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Black holes are inevitable predictions of general relativity—our best theory of space,
time and gravity. But they clash in multiple ways with quantum mechanics,
our equally successful description of the subatomic world. One such clash is the
black hole information paradox—and a proposed solution—black hole complementarity—may forced
us to radically rethink what it even means for something to exist.
We know that our universe is fundamentally self-consistent—otherwise what are we even
doing trying to science it. But that means when a contradiction appears in our scientific
description of the universe, we know something is wrong with that description. It’s really exciting
when this happens because the nature of the inconsistency can point the way to a better, more
encompassing scientific description. Black holes are one of the favourite tools of the theoretical
physicist because they lead to multiple inconsistencies between general relativity
and quantum mechanics, and so may be our best path to the grander theory that unites the two.
The conflict we looked at recently is the black hole information paradox, and it’s not a bad
idea to watch that episode before this. In it, our intrepid heroes of the gedankenexperiment,
Alice and Bob, discovered that black holes must violate either a fundamental principle of general
relativity or of quantum mechanics. When Alice carries a quantum bit—a qubit—into a black hole,
she witnesses the qubit cross the event horizon. She must, because according to
the equivalence principle—a founding axiom of general relativity—Alice can’t sense anything
unusual when crossing that horizon. Meanwhile, Bob, watching from afar, has to either witness
the qubit escape in the Hawking radiation leaked as the black hole evaporates, or the qubit vanish forever
in that evaporation. In the first case, the qubit is duplicated—it’s both inside
and outside the black hole. In the second it’s annihilated. Either way, a foundational principle
of quantum mechanics—conservation of quantum information or unitarity appears to be violated.
Because both the equivalence principle and unitarity are fundamental to their
respective theories, we know something must be wrong with our understanding of what happens
to quantum information in a black hole. There have been various proposed solutions,
but today I’m going to focus on one of the earliest, and perhaps the least intuitive—and
that’s black hole complementarity, formulated by Leonard Susskind and others in the early 90s.
Black hole complementarity states that there actually is no contradiction. It proposes that
it’s fine for the same quantum information to be inside the black hole as measured by one observer,
and frozen on the surface or radiated away in Hawking radiation according to
another. And according to “BHC” this is fine because no one can ever observe both states,
so no one can prove that unitarity was broken, which means it … wasn’t?
To get our heads around this, let’s start by making the conflict much more precise. To do
that we’re going to use the same black hole map that proper black hole theorists like to use—the
Penrose diagram. Without a black hole, a Penrose diagram looks like this. Up is roughly speaking
the forward time direction, and left and right are roughly one spatial dimension. But space and
time are rescaled so they bend into each other and pile up towards the boundaries. Tick marks
are drawn closer together so that the border of the graph represents infinite distance and infinite past or future. And
all of this is done in just such a way so that light will always travel a 45 degree path. All
sub-lightspeed travel has to take a steeper slope—more time taken to travel less space.
Near an event horizon of a black hole we can think of spacetime as being
infinitely stretched from the point of view of a distant observer. That means we can just say
one of these boundaries is our event horizon, and add the interior of the black hole on the
other side. In these coordinates, the central singularity looks like the top of the Penrose
diagram is cut off—that represents the cessation of space and time inside the black hole. For now
you’ll have to take my word that this is a valid way to draw the spacetime of a universe with a
black hole, but we have other videos on the Penrose diagram if you need more convincing.
Let’s see what Alice’s black hole expedition looks like on the Penrose diagram. Both She
and Bob move up in time, while Alice and the qubit also move closer to the event
horizon. Light from the qubit reaches Alice and then Bob, carrying information about the
qubit’s location—its past location by the time Alice or Bob see it.
Approaching the event horizon, those photons still reach Alice quickly but take longer and
longer to reach Bob. Photons traveling from just above the event horizon only reach Bob
in the far future. No photon emitted below the event horizon can ever reach Bob, so to him
the qubit and Alice are frozen just above the event horizon. Those photons emitted
inside the black hole are doomed to hit the singularity, as is the qubit. As is Alice.
The diagram we’ve been using is for a black hole that’s always been there and always
will be. Real black holes typically form from collapsed stars, and they also leak
Hawking radiation until they disappear. Here’s how we might depict such a black hole. We have
a Penrose diagram for the universe where the black hole forms somewhere in space when a
collapsing star forms an event horizon. Then it evaporates by Hawking radiation. We only
need half of this map because, well, nothing going in one side ever comes out the other.
Let’s look at just the qubit’s path. According to both Alice and Bob it falls and reaches the
event horizon. According to just Alice it enters the black hole and hits the singularity. Bob,
on the other hand, sees it freeze on the horizon and emerge again as Hawking radiation.
It emerges only after the black hole is at least half-evaporated because, according to physicist
Don Page, before that point the information in the emitted radiation is hopelessly scrambled.
