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Join us for #YourDailyEquation with Brian Greene. Every Mon – Fri at 3pm EDT, Brian Greene will offer brief and breezy discussions of pivotal equations. Even if your math is a bit rusty, tune in for accessible and exciting stories of nature and numbers that will allow you to see the universe in a new way.
Episode 26 #YourDailyEquation: Albert Einstein’s General Theory of Relativity, phrased in terms of warps and curves in space and time, provides our most refined description of the gravitational force. Join Brian Greene for a visual exploration of Einstein’s most profound discovery and the equations he triumphantly announced in 1915.
Brian Greene is a professor of physics and mathematics at Columbia University, and is recognized for a number of groundbreaking discoveries in his field of superstring theory. His books, The Elegant Universe, The Fabric of the Cosmos, and The Hidden Reality, have collectively spent 65 weeks on The New York Times bestseller list.Read More
Hey everyone. Welcome to this next episode of Your Daily Equation. It may look a little bit different from the place where I’ve done the earlier episodes, but actually I’m in exactly the same spot. It’s just that the rest of the room has gotten to be so incredibly messy with all sorts of stuff that I’ve had to shift my location in order that you don’t have to look at the messy room that would otherwise be behind me.
All right, so with that little detail out of the way today’s episode, I’m going to start in on one of the really big ones, the big ideas, the big equations. Einstein’s general theory of relativity. And just to give a little bit of context to this, let me just note… Bring this up to a different position. I’ve got to angle myself differently. Sorry. I think that’s okay. Up on the screen, good.
All right. So we’re talking about general relativity. And to put this just in context of the other big, vital essential ideas that really revolutionized our understanding of the physical universe starting in the 20th century, I’d like to organize those developments by writing down three axes. And these axes, you can think about, say, as the speed axis, you can think about it as the length axis, and the third you can think about… I cannot believe that Siri just heard me. That is so irritating. Go away Siri! Hey, all right, all right. Back to where I was. Huh, I have to learn how to turn Siri off when I do these things.
Anyway, the third axis is the mass axis. And the way to think about this little diagram is that when you are thinking about how the universe behaves in the realms of extremely high speed, that takes you to Einstein’s special theory of relativity which it just so happens, that is the subject that I started with in this series of Your Daily Equation. When you go to extremes along the length axis, and by extremes here, I really mean the extremes of very small, not very big, that takes you to quantum mechanics, which in some sense really is the second major focus that I had in this, Your Daily Equation series. And now we’re onto the mass axis, where, when you look at how the universe behaves at extremely high masses, that’s where gravity matters, that takes you to the general theory of relativity, our focus today.
Okay, so that’s how things fit in to that overarching organizational scheme, for thinking about the dominant theories of the physical universe. And so let’s now get into the subject of gravity, the force of gravity. And many people believed not far after say the late 1600s, that the issue of gravity had been completely sorted out by Isaac Newton because Newton gave us his famous universal law of gravity. Remember, this is during the Black Death way back in the late 1600s. Newton retreats from Cambridge University, goes to his family’s home in the safety of the countryside there. And in solitude, through really the amazing power of his mental faculties and creative ways of thinking about how the world works, he comes up with this law, the universal law of gravity, that if you have two masses that say, have mass M1 and mass M2, that there is a universal force of attraction between them acting to pull them together. And the formula for that is a constant, Newton’s gravitational constant, M1, M2, divided by the square of their separation. So if their distance are apart, then you divide by r squared. And the direction of the force is along the line connecting their center, center of masses.
And that seemed to be the be-all and end-all of the force of gravity in terms of describing it mathematically. And indeed, just to get us all on the same page, here is a little animation that shows Newton’s law in action. So you have a planet like the earth and orbit around a star like the sun, and using that little mathematical formula, you can predict where the planet should be at any given moment, and you look up into night sky and the planets are just where the math says that they should be. And we take it for granted now, but wow! Right? Think about the power of this little mathematical equation to describe things that are happening out there in space. So understandably, rightly so, there was a general consensus that the force of gravity was understood by Newton and his universal law of gravity.
