Transcription
In book one of the series, "The Biggest Ideas in the Universe," called SpaceTime, motion, you take on classical mechanics and general relativity. Uh, by taking on the main equation of general relativity and making it, uh, accessible, easy to understand. So, um, maybe at a high level, what is general relativity? What's a good way to start to try to explain it?
Probably the best way to start to try to explain it is special relativity, which came first, 1905. Uh, it was the culmination, right, of many decades of people putting things together. But it was Einstein in 1905. In fact, it wasn't even Einstein, I should give more credit to Minkowski in 1907. So Einstein in 1905 figured out that you could get rid of the ether, the idea of a rest frame for the universe, and all the equations of physics would make sense with the speed of light being a maximum.
But then it was Minkowski, who used to be Einstein's professor, in 1907, who realized the most elegant way of thinking about this idea of Einstein's was to blend space and time together into spacetime. To really imagine that there is no hard and fast division of the four-dimensional world in which we live into space and time separately. Einstein was at first dismissive of this. He thought it was just like, "Oh, the mathematicians are over-formalizing again."
But then he later realized that if, if spacetime is a thing, it can have properties. And in particular, it can have a geometry. It can be curved from place to place. And that was what let him solve the problem of gravity. He was always, he had previously been trying to fit in what we knew about gravity from Newtonian, uh, mechanics, the inverse square law of gravity, to his new relativistic theory. It didn't work. So the final leap was to say, gravity is the curvature of spacetime. And that statement is basically just relativity.
And, uh, the tension with Minkowski was, he was a mathematician. Yes. So it's a tension between physics and, and mathematics. In fact, in, uh, your lecture about this equation, one of them, you, uh, say that Einstein is a better physicist than he gets credit for. Yep. I know that's hard. That's, that's a little bit of a joke there, right? Because we all give Einstein a lot of credit. But then we also, partly based on fact, but partly to make ourselves feel better, tell ourselves a story about how later in life Einstein couldn't keep up. Uh, there were younger people doing quantum mechanics and quantum field theory and particle physics, and he was just sort of, uh, unable to really philosophically get over his objections to that.
And I think that that story about the latter part is completely wrong. Like, almost 180 degrees wrong. I think that Einstein understood quantum mechanics as well as anyone, at least up through the 1930s. I think that his philosophical objections to it are correct. So he should actually have been taken much more seriously about that. And what he did, what he achieved in trying to think these problems through, is to really, basically understand the idea of quantum entanglement, which is kind of important these days when it comes to understanding quantum mechanics.
Now, it's true that in the '40s and '50s, uh, he placed his efforts in hopes for unifying electricity and magnetism with gravity. That didn't really work out very well. All of us, you know, try things that don't work out. I don't hold that against him. But in terms of IQ points, in terms of trying to be a clear-thinking physicist, he was really, really great.
What does greatness look like for a physicist? So how difficult is it to take the leap from special relativity to general relativity? How difficult is it to imagine that, to consider spacetime together, and to imagine that, uh, there's a curvature to this whole thing?
Yeah, that's a great question. Um, I think that if you want to make the case for Einstein's greatness, which is not hard to do, there's two things you point at. One is in 1905, his famous miracle year. He writes three different papers on three wildly different subjects, all of which would make you famous just for writing that one paper. Um, special relativity is one of them. Brownian motion is another one. One which is just, you know, the little vibrations of tiny little dust specs in the air. But who cares about that? What matters is it proves the existence of atoms. He explains Brownian motion by imagining there are molecules in the air and deriving their properties. Brilliant. And then he basically starts the world on the road to quantum mechanics with his paper on, which again, is given a boring label of the photoelectric effect. What it really was is he invented photons. He showed that light should be thought of as particles as well as waves. And he did all three of those very different things in one year. Okay.
