The Black Hole Information Paradox

The positions, and who holds them

Position	Claim	Standing
Information is lost	Hawking's original 1976 conclusion; quantum mechanics is modified by gravity	Almost entirely abandoned; the Banks–Susskind–Peskin energy non-conservation argument is the main obstacle
Remnants	Evaporation halts near the Planck scale, leaving a tiny object holding all the information	Widely rejected: an unbounded number of internal states in a Planck-sized object leads to infinite pair-production rates
Information comes out in the radiation	Subtle correlations, invisible to the semiclassical calculation, carry it	The consensus position; the mechanism is what is missing
Baby universes	The interior pinches off into a disconnected universe carrying the information	Rarely defended now; information is still inaccessible, so unitarity is not really saved for us
Firewalls	The horizon is destroyed at the Page time	Taken seriously as a possibility; most hope it is avoided
Fuzzballs	There is no interior at all; the black hole is a horizon-sized string configuration with no singularity	Actively developed by Samir Mathur and collaborators; the hardest problem is constructing enough microstates for realistic black holes
ER = EPR	Entanglement between the interior and the radiation is a wormhole connecting them	Maldacena and Susskind, 2013; suggestive, not established

What the AMPS argument really forces

It is worth being precise about the firewall paper, because it is often described as a claim that black holes have firewalls. Its actual content is an inconsistency proof. Some assumption must be dropped, and every response identifies a different one.

• Drop the smooth horizon. Accept the firewall. Consistent, but it makes the horizon a special place, which general relativity says it is not.
• Drop effective field theory outside. The "$A = R_B$" proposals of Susskind, and of Papadodimas and Raju, hold that the infalling partner mode is not independent of the early radiation — it is a complicated operator built from it. The interior is state-dependent, which is a serious departure from standard quantum mechanics.
• Drop unitarity. Almost nobody takes this route now.
• Deny that the decoding can be done. Harlow and Hayden showed in 2013 that distilling the entanglement from the early radiation to expose the contradiction would take a time exponential in the black hole's entropy — far longer than the black hole lives. Computational complexity may protect the horizon even if nothing else does. This is an unusual kind of physical argument, and it is taken seriously.

The disagreements that persist

What replica wormholes are. In the models where the calculation is cleanest, the gravitational path integral appears to compute an ensemble average over many boundary theories rather than the properties of a single one. For two-dimensional Jackiw–Teitelboim gravity this connection to random matrix ensembles is precise. Whether that carries over to real four-dimensional gravity, where there is presumably one theory and not an ensemble, is unresolved and is the sharpest current technical worry.

Whether AdS/CFT settles anything about our universe. We live in a universe with a small positive cosmological constant, not a negative one. There is no accepted holographic description of de Sitter space. Arguments that unitarity holds in AdS are strong; the extrapolation is an assumption.

What an infalling observer sees. Genuinely open.

Whether the interior exists at all. The fuzzball programme says no.

Whether the singularity matters. Hawking's original argument leans on the singularity destroying information. If quantum gravity resolves the singularity, part of the paradox's motivation weakens — but the entanglement-counting version of the argument survives without any reference to a singularity.

Why this is one of the most productive problems in physics

Notice what the paradox has produced. Black hole entropy. The holographic principle. AdS/CFT, the most-cited framework in modern theoretical physics. Complementarity. Quantum extremal surfaces. Deep links between entanglement, geometry and computational complexity. Not one of these was the goal; all of them came from people trying to work out where the book went.

1973. Jacob Bekenstein, then a graduate student of John Wheeler's, proposes that black holes carry entropy proportional to horizon area, motivated by the worry that dropping a hot cup of tea into one would otherwise decrease the entropy of the universe. Hawking initially thinks he is wrong.

1974–75. Hawking, attempting to refute Bekenstein, finds instead that black holes emit thermal radiation. The 1974 announcement in Nature is followed by the full paper, "Particle Creation by Black Holes", in 1975. This converts Bekenstein's suggestion into a precise thermodynamics and fixes the coefficient of the area law at $1/4$.