But then of course, other folks come into the story. And the person of course that I have in mind here is Einstein. And Einstein begins to think about the force of gravity in roughly 1907 or so. And look, he comes to the conclusion that sure, Newton made great progress in understanding the force of gravity, but the law that he gave us over here can’t really be the full story, right? Why can’t it be the full story? Well, you can immediately catch the gist of Einstein’s reasoning by noting that in this formula that Newton gave us, there’s no time variable. There’s no temporal quality to that law. Why do we care about that? Well, think about it. If I was to change the value of the mass, then according to this formula, the force would immediately change. So the force felt over here at mass M2, given by this formula will immediately change, if say I changed the value of M1 in this equation, or if I change the separation, if I move M1 this way, making r a little smaller, or this way, making r a little bit bigger, this guy over here is going to immediately feel the effect of that change, immediately, instantaneously, faster than the speed of light. And Einstein says there can’t be that kind of influence that exerts a change of force instantaneously. That’s the issue.
Now, a small footnote, some of you may come back at me and say, “What about quantum entanglement?” something that we discussed in an earlier episode when we were focusing our attention on quantum mechanics. You will recall that when I discussed the spooky action of Einstein, we noted that there’s no information that’s traveling from one entangled particle to another. There is an instantaneous, according to a given reference frame, correlation between the properties of the two distant particles. This one’s up and the other one’s down. But there’s no signal, no information that you can extract from that because the sequence of results at the two distant locations are random, and randomness doesn’t contain information. So that’s the end of the footnote. But bear in mind, there really is a sharp distinction between the gravitational version of the instantaneous change in a force, versus the quantum mechanical correlation from entangled particles. All right, let me put that to the side.
So Einstein realizes there’s a real issue here. And just to bring that issue home, let me show you a little example here. So imagine that you’ve got the planets in orbit around the sun, and imagine that somehow I’m able to reach in and I pluck the sun out of space. What will happen according to Newton? Well, Newton’s law says the force drops to zero if the mass at the center goes away. So the planets, as you see, are immediately, instantaneously released from their orbits. So the planets instantaneously feel the absence of the sun, the change in their motion, that is exerted instantaneously from the changing mass at the location of the sun to the location of the plants. No good according to Einstein.
So Einstein says, “Look, perhaps if I understood better what Newton had in mind regarding the mechanism by which gravity exerts its influence from one place to another,” Einstein says, “maybe I would be able to calculate the speed of that influence.” And maybe with hindsight or a better understanding a couple of hundred years later, maybe Einstein says to himself, “I will be able to show that in Newton’s theory, the force of gravity is not instantaneous.”
So Einstein goes to check in on this and he realizes as many scholars had already realized, that Newton himself is kind of embarrassed by his own universal law of gravity because Newton himself realized that he had never specified the mechanism by which gravity exerts influence. He said, “Look, if you have the sun and you have the earth and they’re separated by a distance, there is a force of gravity between them,” and he gives us a formula for it, but he doesn’t tell us how gravity actually exerts that influence. And therefore there was no mechanism that Einstein could analyze to truly figure out the speed with which that mechanism for transmitting gravity operates, and therefore he was stuck. So Einstein sets himself the goal of truly figuring out the mechanism of how gravitational influences are exerted from place to place. And he starts at about 1907, and finally, by 1915, he writes down the final answer in the form of the equations of the general theory of relativity.
And I’m going to now describe the basic idea, which I think many of you are familiar with of what Einstein found, and then I’ll briefly outline the steps by which Einstein came to this realization. And I’ll finish up with the mathematical equation that summarizes the insights that Einstein came to.
All right, so for the general idea, Einstein says, “Look, if you have the sun and the earth, and the sun is exerting an influence on the earth, what could be the source of that influence?” Well, the puzzle is, there’s nothing but empty space between the sun and the earth. So Einstein, ever the capable genius to look at the most obvious answer, if there’s only empty space, then it must be space itself, space itself that communicates the influence of gravity. Now how can space do that? How can space exert any kind of influence at all? Einstein ultimately comes to the realization that space and time can warp and curve, and through their curved shape they can influence the motion of objects.
And so the way to think about it is, imagine that space, this is not a perfect analogy, but imagine space is like a rubber sheet or a piece of spandex, and when there’s nothing in the environment, the rubber sheet is flat. But if you take a bowling ball say, and you put it in the middle of the rubber sheet, the rubber sheet will be curved. And then if you set marbles rolling around on the rubber sheet or on the spandex, the marbles will now get a curved trajectory because they’re rolling in the curved environment that the presence of the bowling ball or the shot put creates.
In fact, you can actually do this. I did a little home experiment with my kids. You can see the full video online if you like. This from a few years ago. But there you see it. We have a piece of spandex in our living room and we have marbles that are rolling around. And that gives you a sense of how the planets are nudged into orbit by virtue of the curved spacetime environment through which they travel, a curved environment that the presence of a massive object like the sun can create.
