Black hole information paradox

In quantum mechanics, the evolution of the state is governed by the Schrödinger equation. The Schrödinger equation obeys two principles that are relevant to the paradox—quantum determinism, which means that given a present wave function, its future changes are uniquely determined by the evolution operator, and reversibility, which refers to the fact that the evolution operator has an inverse, meaning that the past wave functions are similarly unique. The combination of the two means that information must always be preserved. In this context "information" means all the details of the state, and the statement that information must be preserved means that details corresponding to an earlier time can always be reconstructed at a later time. Mathematically, the Schrödinger equation implies that the wavefunction at a time t1 can be related to the wavefunction at a time t2 by means of a unitary operator. |\Psi(t_1)\rangle = U(t_1, t_2)|\Psi(t_2)\rangle. Since the unitary operator is bijective, the wavefunction at t2 can be obtained from the wavefunction at t1 and vice versa. The reversibility of time evolution described above applies only at the microscopic level, since the wavefunction provides a complete description of the state. It should not be conflated with thermodynamic irreversibility. A process may appear irreversible if one keeps track only of the system's coarse-grained features and not of its microscopic details, as is usually done in thermodynamics. But at the microscopic level, the principles of quantum mechanics imply that every process is completely reversible. Starting in the mid-1970s, Stephen Hawking and Jacob Bekenstein put forward theoretical arguments that suggested that black-hole evaporation loses information, and is therefore inconsistent with unitarity. Crucially, these arguments were meant to apply at the microscopic level and suggested that black-hole evaporation is not only thermodynamically but microscopically irreversible. This contradicts the principle of unitarity described above and leads to the information paradox. Since the paradox suggested that quantum mechanics would be violated by black-hole formation and evaporation, Hawking framed the paradox in terms of the "breakdown of predictability in gravitational collapse". This calculation utilized the framework of general relativity and quantum field theory. The calculation of Hawking radiation is performed at the black hole horizon and does not account for the backreaction of spacetime geometry; for a large enough black hole the curvature at the horizon is small and therefore both these theories should be valid. Hawking relied on the no-hair theorem to arrive at the conclusion that radiation emitted by black holes would depend only on a few macroscopic parameters, such as the black hole's mass, charge, and spin, but not on the details of the initial state that led to the formation of the black hole. In addition, the argument for information loss relied on the causal structure of the black hole spacetime, which suggests that information in the interior should not affect any observation in the exterior, including observations performed on the radiation the black hole emits. If so, the region of spacetime outside the black hole would lose information about the state of the interior after black-hole evaporation, leading to the loss of information. Today, some physicists believe that the holographic principle (specifically the AdS/CFT duality) demonstrates that Hawking's conclusion was incorrect, and that information is in fact preserved. Moreover, recent analyses indicate that in semiclassical gravity the information loss paradox cannot be formulated in a self-consistent manner due to the impossibility of simultaneously realizing all of the necessary assumptions required for its formulation. == Black hole evaporation ==

Hawking radiation of a black hole which forms, and then completely evaporates away. Time shown on vertical axis from bottom to top; space shown on horizontal axis from left (radius zero) to right (growing radius). In 1973–1975, Stephen Hawking showed that black holes should slowly radiate away energy, and he later argued that this leads to a contradiction with unitarity. Hawking used the classical no-hair theorem to argue that the form of this radiation—called Hawking radiation—would be completely independent of the initial state of the star or matter that collapsed to form the black hole. He argued that the process of radiation would continue until the black hole had evaporated completely. At the end of this process, all the initial energy in the black hole would have been transferred to the radiation. But, according to Hawking's argument, the radiation would retain no information about the initial state and therefore information about the initial state would be lost. More specifically, Hawking argued that the pattern of radiation emitted from the black hole would be random, with a probability distribution controlled only by the black hole's initial temperature, charge, and angular momentum, not by the initial state of the collapse. The state produced by such a probabilistic process is called a mixed state in quantum mechanics. Therefore, Hawking argued that if the star or material that collapsed to form the black hole started in a specific pure quantum state, the process of evaporation would transform the pure state into a mixed state. This is inconsistent with the unitarity of quantum-mechanical evolution discussed above. The loss of information can be quantified in terms of the change in the fine-grained von Neumann entropy of the state. A pure state is assigned a von Neumann entropy of 0, whereas a mixed state has a finite entropy. The unitary evolution of a state according to Schrödinger's equation preserves the entropy. Therefore Hawking's argument suggests that the process of black-hole evaporation cannot be described within the framework of unitary evolution. Although this paradox is often phrased in terms of quantum mechanics, the evolution from a pure state to a mixed state