So we talked about this question, why do planets of so different masses will follow the same elliptical orbits. So if you can actually experience this in daily life too. So if you fall 2 objects, drop 2 objects of different masses why do follow and why do fall in the same way? So again there is a legend about this famous experiment from the Leaning Tower of Pisa. So this is a picture when I visited with my family in Pisa. And this is my daughter, she was very little. She wasn't allowed to climb up the tower because she was too young. The people worried that little kids might fall in the the cracks or from the tower, I guess. but anyway. So There's this legend that the Galileo Galilei dropped objects of 2 different masses from the top of the leaning tower, and as crowd watched in awe, both of them reached the ground at exactly the same time. So, objects of different masses seem to follow exactly the same motion. So, why is that? Well in, the understanding by Newton, it's actually not very clear why that is the case. So this is actually based on 2 different issues. The first one is, follows this F equals m a. So heavier objects are hard to move. We compared the little kid on a tricycle and a big tank. Smaller objects, lighter objects are easier to move. Heavier objects are more difficult. So, if you think of a heavier object based on this F equal ma. You might think that one would actually move more slowly. On the other hand, force of gravity Is proportional to how much it weighs. A heavier object is pulled more strongly by the Earth. So then you might think that a heavier object would fall faster instead. And it turns out that, no, heavier objects should move more slowly because F equals m a. But it's being pulled more strongly because it's heavier and because of the universal theory of gravitation. The dependence on mass between 2 equations exactly cancel, if you think about the motion. So you just set this F and that F the same. Little m appears in both equations the same way. You can cross them out and find that the force acting on the object would cause an acceleration, which doesn't depend on the mass of the object. So, that's the way different objects end up moving exactly the same way. But you know, if you listen to this explanation, I'm sure you feel, it's a little, not very convincing about it. Somehow, it looks just too miraculous that the same m appears in 2 equations that exactly cancel against each other. And, and who would you think that who actually was worried about this question the most? A famous guy, namely, Albert Einstein. So, he thought about this problem a lot. Why do objects of different masses move in the same way? And Newton's explanation was not quite satisfactory because that relies on cancellation between 2 equations and so, no didn't quite let us believe that, that is exactly what's going on. Einstein came up with this revolutionary idea: The force of gravity in some sense is an illusion, he says. It's not a force. It's the nature of space and time. Gravity is a property of space. And if you have a heavy object like the Earth around us, the Earth actually warps space. So it's not flat anymore. It sort of warps towards the center of the Earth. And when you drop an object, every object thinks it's going around a straight line. But if the space itself is curved like this, and if you think you're going straight... Space is curved, you end up actually curving like this, so you don't go straight anymore. And because it's a property of space, it doesn't matter how much you weigh. It doesn't depend on the mass of the object you're throwing. Everything follows the same trajectory. Because that's, that's the nature of space itself. And this theory is called the theory of general relativity, or general theory of relativity; and either one is fine. And so that's the idea by Einstein. So we would like to demonstrate that with a, again, simple video clip. So, what's going on here is that, here's the space that is warped toward the center. So, that's the ideal of the warped space. And if you throw a ball around it, then none of them actually go straight as you can see. It bends around and goes in a curved motion. So, that's the idea of the Keplerian motion of planets around the sun. The Sun actually warps space. And that's why things don't go straight any more And, and ends up orbiting around the sun. But in some sense, the being pulled by the sun, by the force of gravity, is an illusion. Everything is actually going straight, but space itself is curved. That's why it looks like everybody's undergoing this elliptical motion around the sun. So that's the idea by Einstein. And this way, we understand much, much more easily why objects of different masses end up moving the same way. And this idea of space getting warped actually gave rise to a prediction of something very familiar to you, namely a black hole. You might have actually seen it in a movie or science fiction novels Black holes actually do exist. And they are objects that are so heavy, it warps the space so much that even light can't escape its gravity. So it's eternally trapped. Nothing can come out from black hole. In turns out that there are actually many black holes in our universe. When you have a rather heavy star, let's say 100 times heavier than our sun. And, and they, stars also have the end of their lives. And then they actually end up collapsing into much smaller size. And some of them form black holes at the end of their lives. As a matter of fact there's even a super massive black hole at the center of our own galaxy. And I'll show you evidence of that. But before getting to the evidence, uh,let's talk a little bit about this idea called escape velocity. If you're not familiar with calculus you can skip this slide entirely. So if you try to escape gravity You would like to move away from the gravitational trapping potential, away, all the way to infinity, where your total energy goes back to 0. And the formula for energy is the sum of the kinetic energy depending on how fast you move And the potential energy due to gravity and the sum of them is conserved. So if you managed to escape the force of gravity, this sum must be either 0 or greater, namely positive. If this is negative, you are trapped by gravity. Just by solving this equation, you'll find that your velocity must be this, at least, to be able to escape the force of gravity, and this is what is called escape velocity. But if this required escape velocity is bigger than speed of light -- you know, nothing can go faster than the speed of light. So, you can never do that. So, by setting this combination to be the speed of light, that would be the condition that even light would not be able to escape from the force of gravity. So, that's this equation here. So, what this is telling you is that, if the mass of the object And the size of the object have exactly this combination given by speed of light. Then that's the black hole. Nothing can escape the force of gravity from this object, even when you are going with speed of light. So that tells you [COUGH] how big a black hole is, depending on its mass, and this will be one of the homework problems, if you would like to do it. And if you put in, for example, the mass of the sun into this equation, you'll find the radius is only 3 kilometers, 2.95. And so what it's telling you is that, if you managed to squeeze the entire sun of the sun down to 3 kilometers size, and that would end up being a black hole, so that you'd never be able to escape the force of gravity from it. That sounds crazy - who would be able to crush the entire mass of the sun down to such a small size? It turns out. That you can do that in the universe, and we will look at that later. So would you like to be near a black hole? Probably not.