Let me show you a more precise, well, not more precise, but a more relevant version of this orbit so you can see it at work in space. So here you go. So this grid represents 3D space. It’s a little hard to picture it fully so I’m going to go to a two dimensional version of this picture that shows all the essential ideas. Notice that space is flat when there’s nothing there but if I bring in the sun, the fabric warps. Similarly, if I look in the vicinity of the earth, the earth too also warps the environment. And now focus your attention on the moon, because this is the point. The moon, according to Einstein, is kept in orbit because it’s rolling along a valley in the curved environment that the earth creates. That is the mechanism by which gravity operates. And if you pull back, you see that the earth is kept in orbit around the sun for exactly the same reason. It is rolling around a valley in the warped environment that the sun creates. That’s the basic idea.
Now, look, there are a bunch of subtleties in here. Maybe I’ll quickly address them right now. You could say to me, hey look, with the example of the spandex, which is the at home version of the sun warping the fabric around it, if I put a bowling ball or a shot put on a rubber sheet or a piece of spandex, the reason why it warps the spandex is because the earth is pulling the object downward. But wait, I thought we were trying to explain gravity. So our little example now seems to be using gravity to explain gravity. What are we doing? Well, you’re absolutely right. This metaphor, this analogy really needs to be thought of in the following way. It’s not that we are saying that earth’s gravity is causing the environment to warp, rather, Einstein is telling us that a massive energetic object merely by virtue of its presence in space warps the environment around it. And by warping the environment, I mean warping the full environment around it.
Of course, I have difficulty showing that fully, but actually let me just give you this little visual here that gets partway toward it. Now you see that the full 3D environment is being warped by the sun. It’s harder to picture that one. And the two diversions, pretty good to keep in mind, but the 3D one is really what’s happening. We’re not looking at a slice of space. We’re looking at the entire environment being influenced by the presence of a massive body within it.
All right. That is the basic idea. And now I want to spend just a couple minutes on how it is that Einstein came to this idea. And it’s really a two step process. So step one, Einstein realizes that there is a deep and unexpected connection between accelerated motion, acceleration, and gravity. And then he realizes that there is another unexpected and beautiful relationship between acceleration and curvature. Curvy spacetimes, curvature. And the final step then of course will be, he realizes that there is a connection, therefore, between gravity and curvature. So this link right over here is forged if you will, through acceleration being the common quality that leads you both to an understanding of gravity, an understanding of curvature, therefore, a link between gravity and curvature.
Okay, so let me just quickly explain those links, the first of which happens in… Well, it was always there, but Einstein realized it in 1907. 1907, Einstein is still in the patent office in Bern, Switzerland. He had the great success in 1905 with the special theory of relativity, but he still is working in the patent office. And he has one afternoon what he calls, the happiest thought of his entire life. What is that happiest thought? That happiest thought is he imagines a painter who is painting the exterior of a building on a high ladder. He imagines that painter falling off the ladder, or falling off the roof and going into free fall. He doesn’t take this thought all the way to the impact of the ground. The impact is not his happiest thought. The happiest thought happens during the journey. Why? Einstein realizes that the painter during this descent will not feel his or her, they will not feel their own weight.
What do you mean by that? Well, I like to frame it this way. Imagine that the painter is standing on a scale that’s Velcroed to their shoes, and they are standing on the scale on the ladder, kind of a hard image, but imagine that they’re now falling. As the painter falls, the scale falls at the same rate as the painter, therefore they fall together, which means the painter’s feet don’t exert a push on the scale. They can’t because the scale is moving away at exactly the same rate as the feet are moving downward too. So, looking down at the reading on the scale, the painter will see that the reading drops to zero. The painter feels weightless. The painter does not feel their own weight.
Now, to give you a little example of that, that again, this is sort of an episode of general relativity, but it’s do it at home physics. So this is a DYI version of the general theory of relativity. So how can you establish without falling off the roof of a house in a more safe manner? How can you establish that free fall, this kind of accelerated downward motion, accelerated downward motion can, in some sense, cancel out the force of gravity.
Well, I did an example of that on the Late Show with Stephen Colbert some years ago, and they did a nice job filming it. So let me show you the basic idea. So imagine you have a bottle filled with water and it’s got some holes in it. The water sprays out of the holes of the bottle of course. Why does it do that? Because gravity is pulling on the water and that pull forces the water out of the holes in the bottle. But if you let the bottle go into free fall, like the painter, the water will no longer feel its own weight. That feeling, that force of gravity, nothing will pull the water out of the hole so the water should stop spraying out of the holes. And check this out, it really does work. All right, here we go. During the descent, look in slow-mo, there is no water spraying out of the holes during that accelerated motion, that descent.
