View source for Black hole information paradox

{{short description|Puzzle of disappearance of information in a black hole}}
{{use dmy dates|date=June 2023}}
[[File:Black hole - Messier 87 crop max res.jpg|alt=|thumb|The first image (silhouette or shadow) of a black hole, taken of the supermassive black hole in M87 with the [[Event Horizon Telescope]], released in April 2019]]
The '''black hole information paradox'''<ref>The short form "ínformation paradox" is also used for the [[Arrow information paradox]].</ref> is a [[paradox]] that appears when the predictions of [[quantum mechanics]] and [[general relativity]] are combined. The theory of [[general relativity]] predicts the existence of [[black hole]]s that are regions of spacetime from which nothing—not even light—can escape. In the 1970s, [[Stephen Hawking]] applied the [[Semiclassical gravity|semiclassical]] approach of [[quantum field theory in curved spacetime]] to such systems and found that an isolated black hole would emit a form of radiation (now called [[Hawking radiation]] in his honor). He also argued that the detailed form of the radiation would be independent of the initial state of the black hole,<ref name="hawkbreak">{{cite journal|doi=10.1103/PhysRevD.14.2460|title=Breakdown of predictability in gravitational collapse|year=1976|last1=Hawking|first1=S. W.|author-link1=Stephen Hawking|journal=Physical Review D|volume=14|issue=10|pages=2460–2473|bibcode=1976PhRvD..14.2460H}}</ref> and depend only on its [[mass]], [[electric charge]] and [[angular momentum]].

The information paradox appears when one considers a process in which a black hole is formed through a physical process and then evaporates away entirely through Hawking radiation. Hawking's calculation suggests that the final state of radiation would retain information only about the total mass, electric charge and angular momentum of the initial state. Since many different states can have the same mass, charge and angular momentum, this suggests that many initial [[Quantum state|physical states]] could evolve into the same final state. Therefore, [[information]] about the details of the initial state would be permanently lost; however, this violates a core precept of both classical and quantum physics: that, ''in principle only,'' the state of a system at one point in time should determine its state at any other time.<ref name="SH-2006">{{cite video |people=Hawking, Stephen |title=The Hawking Paradox |url=http://dsc.discovery.com/tv-shows/other-shows/videos/other-shows-into-the-universe-with-stephen-hawking.htm |date=2006 |publisher=[[Discovery Channel]] |access-date=13 August 2013 |url-status=dead |archive-url=https://web.archive.org/web/20130802071021/http://dsc.discovery.com/tv-shows/other-shows/videos/other-shows-into-the-universe-with-stephen-hawking.htm |archive-date=2 August 2013 }}</ref><ref name="NYT-20130812">{{cite news |last=Overbye |first=Dennis |author-link=Dennis Overbye |title=A Black Hole Mystery Wrapped in a Firewall Paradox |url=https://www.nytimes.com/2013/08/13/science/space/a-black-hole-mystery-wrapped-in-a-firewall-paradox.html |date=12 August 2013 |work=[[The New York Times]] |access-date=12 August 2013 }}</ref> Specifically, in quantum mechanics the state of the system is encoded by its [[wave function]]. The evolution of the wave function is determined by a [[unitary operator]], and [[unitarity (physics)|unitarity]] implies that the wave function at any instant of time can be used to determine the wave function either in the past or the future. In 1993, [[Don Page (physicist)|Don Page]] argued that if a black hole starts in a pure quantum state and evaporates completely by a unitary process, the [[von Neumann entropy]] of the Hawking radiation initially increases and then decreases back to zero when the black hole has disappeared.<ref name=1993-page>{{cite journal | volume=71 | last = Page | first = Don N. | title = Information in Black Hole Radiation | journal = Physical Review Letters | pages = 3743–3746 | date = 6 December 1993 | issue = 23 | doi=10.1103/PhysRevLett.71.3743 | arxiv=hep-th/9306083 |bibcode = 1993PhRvL..71.3743P | pmid = 10055062 | s2cid = 9363821 }}</ref> This is called the Page curve.<ref name="CoxForshaw Page-curve">{{cite book |last1=Cox |first1=Brian |authorlink1=Brian Cox (physicist)|last2=Forshaw |first2=Jeff |title=Black Holes: the key to understanding the Universe |date=2022 |publisher=HarperCollins Publishers |location=New York, NY |isbn=9780062936691 |page=220-225 |quote=Page curve }}</ref>

It is now generally believed that information is preserved in black-hole evaporation.<ref name="QT-20201030" /><ref name="entrhawk">{{cite journal |last1=Almheiri |first1=Ahmed |last2=Hartman |first2=Thomas |last3=Maldacena |first3=Juan |last4=Shaghoulian |first4=Edgar |last5=Tajdini |first5=Amirhossein |title=The entropy of Hawking radiation |journal=Reviews of Modern Physics |date=21 July 2021 |volume=93 |issue=3 |pages=035002 |doi=10.1103/RevModPhys.93.035002|arxiv=2006.06872 |bibcode=2021RvMP...93c5002A |s2cid=219635921 }}</ref><ref name="lessons">{{cite journal |last1=Raju |first1=Suvrat |title=Lessons from the information paradox |journal=Physics Reports |date=January 2022 |volume=943 |pages=1–80 |doi=10.1016/j.physrep.2021.10.001 |arxiv=2012.05770 |bibcode=2022PhR...943....1R |s2cid=228083488 }}</ref> For many researchers, deriving the Page curve is synonymous with solving the black hole information puzzle.<ref name=GS2022textbook>{{cite book |last1=Grumiller |first1=Daniel |last2=Sheikh-Jabbari |first2=Mohammad Mehdi |title=Black Hole Physics: From Collapse to Evaporation |date=2022 |publisher=Springer Graduate Texts in Physics |location=Switzerland |doi=10.1007/978-3-031-10343-8 |isbn=978-3-031-10342-1 |s2cid=253372811 |url=https://link.springer.com/book/10.1007/978-3-031-10343-8}}</ref>{{rp|291}} But views differ as to precisely how Hawking's original semiclassical calculation should be corrected.<ref name="entrhawk"/><ref name="lessons"/><ref name="mathurpedag">{{cite journal |last1=Mathur |first1=Samir D |title=The information paradox: a pedagogical introduction |journal=Classical and Quantum Gravity |date=21 November 2009 |volume=26 |issue=22 |pages=224001 |doi=10.1088/0264-9381/26/22/224001|arxiv=0909.1038 |bibcode=2009CQGra..26v4001M |s2cid=18878424 }}</ref><ref name="bhlqg">{{cite journal |last1=Perez |first1=Alejandro |title=Black holes in loop quantum gravity |journal=Reports on Progress in Physics |date=1 December 2017 |volume=80 |issue=12 |pages=126901 |doi=10.1088/1361-6633/aa7e14 |pmid=28696338 |arxiv=1703.09149 |bibcode=2017RPPh...80l6901P |s2cid=7047942 }}</ref> In recent years, several extensions of the original paradox have been explored. Taken together, these puzzles about black hole evaporation have implications for how gravity and quantum mechanics must be combined. The information paradox remains an active field of research in [[quantum gravity]].