1976. "Breakdown of Predictability in Gravitational Collapse". Hawking argues that the evaporation maps pure states to mixed ones, and concludes that predictability breaks down.

1981. At a small conference in San Francisco hosted by Werner Erhard, Hawking presents the argument to an audience including Susskind and 't Hooft. Susskind later described this as the moment the war began, and dated his own alarm precisely to it.

1984. Banks, Susskind and Peskin show that modifying quantum mechanics to allow pure-to-mixed evolution generically produces catastrophic violations of energy-momentum conservation, closing off the most obvious way to accept Hawking's conclusion.

1993. Susskind, Thorlacius and Uglum propose black hole complementarity. Don Page derives the Page curve. 't Hooft proposes dimensional reduction.

1995. Susskind formulates the holographic principle. Polchinski discovers D-branes.

1996. Andrew Strominger and Cumrun Vafa count the microstates of a class of extremal black holes in string theory and reproduce the Bekenstein–Hawking entropy exactly, including the factor of $1/4$. This is powerful evidence that black holes have ordinary quantum microstates.

1997. Maldacena's AdS/CFT conjecture. In the same year, Hawking and Thorne bet Preskill that information is destroyed; the stake is an encyclopedia of the winner's choice.

July 2004. At GR17 in Dublin, Hawking announces that he has changed his mind and concedes the bet, presenting Preskill with Total Baseball: The Ultimate Baseball Encyclopedia. Thorne does not concede. The written paper appears in 2005. The argument persuades few, but the conclusion is by then widely held on AdS/CFT grounds.

2012. Almheiri, Marolf, Polchinski and Sully post "Black Holes: Complementarity or Firewalls?", showing the consensus picture is inconsistent. The field, which had considered the matter closed, reopens.

2013. Maldacena and Susskind propose ER = EPR. Harlow and Hayden give the computational-complexity argument that the firewall paradox cannot be operated in practice.

2019. Penington posts arXiv:1905.08255; Almheiri, Engelhardt, Marolf and Maxfield post arXiv:1905.08762 within days. Both derive the Page curve using quantum extremal surfaces and islands.

2020. Replica wormholes are identified as the gravitational path-integral origin of the island rule, in papers by Penington, Shenker, Stanford and Yang and by Almheiri, Hartman, Maldacena, Shaghoulian and Tajdini.

2021. The Reviews of Modern Physics review consolidates the results and states clearly what remains unexplained.

A note on the bet. There were two. The 1997 information bet was conceded by Hawking in 2004. A separate 1991 bet with Preskill and Thorne concerned naked singularities. The habit of settling theoretical disputes with wagers was Hawking's, and the bets are documented, though the accompanying stories have grown in the retelling.

It is the sharpest available conflict between our two best theories. General relativity and quantum mechanics do not contradict each other in ordinary circumstances; they simply apply to different regimes. The information paradox is one of the very few places where they give incompatible answers to a well-defined question, which is why it has served as the main laboratory for quantum gravity for half a century.

It produced holography. The observation that black hole entropy scales with area, not volume, led to the holographic principle and then to AdS/CFT. That framework is now used far outside its origin — for strongly coupled plasmas studied at RHIC and the LHC, for models of high-temperature superconductivity, and for the whole "it from qubit" programme relating entanglement to spacetime geometry. None of that was the aim.

It made entanglement entropy a tool of gravitational physics. The Ryu–Takayanagi formula, quantum extremal surfaces and the entanglement wedge came out of this problem. The current best guess about what spacetime is — that geometry emerges from the entanglement structure of an underlying quantum system — is largely a product of trying to work out where the book went.

It disciplined speculation about modifying quantum mechanics. The Banks–Susskind–Peskin result is a useful general lesson: you cannot casually relax unitarity, because unitarity is load-bearing for energy conservation and locality. Proposals to weaken quantum mechanics now have to face that calculation.

Quantum information theory as gravitational physics. Monogamy of entanglement, quantum error-correcting codes, the Page curve, computational complexity: these are the working vocabulary of black hole physics now. The Almheiri–Dong–Harlow observation that AdS/CFT is a quantum error-correcting code is one of the most influential ideas of the last decade, and it originated as a technical step in this problem.

