The black hole information paradox has fascinated and perplexed theoretical
physicists for almost 50 years. However, in a succession of ground-breaking
articles, they have tantalizingly close to solving it. They may now
confidently claim that information does indeed escape a black hole. If you
enter one, you won't be lost forever. The knowledge required to reconstruct
your body will emerge, particle by particle. Since string theory was their
top contender for a unified theory of nature, most physicists have long
thought that it would. However, although being motivated by string theory,
the new computations are entirely independent and lack any reference to
strings. Information is transmitted through gravity's natural mechanisms —
merely regular gravity with a thin overlay of quantum effects.
Gravity has taken on an odd new function in this scenario. A black hole's
tremendous gravity prevents anything from escaping, according to Einstein's
general theory of relativity. This idea was uncontested in the more complex
theory of black holes that Stephen Hawking and his associates came up with
in the 1970s. A hybrid technique known as "semiclassical" physics was used
by Hawking and others to describe matter in and around black holes using
quantum theory while continuing to describe gravity using Einstein's
classical theory. The technique suggested fresh effects near the hole's
edge, but the inside remained completely walled off. Scientists believed
Hawking had successfully performed the semiclassical computation. Any future
development would have to include gravity as a quantum phenomenon.
The authors of the new research contest that. They have discovered new
gravitational combinations that Hawking did not include but that Einstein's
theory allows. These are known as semiclassical effects. These effects first
appear muted, but when the black hole ages significantly, they start to take
center stage. The hermit kingdom in the pit changes into a ferociously open
system. Not only does knowledge leak out, but everything new that enters is
often instantly regurgitated. Although the new semiclassical theory hasn't
yet explained how precisely the information escapes, theorists already have
signs of the escape mechanism because to how quickly discoveries have been
made over the previous two years.
One of the co-authors, Donald Marolf from the University of California,
Santa Barbara, remarked, "That is the most fascinating thing that has
happened in this area, in my opinion, since Hawking."
Leading theoretical physicist Eva Silverstein of Stanford University, who
was not directly engaged, called the result "a milestone calculation."
The authors claim that while you may anticipate them to rejoice, they also
feel disappointed. It could have been much more challenging to complete the
computation if it had incorporated the deep characteristics of quantum
gravity rather than only a light dusting, but once completed, it would have
shed light on those depths. They are concerned that they could have found a
solution to this specific issue but not the comprehensive closure they
desired. In reference to a completely quantum theory of gravity, Geoff
Penington of the University of California, Berkeley remarked, "The goal was
that if we could answer this issue — if we could see the information flowing
out — in order to accomplish so we would have had to learn about the
microscopic theory."
In Zoom calls and webinars, there is a heated discussion over what it all
means. The piece is quite mathematical and has a Rube Goldberg feel to it,
tying together one calculational trick after another in a way that is
difficult to understand. Wormholes, the holographic principle, emergent
space-time, quantum entanglement, and quantum computers are just a few of
the modern concepts that occur in the field of basic physics, making it both
fascinating and perplexing.
Moreover, not everyone is persuaded. Some people still believe that Hawking
was correct and that new physics, like as string theory, must be used in
order for information to escape. Nick Warner of the University of Southern
California remarked, "I'm quite skeptical to folks who come in and say,
'I've got a solution with simply quantum mechanics and gravity. "Because it
has already led us in circles."
But it seems like everyone can agree on one thing. Space-time appears to
break apart near a black hole in some way, suggesting that it is not the
foundation of reality but rather an emerging structure from something
deeper. Despite the fact that Einstein defined gravity as the geometry of
space-time, his theory also calls for the breakdown of space-time, which is
ultimately the reason why information may evade the gravitational hold that
it has on it.
The Key Is in the Curve
Don Page and his family spent Christmas 1992 house-sitting in Pasadena,
where they took advantage of the pool and Rose Parade. The break was also
utilized by Page, a physicist at the University of Alberta in Canada, to
reflect on how paradoxical black holes actually are. His early research on
black holes during his doctoral studies in the 1970s was crucial in helping
his advisor Stephen Hawking realize that radiation from black holes is
caused by arbitrary quantum processes near the hole's edge. A black hole
rots from the outside in, to put it simply.
Its shed particles don't seem to have any information about what's within.