== Relevant principles ==


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—[[Determinism#Quantum and classical mechanics|quantum determinism]], which means that given a present wave function, its future changes are uniquely determined by the evolution operator, and [[Time reversibility|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.<ref name="sh 2019 0823">{{cite web |last1=Hossenfelder |first1=Sabine |authorlink1=Sabine Hossenfelder|title=How do black holes destroy information and why is that a problem? |url=http://backreaction.blogspot.com/2019/08/how-do-black-holes-destroy-information.html |website=Back ReAction |access-date=23 November 2019 |date=23 August 2019}}</ref> 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 t<sub>1</sub> can be related to the wavefunction at a time t<sub>2</sub> by means of a unitary operator.
<math display="block"> |\Psi(t_1)\rangle = U(t_1, t_2)|\Psi(t_2)\rangle.</math>
Since the unitary operator is [[Bijection|bijective]], the wavefunction at t<sub>2</sub> can be obtained from the wavefunction at t<sub>1</sub> 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 [[irreversible process|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".<ref name="hawkbreak" />

The arguments for microscopic irreversibility were backed by Hawking's calculation of the spectrum of radiation that isolated black holes emit.<ref name=Hawking>{{cite journal |last=Hawking |first=Stephen |author-link=Stephen Hawking |title=Particle Creation by Black Holes |journal=[[Commun. Math. Phys.]] |volume=43 |issue=3 |pages=199–220 |date=1 August 1975 |doi=10.1007/BF02345020 |bibcode = 1975CMaPh..43..199H |s2cid=55539246 |url=https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-43/issue-3/Particle-creation-by-black-holes/cmp/1103899181.pdf }}</ref> 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 of space time|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.<ref name=Barbon>{{cite journal | doi = 10.1088/1742-6596/171/1/012009 | volume=171 | title=Black holes, information and holography | year=2009 | journal=Journal of Physics: Conference Series | page=012009 | last1 = Barbón | first1 = J L F| issue=1 | bibcode=2009JPhCS.171a2009B | doi-access=free }} http://iopscience.iop.org/1742-6596/171/1/012009 p.1: "The most
important departure from conventional thinking in recent years, the ''holographic principle''...provides a ''definition'' of quantum gravity...[and] guarantees that the whole process is unitary."</ref> 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.<ref>{{cite journal |author1=Luca Buoninfante |author2=Francesco Di Filippo |author3=Shinji Mukohyama |title=On the assumptions leading to the information loss paradox |journal=Journal of High Energy Physics |date=2021 |volume=2021 |issue=10 | page=81| doi=10.1007/JHEP10(2021)081 | arxiv=2107.05662 | bibcode=2021JHEP...10..081B |s2cid=235828913}}</ref><ref>{{cite journal |author1=Robert B. Mann |author2=Sebastian Murk |author3=Daniel R. Terno |title=Surface gravity and the information loss problem |journal=Physical Review D |date=2022 |volume=105 |issue=12 |page=124032 |doi=10.1103/PhysRevD.105.124032 |arxiv=2109.13939 |bibcode=2022PhRvD.105l4032M |s2cid=249799593}}</ref>

== Black hole evaporation ==
=== Hawking radiation ===
{{main|Hawking radiation}}
[[Image:black hole Penrose.png|thumb|The [[Penrose diagram]] 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 (physics)|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 equation|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 (Hamiltonian)|Liouville's theorem]] in classical physics (see e.g.<ref>L. Susskind and J. Lindesay, Black Holes, Information and the String Theory Revolution, World Scientific, 2005, pp. 69-84; {{ISBN|978-981-256-083-4}}.</ref>).

In equations, Hawking showed that if one denotes the [[creation and annihilation operators]] at a frequency <math>\omega</math> for a quantum field propagating in the black-hole background by <math>a_{\omega}</math> and <math>a_{\omega}^{\dagger}</math> then the expectation value of the product of these operators in the state formed by the collapse of a black hole would satisfy <math display="block"> \langle a_{\omega} a_{\omega}^{\dagger} \rangle_{\rm hawk} = {1 \over 1 - e^{-\omega /{kT} }} </math> where {{math|''k''}} is the [[Boltzmann constant]] and {{math|''T''}} is the temperature of the black hole. (See, for example, section 2.2 of.<ref name="lessons" />) This formula has two important aspects. The first is that the form of the radiation depends only on a single parameter, temperature, even though the initial state of the black hole cannot be characterized by one parameter. Second, the formula implies that the black hole radiates mass at a rate given by <math display="block"> {d M \over d t} = -{a T^4}</math> where {{math|''a''}} is constant related to fundamental constants, including the [[Stefan–Boltzmann constant]] and certain properties of the black hole spacetime called its [[black hole greybody factors|greybody factors]].

The temperature of the black hole is in turn dependent on its mass, charge, and angular momentum. For a [[Schwarzschild Black Hole]] the temperature is given by
<math display="block">
T = {\hbar c^3 \over 8 \pi k G M}
</math>
This means that if the black hole starts out with an initial mass <math>M_0</math>, it evaporates completely in a time proportional to <math>M_0^3</math>.

The important aspect of these formulas is that they suggest that the final gas of radiation formed through this process depends only on the black hole's temperature and is independent of other details of the initial state. This leads to the following paradox. Consider two distinct initial states that collapse to form a Schwarzschild black hole of the same mass. Even though the states were distinct at first, since the mass (and hence the temperature) of the black holes is the same, they will emit the same Hawking radiation. Once they evaporate completely, in both cases, one will be left with a featureless gas of radiation. This gas cannot be used to distinguish between the two initial states, and therefore information has been lost.

=== Page curve ===
During the same time period in the 1970s, [[Don Page (physicist)|Don Page]] was a doctoral student of Stephen Hawking. He objected to Hawking's reasoning leading to the paradox above, initially on the basis of violation of [[CPT symmetry]].<ref>{{cite journal |last1=Page |first1=Don N. |title=Is Black-Hole Evaporation Predictable? |journal=Phys. Rev. Lett. |date=4 February 1980 |volume=44 |issue=5 |page=301 |doi=10.1103/PhysRevLett.44.301 |bibcode=1980PhRvL..44..301P |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.44.301}}</ref> 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 [[unitarity (physics)|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.<ref name="1993-page" /> This is known as the Page curve; and the time corresponding to the maximum or turnover point of the curve, which occurs at about half the black-hole lifetime, is called the Page time.<ref name="Almheiri et al RMP2021">{{cite journal |last1=Almheiri |first1=Ahmed |last2=Hartman |first2=Thomas |last3=Maldacena |first3=Juan |last4=Shaghoulian |first4=Edgar |last5=Tajdini |first5=Amirhossein |title=The entropy of Hawking radiation |journal=Reviews of Modern Physics |date=21 July 2021 |volume=93 |issue=3 |pages=035002 |doi=10.1103/RevModPhys.93.035002|arxiv=2006.06872 |bibcode=2021RvMP...93c5002A |s2cid=219635921 |quote=Glossary. Page curve: Consider a spacetime with a black hole formed by the collapse of a pure state. Surround the black hole by an imaginary sphere whose radius is a few Schwarzschild radii. The Page curve is a plot of the fine-grained entropy outside of this imaginary sphere, where we subtract the contribution of the vacuum. Since the black hole Hawking radiates and the Hawking quanta enter this faraway region, this computes the fine-grained entropy of Hawking radiation as a function of time. Notice that the regions inside and outside the imaginary sphere are open systems. The curve begins at zero when no Hawking quanta have entered the exterior region, and ends at zero when the black hole has completely evaporated and all of the Hawking quanta are in the exterior region. The “Page time” corresponds to the turnover point of the curve. }}</ref>  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.<ref name="CoxForshaw Page-curve" />

