysis was based. It disables the retrodiction of the universe's initial state from its final one, and portends the time-asymmetry of quantum gravity. Small wonder that Hawking's paradox has met with considerable resistance. Here we use a simple result for C*-algebras to offer an argument for pure-to-mixed state evolution in black hole evaporation, and review responses to the Hawking paradox with respect to how effectively they rebut this argument.
We analyze a system consisting of an oscillator coupled to a field. With the field traced out as an environment, the oscillator loses coherence on a very short decoherence time scale; but, on a much longer relaxation time scale, it evolves predictably into a unique, pure (ground) state. This example of recoherence has interesting implications both for the interpretation of quantum theory and for the loss of information during black hole evaporation---the so-called black hole information paradox. We examine these implications by investigating the intermediate and final states of the quantum field, treated as an open system coupled to an unobserved oscillator.
We have recently argued that if one introduces a relational time in quantum mechanics and quantum gravity, the resulting quantum theory is such that pure states evolve into mixed states. The rate at which states decohere depends on the energy of the states. There is therefore the question of how this can be reconciled with Galilean invariance. More generally, since the relational description is based on objects that are not Dirac observables, the issue of covariance is of importance in the formalism as a whole. In this note we work out an explicit example of a totally constrained, generally covariant system of nonrelativistic particles that shows that the formula for the relational conditional probability is a Galilean scalar and therefore the decoherence rate is invariant.