The hole's mass increases by 100 kilograms if an astronaut weighing 100
kilograms falls in. However, the radiation that the hole produces—the
equivalent of 100 kilograms—is wholly unstructured. The radiation has no
characteristics that indicate whether it originated from an astronaut or a
lead lump.
That's an issue since the black hole eventually discharges its final bit of
energy and vanishes. All that is left is a sizable cloud of amorphous
particles that are randomly flitting around. Whatever fell in would not be
recoverable. As a result, the creation and dissolution of black holes are
irreversible processes that seem to contradict the principles of quantum
physics.
At the time, Hawking and the majority of other theorists came to the same
conclusion: if irreversibility violated the then-understood physical rules,
so much the worse for those laws. But Page was troubled because
irreversibility would go against time's basic symmetry. In 1980, he diverged
from his old mentor and asserted that information must be released from
black holes, or at the very least preserved. That split the physics
community in two. The majority of general relativists with whom Page spoke
concurred with Hawking. Particle scientists, though, tended to concur with
me.
Page discovered on his Pasadena trip that both groups had overlooked a
crucial detail. The mystery was not simply what happened when the black
hole's life came to an end, but also what occurred in the meantime.
He thought about quantum entanglement, a part of the process that has
received very little attention. The radiation that is released still has a
quantum mechanical connection to where it came from. Both the radiation and
the black hole appear random when measured separately, but when combined,
they show a pattern. It's similar to using a password to secure your data.
Without the password, the data is meaningless. If you picked a strong
password, it will also be meaningless. But by working together, they can
access the data. Maybe information may leave the black hole in a similarly
encrypted fashion, Page reasoned.
Page determined what it would entail for the overall entanglement—also
referred to as the entanglement entropy—between the black hole and the
radiation. Since the black hole has not yet released any radiation to become
entangled with, the entanglement entropy is 0 at the beginning of the whole
process. Since there is no longer a black hole at the conclusion of the
operation, provided information is retained, the entanglement entropy should
be zero once more. Page stated, "I started wondering how the radiation
entropy might alter in between.
Initially, the entanglement entropy increases as radiation slowly leaks
out. Page concluded that this pattern must change. If the entropy is to
reach zero at the terminus, it must cease increasing and begin to decline.
The entanglement entropy should follow a curve with an inverted V form over
time.
At a point now referred to as the Page time, Page determined that this
reversal would have to happen around halfway through the process. This
happened a lot sooner than scientists anticipated. At that time, the black
hole is still massive; it is undoubtedly a long way from the subatomic level
at which any alleged exotic effects would manifest. The recognized rules of
physics need to remain applicable. Furthermore, nothing in those regulations
can make the slope bend downward.
That made the issue even worse. Scientists had previously believed that a
quantum theory of gravity only applied in absurdly extreme circumstances,
such as when a star shrank to the size of a proton. Now Page was telling
them that under some circumstances, conditions similar to those in your
kitchen, quantum gravity mattered.
According to Page's view, the black hole information problem should be
classified as a paradox rather than just a conundrum. A contradiction in the
semiclassical approximation was made clear. David Wallace, a philosopher of
physics at the University of Pittsburgh, stated that the Page-time paradox
"seems to hint to a breakdown of low-energy physics at a place where it has
no business breaking down, because the energies are still low."
On the plus side, Page's explanation of the issue opened the door to a fix.
He demonstrated that information leaves the black hole if entanglement
entropy follows the Page curve. He converted a discussion into a computation
by doing this. Scientists aren't usually great writers, according to Harvard
University's Andrew Strominger. "Sharp equations are ideal for us,"
It was now just necessary for physicists to compute the entanglement
entropy. They would receive a direct response if they were successful. Does
the entropy of entanglement exhibit an inverted V pattern or not? If it
occurs, the black hole keeps the data, proving the particle scientists
correct. If not, information is either destroyed or trapped in the black
hole, and general relativists are welcome to take the first doughnut during
faculty meetings.
It took theorists over three decades to find out how to accomplish it,
despite the fact that Page clearly outlined what physicists needed to
do.
The Black Hole From the Inside
In the past two years, scientists have demonstrated that black holes'
entanglement entropy truly does follow the Page curve, proving that
information may leave the black hole. The analysis was carried out in
phases. Initially, they used concepts from string theory to demonstrate how
it might function. Then, in articles released last November, physicists
completely severed their ties to string theory.