Recent progress in deriving the Page curve for unitary black hole evaporation is a significant step towards finding both a resolution to the information paradox and a more general understanding of unitarity in quantum gravity.<ref name="Imseis 2021">{{cite web |last1=Imseis |first1=Michael T.N. |title=A Pedagogical Review of Black Holes, Hawking Radiation and the Information Paradox |url=https://www.researchgate.net/publication/356129307 |website=researchgate.net |date=October 30, 2021}}</ref> Many researchers consider deriving the Page curve as synonymous with solving the black hole information paradox.<ref name="GS2022textbook" />{{rp|291}}

== 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 [[Thorne–Hawking–Preskill bet|bet]] made in 1997 between [[John Preskill]] on the one hand with [[Stephen Hawking|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.<ref>{{Cite book|title = The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics|url = https://books.google.com/books?id=f3_rnRVxGIsC|publisher = Little, Brown|date = 2008-07-07|access-date = 2015-04-07|isbn = 9780316032698|first = Leonard|last = Susskind|page = 10|quote = It was not a war between angry enemies; indeed the main participants are all friends. But it was a fierce intellectual struggle of ideas between people who deeply respected each other but also profoundly disagreed.}}</ref>) Susskind writes that Hawking was eventually persuaded that black-hole evaporation was unitary by the [[holographic principle]], which was first proposed by 't Hooft, further developed by Susskind, and later given a precise [[string theory]] interpretation by the AdS/CFT correspondence.<ref>{{cite web|url=http://richarddawkins.net/articles/2846|title=Susskind Quashes Hawking in Quarrel Over Quantum Quandary|publisher=CALIFORNIA LITERARY REVIEW |date=2008-07-09|archive-url=https://web.archive.org/web/20120402185759/http://richarddawkins.net/articles/2846|archive-date=2012-04-02|url-status=dead}}<!--The website is now changed to irrelevant content, but the archived one is correct.--></ref> 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.<ref>{{cite web |title=July 21, 2004: Hawking concedes bet on black hole information loss |url=https://www.aps.org/publications/apsnews/201307/physicshistory.cfm |website=www.aps.org |publisher=American Physical Society |access-date=5 January 2022 |language=en}}</ref>

== 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.<ref name="lessons" /> This viewpoint has been the subject of extensive recent research and received further support in 2019 when researchers amended the computation of the entropy of the Hawking radiation in certain models and showed that the radiation is in fact dual to the black hole interior at late times.<ref name="BHInterior">{{cite arXiv|eprint=1911.11977|title=Replica wormholes and the black hole interior|first1=G.|last1=Penington|first2=S.|last2=Shenker|first3=D.|last3=Stanford|first4=Z.|last4=Yang|year=2019|class=hep-th}}</ref><ref name="HREntropy">{{cite journal|arxiv=1911.12333|title=Replica Wormholes and the Entropy of Hawking Radiation|first1=A.|last1=Almheiri|first2=T.|last2=Hartman|first3=J.|last3=Maldacena|first4=E.|last4=Shaghoulian|first5=A.|last5=Tajdini|year=2019|doi=10.1007/JHEP05(2020)013|volume=2020|journal=Journal of High Energy Physics|issue=5|s2cid=208310010}}</ref> 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.<ref name="baezweek207">{{cite web |url=http://math.ucr.edu/home/baez/week207.html|title=This Week's Finds in Mathematical Physics (Week 207)|last1=Baez |first1=John |access-date=2011-09-25}}</ref> In this perspective, it is the [[event horizon]] of the black hole that is important and not the [[Gravitational singularity|black-hole singularity]]. The GISR (Gravity Induced Spontaneous Radiation) mechanism of references<ref name="dfzeng2021">{{cite journal |last1=Ding-fang |first1=Zeng |title=Spontaneous Radiation of Black Holes |journal=Nuclear Physics B |year=2022 |volume=977 |page=115722 |doi=10.1016/j.nuclphysb.2022.115722 |arxiv=2112.12531 |bibcode=2022NuPhB.97715722Z |s2cid=245425064 |url=https://www.sciencedirect.com/science/article/pii/S0550321322000736 |ref=dfzeng2021}}</ref><ref name="dfzeng2022">{{cite journal |last1=Ding-fang |first1=Zeng |title=Gravitation Induced Spontaneous Radiation |journal=Nuclear Physics B |date=2022 |volume=990 |page=116171 |doi=10.1016/j.nuclphysb.2023.116171 |s2cid=257840729 |url=https://www.sciencedirect.com/science/article/pii/S0550321323001001 |arxiv=2207.05158 }}</ref> can be considered an implementation of this idea but with the quantum perturbations of the event horizon replaced by the microscopic states of the black hole.

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.<ref name="bhlqg" />

Researchers also study other possibilities, including a modification of the laws of quantum mechanics to allow for non-unitary time evolution.

Some of these solutions are described at greater length below.

=== GISR mechanism resolution to the paradox ===

This resolution takes GISR as the underlying mechanism for Hawking radiation, considering the latter only as a resultant effect. The physics ingredients of GISR are reflected in the following explicitly hermitian hamiltonian

<math display="block">\begin{align}
                    H &= \begin{pmatrix}w^i\\&w_{-}^{j}\\&&\ddots\\&&&{\scriptstyle\it0}^{\scriptscriptstyle\it1}\end{pmatrix} + \sum_q\hbar\omega_qa^\dagger_qa_q + \sum_{u,v}^{|u-v|=\hbar\omega_q}g_{u^n v^\ell}b^\dagger_{u^n v^\ell}a_q \\
  g_{u\;\!\!^n v^\ell} &\propto -\frac{\hbar}{G\{M_u,M_v\}^\mathrm{max}}\mathrm{Siml}\{\Psi[M_{u\;\!\!^n}\!(r)],\Psi[M_{v\;\!\!^\ell}\!(r)]\}
\end{align}</math>

The first term of <math>H</math> is a diagonal matrix representing the microscopic state of black holes no heavier than the initial one. The second term describes vacuum fluctuations of particles around the black hole and is represented by many harmonic oscillators. The third term couples the vacuum fluctuation modes to the black hole, such that for each mode whose energy matches the difference between two states of the black hole, the latter transitions with an amplitude proportional to the similarity factor of their microscopic wave functions. Transitions between higher energy state <math>u</math> to lower energy state <math>v</math> and vice versa, are equally permitted at the Hamiltonian level. This coupling mimics the photon-atom coupling in the [[Jaynes–Cummings]] model of atomic physics, replacing the photon's vector potential with the binding energy of particles to be radiated in the black hole case, and the dipole moment of initial-to-final state transitions in atoms with the similarity factor of the initial and final states' wave functions in black holes. Despite its ad hoc nature, this coupling introduces no new interactions beyond gravity, and it is deemed necessary irrespective of the future development of quantum gravitational theories.

