The Most Famous Paradox in Physics Nears Its End
In a landmark series of calculations, physicists have proved that black holes can shed information, which seems impossible by definition. The work appears to resolve a paradox that Stephen Hawking first described five decades ago.
In a series of breakthrough papers, theoretical physicists have come tantalizingly close to resolving the black hole information paradox that has entranced and bedeviled them for nearly 50 years. Information, they now say with confidence, does escape a black hole. If you jump into one, you will not be gone for good. Particle by particle, the information needed to reconstitute your body will reemerge. Most physicists have long assumed it would; that was the upshot of string theory, their leading candidate for a unified theory of nature. But the new calculations, though inspired by string theory, stand on their own, with nary a string in sight. Information gets out through the workings of gravity itself — just ordinary gravity with a single layer of quantum effects.
This is a peculiar role reversal for gravity. According to Einstein’s general theory of relativity, the gravity of a black hole is so intense that nothing can escape it. The more sophisticated understanding of black holes developed by Stephen Hawking and his colleagues in the 1970s did not question this principle. Hawking and others sought to describe matter in and around black holes using quantum theory, but they continued to describe gravity using Einstein’s classical theory — a hybrid approach that physicists call “semiclassical.” Although the approach predicted new effects at the perimeter of the hole, the interior remained strictly sealed off. Physicists figured that Hawking had nailed the semiclassical calculation. Any further progress would have to treat gravity, too, as quantum.
That is what the authors of the new studies dispute. They have found additional semiclassical effects — new gravitational configurations that Einstein’s theory permits, but that Hawking did not include. Muted at first, these effects come to dominate when the black hole gets to be extremely old. The hole transforms from a hermit kingdom to a vigorously open system. Not only does information spill out, anything new that falls in is regurgitated almost immediately. The revised semiclassical theory has yet to explain how exactly the information gets out, but such has been the pace of discovery in the past two years that theorists already have hints of the escape mechanism.
“That is the most exciting thing that has happened in this subject, I think, since Hawking,” said one of the co-authors, Donald Marolf of the University of California, Santa Barbara.
“It’s a landmark calculation,” said Eva Silverstein of Stanford University, a leading theoretical physicist who was not directly involved.
You might expect the authors to celebrate, but they say they also feel let down. Had the calculation involved deep features of quantum gravity rather than a light dusting, it might have been even harder to pull off, but once that was accomplished, it would have illuminated those depths. So they worry they may have solved this one problem without achieving the broader closure they sought. “The hope was, if we could answer this question — if we could see the information coming out — in order to do that we would have had to learn about the microscopic theory,” said Geoff Penington of the University of California, Berkeley, alluding to a fully quantum theory of gravity.
What it all means is being intensely debated in Zoom calls and webinars. The work is highly mathematical and has a Rube Goldberg quality to it, stringing together one calculational trick after another in a way that is hard to interpret. Wormholes, the holographic principle, emergent space-time, quantum entanglement, quantum computers: Nearly every concept in fundamental physics these days makes an appearance, making the subject both captivating and confounding.
And not everyone is convinced. Some still think that Hawking got it right and that string theory or other novel physics has to come into play if information is to escape. “I’m very resistant to people who come in and say, ‘I’ve got a solution in just quantum mechanics and gravity,’” said Nick Warner of the University of Southern California. “Because it’s taken us around in circles before.”
But almost everyone appears to agree on one thing. In some way or other, space-time itself seems to fall apart at a black hole, implying that space-time is not the root level of reality, but an emergent structure from something deeper. Although Einstein conceived of gravity as the geometry of space-time, his theory also entails the dissolution of space-time, which is ultimately why information can escape its gravitational prison.
The Curve Becomes the Key
In 1992, Don Page and his family spent their Christmas vacation house-sitting in Pasadena, enjoying the swimming pool and watching the Rose Parade. Page, a physicist at the University of Alberta in Canada, also used the break to think about how paradoxical black holes really are. His first studies of black holes, when he was a graduate student in the ’70s, were key to his adviser Stephen Hawking’s realization that black holes emit radiation — the result of random quantum processes at the edge of the hole. Put simply, a black hole rots from the outside in.