When Ahmed Almheiri of the Institute for Advanced Study described a method
for examining how black holes evaporate, the work really got going.
Almheiri, shortly joined by a number of colleagues, put into practice a
proposal created in 1997 by Juan Maldacena, who is currently employed at
IAS. Penington was simultaneously working.
Think of a cosmos that is enclosed in a border, much like a snow globe. The
interior is essentially identical to our universe, with the exception of a
large wall surrounding it. It possesses gravity, matter, and other
properties. The border is a form of universe as well. It lacks depth and
gravity since it is only a surface. However, lively quantum physics makes up
for that, and overall, it is just as intricate as the inside. Despite how
unlike these two universes appear, they are a great complement. Everything
on the boundary has a counterpart in the inside, or "bulk." And since since
Maldacena proposed it, this "AdS/CFT" duality has been string theorists'
favorite playground, despite the fact that the geometry of the bulk differs
from the geometry of our own world.
According to the logic of this duality, a black hole has a counterpart on
the boundary if it exists in the bulk. It clearly maintains information
since the border is determined by quantum physics without the difficulties
of gravity. The dark hole must also.
In order to study how black holes evaporate in AdS/CFT, researchers first
had to solve a little issue: black holes don't really evaporate in AdS/CFT.
Like steam in a pressure cooker, radiation fills the small space, and
whatever the hole emits ultimately absorbs it again. Jorge Varelas da Rocha,
a theoretical physicist at the University Institute of Lisbon, predicted
that the system will approach a stable state.
Almheiri and his coworkers followed Rocha's proposal to install the
equivalent of a steam valve on the border to drain off the radiation and
stop it from coming back in to address this issue. One of Almheiri's
co-authors, Netta Engelhardt of the Massachusetts Institute of Technology,
remarked, "It suctions the radiation out." The scientists created a black
hole in the center of the large volume of space, started to release
radiation, and then watched what occurred.
They used on the more detailed knowledge of AdS/CFT that Engelhardt and
others, notably Aron Wall at the University of Cambridge, had acquired in
the last ten years to track the entanglement entropy of the black hole.
Today, physicists are able to identify which portion of the bulk and which
portion of the boundary correspond to one another, as well as which
characteristics of the bulk and which characteristics of the frontier.
The quantum extremal surface, as it is known in physics, is the connection
between the two halves of the duality. (These surfaces are generic
characteristics; a black hole is not necessary for them to exist.) In
essence, picture blowing a soap bubble in the large. Naturally, the bubble
takes on a form that reduces its surface area. Because the laws of geometry
might be different from those we are accustomed with, the form need not be
circular, like the bubbles at a kid's birthday party; instead, the bubble is
a probe of that geometry. Quantum effects can also lengthen it.
The location of the quantum extremal surface can provide researchers with
two crucial pieces of knowledge. The bulk is split into two pieces by the
surface, which then aligns each with a different section of the boundary.
Second, a fraction of the entanglement entropy between those two regions of
the border is proportional to the surface's area. By connecting a geometric
idea (area) with a quantum concept (entanglement), the quantum extremal
surface offers a look into how gravity and quantum theory can merge.
But an odd thing happened when scientists utilized these quantum extremal
surfaces to analyze an evaporating black hole. They discovered that the
boundary's entanglement entropy increased early on in the evaporation
process, as they had predicted. The authors concluded that the hole's
entanglement entropy was increasing since it was the solitary object inside
of space. So far, so good for Hawking's initial estimates.
That abruptly changed. Just inside the black hole's horizon, a quantum
extremal surface suddenly appeared. At first, the remainder of the system
was unaffected by this surface. But ultimately it took over as the entropy
determining factor, causing a decline. It is compared to a transition like
boiling or freezing by the researchers. According to Engelhardt, "We think
of this as a shift in phase akin to thermodynamic phases—between gas and
liquid.
It had three meanings. First, Hawking's estimate did not account for the
unexpected change, which indicated the emergence of new physics. Second, the
cosmos was split in half by the extremal surface. The border was represented
by one component. The other was a world of here-be-dragons, about which the
border had no information, suggesting that radiation leaking from the system
was having an impact on the informational content of the realm.
Third, the quantum extremal surface's location was very important. It was
situated just within the black hole's horizon. The quantum extremal surface
and entanglement entropy decreased together with the hole's reduction. The
first time a computation has done that was when it produced the downward
slope Page expected.