From the hamiltonian of GISR and the standard Schrodinger equation controlling the evolution of wave function of the system

<math>|\psi(t)\rangle=\sum_{u=w}^0\sum_{n=1}^{u}\sum_{\omega{}s}^{\omega+u=w}
e^{-iut-i\omega{}t}c_{u\;\!\!^n}^{\omega{}s}(t)|u\;\!\!^n\otimes\omega{}s\rangle
</math>

<math>i\hbar{\partial}_t|\psi(t)\rangle = H|\psi(t)\rangle</math>

here <math>\omega{}s</math> is the index of the radiated particles set with total energy <math>\omega</math>. In the case of short time evolution or single quantum emission, Wigner-Wiesskopf approximation allows one<ref name="dfzeng2021"/><ref name="dfzeng2022"/> to show that the power spectrum of GISR is exactly of thermal type and the corresponding temperature equals that of Hawking radiation. However, in the case of long time evolution or continuous quantum emission, the process is off-equilibrium and is characterised by an initial state dependent black hole mass or temperature vs. time curve. The observers far away can retrieve the information stored in the initial black hole from this mass or temperature versus time curve.

The hamiltonian and wave function description of GISR allows one to calculate the entanglement entropy between the black hole and its Hawking particles explicitly.

<math>s_{BR} = -\operatorname{tr}_{B}\rho_{B}\log\rho_{B}=-tr_{R} \rho_{R}\log\rho_{R}</math>

<math>
\rho_{B} = \operatorname{tr}_{R}\sum_{u=w}^0\sum_{n=1}^{u}\sum_{\omega{}s}^{\omega+u=w}|c_{u^n}^{\omega{}s}\rangle\langle c_{u^n}^{\omega{}s}|
</math>

<math>
\rho_{R} = \operatorname{tr}_{B}\sum_{u=w}^0\sum_{n=1}^{u}\sum_{\omega{}s}^{\omega+u=w}|c_{u^n}^{\omega{}s}\rangle\langle c_{u^n}^{\omega{}s}|
</math>

Since the Hamiltonian of GISR is explicitly Hermitian, the resulting Page curve is naturally expected, except for some late-time Rabi-type oscillations. These oscillations arise from the equal probability of emission and absorption transitions as the black hole approaches the vanishing stage. The most important lesson from this calculation is that the intermediate state of an evaporating black hole cannot be considered a semiclassical object with a time-dependent mass. Instead, it must be viewed as a superposition of many different mass ratio combinations of the black hole and Hawking particles. References<ref name="dfzeng2021"/><ref name="dfzeng2022"/> designed a Schrödinger cat-type thought experiment to illustrate this fact, where an initial black hole is bounded with a group of living cats and each Hawking particle kills on from the group. In the quantum description, because the exact timing and number of particles radiated by a black hole cannot be determined definitively, the intermediate state of the evaporating black hole must be considered a superposition of many cat groups, each with a different ratio of dead members. The biggest flaw in the argument for the information loss paradox is ignoring this superposition.

=== Small-corrections resolution to the paradox ===
This idea suggests that Hawking's computation fails to keep track of small corrections that are eventually sufficient to preserve information about the initial state.<ref name="giddings1995"/><ref name="preskill1992"/><ref name="lessons" /> This can be thought of as analogous to what happens during the mundane process of "burning": the radiation produced appears to be thermal, but its fine-grained features encode the precise details of the object that was burnt. This idea is consistent with reversibility, as required by quantum mechanics. It is the dominant idea in what might broadly be termed the string-theory approach to quantum gravity.

More precisely, this line of resolution suggests that Hawking's computation is corrected so that the two point correlator computed by Hawking and described above becomes
<math display="block"> \langle a_{\omega} a_{\omega}^{\dagger} \rangle_{\rm exact} = \langle a_{\omega} a_{\omega}^{\dagger} \rangle_{\rm hawk} (1 + \epsilon_2) </math>
and higher-point correlators are similarly corrected
<math display="block"> \langle a_{\omega_1} a_{\omega_1}^{\dagger} a_{\omega_2} a_{\omega_2}^{\dagger} \ldots a_{\omega_n} a_{\omega_n}^{\dagger} \rangle_{\rm exact} = \langle a_{\omega} a_{\omega}^{\dagger} \rangle_{\rm hawk} (1 + \epsilon_n) </math>
The equations above utilize a concise notation and the correction factors <math>\epsilon_i</math> may depend on the temperature, the frequencies of the operators that enter the correlation function and other details of the black hole.

[[Juan Maldacena|Maldacena]] initially explored such corrections in a simple version of the paradox.<ref name="maldetern">{{cite journal |last1=Maldacena |first1=Juan |title=Eternal black holes in anti-de Sitter |journal=Journal of High Energy Physics |date=12 April 2003 |volume=2003 |issue=4 |pages=021 |doi=10.1088/1126-6708/2003/04/021|arxiv=hep-th/0106112 |bibcode=2003JHEP...04..021M |s2cid=7767700 }}</ref> They were then analyzed by [[Kyriakos Papadodimas|Papadodimas]] and [[Suvrat Raju|Raju]],<ref name ="infalling">{{cite journal |last1=Papadodimas |first1=Kyriakos |last2=Raju |first2=Suvrat |title=An infalling observer in AdS/CFT |journal=Journal of High Energy Physics |date=30 October 2013 |volume=2013 |issue=10 |pages=212 |doi=10.1007/JHEP10(2013)212|arxiv=1211.6767 |bibcode=2013JHEP...10..212P |s2cid=53650802 }}</ref><ref>{{cite journal |last1=Papadodimas |first1=Kyriakos |last2=Raju |first2=Suvrat |title=State-dependent bulk-boundary maps and black hole complementarity |journal=Physical Review D |date=29 April 2014 |volume=89 |issue=8 |pages=086010 |doi=10.1103/PhysRevD.89.086010|arxiv=1310.6335 |bibcode=2014PhRvD..89h6010P |s2cid=119118804 }}</ref><ref>{{cite journal |last1=Papadodimas |first1=Kyriakos |last2=Raju |first2=Suvrat |title=Black Hole Interior in the Holographic Correspondence and the Information Paradox |journal=Physical Review Letters |date=5 February 2014 |volume=112 |issue=5 |pages=051301 |doi=10.1103/PhysRevLett.112.051301|pmid=24580584 |arxiv=1310.6334 |bibcode=2014PhRvL.112e1301P |s2cid=118867229 }}</ref> who showed that corrections to low-point correlators (such as <math>\epsilon_2</math> above ) that were exponentially suppressed in the [[Black hole entropy|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.<ref name="entrhawk" /> See [[#Recent developments|§ Recent developments]].