From a perspective outside space and time, the quantum bit in some sense exists at all of these
spacetime points, but does it ever exist in two physical locations simultaneously—at the
same instant time? Well there’s no absolute definition of “simultaneous” in Einstein’s
relative universe. But these lines on the Penrose diagram could be considered to describe different
spatial locations at the same moment in time. Therefore, for anything duplicated
on one of these lines, the copies can be thought of as existing at the same time.
So, before it hits the event horizon there’s only one qubit. After the black
hole evaporates there’s only one qubit—the one leaked out in Hawking photons. But
between its entry into the black hole and the black hole’s evaporation we can argue
that the qubit exists simultaneously in two places, violating unitarity.
The key to this is to really dig into what we mean by “existing simultaneously”. Due
to the finite travel time of light, we can only confirm simultaneous existence at two
spacetime points after the light from both reaches us. On our original Penrose diagram,
we only have information about the parts of the universe from which signals traveling at the
speed of light or lower could reach us—that’s our past light-cone. This is the only region in which
we can verify simultaneity—and we can only verify that things existed simultaneously after the fact.
But if we try to do that for our duplicated qubits, we see that there is no past light
cone—no possible observer—who can ever verify that both exist at the same time. Alice sees one,
Bob sees the other, but no one can ever see both. Black hole complementarity argues that
the impossibility of any one observer measuring both qubits means that there’s
no violation of unitarity, so there’s no contradiction. Before we pick that apart,
let’s make sure it’s really impossible for any one observer to see both the Hawking-radiated
and the swallowed versions of the qubit. Physicists Bill Hayden and John Preskill
figured out the best chance of one observer seeing both. The thought experiment goes like this:
Alice jumps into the black hole with the quantum bit just before the black hole is half evaporated
because she knows that only after this so-called Page time can quantum information get back out.
Below the event horizon she tries to send the qubit upwards. She knows it can’t re-cross the
event horizon, but it will slow the qubit’s descent to give Bob more time to catch it.
And now Bob also drops into the black hole. He times the leap exquisitely so that he catches
the Hawking-radiated qubit on its way out, and hopes to also see the swallowed
qubit once inside. And … he misses it. Even with the most perfectly timed experiment,
Hayden and Preskill show that Bob will always barely miss being able to see both qubits.
So it seems that nature is working awfully hard to make it impossible for anyone to
see both versions of the qubit. So maybe the unobservability of the cloned qubits
is telling us something fundamental. That would be the argument of black hole complementarity,
which states that, because it’s impossible for anyone to observe
both qubits, there’s no contradiction—no violation of unitarity—for both to exist.
This sounds like some sort of weird quantum stuff. And complementarity
is indeed fundamental to quantum mechanics. For example, there are complementary quantum
properties like position and momentum that can never be measured perfectly at the same
time. Or complementary descriptions like the wave-like versus particle-like behavior of
a quantum object. The word complementarity implies a connection to quantum mechanics,
but the connection isn’t clear. For black hole complementarity there are different
interpretations, which are still argued over, and which aren’t even necessarily mutually exclusive.
So interpretation 1): If black hole complementarity is right,
it may be telling us that, while unitarity and the conservation of quantum information always hold,
the way they hold is relative to a given observer. Alice will always find that quantum
mechanics works perfectly and that there are never any contradictions. So will Bob,
but for him quantum mechanics might work perfectly in a different way. The key is
that if Alice and Bob can never communicate, no contradiction is ever discovered. If this is
right then it’s telling us something about what the wavefunction and quantum information really
represent and that our description of the world depends quite radically on our reference frame.
Even a concept as basic as the existence of a quantum state may be relative to the observer.
It’s not the only hint at this uncomfortable idea—for example, the existence of a particle can
depend on the acceleration of an observer, as we saw in our Unruh radiation episode. Interestingly,
both Unruh radiation and black hole complementarity involve uncrossable horizons.
So interpretation 2) for black hole complementarity is that the interior and exterior descriptions of
the quantum information are, in a sense, equivalent. Or rather they are different
descriptions of what is fundamentally the one quantum system. There’s no duplication because
the interior and exterior of the black hole are different descriptions of the same abstract
quantum system. This is a form of holography, in which a lower dimensional system can be
equally-well described as a system one dimension higher—a boundary and its interior become
different ways to talk about the same thing. We’ve talked about holographic principle before—it’s an
idea that can be extended well beyond black holes, even suggesting that the interior of
our universe may have a dual and complementary description on its infinitely distant surface.
Black hole complementarity is by no means the accepted solution to the black hole
information paradox. We haven’t talked about black hole firewalls yet—in which
an extreme energy screen just above the event horizon fries anything that tries to enter,
eliminating any duplication of qubits but also violating the equivalence principle. In an episode
coming very soon we’ll see why some physicists think the firewall must exist, and also why the
firewall may not free us from the strangeness of black hole complementarity or vice versa.
So, yeah, black holes are contradictions. They are holes in the universe and in our
understanding of it. But through those holes we’re glimpsing grander visions of what our
universe might really be. For example, that there’s a sort of extreme relativity to our
description of the world, or that the interiors of black holes and of universes may be in a
weird way equivalent to their surfaces—each the warped reflection of a complementary spacetime.
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