The particles it sheds appear to carry no information about the interior contents. If a 100-kilogram astronaut falls in, the hole grows in mass by 100 kilograms. Yet when the hole emits the equivalent of 100 kilograms in radiation, that radiation is completely unstructured. Nothing about the radiation reveals whether it came from an astronaut or a lump of lead.
That’s a problem because, at some point, the black hole emits its last ounce and ceases to be. All that’s left is a big amorphous cloud of particles zipping here and there at random. It would be impossible to recover whatever fell in. That makes black hole formation and evaporation an irreversible process, which appears to defy the laws of quantum mechanics.
Hawking and most other theorists at the time accepted that conclusion — if irreversibility flouted the laws of physics as they were then understood, so much the worse for those laws. But Page was perturbed, because irreversibility would violate the fundamental symmetry of time. In 1980 he broke with his former adviser and argued that black holes must release or at least preserve information. That caused a schism among physicists. “Most general relativists I talked to agreed with Hawking,” said Page. “But particle physicists tended to agree with me.”
On his Pasadena vacation, Page realized that both groups had missed an important point. The puzzle wasn’t just what happens at the end of the black hole’s life, but also what leads up to it.
He considered an aspect of the process that had been relatively neglected: quantum entanglement. The emitted radiation maintains a quantum mechanical link to its place of origin. If you measure either the radiation or the black hole on its own, it looks random, but if you consider them jointly, they exhibit a pattern. It’s like encrypting your data with a password. The data without the password is gibberish. The password, if you have chosen a good one, is meaningless too. But together they unlock the information. Maybe, thought Page, information can come out of the black hole in a similarly encrypted form.
Page calculated what that would mean for the total amount of entanglement between the black hole and the radiation, a quantity known as the entanglement entropy. At the start of the whole process, the entanglement entropy is zero, since the black hole has not yet emitted any radiation to be entangled with. At the end of the process, if information is preserved, the entanglement entropy should be zero again, since there is no longer a black hole. “I got curious how the radiation entropy would change in between,” Page said.
Initially, as radiation trickles out, the entanglement entropy grows. Page reasoned that this trend has to reverse. The entropy has to stop rising and start dropping if it is to hit zero by the endpoint. Over time, the entanglement entropy should follow a curve shaped like an inverted V.
Page calculated that this reversal would have to occur roughly halfway through the process, at a moment now known as the Page time. This is much earlier than physicists assumed. The black hole is still enormous at that point — certainly nowhere near the subatomic size at which any putative exotic effects would show up. The known laws of physics should still apply. And there is nothing in those laws to bend the curve down.
With that, the problem got much more acute. Physicists had always figured that a quantum theory of gravity came into play only in situations so extreme that they sound silly, such as a star collapsing to the radius of a proton. Now Page was telling them that quantum gravity mattered under conditions that, in some cases, are comparable to those in your kitchen.
Page’s analysis justified calling the black hole information problem a paradox as opposed to merely a puzzle. It exposed a conflict within the semiclassical approximation. “The Page-time paradox seems to point to a breakdown of low-energy physics in a place where it has no business breaking down, because the energies are still low,” said David Wallace, a philosopher of physics at the University of Pittsburgh.
On the bright side, Page’s clarification of the problem paved the way to a solution. He established that, if entanglement entropy follows the Page curve, then information gets out of the black hole. In doing so, he transformed a debate into a calculation. “Physicists are not always so good at words,” said Andrew Strominger of Harvard University. “We do best with sharp equations.”
Now physicists just had to calculate the entanglement entropy. If they could pull it off, they’d get a straight answer. Does the entanglement entropy follow an inverted V or not? If it does, the black hole preserves information, which means particle physicists were right. If it doesn’t, the black hole destroys or bottles up information, and general relativists can help themselves to the first doughnut at faculty meetings.
Yet even though Page spelled out what physicists had to do, it took theorists nearly three decades to figure out how.
The Inside-Out Black Hole
Over the past two years, physicists have shown that the entanglement entropy of black holes really does follow the Page curve, indicating that information gets out. They did the analysis in stages. First, they showed how it would work using insights from string theory. Then, in papers published last fall, researchers cut the tether to string theory altogether.