=== Fuzzball resolution to the paradox ===

Some researchers, most notably [[Samir Mathur]], have argued<ref name="mathurpedag" /> that the small corrections required to preserve information cannot be obtained while preserving the semiclassical form of the black-hole interior and instead require a modification of the black-hole geometry to a [[Fuzzball (string theory)|fuzzball]].<ref>{{cite journal |last1=Skenderis |first1=Kostas |last2=Taylor |first2=Marika |title=The fuzzball proposal for black holes |journal=Physics Reports |date=October 2008 |volume=467 |issue=4–5 |pages=117–171 |doi=10.1016/j.physrep.2008.08.001|arxiv=0804.0552 |bibcode=2008PhR...467..117S |s2cid=118403957 }}</ref><ref>{{cite journal |last1=Lunin |first1=Oleg |last2=Mathur |first2=Samir D. |title=AdS/CFT duality and the black hole information paradox |journal=Nuclear Physics B |date=February 2002 |volume=623 |issue=1–2 |pages=342–394 |doi=10.1016/S0550-3213(01)00620-4|arxiv=hep-th/0109154 |bibcode=2002NuPhB.623..342L |s2cid=12265416 }}</ref><ref>{{cite journal |last1=Mathur |first1=S.D. |title=The fuzzball proposal for black holes: an elementary review |journal=Fortschritte der Physik |date=15 July 2005 |volume=53 |issue=7–8 |pages=793–827 |doi=10.1002/prop.200410203|arxiv=hep-th/0502050 |bibcode=2005ForPh..53..793M |s2cid=15083147 }}</ref>

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 force|tidal effects]] are very small at the [[Event horizon|black-hole horizon]] and remain small in the interior until one approaches the [[Gravitational singularity|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.<ref>{{cite journal |last1=Mathur |first1=Samir D. |last2=Saxena |first2=Ashish |last3=Srivastava |first3=Yogesh K. |title=Constructing "hair" for the three charge hole |journal=Nuclear Physics B |date=March 2004 |volume=680 |issue=1–3 |pages=415–449 |doi=10.1016/j.nuclphysb.2003.12.022 |arxiv=hep-th/0311092|bibcode=2004NuPhB.680..415M |s2cid=119490735 }}</ref><ref>{{cite journal |last1=Kanitscheider |first1=Ingmar |last2=Skenderis |first2=Kostas |last3=Taylor |first3=Marika |title=Fuzzballs with internal excitations |journal=Journal of High Energy Physics |date=15 June 2007 |volume=2007 |issue=6 |pages=056 |doi=10.1088/1126-6708/2007/06/056 |arxiv=0704.0690 |bibcode=2007JHEP...06..056K |s2cid=18638163 |issn=1029-8479}}</ref><ref>{{cite book |last1=Bena |first1=Iosif |last2=Warner |first2=Nicholas P. |title= Supersymmetric Mechanics - Vol. 3|chapter=  Black Holes, Black Rings, and their Microstates|journal=<!----> |series=Lecture Notes in Physics |date=2008 |volume=755 |pages=1–92 |doi=10.1007/978-3-540-79523-0_1 |arxiv=hep-th/0701216|isbn=978-3-540-79523-0 |s2cid=119096225 }}</ref><ref>{{cite journal |last1=Bena |first1=Iosif |last2=Giusto |first2=Stefano |last3=Russo |first3=Rodolfo |last4=Shigemori |first4=Masaki |last5=Warner |first5=Nicholas P. |title=Habemus Superstratum! A constructive proof of the existence of superstrata |journal=Journal of High Energy Physics |date=May 2015 |volume=2015 |issue=5 |pages=110 |doi=10.1007/JHEP05(2015)110 |arxiv=1503.01463 |bibcode=2015JHEP...05..110B |s2cid=53476809 |issn=1029-8479}}</ref><ref>{{cite journal |last1=Bena |first1=Iosif |last2=Giusto |first2=Stefano |last3=Martinec |first3=Emil J. |last4=Russo |first4=Rodolfo |last5=Shigemori |first5=Masaki |last6=Turton |first6=David |last7=Warner |first7=Nicholas P. |title=Smooth horizonless geometries deep inside the black-hole regime |journal=Physical Review Letters |date=8 November 2016 |volume=117 |issue=20 |pages=201601 |doi=10.1103/PhysRevLett.117.201601 |pmid=27886509 |arxiv=1607.03908 |bibcode=2016PhRvL.117t1601B |s2cid=29536476 |issn=0031-9007}}</ref>

The [[Firewall (physics)|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 [[Samir Mathur|Mathur's]] argument that small corrections are insufficient to resolve the information paradox.<ref name="mathurpedag" />

The fuzzball and firewall proposals have been questioned for lacking an appropriate mechanism that can generate structure at the horizon scale.<ref name="lessons" />

===Strong-quantum-effects resolution to the paradox===

In the final stages of black-hole evaporation, quantum effects become important and cannot be ignored. The precise understanding of this phase of black-hole evaporation requires a complete theory of quantum gravity. Within what might be termed the loop-quantum-gravity approach to black holes, it is believed that understanding this phase of evaporation is crucial to resolving the information paradox.

This perspective holds that Hawking's computation is reliable until the final stages of black-hole evaporation, when information suddenly escapes.<ref name="giddings1995"/><ref name="preskill1992"/><ref>{{cite journal |last1=Ashtekar |first1=Abhay |title=Black Hole Evaporation: A Perspective from Loop Quantum Gravity |journal=Universe |date=24 January 2020 |volume=6 |issue=2 |pages=21 |doi=10.3390/universe6020021|arxiv=2001.08833 |bibcode=2020Univ....6...21A |doi-access=free }}</ref><ref name="bhlqg" /> Another possibility along the same lines is that black-hole evaporation simply stops when the black hole becomes Planck-sized. Such scenarios are called "remnant scenarios".<ref name="giddings1995"/><ref name="preskill1992"/>

An appealing aspect of this perspective is that 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. On the other hand, this idea implies that just before the sudden escape of information, a very small black hole must be able to store an arbitrary amount of information and have a very large number of internal states. Therefore, researchers who follow this idea must take care to avoid the common criticism of remnant-type scenarios, which is that they might may violate the [[Bekenstein bound]] and lead to a violation of effective field theory due to the production of remnants as virtual particles in ordinary scattering events.<ref>{{cite journal|author1-link=Steven Giddings |last1=Giddings |first1=Steven B. |title=Constraints on black hole remnants |journal=Physical Review D |date=15 January 1994 |volume=49 |issue=2 |pages=947–957 |doi=10.1103/PhysRevD.49.947|pmid=10017053 |arxiv=hep-th/9304027 |bibcode=1994PhRvD..49..947G |s2cid=10123547 }}</ref><ref>{{cite journal|last=Giddings |first=Steven B.|year=1998|title=Comments on information loss and remnants|arxiv=hep-th/9310101|bibcode = 1994PhRvD..49.4078G |doi = 10.1103/PhysRevD.49.4078|pmid=10017412|volume=49|issue=8|journal=Physical Review D|pages=4078–4088|s2cid=17746408}}</ref>

