Can Black Holes Unify General Relativity & Quantum Mechanics? (Script) (Patreon)

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 to say that something to exists.

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We know that our universe is fundamentally self-consistent—otherwise what are we even doing trying to science it. 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 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 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 necessarily mutually exclusive. 

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.

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 same 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 extremely 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.