But the other thing that gets him genius status is, like you say, general relativity. So this takes 10 years, from 1905 to 1915. He wasn't only doing general relativity. He was working on other things. He wrote, he invented a refrigerator. He did various interesting things. And he wasn't even the only one working on the problem. There were other people who suggested relativistic theories of gravity. But he really applied himself to it. And I think, as your question suggests, the solution was not a matter of turning a crank. It was something fundamentally creative. You know, in his own telling of the story, his greatest moment, his happiest moment, was when he realized that if the way that we would modernly say it, in modern terms, if you were in a rocket ship accelerating at one G, at one, uh, acceleration due to gravity, if the rocket ship were very quiet, you wouldn't be able to know the difference between being in a rocket ship and being on the surface of the Earth. Gravity is sort of not detectable, or at least not distinguishable from acceleration.
So number one, that's a pretty clever thing to think. But number two, if you or I had that thought, we would have gone, "Huh, we're pretty clever." He reasons from there to say, "Okay, if gravity is not detectable, then it can't be like an ordinary force, right? The electromagnetic force is detectable. We can put charged particles around positively charged particles, and negatively charged particles respond differently to an electric field or to a magnetic field." He realizes that what his thought experiment showed, or at least suggested, is that gravity isn't like that. Everything responds in the same way to gravity. How could that be the case? And then this other leap he makes is, "Oh, it's because it's the curvature of spacetime." Right? It's a feature of spacetime. It's not a force on top of it. And the feature that it is, is curvature. And then finally, he says, "Okay, clearly I'm going to need the mathematical tools necessary to describe curvature. I don't know them. So I will learn them." And they didn't have Mook's or AI helpers back in those days. He had to sit down and read the math papers, and he taught himself differential geometry and invented general relativity.
What about the step of including time as just another dimension? So combining space and time, is that a simple mathematical leap as Minkowski suggested?
It's certainly not simple. Actually, um, it's a, it's a profound insight. That's why I said I think we should give Minkowski more credit than we do. You know, he's the one who really put the finishing touches on special relativity. Again, many people had talked about how things change when you move close to the speed of light, uh, what Maxwell's equations of electromagnetism predict, and so forth, what their symmetries are. So people like Lorentz and Fitzgerald and Poincaré, there's a story that goes there. And in, in the usual telling, Einstein sort of puts the capstone on it. He's the one who says, "All of this makes much more sense if there just is no ether. It is undetectable. We don't know how fast everything is relative." Thus the name relativity. But he didn't take the actual final step, which was to realize that the underlying structure that he had invented is best thought of as unifying space and time together. I honestly don't know what was going through Minkowski's mind when he thought that. And not sure if he was, you know, so mathematically adept that it was just clear to him, uh, or he was really struggling and he did trial and error for a while. I'm not sure.
I mean, do you, for him or for Einstein, visualize the four-dimensional space? Try to play with the idea of time as just another dimension?
Oh, yeah, all the time. I mean, we, of course, make our lives easy by ignoring two of the dimensions of space. So instead of four-dimensional spacetime, we just draw pictures of one dimension of space, one dimension of time, so-called spacetime diagram. But, you know, I mean, maybe this is lurking underneath your question, but even the best physicists will draw, you know, a hor a vertical axis and a horizontal axis, and they'll go, "Spacetime." But deep down, that's wrong because you're sort of preferring one direction of space and one direction of time. And it's really the whole two-dimensional thing that is spacetime. The more legitimate thing to draw on that picture are rays of light, are light cones from every point. There is a fixed direction at which the speed of light would represent, and that is actually inherent in the structure. The division into space and time is something that's easy for us human beings.
What is the difference, general space and time, from the perspective of general relativity?
It's the difference between X and Y when you draw axes on a piece of paper. So there really is no difference. There is almost no difference. There's one difference that is kind of important, which is the following. If you have a curve in space, I'm going to draw it horizontally because that's usually what we do in spacetime diagrams. You have a curve in space. You've heard the motto before that the shortest distance between two points is a straight line. If you have a curve in time, which is, by the way, literally all of our lives, right? We all evolve in time. So you can start with one event in spacetime and another event in spacetime. What Minkowski points out is that the time you measure along your trajectory in the universe is precisely analogous to the distance you travel on a curve through space. And by precisely, I mean, it is also true that the actual distance you travel depends on your path, right? You go a straight line, shortest distance, and a curvy line would be longer. The time you measure in spacetime, the literal time that takes off on your clock, also depends on your path. But it depends on it the other way. So that the longest time between two points is a straight line. And if you zig back and forth in spacetime, you take less and less time to go from point A to point B.
