Transcription
So basically, what seems to be required to predict the fate of a black hole is to say something about the future as well as the past. Now, that, at first sight, sounds crazy and radical and so on, which it is. But in this two-sided cosmology, it's absolutely natural.
Because in the two-sided cosmology, we have the future coming out of the Big Bang, the future universe, the past universe coming out in the opposite direction. Now, really, these two are mirror images of each other, because the final condition is the same by CPT symmetry, or it's related by CPT. So the one is literally the mirror image of the other.
So what I can do is fold the lower universe, think about it as a sort of cone coming out of the Big Bang. So, fold it up so that it doubles the upper cone. Now what I have is what we call a two-sheeted universe. We've got this, and it's just like the particle-antiparticle pair. Imagine if you really put those two things on top of each other. This double-sided universe is like the universe-anti-universe pair, and they're parallel. You can think of them as being parallel to each other.
You see, the picture is very beautiful. It says that the future universe you should think about as a sheet, as one of two sheets. And there's, if you like, the past universe is the other sheet. Now, what goes on when you make a black hole? Well, literally, you cut a triangle out of the future sheet, and the same thing happens on the past sheet. And those two cut triangles are put on top of each other like this, and there's nothing in between. There's just a seam where they join, where the two sheets join. So the black hole horizon is the seam. There's nothing inside the black hole. There's a hole in this double universe. And then when the black hole evaporates, the whole thing rejoins and the black hole goes away, and we're left with two sheets again. So the formation of a black hole is literally just the sticking together of the past and the future universe in which the section that's stuck together is just eliminated. It doesn't exist. It's literally a hole in this double picture. But all you have on the sides of the hole are a seam.
Okay, I have some technical questions, but people who are watching, before I get to them, they may be wondering, what happens to me as I fall toward the black hole? So what happens to me in the traditional picture prior to this paper? Right. And then what happens in your view, or in you and your collaborators' view? Brilliant. Yes, exactly.
So the traditional picture is that you would experience nothing special at all. As you cross the horizon, you're sitting in your spaceship. You see, the matter of when you cross the horizon is, there are actually different definitions of when you cross the horizon. Because the horizon is a somewhat subjective notion, in the sense that if I'm trying to communicate from my spaceship to another spaceship that's, let's say, further out from the black hole, depending on exactly where that spaceship is, I may or may not be able to send signals. So when I cross the horizon, the usual definition of what's called the event horizon is that when I cross the surface, I cannot communicate to someone at infinity, infinitely far away from the black hole. No signal I send will ever reach infinity. But if someone's nearer, you may be able to communicate with them. And so there's something called the event horizon, there's something called the apparent horizon. This is a surface which Roger Penrose defined in his proof that black hole formation is inevitable. And his definition was much more physical. It was that if you imagine sending out light rays in this space-time where the black hole is forming, there will be some of them, some of those shells of light rays will start reconverging. And when they reconverge, they can never diverge again. So basically, when the outgoing light rays start to converge, you can call that when the black hole is formed locally. And so that's called the apparent horizon. So there's still this ambiguity about exactly where the horizon would be. Our best guess would be, so in the conventional picture, nothing happens at all. You just fall across the horizon. Okay, some of your signals... Both horizons. It doesn't matter if it's apparent or if it's an event. It doesn't matter. In the standard picture, it doesn't matter at all because locally you have no idea whether your signals are ever going to reach somebody. It's not something that concerns you at all. You might send a signal and nobody ever receives it, but so what? You don't experience anything in the standard picture. You just fall across the horizon and nothing happens to you at all. What happens next is very dramatic because you then inevitably fall into the singularity and get crushed. So that's the standard picture. Nothing exceptional happens at the horizon, at either horizon at all. The horizons, by definition, are just where light either fails to make it off to infinity or the outgoing light rays start to reconverge. And in fact, that doesn't really affect you at all either, because it's a very global property. It's not something you could measure locally.