is also inconsistent with Liouville's theorem in classical physics (see e.g.). In equations, Hawking showed that if one denotes the creation and annihilation operators at a frequency \omega for a quantum field propagating in the black-hole background by a_{\omega} and a_{\omega}^{\dagger} then the expectation value of the product of these operators in the state formed by the collapse of a black hole would satisfy \langle a_{\omega} a_{\omega}^{\dagger} \rangle_{\rm hawk} = {1 \over 1 - e^{-\omega /{kT} }} where is the Boltzmann constant and is the temperature of the black hole. (See, for example, section 2.2 of. In 1993, Page focused on the combined system of a black hole with its Hawking radiation as one entangled system, a bipartite system, evolving over the lifetime of the black hole evaporation. Lacking the ability to make a full quantum analysis, he nonetheless made a powerful observation: If a black hole starts in a pure quantum state and evaporates completely by a unitary process, the von Neumann entropy or entanglement entropy of the Hawking radiation initially increases from zero and then must decrease back to zero when the black hole to which the radiation is entangled has totally evaporated. In short, if black hole evaporation is unitary, then the radiation entanglement entropy follows the Page curve. After the Page time, correlations appear and the radiation becomes increasingly information rich. Many researchers consider deriving the Page curve as synonymous with solving the black hole information paradox. == Popular culture ==

The information paradox has received coverage in the popular media and has been described in popular-science books. Some of this coverage resulted from a widely publicized bet made in 1997 between John Preskill on the one hand with Hawking and Kip Thorne on the other that information was not lost in black holes. The scientific debate on the paradox was described in Leonard Susskind's 2008 book The Black Hole War. (The book carefully notes that the 'war' was purely a scientific one, and that, at a personal level, the participants remained friends.) Susskind writes that Hawking was eventually persuaded that black-hole evaporation was unitary by the holographic principle, which was first proposed by Gerard 't Hooft, further developed by Susskind, and later given a precise string theory interpretation by the AdS/CFT correspondence. In 2004, Hawking also conceded the 1997 bet, paying Preskill with a baseball encyclopedia "from which information can be retrieved at will". Thorne refused to concede. == Solutions ==

Since the 1997 proposal of the AdS/CFT correspondence, the predominant belief among physicists is that information is indeed preserved in black hole evaporation. There are broadly two main streams of thought about how this happens. Within what might broadly be termed the "string theory community", the dominant idea is that Hawking radiation is not precisely thermal but receives quantum correlations that encode information about the black hole's interior. Hawking himself was influenced by this view and in 2004 published a paper that assumed the AdS/CFT correspondence and argued that quantum perturbations of the event horizon could allow information to escape from a black hole, which would resolve the information paradox. In this perspective, it is the event horizon of the black hole that is important and not the black-hole singularity. The GISR of references is an implementation of this idea, with the quantum perturbation of event horizon replaced by the microscopic state of black holes. On the other hand, within what might broadly be termed the "loop quantum gravity community", the dominant belief is that to resolve the information paradox, it is important to understand how the black-hole singularity is resolved. These scenarios are broadly called "remnant scenarios" since information does not emerge gradually but remains in the black-hole interior only to emerge at the end of black-hole evaporation. They were then analyzed by Papadodimas and Raju, who showed that corrections to low-point correlators (such as \epsilon_2 above ) that were exponentially suppressed in the black-hole entropy were sufficient to preserve unitarity, and significant corrections were required only for very high-point correlators. The mechanism that allowed the right small corrections to form was initially postulated in terms of a loss of exact locality in quantum gravity so that the black-hole interior and the radiation were described by the same degrees of freedom. Recent developments suggest that such a mechanism can be realized precisely within semiclassical gravity and allows information to escape. The defining characteristic of the fuzzball is that it has structure at the horizon scale. This should be contrasted with the conventional picture of the black-hole interior as a largely featureless region of space. For a large enough black hole, tidal effects are very small at the black-hole horizon and remain small in the interior until one approaches the black-hole singularity. Therefore, in the conventional picture, an observer who crosses the horizon may not even realize they have done so until they start approaching the singularity. In contrast, the fuzzball proposal suggests that the black hole horizon is not empty. Consequently, it is also not information-free, since the details of the structure at the surface of the horizon preserve information about the black hole's initial state. This structure also affects the outgoing Hawking radiation and thereby allows information to escape from the fuzzball. The fuzzball proposal is supported by the existence of a large number of gravitational solutions called microstate geometries. The firewall proposal can be thought of as a variant of the fuzzball proposal that posits that the black-hole interior is replaced by a firewall rather than a fuzzball. Operationally, the difference between the fuzzball and the firewall proposals has to do with whether an observer crossing the horizon of the black hole encounters high-energy matter, suggested