The team was able to demonstrate that the entanglement entropy matched the
Page curve and so proved that black holes do indeed emit information. The
quantum entanglement that makes this possible allows it to leak out in a
highly encrypted form. In fact, the information is so securely encrypted
that it appears the black hole has not divulged anything. However, the black
hole ultimately reaches a breaking point where the data can be deciphered.
The study, which was published in May 2019, demonstrated all of this using
new theoretical techniques that geometrically quantify entanglement.
The math has to be simplified to the bare minimum even using these tools.
For instance, there was just one spatial dimension in the majority of this
AdS/CFT world. The black hole was a brief line segment rather than a large
black ball. However, the scholars maintained that gravity is gravity and
that what applies to this underdeveloped Lineland should apply to the rest
of the universe. (In April 2020, researchers from Osaka University Koji
Hashimoto, Norihiro Iizuka, and Yoshinori Matsuo examined black holes in a
more accurate flat geometry and proved the findings are still true.)
Almheiri and a different group of colleagues moved on to studying the
radiation in August 2019. They discovered that the black hole and the
radiation it emits both exhibit the same Page curve, indicating that data
must be sent from one to the other. The computation just states that it is
transferred; it makes no mention of how.
They learned as part of their research that the cosmos goes through a
perplexing reconfiguration. The black hole is in the center of space at this
point, and radiation is escaping. The calculations predict that once enough
time has elapsed, particles located inside the black hole's interior become
a component of the radiation rather than the hole itself. They have just
been transferred, not flown overseas.
This is noteworthy because normally these inner particles would increase
the entropy of the entanglement between the radiation and the black hole.
They no longer contribute to the entropy if they are no longer a member of
the black hole, which explains why it starts to fall.
The inner core of radiation was referred to by the authors as a "island"
and they found its existence to be "surprising." What does it imply that
particles may exist in a black hole but not be a part of it? The physicists
were able to establish that information is kept, but in doing so they simply
managed to make the riddle greater. Almheiri and the others would gaze off
into the distance whenever I questioned them about what it meant since they
were at a loss for words.
Go through the Wormholes
The AdS/CFT duality, often known as the snow globe world, has been assumed
in the computations thus far. While this is a vital test case, it is
ultimately somewhat fabricated. The next stage was to have a broader view of
black holes.
The scientists used a theory that Richard Feynman created in the 1940s. It
is the mathematical representation of a fundamental quantum mechanical
concept and is known as the path integral: Everything that is possible does
occur. A particle traveling from point A to point B in quantum physics
travels all conceivable pathways, which are added together to form a
weighted total. Generally speaking, but not always, the highest-weighted
path is the one you would anticipate from standard classical physics. The
particle can abruptly switch from one route to another if the weights are
altered, which is not feasible according to conventional physics.
Because the path integral predicts particle motion so well, it was proposed
as a quantum theory of gravity in the 1950s. That required swapping out a
single space-time geometry for a variety of potential forms. To humans,
space-time appears to have a single, well defined geometry; for instance, it
is just curved enough near Earth such that things often orbit its center.
However, alternative forms, including ones that are more curvier, are latent
in quantum gravity and can emerge under the correct situations. This theory
was first advanced by Feynman in the 1960s, and by Hawking in the 1970s and
1980s. The gravitational path integral was difficult to implement even with
their considerable genius, so physicists abandoned it in favor of other
theories of quantum gravity. John Preskill of the California Institute of
Technology remarked, "We never really understood how to explain exactly what
it is—and guess what, we still don't.
What are "all" potential forms, to begin with? That meant all topologies to
Hawking. It's possible that space-time can form knots that resemble
doughnuts or pretzels. The increased connectedness makes "wormholes" or
tunnels connecting otherwise distant places and times. There are several
varieties of them.
Similar to the beloved science fiction authors' portals, spatial wormholes
connect different star systems. Little universes that sprout from our own
and eventually reconnect with it are known as space-time wormholes. Although
general relativity allows for these structures, astronomers have never
observed either type. However, the theory has a strong track record of
making predictions that later turn out to be correct, such as those
regarding black holes and gravitational waves. Although the experts doing
the new analysis of black holes did not unanimously agree with Hawking that
these strange forms belonged in the mix, they took the notion proviso.