So this is what we mean here about this relationship between acceleration and gravity. This is a version where the accelerated downward motion, faster and faster as the bottle of water or the painter falls, the force of gravity is canceled if you will, by that downward motion. And you might say, well what do you mean canceled? Why is the bottle falling? Why is the painter falling? That’s gravity. But I’m saying not from our experience watching the painter fall, not from our experience watching the bottle of water fall. I’m saying that if you put yourselves in the shoes of the painter, or you put yourselves in the shoes of a bottle of water, whatever that means, then from that perspective, the freely falling perspective, from your perspective in that accelerated trajectory, you don’t feel the force of gravity. That’s what I mean.
Now the important point is that there’s also a reverse to this situation. Accelerated motion cannot only cancel out gravity, but accelerated motion can mock up, it can sort of fake a version of gravity. And it’s a perfect fake. Again, what do I mean by that? Well, imagine that you are floating in outer space, so you really are completely weightless. And then imagine that someone causes you to accelerate. They tie a rope to you and they accelerate you. Let’s say they accelerate you like this. They accelerate you upward, and imagine that they do that by putting a platform under your feet. So you’re standing on this platform in empty space, feeling weightless. Now they attach a rope or a crane, whatever, to a hook on the platform on which you’re standing, and that crane, that hook, that rope pulls you upward. As you’re accelerating upward the board under your feet, you’re going to feel it pressing against your feet. And if you close your eyes and if the acceleration is correct, you’ll feel like you are in a gravitational field. Because, how does a gravitational field on planet earth feel? How do you feel it? You feel it by virtue of the floor pushing up against your feet. And if that platform accelerates upward, you will feel it pressing against your feet in the same way if the acceleration is correct.
So that’s a version where accelerated motion creates a force that feels just like the force of gravity. You’ve experienced this on an airplane that’s beginning to taxi and it’s about to take off. As it accelerates, you feel pressed back in your seat. That feeling of being pressed back, you close your eyes and it can sort of feel like you’re lying down. The force of the seat on your back is almost like the force you’d feel if you were just lying on your back on a couch. So that’s the link between accelerated motion and gravity.
Now, for part two of this. So that’s 1907. So for part two, we need the connection between acceleration and curvature. And there are many ways, I mean the history is fascinating. And again, as I mentioned it before, because I love the piece, we have this stage piece, light falls, you can check it out, where we go through the whole history of these ideas in a stage presentation. But there are actually a number of people who contributed to thinking about gravity in terms of curves, or at least Einstein’s recognition of this. And there’s one particularly beautiful way of thinking about it that I like. It’s called the Ehrenfest paradox. It’s not actually a paradox at all. Paradoxes are usually when we don’t understand things at first and there is a seeming paradox, but ultimately we sort it all out. But sometimes the word paradox isn’t removed from the description. And let me give you this example that gives us a link between acceleration and curvature. How does it go?
Remember, accelerated motion means a change in velocity. Velocity is something that has a speed and a direction. So there’s a special kind of accelerated motion where the speed, the magnitude doesn’t change but the direction does. And what I have in mind here is circular motion. Circular motion is a kind of acceleration. And what I’d now like to show you is that circuit emotion, that accelerated motion, naturally gives us the recognition that curvature must come into play. And the example I’m going to show you is a familiar ride. You may have been on it at an amusement park or a carnival. It’s often called the Tornado ride. I described this in the Elegant Universe but I’ll show you a visual in just a moment. You know, it’s a ride, you stand on this circular platform that spins around and you actually feel your body pressed against a circular cage that is moving, it’s attached to the circular platform. And that outward force that you feel, it can be strong enough that sometimes they actually drop the bottom of the ride out, that you’re standing on. So you’re just hovering there and sometimes in mid-air, but your body is pressed by the circular motion against the cage. And there’s enough friction, hopefully, that you don’t slip away and fall.
All right, that’s the setup. Here’s the issue. Here is this circular ride. Imagine that you measure the circumference of this ride from the outside, not on the ride itself. So you lay out these rulers and whatever you find, I think in this case, there were 24 rulers, 24 feet. You can also measure the radius and you can get a number for that as well. And indeed, if you look at the relationship between the circumference and the radius, you will find that C equals 2πr, just as we all learned in junior high school. But now, imagine measuring this from the perspective of someone on the ride itself, the accelerated observer. Well, when they measure the radius, they’re going to get the exact same answer because that’s moving perpendicular to the motion, no Lorentz contraction. But if you measure the circumference, look what happens. The rulers are all instantaneously moving in the direction of the motion so they are all shrunken, contracted. Therefore, it takes more of those rulers to go all the way around.