by the firewall proposal, or merely low-energy structure, suggested by the fuzzball proposal. The firewall proposal also originated with an exploration of Mathur's argument that small corrections are insufficient to resolve the information paradox. Soft-hair resolution to the paradox In 2016, Hawking, Perry and Strominger noted that black holes may contain "soft hair". Particles that have no rest mass, like photons and gravitons, can exist with arbitrarily low-energy and are called soft particles. The soft-hair resolution posits that information about the initial state is stored in such soft particles. The existence of such soft hair is a peculiarity of four-dimensional asymptotically flat space and therefore this resolution to the paradox does not carry over to black holes in Anti-de Sitter space or black holes in other dimensions. Information is irretrievably lost A minority view in the theoretical physics community is that information is genuinely lost when black holes form and evaporate. This conclusion follows if one assumes that the predictions of semiclassical gravity and the causal structure of the black-hole spacetime are exact. But this conclusion leads to the loss of unitarity. Banks, Susskind and Peskin argue that, in some cases, loss of unitarity also implies violation of energy–momentum conservation or locality, but this argument may possibly be evaded in systems with a large number of degrees of freedom. 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 Penrose advocates critically depends on the condition that information is in fact lost in black holes. This new cosmological model might 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. The significance of these findings was debated. Along similar lines, Modak, Ortíz, Peña, and Sudarsky have argued that the paradox can be dissolved by invoking foundational issues of quantum theory often called the measurement problem of quantum mechanics. This work built on an earlier proposal by Okon and Sudarsky on the benefits of objective collapse theory in a much broader context. The original motivation of these studies was Penrose's long-standing proposal wherein collapse of the wave-function is said to be inevitable in the presence of black holes (and even under the influence of gravitational field). Experimental verification of collapse theories is an ongoing effort. Black hole complementarity One attempt to resolve the black hole information paradox is known as black hole complementarity. Black hole complementarity suggests that infalling information would be cloned, with one copy falling into the black hole and one copy escaping as Hawking radiation. This would seem to violate the no-cloning theorem of quantum mechanics, which states that information cannot be cloned. However, the creators of black hole complementarity argued that, since the infalling copy of the information is only accessible to an infalling observer and the escaping copy of the information is only accessible to an outside observer, it is impossible to observe both copies of the information and therefore the no-cloning theorem is not violated. However, modern physics has discovered that black hole complementarity is still problematic, as it is still possible to observe both copies of the information. For a Schwarzschild black hole, it is true that only one copy of the information could be observed, because if the observer waited outside the black hole until the outgoing radiation escaped, they would not be able to reach the infalling radiation before it hit the singularity. However, in a spinning or charged black hole, the singularity is timelike, meaning that a piece of information could orbit around the singularity indefinitely without ever hitting it, allowing an infalling observer who already saw the outgoing radiation to also observe the ingoing radiation. The firewall paradox suggests that a "firewall" of intense energy destroys incoming particles at the event horizon, eliminating the problem of violating unitary of monogamy of entanglement. However, this still violates the equivalence principle of general relativity, which requires that the event horizon of a black hole cannot be locally detectable. In general, which—if any—of these principles should be abandoned remains a topic of debate. Other proposed resolutions Some other resolutions to the paradox have also been explored. These are listed briefly below. • Information is stored in a large remnantThis idea suggests that Hawking radiation stops before the black hole reaches the Planck size. Since the black hole never evaporates, information about its initial state can remain inside the black hole and the paradox disappears. But there is no accepted mechanism that would allow Hawking radiation to stop while the black hole remains macroscopic. • Information is stored in a baby universe that separates from our own universe.Some models of gravity, such as the Einstein–Cartan theory of gravity, which extends general relativity to matter with intrinsic angular momentum (spin), predict the formation of such baby universes. No violation of known general principles of physics is needed. There are no physical constraints on the number of the universes, even though only one remains observable.The Einstein–Cartan theory is difficult to test because its predictions are significantly different from general-relativistic ones only at extremely high densities. • Information is encoded in the correlations between future and pastThe final-state proposal suggests that boundary conditions must be imposed at the black-hole singularity, which, from a causal perspective, is to the future of all events in the black-hole interior. This helps reconcile black-hole evaporation with unitarity but contradicts the intuitive idea of causality and locality of time-evolution. • Quantum-channel theoryIn 2014, Chris Adami argued that analysis using quantum channel theory causes any apparent paradox to disappear; Adami rejects black hole complementarity, arguing instead that no space-like surface contains duplicated quantum information. ==See also==