How do I make sense of that? The, uh, difference between the observed reality and the objective reality underneath it? Or is objective reality a silly notion given general relativity?
I'm a huge believer in objective reality. I think that objective reality, objective is real. Um, but I do think that people are kind of a little overly casual about the relationship between what we observe and objective reality in the following sense. Of course, in order to explain the world, our starting point and our ending point is our observations, our experimental input, the phenomena we experience and see around us in the world. But in between, there's a theory. There's a mathematical formalization of our ideas about what is going on. And if a theory fits the data and is very simple and makes sense in its own terms, then we say that the theory is right. And that means that we should attribute some reality to the entities that play an important role in that theory, at least provisionally, until we come up with a better theory down the road.
I think a nice way to test the difference between objective reality and the observed reality is what happens at the, uh, at the edge of the horizon of a black hole. So technically, as you get closer to that horizon, time stands still.
Yes and no. It depends on exactly how careful we're being. So here is a, a bunch of things I think are correct. If you imagine there is a black hole spacetime, so like the whole solution to Einstein's equation, and, and you treat you and me as what we call test particles, so we don't have any gravitational fields ourselves, we just move around in the gravitational field. That's obviously an approximation, okay? But let's, let's imagine that. And you stand outside the black hole and I fall in. And as I'm falling in, I'm waving to you, you know, because I'm going into the black hole. You will see me move more and more slowly, and also the light from me is red-shifted. So I kind of look embarrassed because I'm falling into a black hole. And there is a limit. There's a last moment, moment that light will be emitted from me from your perspective, forever. Okay. Now, you don't literally see it because I'm emitting photons more and more slowly, right? Because from your point of view. So it's not like I'm equally bright. I basically fade from view in that picture. Okay. So that's one approximation.
The other approximation is I do have a gravitational field of my own. And therefore, as I approach the black hole, the black hole doesn't just sit there and let me pass through. It kind of moves out to eat me up because its net energy mass is going to be mine plus its. But roughly speaking, yes, I think so. I don't like to go to the dramatic extremes because that's where the approximations break down. But if you see something falling into a black hole, you see its clock ticking more and more slowly. How do we know it fell in? We don't. I mean, how would we? Because it's always possible that right at the last minute, it had a change of heart and starts accelerating away. Right? If you don't see it pass in, you don't know.
And let's point out that as smart as Einstein was, he never figured out black holes. And he could have. It's kind of embarrassing. It took decades for people thinking about general relativity to understand that there are such things as black holes because basically Einstein comes up with general relativity, 1915. Two years later, Schwarzschild, Carl Schwarzschild, derives the solution to to Einstein's equation that represents a black hole, the Schwarzschild solution. No one recognized it for what it was until the '50s. David Finkelstein and other people. And that's just, you know, one of these examples of physicists not being as clever as they should have been.
Well, that's the singularity. That's the kind of the edge of the theory, the limit. So it's understandable that it's difficult to imagine the limit of things.
It is absolutely hard to imagine. And a black hole is very different in many ways from what we're used to. On the other hand, I mean, I mean, the real reason, of course, is that between 1915 and 1955, there's a bunch of other things that are really interesting going on in physics, all of particle physics and quantum field theory. So many of the greatest minds were focused on that. But still, if the universe hands you a solution to general relativity in terms of curved spacetime, and it's kind of mysterious, certain features of it, I would put some effort in trying to figure it out.
So how does a black hole work? Put yourself in the shoes of Einstein and take general relativity to its natural conclusion about these massive things.
