Black hole information paradox

Artist's representation of a black hole

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. 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 things to keep in mind here: 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.

Hawking radiation

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. . The solution to the problem that concluded the battle is the holographic principle, which was first proposed by Gerardus '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".

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 2005, 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. 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 unitarilly as soon as gravitation comes into play... like 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.

Main approaches to the solution of the paradox

Information is irretrievably lost:

• Advantage: Seems to be a direct consequence of relatively non-controversial calculation based on semiclassical gravity.
• Disadvantage: Violates unitarity, as well as energy conservation or causality.

Information gradually leaks out during the black-hole evaporation:

• Advantage: Intuitively appealing because it qualitatively resembles information recovery in a classical process of burning.
• Disadvantage: Requires a large deviation from classical and semiclassical gravity (which do not allow information to leak out from the black hole) even for macroscopic black holes for which classical and semiclassical approximations are expected to be good approximations.

Information suddenly escapes out during the final stage of black-hole evaporation:

• Advantage: A significant deviation from classical and semiclassical gravity is needed only in the regime in which the effects of quantum gravity are expected to dominate.
• Disadvantage: Just before the sudden escape of information, a very small black hole must be able to store an arbitrary amount of information, which violates the Bekenstein bound.

Information is stored in a Planck-sized remnant:

• Advantage: No mechanism for information escape is needed.
• Disadvantage: A very small object must be able to store an arbitrary amount of information, which violates the Bekenstein bound.

Information is stored in a massive remnant:

• Advantage: No mechanism for information escape is needed and a large amount of information does not need to be stored in a small object.
• Disadvantage: No appealing mechanism that could stop Hawking evaporation of a macroscopic black hole is known.

Information is stored in a baby universe that separates from our own universe:

• Advantage: No violation of known general principles of physics is needed.
• Disadvantage: It is difficult to find an appealing concrete theory that would predict such a scenario.

Information is encoded in the correlations between future and past:

• Advantage: Semiclassical gravity is sufficient, i.e., the solution does not depend on details of (still not well understood) quantum gravity.
• Disadvantage: Contradicts the intuitive view of nature as an entity that evolves with time.

The equation

The entropy of a black hole is given by the equation:

$S={\frac {c^{3}kA}{4\hbar G}}$

where S is the entropy, c is the speed of light, k is Boltzmann's constant, A is the surface area of the event horizon, ħ ("h-bar") is the reduced Planck's Constant and G is the gravitational constant.

See also

• Cosmic censorship hypothesis
• Fuzzball (string theory)
• Holographic principle
• Maxwell's Demon

External links

• Black Hole Information Loss Problem, a USENET physics FAQ page
• "Do black holes destroy information?", John Preskill (1992), hep-th/9209058. Discusses methods of attack on the problem, and their apparent shortcomings.
• Report on Hawking's 2004 theory at New Scientist
• Report on Hawking's 2004 theory at Nature
• Hawking, S. W. (July 2005), Information Loss in Black Holes, arxiv:hep-th/0507171. Stephen Hawking's purported solution to the black hole unitarity paradox.
• Hawking and unitarity: an up-to-date discussion of the information loss paradox and Stephen Hawking's role in it
• The Hawking Paradox - BBC Horizon documentary (2005)