=== Soft-hair resolution to the paradox ===
In 2016, [[Stephen Hawking|Hawking]], [[Malcolm Perry (physicist)|Perry]] and [[Andrew Strominger|Strominger]] noted that black holes must contain "soft hair".<ref name=SciAm2016>{{Cite web|title = Stephen Hawking's New Black-Hole Paper, Translated: An Interview with Co-Author Andrew Strominger|url = http://blogs.scientificamerican.com/dark-star-diaries/stephen-hawking-s-new-black-hole-paper-translated-an-interview-with-co-author-andrew-strominger/|website = Scientific American Blog Network|access-date = 2016-01-09}}</ref><ref name=Hawking2016>{{Cite journal |title = Soft Hair on Black Holes |arxiv = 1601.00921 |date = 2016-01-05|first1 = Stephen W.|last1 = Hawking|first2 = Malcolm J.|last2 = Perry|first3 = Andrew|last3 = Strominger |doi=10.1103/PhysRevLett.116.231301 |pmid = 27341223 |volume=116 |issue = 23 |pages = 231301 |journal=Physical Review Letters |bibcode=2016PhRvL.116w1301H|s2cid = 16198886 }}</ref><ref>{{Cite web|last=Castelvecchi |first=Davide |title = Hawking's Latest Black Hole Paper Splits Physicists (Nature) |url=http://www.scientificamerican.com/article/hawking-s-latest-black-hole-paper-splits-physicists/ |date=27 January 2016 |work=[[Scientific American]] |access-date=31 October 2020}}</ref> 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.<ref name="giddings1995">{{cite conference |first=Steven B. |last=Giddings |title=The black hole information paradox |arxiv=hep-th/9508151 |book-title=Particles, Strings and Cosmology |year=1995 |conference=Johns Hopkins Workshop on Current Problems in Particle Theory 19 and the PASCOS Interdisciplinary Symposium 5|bibcode = 1995hep.th....8151G }}</ref><ref name="preskill1992">{{cite conference |first=John |last=Preskill |title=Do Black Holes Destroy Information? |conference=International Symposium on Black Holes, Membranes, Wormholes, and Superstrings |arxiv=hep-th/9209058 |year=1992|bibcode = 1993bhmw.conf...22P }}</ref> 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.<ref name="nikolic_JCAP2015">{{cite journal|last=Nikolic|first=Hrvoje|title=Violation of unitarity by Hawking radiation does not violate energy-momentum conservation|volume=2015|issue=4|year=2015|pages=002|arxiv=1502.04324|bibcode = 2015JCAP...04..002N |doi = 10.1088/1475-7516/2015/04/002|journal=Journal of Cosmology and Astroparticle Physics|s2cid=44000069}}</ref> 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]].<ref>{{cite arXiv|eprint=1011.3706|title=Concentric circles in WMAP data may provide evidence of violent pre-Big-Bang activity|first1=V. G.|last1=Gurzadyan|first2=R.|last2=Penrose|author2-link=Roger Penrose|year=2010|class=astro-ph.CO}}</ref> The significance of these findings was debated.<ref>{{cite journal|arxiv=1012.1268|title=A search for concentric circles in the 7-year WMAP temperature sky maps|journal=The Astrophysical Journal|volume=733|issue=2|pages=L29|first1=I. K.|last1=Wehus|first2=H. K.|last2=Eriksen|year=2010|doi=10.1088/2041-8205/733/2/L29|bibcode=2011ApJ...733L..29W|s2cid=119284906}}</ref><ref>{{cite journal|arxiv=1012.1305|title=No evidence for anomalously low variance circles on the sky|journal=Journal of Cosmology and Astroparticle Physics|volume=2011|issue=4|pages=033|first1=A. |last1=Moss|first2=D.|last2=Scott|first3=J. P.|last3=Zibin|year=2010|doi=10.1088/1475-7516/2011/04/033|bibcode=2011JCAP...04..033M|s2cid=118433733}}</ref><ref>{{cite journal|arxiv=1012.1656|title=Are There Echoes From The Pre-Big Bang Universe? A Search for Low Variance Circles in the CMB Sky|journal=The Astrophysical Journal|volume=740|issue=2|pages=52|first1=A. |last1=Hajian|year=2010|doi=10.1088/0004-637X/740/2/52|bibcode=2011ApJ...740...52H|s2cid=118515562}}</ref><ref>{{cite arXiv|eprint=1105.1081|title=Comment on "CCC-predicted low-variance circles in CMB sky and LCDM"|first1=H. K.|last1=Eriksen|first2=I. K.|last2=Wehus|year=2010|class=astro-ph.CO}}</ref>

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.<ref>{{Cite journal|title = Black hole evaporation: information loss but no paradox|journal = General Relativity and Gravitation|issn = 1572-9532|volume = 47|issue = 10|pages = 120|doi = 10.1007/s10714-015-1960-y|first1 = Sujoy K.|last1 = Modak|first2 = Leonardo|last2 = Ortíz|first3 = Igor|last3 = Peña|first4 = Daniel|last4 = Sudarsky|arxiv = 1406.4898|bibcode = 2015GReGr..47..120M |year = 2015|s2cid = 118447230}}</ref> This work built on an earlier proposal by Okon and Sudarsky on the benefits of [[objective collapse theory]] in a much broader context.<ref>{{Cite journal|title = Benefits of Objective Collapse Models for Cosmology and Quantum Gravity|journal = Foundations of Physics|issn = 1572-9516|volume = 44|issue = 2|doi = 10.1007/s10701-014-9772-6|first1 = Elias|last1 = Okon|first2 = Daniel|last2 = Sudarsky|arxiv = 1309.1730|pages=114–143|bibcode = 2014FoPh...44..114O |year = 2014|s2cid = 67831520}}</ref> 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).<ref>{{Cite book|title=Three Hundred Years of Gravitation |chapter=Newton, quantum theory and reality|first1=Roger |last1=Penrose |publisher=Cambridge University Press |date=1989 |isbn=9780521379762 |page=17 |chapter-url=http://www.cambridge.org/jp/academic/subjects/astronomy/cosmology-and-relativity/three-hundred-years-gravitation?format=PB&isbn=9780521379762}}</ref><ref>{{Cite journal|title=On Gravity's role in Quantum State Reduction|journal = General Relativity and Gravitation|issn = 1572-9532|volume = 28|issue = 5|doi= 10.1007/BF02105068|first = Roger|last = Penrose|pages=581–600|bibcode = 1996GReGr..28..581P |year=1996|citeseerx=10.1.1.468.2731|s2cid = 44038399}}</ref> Experimental verification of collapse theories is an ongoing effort.<ref>{{Cite journal|title = Models of wave-function collapse, underlying theories, and experimental tests|journal = Rev. Mod. Phys.|issn =1539-0756|volume = 85|issue = 2|first = Angelo|last = Bassi |display-authors=et al.|doi=10.1103/RevModPhys.85.471|pages=471–527|bibcode=2013RvMP...85..471B|arxiv = 1204.4325 |year = 2013|s2cid = 119261020}}</ref>

=== Other proposed resolutions ===
Some other resolutions to the paradox have also been explored. These are listed briefly below.