It's best to think of a black hole as not an object so much as a region of spacetime. Okay? It's a region with the property, at least in classical general relativity. Quantum mechanics makes everything harder. But let's imagine we're being classical for the moment. It's a region of spacetime with the property that if you enter, you can't leave. Literally, the equivalent of escaping a black hole would be moving faster than the speed of light. They're both precisely equally difficult. You would have to move faster than the speed of light to escape from the black hole. So once you're in, that's fine. You know, in principle, uh, you don't even notice when you cross the event horizon, as we call it. The event horizon is that point of no return where once you're inside, you can't leave. But meanwhile, the spacetime is sort of collapsing around you, uh, to ultimately a singularity in your future, which means that the gravitational forces are so strong they tear your body apart. Um, and you will die in a amount of time. The time it takes, if the, if the black hole is about the mass of the sun, to go from the event horizon to the singularity takes about one millionth of a second.
And what happens to you if you fall into the black hole? Like, if we think of an object as, uh, information, that information gets destroyed.
Well, you've raised a crucially difficult point. So that's why I keep needing to distinguish between black holes according to Einstein's theory of general relativity, which is book one of spacetime and geometry, which is perfectly classical. And then come the 1970s, we start asking about quantum mechanics. And what happens in quantum mechanics? According to classical general relativity, the information that makes up you when you fall into the black hole is lost to the outside world. It's there, it's inside the black hole, but we can't get it anymore.
In the 1970s, Stephen Hawking comes along and points out that black holes radiate. They give off photons and other particles to the universe around them. And as they radiate, they lose mass and eventually they evaporate, they disappear. So once that happens, I can no longer say the information about you or a book that I threw in the black hole or whatever, is still there as hidden behind the black hole because the black hole has gone away. So either that information is destroyed, like you said, or it is somehow transferred to the radiation that is coming out, the Hawking radiation. The large majority of people who think about this believe that the information is somehow transferred to the radiation, and information is conserved. That is a feature both of general relativity by itself and of quantum mechanics by itself. So when you put them together, that should still be a feature. We don't know that for sure. There are people who have doubted it, including Stephen Hawking for a long time. But that's what most people think. And so what we're trying to do now, in a topic which has generated many, many hundreds of papers called the black hole information loss puzzle, is figure out how to get the information from you or the book into the radiation that is escaping the black hole.
Is there any way to observe Hawking radiation to a degree where you can start getting insight, or is this all just in the space of theory right now?
Right now, we are nowhere close to observing Hawking radiation. Here's the sad fact: the larger the black hole is, the lower its temperature is. So a small black hole, like a microscopically small black hole, might be very visible. It's giving off light. But something like the black hole at the center of our galaxy, 3 million times the mass of the sun, or something like that, Sagittarius A star, uh, that is so cold and low temperature that its radiation will never be observable. Black holes are hard to make. We don't have any nearby. The ones we have out there in the universe are very, very faint. So there's no immediate hope for detecting Hawking radiation. Allegedly, we don't have any nearby. As far as we know, we don't have any nearby. Could tiny ones be hard to detect somewhere at the edges of the solar system, maybe? So you don't want them to be too tiny, or they're exploding, right? They're, they're very bright and then they'll be visible. But there's an absolutely regime where black holes are large enough not to be visible because the larger ones are fainter, right, not giving off radiation, but small enough to not have been detected through their gravitational effect.
Yeah, psychologically, just emotionally, how do you feel about black holes? Do they scare you?
I love them. I love black holes. But the universe weirdly makes it hard to make a black hole, right? Because you really need to squeeze an enormous amount of matter and energy into a very, very small region of space. So we know how to make stellar black holes. A supermassive star can collapse to make a black hole. We know. We also have these supermassive black holes at the center of galaxies. We're a little unclear where they came from. I mean, maybe stellar black holes that got together, uh, and combined. But that's, you know, one of the exciting things about new data from the James Webb Space Telescope is that quite large black holes seem to exist relatively early in the history of the universe. So it was already difficult to figure out where they came from. Now it's an even tougher puzzle.