They only focused on the topologies that were crucial to an evaporating
black hole since they could not practically take into account all potential
topologies, which are figuratively uncountable. These are referred to as
saddle points for mathematical reasons, and they have a calm-appearing
geometry. In the end, the teams were unable to complete the whole summation
of forms since it was above their capabilities. To find the saddle points,
they mainly employed the path integral as a tool.
The entanglement entropy was calculated after the route integral was
applied to the black hole and its radiation. The logarithm of a matrix, or
collection of numbers, is what this quantity is described as. Even under the
best of circumstances, the computation is challenging, but in this instance
the scientists didn't really have the matrix, which would have necessitated
determining the route integral. As a result, they were forced to carry out a
surgery on an unknown amount. They used another mathematical ruse to do
it.
They discovered that entropy can be calculated without knowing the entire
matrix. Instead, they can consider repeatedly measuring the black hole and
integrating those data in a way that preserves the information they want.
This "replica trick" was first used in relation to gravity in 2013 and dates
back to the 1970s' research of magnets.
Tom Hartman of Cornell University, one of the authors of the new study,
compared the replica technique to determining if a coin is fair. Normally,
you'd throw it repeatedly to determine whether it had a 50/50 chance of
landing on each side. But imagine you're unable to accomplish it for any
reason. The "replicas" are two identical coins that you toss in its place,
and you observe how frequently they land on the same side. The coins are
fair if it occurs 50% of the time or less. Even if you are still unsure
about each probability, you can still generalize about randomness. This is
comparable to not knowing the entire black hole matrix but nevertheless
calculating its entropy.
Despite being trick, it uses actual physics. The gravitational path
integral cannot tell fake black holes apart from real ones. They are taken
literally. Some of the latent topologies included in the gravitational route
integral are therefore activated. The outcome is a brand-new saddle point
with numerous black holes connected by wormholes in space-time. It faces up
against the regular geometry of a solitary black hole encircled by a Hawking
radiation mist.
In essence, the amount of entanglement entropy determines how heavily
weighted the wormholes and the single black hole are. Wormholes are given a
low weighting since they have a lot, making them initially uninteresting.
However, compared to the Hawking radiation, their entropy is decreasing. The
dynamics of the black hole are eventually controlled by the wormholes, which
eventually emerge as the dominant of the two. In classical general
relativity, changing from one geometry to another is not conceivable since
it is a fundamentally quantum process. The two primary findings of the
investigation are the additional geometric configuration and the transition
procedure that accesses it.
Due to their geographical ties, the West Coast and East Coast groups of
physicists published their research in November 2019 demonstrating how they
were able to replicate the Page curve using this technique. This allowed
them to establish that anything enters the black hole loses its
informational content due to radiation. Even a fervent opponent of string
theory can agree with the gravitational path integral; string theory need
not be true. Even so, the analysis doesn't yet explain how the information
escapes, despite how sophisticated it is.
Building Blocks of Space-Time
These calculations show that the radiation contains a wealth of data. You
ought to be able to determine what dropped into the black hole some way by
measuring it. Yet how?
The West Coast group of theorists conjured up the idea of directing the
radiation into a quantum computer. A computer simulation is a physical
system in and of itself; a quantum simulation is particularly similar to the
phenomenon it is modeling. Therefore, the scientists conjured up the idea of
gathering all the radiation, putting it into a huge quantum computer, and
then simulating the black hole in its entirety.
And the narrative took an amazing turn as a result. The quantum computer
also gets extremely entangled with the black hole since the radiation is so
intricately linked to the source of it. Entanglement between the real and
simulated black holes in the simulation results in a geometric connection.
Simply said, a wormhole connects the two. According to Douglas Stanford, a
theoretical physicist at Stanford and a member of the West Coast team,
"there's the actual black hole and then there's the simulated one in the
quantum computer, and there can be a replica wormhole linking those." This
concept is an illustration of the hypothesis put out by Stanford researchers
Maldacena and Leonard Susskind in 2013 that quantum entanglement may be
viewed as a wormhole. In turn, the wormhole offers a covert passageway via
which information can leave the interior.
How literally to interpret all of these wormholes has been a hotly
contested topic among theorists. Although the wormholes seem to have a
fragile relationship to reality since they are so deeply buried in the
equations, they do have real-world repercussions. "There's something plainly
correct about these wormholes," said Raghu Mahajan, a physicist at Stanford,
"but it's impossible to say what's real and what's unphysical."