The work began in earnest in October 2018, when Ahmed Almheiri of the Institute for Advanced Study laid out a procedure for studying how black holes evaporate. Almheiri, joined soon by several colleagues, applied a concept first developed by Juan Maldacena, now at IAS, in 1997. (Penington was working in parallel.)
Consider a universe encased in a boundary like a snow globe. Apart from having a big wall around it, the interior is basically like our universe: It has gravity, matter, and so forth. The boundary, too, is a kind of universe. It does not have gravity and, being just a surface, lacks depth. But it makes up for that with vibrant quantum physics, and all in all it’s exactly as complex as the interior. Different though these two universes may look, they are perfectly matched. Everything in the interior, or “bulk,” has a counterpart on the boundary. And even though the geometry of the bulk is unlike the geometry of our own universe, this “AdS/CFT” duality has been string theorists’ favorite playground ever since Maldacena introduced it.
By the logic of this duality, if you have a black hole in the bulk, it has a simulacrum on the boundary. Because the boundary is governed by quantum physics without the complications of gravity, it unequivocally preserves information. So must the black hole.
When researchers set out to analyze how black holes evaporate in AdS/CFT, they first had to overcome a slight problem: In AdS/CFT, black holes do not, in fact, evaporate. Radiation fills the confined volume like steam in a pressure cooker, and whatever the hole emits it eventually reabsorbs. “The system will reach a steady state,” said Jorge Varelas da Rocha, a theoretical physicist at the University Institute of Lisbon.
To deal with that, Almheiri and his colleagues adopted a suggestion of Rocha’s to put the equivalent of a steam valve on the boundary to bleed off the radiation and prevent it from falling back in. “It sucks the radiation out,” said Netta Engelhardt of the Massachusetts Institute of Technology, one of Almheiri’s co-authors. The researchers plopped a black hole at the center of the bulk space, began bleeding off radiation, and watched what happened.
To track the entanglement entropy of the black hole, they drew on the more granular understanding of AdS/CFT that Engelhardt and others, including Aron Wall at the University of Cambridge, have developed in the past decade. Physicists are now able to pinpoint which part of the bulk corresponds to which part of the boundary, and which properties of the bulk correspond to which properties of the boundary.
The key to relating the two sides of the duality is what physicists call a quantum extremal surface. (These surfaces are general features — you don’t need a black hole to have one.) Basically you imagine blowing a soap bubble in the bulk. The bubble naturally assumes a shape that minimizes its surface area. The shape need not be round, like the bubbles at a child’s birthday party, because the rules of geometry can differ from the ones we are familiar with; thus the bubble is a probe of that geometry. Quantum effects can distend it, too.
By calculating where the quantum extremal surface lies, researchers obtain two important pieces of information. First, the surface carves the bulk into two pieces and matches each to a portion of the boundary. Second, the area of the surface is proportional to part of the entanglement entropy between those two portions of the boundary. Thus the quantum extremal surface relates a geometric concept (area) to a quantum one (entanglement), providing a glimpse into how gravity and quantum theory might become one.
But when researchers used these quantum extremal surfaces to study an evaporating black hole, a strange thing happened. Early in the evaporation process, they found, as expected, that the entanglement entropy of the boundary rose. Because the hole was the only thing inside space, the authors deduced that its entanglement entropy was rising. In terms of Hawking’s original calculations, so far so good.
Suddenly that changed. A quantum extremal surface abruptly materialized just inside the horizon of the black hole. Initially this surface had no effect on the rest of the system. But eventually it became the deciding factor for entropy, leading to a drop. The researchers compare it to a transition like boiling or freezing. “We think of this as a change in phase analogous to thermodynamic phases — between gas and liquid,” Engelhardt said.
It meant three things. First, the sudden shift signaled the onset of new physics not covered by Hawking’s calculation. Second, the extremal surface split the universe in two. One part was equivalent to the boundary. The other was a here-be-dragons realm about which the boundary had no information, indicating that bleeding radiation from the system was having an effect on its information content.
Third, the position of the quantum extremal surface was highly significant. It was located just inside the horizon of the black hole. As the hole shrank, so did the quantum extremal surface and, with it, the entanglement entropy. That would produce the downward slope that Page predicted — the first time any calculation had done that.
