Can Black Holes Unify General Relativity & Quantum Mechanics?

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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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along with this shout out to be included on the  event horizon of the Cygnus X-1 black hole. Now,  

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