Scrambling and chaos. Black holes are conjectured to be the fastest scramblers in nature, saturating a bound on the growth of chaos found by Maldacena, Shenker and Stanford in 2016. The out-of-time-order correlators used to quantify this are now measured in cold-atom and superconducting-qubit experiments — an idea that started with the question of how quickly a black hole hides information.

Analogue horizons. Jeff Steinhauer's group and others have created acoustic horizons in Bose–Einstein condensates and reported the analogue of Hawking radiation, including correlations between the paired quanta. These test the kinematics of Hawking's mechanism, not the fate of information in real black holes, and that distinction should be kept firmly in view.

Gravitational wave astronomy. LIGO and Virgo observe black hole mergers directly. Proposals in which the horizon is replaced by structure — fuzzballs, firewalls, gravastars — generically predict "echoes" in the ringdown signal after merger. Searches have so far found no convincing evidence, which places observational limits on how radically the horizon can differ from the classical picture. It is a thin thread connecting a fifty-year theoretical argument to data, but it is a real one.

What it teaches about paradoxes. This one has been declared resolved at least three times — by complementarity, by AdS/CFT, by Hawking himself — and each time a sharper question reopened it. It is a standing reminder that "the community agrees on the answer" and "the community can explain the answer" are different achievements, and that only the second one is understanding.

The setup

Let $|\psi\rangle$ be the pure state of collapsing matter. Semiclassical evolution to the final state gives, schematically,

$$|\psi\rangle \longrightarrow \sum_n e^{-\beta E_n/2}, |n\rangle_{\text{in}} \otimes |n\rangle_{\text{out}},$$

an entangled pair state across the horizon with $\beta = 1/T_H$. Tracing out the interior gives the outside observer

$$\rho_{\text{out}} = \operatorname{Tr}_{\text{in}} |\Psi\rangle\langle\Psi| = \frac{1}{Z}\sum_n e^{-\beta E_n} |n\rangle\langle n|,$$

exactly a thermal density matrix. Once the interior is gone, this mixed state is the entire final state. The map $|\psi\rangle\langle\psi| \mapsto \rho_{\text{out}}$ is not unitary; Hawking described it as a superscattering operator $$$ that does not factorise as $S \rho S^\dagger$.

Temperature, entropy, lifetime

For a Schwarzschild black hole of mass $M$:

$$T_H = \frac{\hbar c^3}{8\pi G M k_B}, \qquad S_{BH} = \frac{4\pi G M^2 k_B}{\hbar c} = \frac{A}{4\ell_P^2}k_B .$$

The negative heat capacity, $C = dM/dT < 0$, is what makes evaporation run away. Treating the hole as a black body of area $A \propto M^2$ radiating at $T_H \propto 1/M$ gives

$$\frac{dM}{dt} \propto -\frac{1}{M^2} \quad \Longrightarrow \quad t_{\text{evap}} \sim \frac{5120\pi G^2 M^3}{\hbar c^4},$$

roughly $10^{67}$ years for a solar mass, and about $10^{10}$ years — comparable to the age of the universe — for a primordial black hole of around $10^{11},$kg.

The Page curve, quantitatively

Model the black hole as a system of dimension $d_{BH}$ and the radiation as dimension $d_R$, with the joint state random and pure. Page's theorem gives the average entanglement entropy of the smaller subsystem as very close to maximal:

$$\langle S_R \rangle \approx \log d_R - \frac{d_R}{2 d_{BH}}, \qquad d_R \le d_{BH}.$$

So the entropy of the radiation follows

$$S_R(t) \approx \min\big[S_{\text{rad}}^{\text{thermal}}(t),; S_{BH}(t)\big].$$

Early on the radiation is small and $S_R$ tracks the thermal (Hawking) answer, growing. Late on it is bounded by the shrinking black hole entropy and must decline to zero. The crossover is the Page time, at roughly $t_{\text{evap}}/2$, when about half the initial entropy has been radiated.