Okay, so when does this crushing occur that people see in sci-fi movies? And where's the hypercube from Interstellar? Oh, it's at the singularity. Okay, so in Interstellar, the assumption was that they went into the black hole, and then something very spectacular happens at the singularity itself. Now, the truth is that no one has a clue how to make sense of a curvature singularity in general relativity. What happens is that space shrinks in one direction and blows up in orthogonal directions. So typically it shrinks in one and blows up in two, or shrinks in two and blows up in one. And that's just sort of a catastrophic failure of a theory. The whole picture of space-time gets stretched and crushed alternately. In fact, there's something that happens there called Mixmaster Chaos. And the Mixmaster was a machine in the 1960s, which is a food blender. Okay? So, some company, I'm not sure which, maybe it was General Electric, made Mixmasters. And so, this phenomenon of this space-time in which things get crushed and stretched and crushed and stretched alternately is called Mixmaster Behavior. So that is the classical expectation. And in Interstellar, that doesn't make any sense. Everything goes haywire. So in Interstellar, they replace this by somehow time travel and the ability to communicate, let's say, across time. But nobody really has, I would say, a good physics idea for how to make sense of what happens to you. There are notable attempts by people who study holography and they have a much more radical picture than ours, which is that there are wormholes. I guess it's a little bit like Interstellar. There are wormholes which connect the interior of different black holes and share information across these two black holes. But to be honest, I've never been able to make sense of that picture. It's far more radical than ours.
Okay, so in the traditional picture, you pass these so-called horizons. You don't notice anything as you're passing through, and then eventually you get squeezed into a tube, and then you reach what is called the singularity, the curvature singularity, because just like there are different forms of horizons, there are different types of singularities. Curvature singularity, that's right. So you meet that, and then no one knows what occurs once you meet that. Okay, that's the traditional approach since the 20s, 30s? That's the traditional approach. I would say no, it became accepted after Kruskal analyzed the Schwarzschild metric, which is the metric of a non-rotating, non-charged black hole, the simplest black hole. Kruskal analyzed it and realized that there was a way to analytically continue across the horizon, which left the space-time locally Minkowski everywhere, except at the singularity. So, yeah, the conventional picture was only really, began to be accepted in the 60s. But since then, all the general relativity community has essentially bought the standard picture.
Okay, and now you come in. So, the person listening is wondering, they are falling toward a black hole. What do they see as they're going toward it? And what occurs as they move past the horizon, if they can even move past it? Yes, good.
So essentially nothing happens in this picture until you encounter this special surface. And then something extremely dramatic happens. And this is well before anything would happen in the standard picture. What happens is that you encounter antimatter. You encounter an anti-version of your spaceship containing an anti-version of you. Of yourself. Yes. And the two spaceships would meet, annihilate into radiation, which would then fly up the horizon and off to infinity. So it's extremely dramatic. It could not be more different than the standard picture.
Now, would you even see that other person? Let's say there is no... No, no, no, you can't. You don't have a chance because the way light travels in the space-time forbids you from actually seeing any signal from the other side. Until you hit the horizon. The horizon is the first surface at which I could actually see something coming from the other side. I cannot see it before I hit the horizon. Yeah, in your paper you joined two boundaries, one of sigma plus zero and one of sigma minus zero? Exactly. Exactly. So sigma equals zero is where the two join. And neither side knows anything about the existence of the other side until you hit that special surface. So, yeah, it's a very different picture.
By the way, some ideas which in a certain way anticipated what we did also became popular briefly in the string theory community in the, I guess, 2000s, which was called the firewall. People argued, and this was Peter Polchinski and Don Marolf and others, they argued that because black hole formation violated quantum mechanics so badly in the conventional picture, there had to be a different resolution. So they argued there must be a firewall. There must be something which prevents you from going into the interior. And there was a lot of... These are very smart people and there was a lot of debate about it, but I think it was inconclusive. So our picture, I think, is a better, I would claim, a better motivated mathematical description than a firewall. But something very dramatic is going to happen when you hit the horizon. And it's important to realize that process is quantum. As you hit the horizon, the process of pair annihilation, as I described at the beginning, it cannot happen classically. It's not allowed. It depends on the particles going faster than light for a brief quantum moment. This curve turns around. That's pair annihilation. And what we're claiming is that is exactly the process which saves the black hole in the sense of making it compatible with quantum mechanics. Is that the particles come in from one side, the antiparticles from the other side, they annihilate and sail off as radiation, and there is no interior to the black hole.