Mahajan and others believe that the wormholes are a result of new, nonlocal
physics rather than being actual portals that exist somewhere in the cosmos.
Wormholes allow events at one place to directly affect another place,
without a particle, force, or other influence having to travel across the
intervening distance. This is an example of what physicists refer to as
nonlocality. Almheiri said, "They seem to imply that you have nonlocal
effects that come in. The island and radiation are treated as one system in
the black hole computations, which amounts to a failure of the notion of
"location." We've long known that nonlocal effects of some type must play a
role in gravity, and this is one of them, according to Mahajan. "What you
believed to be independent is not actually independent."
This seems quite shocking at first. General relativity was developed by
Einstein specifically to do away with nonlocality in physics. It takes time
for gravity to extend throughout space. Like every other interaction in
nature, it must spread from one area to another at a finite rate. But over
the years, physicists have realized that the symmetries that relativity is
built on give rise to a new type of nonlocal phenomena.
Marolf and Henry Maxfield, both of Santa Barbara, explored the nonlocality
suggested by the new black hole estimates this past February. They
discovered that the general relativity's symmetries have considerably more
significant impacts than previously thought, which may explain why
space-time has the hall-of-mirrors appearance observed in black hole
investigations.
All of this supports the hypothesis of many physicists that space-time is
not the fundamental unit of existence but rather develops from a deeper
process that is neither spatial nor temporal. That was the main takeaway
from the AdS/CFT duality, according to many. Although they avoid endorsing
either string theory or the duality, the new calculations essentially say
the same thing. Wormholes appear because they are the only way for the path
integral to express how space is collapsing. They serve as geometry's way of
expressing the nongeometric nature of the universe.
The conclusion of the first
Physicists who are not involved in the project or even with string theory
claim to be impressed, albeit cautiously. Since those computations are
extremely difficult, Daniele Oriti of the Ludwig Maximilian University of
Munich stated, "hats off to them.
However, some people find the analysis's shaky foundation of idealizations
unsettling, particularly the limitation of the cosmos to less than three
spatial dimensions. The 1980s enthusiasm surrounding the path integral,
which was sparked by Hawking's work, eventually died down in part due to
theorists' unease with the growing body of approximations. Are modern
physicists making the same mistake? Expert on the gravitational path
integral Renate Loll of Radboud University in the Netherlands remarked, "I
see people making the same hand-waving arguments that were made 30 years
ago." She has claimed that in order for the integral to produce accurate
findings, wormholes must be explicitly disallowed.
Additionally, some are concerned that the writers have misunderstood the
replica technique. The authors go farther than earlier references to the
move by proposing that copies might be gravitationally coupled. Steve
Giddings of Santa Barbara remarked, "It's unclear how that fits into the
context of quantum principles, but they are postulating that any geometries
linking distinct copies are acceptable.
Some don't believe that semiclassical theory can provide a solution because
of the calculation's uncertainties. If you limit yourself to quantum physics
and gravity, there is no good option, Warner stated. He has defended
theories in which stringy effects stop black holes from ever forming. But
the result is essentially the same: Space-time changes phases and assumes a
significantly different composition.
If for no other reason than the complexity and rawness of the current work,
skepticism is justified. It will take some time for physicists to process it
and determine if the arguments have a fatal defect or whether they are
sound. Even the scientists who led the project didn't think it was possible
to solve the information dilemma without a complete quantum theory of
gravity. In fact, they believed that the paradox would serve as their pivot
point for obtaining that more in-depth theory. The Page curve is far off, I
would have said if you had asked me two years ago, Engelhardt added. We'll
require some sort of [deeper] comprehension of quantum gravity.
Do the new calculations, provided they pass examination, really put an end
to the black hole information paradox? The most current research
demonstrates how to compute the Page curve precisely, demonstrating how
information might escape a black hole. It appears that the information
dilemma has been solved as a result. There is no longer a logical
inconsistency that renders the black hole hypothesis absurd.
But in terms of understanding black holes, this is only the beginning of
the end. The exact steps used by information to spread have still not been
sketched out by theorists. Raphael Bousso from Berkeley said, "I don't know
why, but we can now compute the Page curve. Astronauts who inquire as to
whether they can escape a black hole can receive the affirmative from
physicists. However, the unsettling response will be "No clue" if the
astronauts inquire as to how to proceed.