By showing that the entanglement entropy tracked the Page curve, the team was able to confirm that black holes release information. It dribbles out in a highly encrypted form made possible by quantum entanglement. In fact, it is so encrypted that it doesn’t look as if the black hole has given up anything. But eventually the black hole passes a tipping point where the information can be decrypted. The research, posted in May 2019, showed all this using new theoretical tools that quantify entanglement in a geometric way.
Even with these tools, the calculation had to be stripped to its essence to be doable. The bulk in this AdS/CFT universe had just a single dimension of space, for example. The black hole was not a big black ball but a short line segment. Still, the researchers argued, gravity is gravity, and what goes for this impoverished Lineland should hold for the real universe. (In April 2020, Koji Hashimoto, Norihiro Iizuka and Yoshinori Matsuo of Osaka University analyzed black holes in a more realistic flat geometry and confirmed that the findings still hold.)
In August 2019 Almheiri and another set of colleagues took the next step and turned their attention to the radiation. They found that the black hole and its emitted radiation both follow the same Page curve, so that information must be transferred from one to the other. The calculation does not say how it is transferred, only that it is.
As part of the work, they discovered that the universe undergoes a baffling rearrangement. At the outset, the black hole is at the center of space and the radiation is flying out. But after enough time has passed, the equations say, particles deep inside the black hole are no longer part of the hole anymore, but part of the radiation. They have not flown outward, but simply been reassigned.
This is significant because these interior particles would ordinarily contribute to the entanglement entropy between the black hole and the radiation. If they are not part of the black hole anymore, they no longer contribute to the entropy, explaining why it begins to decrease.
The authors dubbed the inner core of radiation the “island” and called its existence “surprising.” What does it mean for particles to be in the black hole, but not of the black hole? In confirming that information is retained, the physicists eliminated one puzzle only to create an even bigger one. Whenever I asked Almheiri and others what it meant, they looked off into the distance, momentarily lost for words.
Enter the Wormholes
So far the calculations presumed the AdS/CFT duality — the snow globe world — which is an important test case but ultimately somewhat contrived. The next step was to consider black holes more generally.
The researchers drew on a concept that Richard Feynman had developed in the 1940s. Known as the path integral, it is the mathematical expression of a core quantum mechanical principle: Anything that can happen does happen. In quantum physics, a particle going from point A to point B takes all possible paths, which are combined in a weighted sum. The highest-weighted path is generally the one you’d expect from ordinary classical physics, but not always. If the weights change, the particle can abruptly lurch from one path to another, undergoing a transition that would be impossible in old-fashioned physics.
The path integral works so well for particle motion that theorists in the ’50s proposed it as a quantum theory of gravity. That meant replacing a single space-time geometry with a mélange of possible shapes. To us, space-time appears to have a single well-defined shape — near Earth, it is curved just enough that objects tend to orbit the center of our planet, for example. But in quantum gravity, other shapes, including much curvier ones, are latent, and they can make an appearance under the right circumstances. Feynman himself took up this idea in the ’60s, and Hawking championed it in the ’70s and ’80s. But even their considerable genius struggled with how to execute the gravitational path integral, and physicists set it aside in favor of other approaches to quantum gravity. “We never really knew how to define exactly what it is — and guess what, we still don’t,” said John Preskill of the California Institute of Technology.
For starters, what are “all” possible shapes? For Hawking, that meant all topologies. Space-time might knot itself into doughnut- or pretzel-like shapes. The extra connectivity creates tunnels, or “wormholes,” between otherwise far-flung places and moments. These come in different types.
Spatial wormholes are like the portals beloved of science-fiction writers, linking one star system to another. So-called space-time wormholes are little universes that bud off our own and reunite with it sometime later. Astronomers have never seen either type, but general relativity permits these structures, and the theory has a good track record of making seemingly bizarre predictions, such as black holes and gravitational waves, that are later vindicated. Not everyone agreed with Hawking that these exotic shapes belong in the mix, but the researchers doing the new analyses of black holes adopted the idea provisionally.