* Information is stored in a large remnant<ref name="giddings1992">{{cite journal|last=Giddings |first=Steven |title=Black Holes and Massive Remnants|volume=46|issue=4 |year=1992|pages=1347–1352|arxiv=hep-th/9203059|bibcode = 1992PhRvD..46.1347G |doi = 10.1103/PhysRevD.46.1347 |pmid=10015052 |journal=Physical Review D|s2cid=1741527 }}</ref><ref name="nikolic_MPLA2015">{{cite journal|last=Nikolic |first=Hrvoje |title=Gravitational crystal inside the black hole|volume=30|issue=37 |year=2015|pages=1550201|arxiv=1505.04088|bibcode = 2015MPLA...3050201N |doi = 10.1142/S0217732315502016 |journal=Modern Physics Letters A|s2cid=62789858 }}</ref>{{paragraph}}This 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.<ref name="preskill1992"/><ref>{{cite journal | first=Nikodem J. | last=Popławski | year=2010 | title=Cosmology with torsion: An alternative to cosmic inflation | journal=Physics Letters B | volume=694 | issue=3 | pages=181–185 | doi=10.1016/j.physletb.2010.09.056 | arxiv=1007.0587|bibcode = 2010PhLB..694..181P }}</ref>{{paragraph}}Some models of gravity, such as the [[Einstein–Cartan theory]] of gravity, which extends general relativity to matter with intrinsic angular momentum ([[Spin (physics)|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.{{paragraph}}The Einstein&ndash;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 past<ref>{{cite journal|last=Hartle |first=James B.|year=1998|title=Generalized Quantum Theory in Evaporating Black Hole Spacetimes|journal=Black Holes and Relativistic Stars|arxiv=gr-qc/9705022|bibcode = 1998bhrs.conf..195H|pages=195 }}</ref><ref>{{cite journal|last=Nikolic|first=Hrvoje|title=Resolving the black-hole information paradox by treating time on an equal footing with space|volume=678|year=2009|pages=218–221|arxiv=0905.0538|bibcode = 2009PhLB..678..218N |doi = 10.1016/j.physletb.2009.06.029|journal=Physics Letters B|issue=2 |s2cid=15074164}}</ref>{{paragraph}}The final-state proposal<ref>{{cite journal |last1=Horowitz |first1=Gary T |last2=Maldacena |first2=Juan |title=The black hole final state |journal=Journal of High Energy Physics |date=6 February 2004 |volume=2004 |issue=2 |pages=008 |doi=10.1088/1126-6708/2004/02/008|arxiv=hep-th/0310281 |bibcode=2004JHEP...02..008H |s2cid=1615746 }}</ref> 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 theory{{paragraph}}In 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]].<ref name="Bradler">{{Cite journal|title = The capacity of black holes to transmit quantum information|journal = Journal of High Energy Physics|issn = 1029-8479|volume = 2014|issue = 5|pages = 95|doi = 10.1007/JHEP05(2014)095|first1 = Kamil|last1 = Bradler|first2 = Christoph|last2 = Adami|arxiv = 1310.7914 |bibcode = 2014JHEP...05..095B |year = 2014|s2cid = 118353646}}</ref><ref>{{cite journal | last1 = Gyongyosi | first1 = Laszlo | year = 2014 | title = A statistical model of information evaporation of perfectly reflecting black holes | journal = [[International Journal of Quantum Information]] | volume = 12 | issue = 7n08| page = 1560025 | doi=10.1142/s0219749915600254|arxiv = 1311.3598 |bibcode = 2014IJQI...1260025G | s2cid = 5203875 }}</ref>
* Topological Invariants and Recursive Dynamics<ref>{{cite web | last = Austin | first = Peter M. | year = 2024 | title=Resolving the Black Hole Information Paradox with the K-Line Framework|url=https://reason.academia.edu/PeterAustin}}</ref>{{paragraph}}The K-Line Theory offers a novel resolution to the black hole information paradox through a framework based on [[Recursion_(computer_science)|recursive]] state evolution, [[topological invariant]] dynamics, and [[Probabilistic_logic_programming|probabilistic]] [[Equivalence_(measure_theory)|equivalence]]. By employing symmetry indices and dynamic constants, the theory ensures that information is preserved throughout the black hole's lifecycle, including during its evaporation via [[Hawking radiation]]. Unlike previous proposals, which often fail to provide a mechanism for information preservation without introducing confounding drawbacks, K-Line Theory addresses these issues by maintaining a hierarchical structure of states and encoding quantum correlations coherently. Numerical simulations within the K-Line framework produce [[entropy]] behaviours consistent with the Page curve, demonstrating information retrieval and coherence. This approach bridges [[quantum mechanics]] and [[general relativity]] without violating established physical principles, providing a comprehensive mechanism for information conservation in black hole physics.

== Recent developments ==

Significant progress was made in 2019, when, starting with work by Penington<ref>{{cite journal |last1=Penington |first1=Geoffrey |title=Entanglement wedge reconstruction and the information paradox |journal=Journal of High Energy Physics |date=September 2020 |volume=2020 |issue=9 |pages=2 |doi=10.1007/JHEP09(2020)002|arxiv=1905.08255 |bibcode=2020JHEP...09..002P |s2cid=160009640 }}</ref> and Almheiri, Engelhardt, Marolf and Maxfield,<ref>{{cite journal |last1=Almheiri |first1=Ahmed |last2=Engelhardt |first2=Netta |last3=Marolf |first3=Donald |last4=Maxfield |first4=Henry |title=The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole |journal=Journal of High Energy Physics |date=December 2019 |volume=2019 |issue=12 |pages=63 |doi=10.1007/JHEP12(2019)063|arxiv=1905.08762 |bibcode=2019JHEP...12..063A |s2cid=160009599 }}</ref> researchers were able to compute the [[von Neumann entropy]] of the radiation black holes emit in specific models of quantum gravity.<ref name="entrhawk" /><ref name="BHInterior" /><ref name="HREntropy" /><ref>{{cite arXiv |last1=Bousso |first1=Raphael |last2=Dong |first2=Xi |last3=Engelhardt |first3=Netta |last4=Faulkner |first4=Thomas |last5=Hartman |first5=Thomas |last6=Shenker |first6=Stephen H. |last7=Stanford |first7=Douglas |title=Snowmass White Paper: Quantum Aspects of Black Holes and the Emergence of Spacetime |date=2 March 2022 |class=hep-th |eprint=2201.03096}}</ref> These calculations showed that, in these models, the entropy of this radiation first rises and then falls back to zero. As explained above, one way to frame the information paradox is that Hawking's calculation appears to show that the [[von Neumann entropy]] of Hawking radiation increases throughout the lifetime of the black hole. But if the black hole formed from a pure state with zero entropy, unitarity implies that the entropy of the Hawking radiation must decrease back to zero once the black hole evaporates completely, i.e., the Page curve.<ref name="CoxForshaw Page-curve" /> Therefore, the results above provide a resolution to the information paradox, at least in the specific models of gravity considered in these models.

These calculations compute the entropy by first analytically continuing the spacetime to a [[Wick rotation|Euclidean spacetime]] and then using the [[replica trick]]. The [[Path integral formulation|path integral]] that computes the entropy receives contributions from novel Euclidean configurations called "replica wormholes". (These wormholes exist in a [[Wick rotation|Wick rotated spacetime]] and should not be conflated with wormholes in the original spacetime.) The inclusion of these wormhole geometries in the computation prevents the entropy from increasing indefinitely.<ref name="QT-20201030">{{cite news |last=Musser |first=Gerge |title=The Most Famous Paradox in Physics Nears Its End|url=https://www.quantamagazine.org/the-black-hole-information-paradox-comes-to-an-end-20201029/ |date=30 October 2020 |work=[[Quanta Magazine]] |access-date=31 October 2020 }}</ref>

These calculations also imply that for sufficiently old black holes, one can perform operations on the Hawking radiation that affect the black hole interior. This result has implications for the related [[firewall (physics)|firewall paradox]], and provides evidence for the physical picture suggested by the [[ER=EPR]] proposal,<ref name="QT-20201030" /> [[black hole complementarity]], and the [[Papadodimas–Raju proposal.]]