So, I imagine that you checked other invariants to make sure there's no other form of curvature like the Kretschmann scalar? Exactly. No, everything is completely regular. All curvature invariants are regular at the horizon. There's nothing new, but all we're saying is actually we found an analytic solution of the Einstein equations which extends as I said up to the horizon of the first exterior and continues onto the horizon of the second exterior without including any interior. I mean, I must say it was very surprising to us that the solution works. We were expecting to find something on the horizon, like a kink in the geometry which forced you to have some kind of stress energy source. This is typically what happens in general relativity. If you try to make a spaceship, for example, which goes faster than light or violates any of the classic principles, you generally find you have to introduce weird forms of matter which kind of allow this behavior. What we found is we didn't have to introduce anything. This is just naturally there in the Einstein theory.
So you don't introduce any odd forms of matter, but there is an odd metric. Is that what psychologically prevented people from coming up with this solution? Yes. Because CPT symmetry is known, and analytical continuation is known. Combining them has this... What is it, an eigenvalue degeneration on the surface? Exactly. Yes. It's a swapping over of eigenvalues. So in the space-time metric, one of the eigenvalues is, let's say, negative and three are positive. Okay? It's a conventional choice whether you make one positive and three negative or one negative and three positive. But let's stick with one negative and three positive. Sure. So what happens when you hit the horizon? The horizon is a two-sphere. And it's completely regular. So that has two positive eigenvalues. And they're all fine. There's nothing weird in those two dimensions. They're perfectly regular geometry. There are two dimensions left. And you can think of them as one of them is the radius and the other one is the time. And what happens is that the eigenvalue of the metric in the time-time direction and the space-space direction... So one was negative, one was positive. What happens is, at the same time, on the horizon the positive one goes negative and the negative one goes positive simultaneously. So space and time effectively switch roles. And that's what happens. And indeed, I think the reason people missed this, though with hindsight Einstein did not miss it as it turns out, it's in his paper. But the reason people missed it starting in the 60s is that they treated the space metric as sacrosanct. It had to be a 4x4 matrix which is symmetric and invertible. And that fails.
Now actually you could say why does the space metric have to have an inverse? I mean it's something we normally use in the mathematics of GR. But I realized, and it was only last week, that actually when you... So, one sort of derivation of general relativity from, let's say, quantum field theory principles is that all you assume is a spin-two particle. Okay? And actually, this derivation goes back to Feynman. Feynman said people are making all this fuss about curved space and geometry. But actually, if we have a spin-2 particle, it travels along and it's spinning around with double the spin of a photon, and we have energy and momentum conservation and relativity, and then we try to see what is the most general possible interaction between these spin-2 particles. You can go through various calculations and you discover, basically, general relativity. That, although Einstein had this amazing picture which gave the full nonlinear theory out of geometry, general relativity is all about geometry, Feynman said, actually, this is completely compatible with particle physics, as long as we have spin two particles. And we would end up with a similar conclusion to Einstein, but on a sort of much more nuts and bolts point of view. From that Feynman point of view, it turns out that to derive general relativity from spin-two and special relativity, what you use... and this is a little bit technical, I'm sorry, but what you use in the action is what's called the densitized inverse metric. Okay, what does that mean? Basically you have root minus g, you might remember from the volume element, gets multiplied by the inverse metric. And that's the only thing which occurs in the derivation. And it turns out that quantity is not singular in our description of GR. Ah, okay. Interesting.
As well as the freedom to change coordinates, you have freedom to change the variables which depend on those coordinates. So in E&M we have electric fields and magnetic fields and we also have the space coordinates. And we never think of any particular choice of those coordinates as being better than any other choice. You're free to change variables. If you want to make the equation... you know, if you discover the equations are not well-defined or have a singularity, what you should do is change coordinates, either on space-time or on your field variables, to try to make the equations make sense. And if you can do that, that's perfectly fine. So what we are claiming is that there is a choice of variables on space-time, at least as far as the metric is concerned, which leaves everything regular. I believe what happens is that there's something else in gravity called the Christoffel symbol. And the Christoffel symbol actually is singular. And that tells you that as a particle hits the horizon, it experiences a sudden force. And the sudden force forces it to travel up the horizon. In other words, forces it to go at the speed of light. Because the only way to escape falling into the black hole is to travel at the speed of light, because the horizon is a light-like surface. And the only way you're going to travel at the speed of light is if you encounter this antiparticle with whom you annihilate. So there is a singularity, but it is not as simple as just saying, “oh, the metric is no good on the horizon.” That's too simplistic. Because the metric itself is not a... The inverse metric, I should say, is not a... Our metric is actually fine. It's the inverse metric, which doesn't exist. I see. But there's nothing sort of sacrosanct about the inverse metric.
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