They couldn’t realistically consider all possible topologies, which are literally uncountable, so they looked only at those that were most important to an evaporating black hole. These are known, for mathematical reasons, as saddle points, and they look like fairly placid geometries. In the end, the teams didn’t actually perform the full summation of shapes, which was beyond them. They used the path integral mostly as a vehicle to identify the saddle points.
The next step, after applying the path integral to the black hole and its radiation, was to calculate the entanglement entropy. This quantity is defined as the logarithm of a matrix — an array of numbers. The calculation is difficult in the best of times, but in this case the physicists didn’t actually have the matrix, which would have required evaluating the path integral. So they had to perform an operation they couldn’t do on a quantity they didn’t know. For that, they busted out another mathematical trick.
They noticed that entropy doesn’t require knowledge of the full matrix. They could instead imagine performing a repeated series of measurements on the black hole and then combining those measurements in a way that retained the knowledge they needed. This so-called replica trick goes back to the study of magnets in the ’70s and was first applied to gravity in 2013.
One of the authors of the new work, Tom Hartman of Cornell University, compared the replica trick to checking whether a coin is fair. Normally you’d toss it many times and see whether it lands on each side with 50-50 probability. But suppose for some reason you can’t do that. So instead you toss two identical coins — the “replicas” — and note how often they land on the same side. If this happens half the time, the coins are fair. Even though you still don’t know the individual probabilities, you can make a basic judgment about randomness. This is analogous to not knowing the full matrix for the black hole, yet still evaluating its entropy.
Trick though it is, it has real physics in it. The gravitational path integral doesn’t distinguish replicas from a real black hole. It takes them literally. This activates some of the latent topologies that the gravitational path integral includes. The result is a new saddle point containing multiple black holes linked by space-time wormholes. It competes for influence with the regular geometry of a single black hole surrounded by a mist of Hawking radiation.
The wormholes and the single black hole are inversely weighted by, basically, how much entanglement entropy they have. Wormholes have a lot, so they receive a low weighting and are thus unimportant at first. But their entropy decreases, whereas that of the Hawking radiation keeps climbing. Eventually the wormholes become the dominant of the two, and they take over the dynamics of the black hole. The shift from one geometry to the other is impossible in classical general relativity — it is an inherently quantum process. The extra geometric configuration and the transition process that accesses it are the two main discoveries of the analysis.
In November 2019, two teams of physicists — known as the West Coast and East Coast groups for their geographical affiliations — posted their work showing that this trick allows them to reproduce the Page curve. In this way, they confirmed that the radiation spirits away the informational content of whatever falls into the black hole. String theory needn’t be true; even a staunch critic of string theory can get on board with the gravitational path integral. Still, as sophisticated as the analysis is, it doesn’t yet say how the information makes its getaway.
The Construction of Space-Time
By these calculations, the radiation is rich in information. Somehow, by measuring it, you should be able to learn what fell into the black hole. But how?
Theorists in the West Coast group imagined sending the radiation into a quantum computer. After all, a computer simulation is itself a physical system; a quantum simulation, in particular, is not altogether different from what it is simulating. So the physicists imagined collecting all the radiation, feeding it into a massive quantum computer, and running a full simulation of the black hole.
And that led to a remarkable twist in the story. Because the radiation is highly entangled with the black hole it came from, the quantum computer, too, becomes highly entangled with the hole. Within the simulation, the entanglement translates into a geometric link between the simulated black hole and the original. Put simply, the two are connected by a wormhole. “There’s the physical black hole and then there’s the simulated one in the quantum computer, and there can be a replica wormhole connecting those,” said Douglas Stanford, a theoretical physicist at Stanford and a member of the West Coast team. This idea is an example of a proposal by Maldacena and Leonard Susskind of Stanford in 2013 that quantum entanglement can be thought of as a wormhole. The wormhole, in turn, provides a secret tunnel through which information can escape the interior.
Theorists have been intensely debating how literally to take all these wormholes. The wormholes are so deeply buried in the equations that their connection to reality seems tenuous, yet they do have tangible consequences. “It’s hard to answer what’s physical and what’s unphysical,” said Raghu Mahajan, a physicist at Stanford, “because there’s something clearly right about these wormholes.”
