The black hole information paradox results from the combination of quantum mechanics and general relativity. It suggests that physical information could disappear in a black hole, allowing many physical states to evolve into the same state. This is a contentious subject since it violates a commonly assumed tenet of science—that in principle complete information about a physical system at one point in time should determine its state at any other time.[1] A postulate of quantum mechanics is that complete information about a system is encoded in its wave function, an abstract concept not present in classical physics. The evolution of the wave function is determined by a unitary operator, and unitarity implies that information is conserved in the quantum sense.
There are two main principles at work: quantum determinism, and reversibility. Quantum determinism means that given a present wave function, its future changes are uniquely determined by the evolution operator. Reversibility refers to the fact that the evolution operator has an inverse, meaning that the past wave functions are similarly unique. With quantum determinism, reversibility, and a conserved Liouville measure, the von Neumann entropy ought to be conserved, if coarse graining is ignored.
Stephen Hawking presented rigorous theoretical arguments based on general relativity and thermodynamics which threatened to undermine these ideas about information conservation in the quantum realm. Several proposals have been put forth to resolve this paradox.
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In 1975, Stephen Hawking and Jacob Bekenstein showed that black holes should slowly radiate away energy, which poses a problem. From the no hair theorem, one would expect the Hawking radiation to be completely independent of the material entering the black hole. Nevertheless, if the material entering the black hole were a pure quantum state, the transformation of that state into the mixed state of Hawking radiation would destroy information about the original quantum state. This violates Liouville's theorem and presents a physical paradox.
More precisely, if there is an entangled pure state, and one part of the entangled system is thrown into the black hole while keeping the other part outside, the result is a mixed state after the partial trace is taken over the interior of the black hole. But since everything within the interior of the black hole will hit the singularity within a finite time, the part which is traced over partially might disappear completely from the physical system.
Hawking remained convinced that the equations of black hole thermodynamics together with the no-hair theorem led to the conclusion that quantum information may be destroyed. This annoyed many physicists, notably John Preskill, who in 1997 bet Hawking and Kip Thorne that information was not lost in black holes. The implications Hawking had opined led to the Susskind-Hawking battle, where Leonard Susskind and Gerard 't Hooft publicly 'declared war' on Hawking's solution, with Susskind publishing a popular book about the debate in 2008 (The Black Hole War: My battle with Stephen Hawking to make the world safe for quantum mechanics, ISBN 9780316016407). The book carefully notes that the "war" was purely a scientific one, and that at a personal level, the participants remained friends.[2] The solution to the problem that concluded the battle is the holographic principle, which was first proposed by 't Hooft but was given a precise string theory interpretation by Susskind. With this, as the title of an article puts it, "Susskind quashes Hawking in quarrel over quantum quandary".[3]
There are various ideas about how the paradox is solved. Since the 1997 proposal of the AdS/CFT correspondence, the predominant belief among physicists is that information is preserved and that Hawking radiation is not precisely thermal but receives quantum corrections. Other possibilities include the information being contained in a Planckian remnant left over at the end of Hawking radiation or a modification of the laws of quantum mechanics to allow for non-unitary time evolution.
In July 2004, Stephen Hawking published a paper and announced a theory that quantum perturbations of the event horizon could allow information to escape from a black hole, which would resolve the information paradox[4]. His argument assumes the unitarity of the AdS/CFT correspondence which implies that an AdS black hole that is dual to a thermal conformal field theory. When announcing his result, Hawking also conceded the 1997 bet, paying Preskill with a baseball encyclopedia "from which information can be retrieved at will." However, Thorne remains unconvinced of Hawking's proof and declined to contribute to the award.
According to Roger Penrose, loss of unitarity in quantum systems is not a problem: quantum measurements are by themselves already non-unitary. Penrose claims that quantum systems will in fact no longer evolve unitarily as soon as gravitation comes into play, precisely as in black holes. The Conformal Cyclic Cosmology advocated by Penrose critically depends on the condition that information is in fact lost in black holes. This new cosmological model might in future be tested experimentally by detailed analysis of the cosmic microwave background radiation (CMB): if true the CMB should exhibit circular patterns with slightly lower or slightly higher temperatures. In November 2010, Penrose and V. G. Gurzadyan announced they had found evidence of such circular patterns, in data from the Wilkinson Microwave Anisotropy Probe (WMAP) corroborated by data from the BOOMERanG experiment.[5] The significance of the findings was subsequently debated by others.[6]
Information is irretrievably lost:
Information gradually leaks out during the black-hole evaporation:
Information suddenly escapes out during the final stage of black-hole evaporation:
Information is stored in a Planck-sized remnant:
Information is stored in a massive remnant:
Information is stored in a baby universe that separates from our own universe:
Information is encoded in the correlations between future and past:[7][8]
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