It has been noted that the models used to perform the Page curve computations above have consistently involved theories where the graviton has mass, unlike the real world, where the graviton is massless.<ref>{{cite journal |last1=Geng |first1=Hao |last2=Karch |first2=Andreas |title=Massive islands |journal=Journal of High Energy Physics |date=September 2020 |volume=2020 |issue=9 |pages=121 |doi=10.1007/JHEP09(2020)121|arxiv=2006.02438 |bibcode=2020JHEP...09..121G |s2cid=219304676 }}</ref> These models have also involved a "nongravitational bath", which can be thought of as an artificial interface where gravity ceases to act. It has also been argued that a key technique used in the Page-curve computations, the "island proposal", is inconsistent in standard theories of gravity with a [[Mass in general relativity|Gauss law]].<ref>{{cite journal |last1=Geng |first1=Hao |last2=Karch |first2=Andreas |last3=Perez-Pardavila |first3=Carlos |last4=Raju |first4=Suvrat |last5=Randall |first5=Lisa |last6=Riojas |first6=Marcos |last7=Shashi |first7=Sanjit |title=Inconsistency of islands in theories with long-range gravity |journal=Journal of High Energy Physics |date=January 2022 |volume=2022 |issue=1 |pages=182 |doi=10.1007/JHEP01(2022)182|arxiv=2107.03390 |bibcode=2022JHEP...01..182G |s2cid=235765761 }}</ref> This would suggest that the Page curve computations are inapplicable to realistic black holes and work only in special toy models of gravity. The validity of these criticisms remains under investigation; there is no consensus in the research community.<ref>{{cite arXiv |last1=Bousso |first1=Raphael |last2=Dong |first2=Xi |last3=Engelhardt |first3=Netta |last4=Faulkner |first4=Thomas |last5=Hartman |first5=Thomas |last6=Shenker |first6=Stephen H. |last7=Stanford |first7=Douglas |title=Snowmass White Paper: Quantum Aspects of Black Holes and the Emergence of Spacetime |eprint=2201.03096 |date=2 March 2022|class=hep-th }}</ref><ref>{{cite arXiv |last1=Agrawal |first1=Prateek |last2=Cesarotti |first2=Cari |last3=Karch |first3=Andreas |last4=Mishra |first4=Rashmish K. |last5=Randall |first5=Lisa |last6=Sundrum |first6=Raman |title=Warped Compactifications in Particle Physics, Cosmology and Quantum Gravity |eprint=2203.07533 |date=14 March 2022|class=hep-th }}</ref>

In 2020, Laddha, Prabhu, Raju, and Shrivastava argued that, as a result of the effects of quantum gravity, information should always be available outside the black hole.<ref>{{cite journal |last1=Laddha |first1=Alok |last2=Prabhu |first2=Siddharth |last3=Raju |first3=Suvrat |last4=Shrivastava |first4=Pushkal |title=The Holographic Nature of Null Infinity |journal=SciPost Physics |date=18 February 2021 |volume=10 |issue=2 |pages=041 |doi=10.21468/SciPostPhys.10.2.041|bibcode=2021ScPP...10...41L |s2cid=211044141 |doi-access=free |arxiv=2002.02448 }}</ref> This would imply that the von Neumann entropy of the region outside the black hole always remains zero, as opposed to the proposal above, where the von Neumann entropy first rises and then falls. Extending this, Raju argued that Hawking's error was to assume that the region outside the black hole would have no information about its interior. <ref>{{cite journal |last1=Raju |first1=Suvrat |title=Failure of the split property in gravity and the information paradox |journal=Classical and Quantum Gravity |year=2022 |volume=39 |issue=6 |page=064002 |arxiv=2110.05470 |doi=10.1088/1361-6382/ac482b |bibcode=2022CQGra..39f4002R |s2cid=238583439 }}</ref>

Hawking formalized this assumption in terms of a "principle of ignorance".<ref name="hawkbreak" /> The principle of ignorance is correct in classical gravity, when quantum-mechanical effects are neglected, by virtue of the [[no-hair theorem]]. It is also correct when only quantum-mechanical effects are considered and gravitational effects are neglected. But Raju argued that when both quantum mechanical and gravitational effects are accounted for, the principle of ignorance should be replaced by a "principle of holography of information"<ref name="lessons" /> that would imply just the opposite: all the information about the interior can be regained from the exterior through suitably precise measurements.

The two recent resolutions of the information paradox described above—via replica wormholes and the holography of information—share the feature that observables in the black-hole interior also describe observables far from the black hole. This implies a loss of exact locality in quantum gravity. Although this loss of locality is very small, it persists over large distance scales. This feature has been challenged by some researchers.<ref>{{cite journal |last1=Guo |first1=Bin |last2=Hughes |first2=Marcel R. R. |last3=Mathur |first3=Samir D |last4=Mehta |first4=Mathur |title=Contrasting the fuzzball and wormhole paradigms for black holes |journal=[[Turkish Journal of Physics]] |date=28 December 2021 |volume=45 |issue=6 |pages=281–365 |doi=10.3906/fiz-2111-13 |arxiv=2111.05295 |s2cid=243861186 }}</ref>

==See also==
{{cols|colwidth=20em}}
* [[AdS/CFT correspondence]]
* [[Black hole thermodynamics#Beyond black holes|Beyond black holes]]
* [[Black hole complementarity]]
* [[Cosmic censorship hypothesis]]
* [[Firewall (physics)]]
* [[Fuzzball (string theory)]]
* [[Holographic principle]]
* [[List of paradoxes]]
* [[Maxwell's demon]]
* [[No-hair theorem]]
* [[No-hiding theorem]]
* [[Thorne–Hawking–Preskill bet]]
{{colend}}

==References==
{{reflist|30em}}

==External links==
*[https://web.archive.org/web/20090122223839/http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/info_loss.html Black Hole Information Loss Problem], a USENET physics FAQ page
*{{Cite arXiv|eprint=hep-th/9209058|title=Do black holes destroy information?|first=John|last=Preskill|year=1992}}. Discusses methods of attack on the problem, and their apparent shortcomings.
*{{cite journal |doi=10.1038/news040712-12|title=Hawking changes his mind about black holes|year=2004|last1=Peplow|first1=Mark|journal=Nature}} Report on Hawking's 2004 theory in [[Nature (journal)|Nature]].
* {{cite journal |arxiv=hep-th/0507171|doi=10.1103/PhysRevD.72.084013|title=Information loss in black holes|year=2005|last1=Hawking|first1=S. W.|journal=Physical Review D|volume=72|issue=8|page=084013|bibcode=2005PhRvD..72h4013H|s2cid=118893360}} Stephen Hawking's purported solution to the black hole [[unitarity]] paradox.
* [http://motls.blogspot.com/2005/07/hawking-and-unitarity.html Hawking and unitarity]: a July 2005 discussion of the information loss paradox and Stephen Hawking's role in it
* [http://www.bbc.co.uk/sn/tvradio/programmes/horizon/hawking_prog_summary.shtml The Hawking Paradox - BBC Horizon documentary (2005)]
* {{IMDb title|qid=Q123517619|title="Horizon" The Hawking Paradox}}
* [https://web.archive.org/web/20130829202947/http://www.nytimes.com/2013/08/13/science/space/a-black-hole-mystery-wrapped-in-a-firewall-paradox.html A Black Hole Mystery Wrapped in a Firewall Paradox ]

{{black holes}}
{{quantum gravity}}
{{Stephen Hawking}}

[[Category:Black holes]]
[[Category:Physical paradoxes]]
[[Category:Relativistic paradoxes]]
[[Category:Leonard Susskind]]
[[Category:Stephen Hawking]]