But rather than think of the wormholes as actual portals sitting out there in the universe, Mahajan and others speculate that they are a sign of new, nonlocal physics. By connecting two distant locations, wormholes allow occurrences at one place to affect a distant place directly, without a particle, force or other influence having to cross the intervening distance — making this an instance of what physicists call nonlocality. “They seem to suggest that you have nonlocal effects that come in,” Almheiri said. In the black hole calculations, the island and radiation are one system seen in two places, which amounts to a failure of the concept of “place.” “We’ve always known that some kind of nonlocal effects have to be involved in gravity, and this is one of them,” Mahajan said. “Things you thought were independent are not really independent.”
At first glance, this is very surprising. Einstein constructed general relativity with the express purpose of eliminating nonlocality from physics. Gravity does not reach out across space instantly. It has to propagate from one place to another at finite speed, like any other interaction in nature. But over the decades it has dawned on physicists that the symmetries on which relativity is based create a new breed of nonlocal effects.
This past February, Marolf and Henry Maxfield, also at Santa Barbara, studied the nonlocality implied by the new black hole calculations. They found that the symmetries of relativity have even more extensive effects than commonly supposed, which may give space-time the hall-of-mirrors quality seen in the black hole analyses.
All this reinforces many physicists’ hunch that space-time is not the root level of nature, but instead emerges from some underlying mechanism that is not spatial or temporal. To many, that was the main lesson of the AdS/CFT duality. The new calculations say much the same thing, but without committing to the duality or to string theory. Wormholes crop up because they are the only language the path integral can use to convey that space is breaking down. They are geometry’s way of saying the universe is ultimately nongeometric.
The End of the Beginning
Physicists not involved in the work, or even in string theory, say they are impressed, if duly skeptical. “Hats off to them, since those calculations are highly nontrivial,” said Daniele Oriti of the Ludwig Maximilian University of Munich.
But some feel uneasy about the tottering pile of idealizations used in the analysis, such as the restriction of the universe to less than three spatial dimensions. The previous wave of excitement over the path integral in the ’80s, driven by Hawking’s work, fizzled out in part because theorists were unnerved by the accumulation of approximations. Are today’s physicists falling into the same trap? “I see people make the same hand-waving arguments that were made 30 years ago,” said Renate Loll of Radboud University in the Netherlands, an expert on the gravitational path integral. She has argued that wormholes need to be expressly forbidden if the integral is to give sensible results.
Skeptics also worry that the authors have overinterpreted the replica trick. In supposing that replicas can be connected gravitationally, the authors go beyond past invocations of the maneuver. “They are postulating that all geometries connecting different replicas are allowed, but it’s not clear how that fits into the framework of quantum rules,” said Steve Giddings of Santa Barbara.
Given the uncertainties of the calculation, some are unconvinced that a solution is available within semiclassical theory. “There’s no good choice if you restrict to quantum mechanics and gravity,” Warner said. He has championed models in which stringy effects prevent black holes from forming in the first place. But the upshot is broadly similar: Space-time undergoes a phase transition to a very different structure.
Skepticism is warranted if for no other reason than because the recent work is complicated and raw. It will take time for physicists to digest it and either find a fatal flaw in the arguments or become convinced that they work. After all, even the physicists behind the efforts didn’t expect to resolve the information paradox without a full quantum theory of gravity. Indeed, they thought the paradox was their fulcrum for prying out that more detailed theory. “If you had asked me two years ago, I would have said: ‘The Page curve — that’s a long way away,’” Engelhardt said. “We’re going to need some kind of [deeper] understanding of quantum gravity.’”
But assuming that the new calculations stand up to scrutiny, do they in fact close the door on the black hole information paradox? The recent work shows exactly how to calculate the Page curve, which in turn reveals that information gets out of the black hole. So it would seem as though the information paradox has been overcome. The theory of black holes no longer contains a logical contradiction that makes it paradoxical.
But in terms of making sense of black holes, this is at most the end of the beginning. Theorists still haven’t mapped the step-by-step process whereby information gets out. “We now can compute the Page curve, and I don’t know why,” said Raphael Bousso at Berkeley. To astronauts who ask whether they can get out of a black hole, physicists can answer, “Sure!” But if the astronauts ask how to do it, the disquieting reply will be: “No clue.”
This article was reprinted on Wired.com.