In this particular case I have just imagined that it’s 48 of those rulers, 48 rulers for the circumference equals 48. The radius is unchanged. Again, that’s moving perpendicular to the instantaneous direction of the motion, which is all in the circumferential direction, right? The radius is going this way, the circumference is going that way so there’s no change in the measurement of the radius, which means C will no longer equal 2πr.
Are you say to yourself, what? How can C not equal 2πr? What does that mean? Well, when you learned that C equals 2πr, you were talking about circles that were drawn on a flat surface. Therefore, it’d be the case that from the perspective of the person on the ride, laying out those little rules and feeling that gravitational force, they’re accelerating, they feel that force pulling them outward from their perspective, it must be that the circle is not flat. It must be curved. It must be the case, you know, sort of a poetic image of this, if you will, over here, kind of a Dali-esque picture, those circles are warped. They’re curved. Clearly, C will not equal 2πr for those particular warped shapes. So that’s kind of an artistic version of it. But the conclusion is that the accelerated motion of the ride, which we know gives a connection to gravity, also gives a connection to curvature.
So then, that is the linkage that we were looking at. The accelerated motion from the circle gives rise to the feeling of the gravitational-like force, that accelerated motion gives rise to measurements from the perspective of the person experiencing that acceleration, that do not satisfy the usual rules of flat Euclidean so-called geometry, and therefore we learned that there is a connection between gravity and curvature.
And now I can bring back the image that we had before with a little more insight from that description. So again, here is flat 3D space when there’s no matter. Go to the two dimensional version just so we can picture it. Bring in a massive body like the sun, and now that gravity gives rise to this curvature. And again, the moon, why does it move around? The moon in some senses is being nudged around by the curvature in the environment. Or said another way, the moon is seeking out the shortest possible trajectory, what we called geodesics. We’ll come to this. And that short as possible trajectory in that curved environment yields to curved paths that we would call a planet going into orbit. That’s the basic chain of reasoning that leads Einstein to this picture.
All right, so then what is the equation? I’m just going to write down the equation, and subsequent episodes, I’m going to, just in this episode, be satisfied to just give you the basic idea and show you the equation. I will unpack the equation later on. But what is the equation? Well, Einstein, in November of 1915 at a lecture at the Prussian Academy of Science, writes down the final equation, which is Rµν-1/2gµνR = 8πG/C4 x Tµν. What in the world does that all mean? Well, this part over here is the mathematical way of talking about curvature. And this fellow over here is the way you talk about energy and mass, also momentum, but we can call it mass energy. Once we learned in special relativity that mass and energy are two sides of the same coin, you recognize that mass is not the only source. A clumpy object like the earth is not the only source for gravity. Energy, more generally is a source for gravity and that is captured by that expression over here, Tµν. I’ll describe this not today but in a subsequent episode. And that is Einstein’s equation for the general theory of relativity.
Now, to really understand this equation, you need to understand all of these gadgets that we have here, the Ricci tensor, the scalar curvature. You need to understand the Riemann curvature tensor to understand those. This is the metric on spacetime. You need to understand that. And I really do mean spacetime. In fact, when we’re talking about the gravitational pull of a planet like the earth or the sun, the imagery that I showed you with the warped environment, it helps your mental thinking about things. But in the usual way that we set up our coordinates, it’s actually the warping of time, not really the warping of space that’s the dominant influence in causing an object to fall. Whether I drop an object here, or whether it’s the moon perpetually falling toward the earth, as it moves in the tangential direction, thereby keeping itself in orbit. So time is really quite important to this. You can’t just think in spacial terms at all.
But to understand all those mathematical details, we have to unpack the mathematics if you will of differential geometry. I will do a little bit of that in subsequent episodes, but I hope this gives you a feel for the basic insight of the general theory of relativity, why it is that Einstein came to this realization that gravity necessarily involved a curvature of spacetime, keep that Tornado ride in mind. Again, no analogy is perfect, but it does help you catch the essential links between accelerated motion and gravity, the water drop, the painter, between accelerated motion and curvature, the Tornado ride. And then it’s the genius of Einstein that puts it all together as we will see and unpack in subsequent episodes.
Okay, that’s all I wanted to do today. That is Your Daily Equation. Until we meet next time, looking forward to that. Until then, take care.