Hawking's calculation gives only the rising branch. The monotonic growth of $S_R$ past $S_{BH}$ is the precise statement of the paradox: the radiation is carrying more entanglement entropy than the object it is entangled with can supply.

Monogamy and the firewall

Let $B$ be a late outgoing Hawking mode, $A$ its interior partner, and $R$ the early radiation.

• Unitarity past the Page time requires $B$ to be nearly maximally entangled with a subsystem of $R$ — otherwise $S_R$ cannot decrease.
• A smooth horizon requires the state near the horizon to be the local vacuum, which means $B$ is nearly maximally entangled with $A$.

Monogamy of entanglement forbids both. Formally, strong subadditivity gives

$$S_{AB} + S_{BR} \ge S_{ABR} + S_B,$$

and the two requirements above ($S_{AB} \approx 0$ from purity of the local vacuum pair, $S_{BR} \approx S_R - S_B$ from unitarity) drive this to a contradiction of order $S_B$.

If $B$ is purified by $R$, then the $AB$ pair is not in its vacuum state. Deviation from the vacuum near the horizon means the infalling observer encounters excited high-energy modes — a firewall at the horizon, with energy density set by the cutoff.

The island formula

The resolution replaces the naive entropy with the quantum extremal surface prescription. The generalised entropy of the radiation region $R$ is

$$S(R) = \min_{I} ; \operatorname{ext}{I} \left[ \frac{\text{Area}(\partial I)}{4 G_N} + S{\text{bulk}}(R \cup I) \right],$$

extremised and then minimised over candidate island regions $I$ lying inside the black hole.

Before the Page time the empty island dominates: $I = \emptyset$, the area term vanishes, and the answer is Hawking's rising bulk entropy.

After the Page time a nontrivial island — a region just inside the horizon — gives a smaller value. Now $S_{\text{bulk}}(R \cup I)$ is small, because the island contains precisely the interior partners of the modes in $R$, and the answer is dominated by the area term $\text{Area}(\partial I)/4G_N \approx S_{BH}$, which decreases as the black hole shrinks.

The minimum of the two branches is exactly the Page curve. The turnover is a change of which saddle dominates.

The interpretation is genuinely radical: after the Page time, the interior of the black hole is part of the entanglement wedge of the distant radiation. The interior is not a separate place holding independent information. It is already encoded in the radiation, in a strongly non-local way.

Replica wormholes

Computing $S = -\operatorname{Tr}\rho\log\rho$ via the replica trick requires the Rényi entropies

$$S_n = \frac{1}{1-n}\log \operatorname{Tr}\rho_R^n ,$$

and $\operatorname{Tr}\rho_R^n$ is given by a gravitational path integral over $n$ replicas of the spacetime, with boundary conditions cyclically identified.

The obvious saddle keeps the $n$ replicas disconnected and reproduces Hawking's answer. Penington, Shenker, Stanford and Yang, and independently Almheiri, Hartman, Maldacena, Shaghoulian and Tajdini, showed in 2019–2020 that there is a second family of saddles in which the replicas are joined by wormholes in the bulk. These replica wormholes dominate after the Page time and, on analytic continuation $n \to 1$, produce exactly the island formula.

The essential point is that nothing new was added. These saddles were always present in the gravitational path integral. They were simply not included.

What remains open

• Mechanism. The island formula computes an entropy. It does not exhibit an operator that carries a specific bit from the interior to infinity. The relation between the two is not understood.
• Ensembles. In Jackiw–Teitelboim gravity the path integral including wormholes computes averages over a random matrix ensemble rather than a single theory. What "averaging over theories" means for a universe with one set of laws — the factorisation puzzle — is unsolved.
• Flat space and de Sitter. Most controlled calculations are in AdS or in two-dimensional models with an auxiliary bath. Our universe is neither.
• Firewalls. Whether the horizon is smooth for an infalling observer is not settled by the island results.
• State dependence. The proposals that reconstruct interior operators from the radiation appear to require operators that depend on the state, which is not how observables work in ordinary quantum mechanics.

Progress, not closure.