Welcome back. Today we're going to talk about Einstein, his theory of General Relativity, and how we can use it to search for extra-solar planets. We're going to begin by talking about General Relativity. Then talk about how one of the predictions of General Relativity, the fact that matter curves space and light is deflected by stars, can be used to discover extra-solar planets. And then in the final section, we'll look about what these new observations of planets might be telling us about stability of planetary systems and how the planets form. So let's begin by reminding ourselves about what we learned already about gravity. And so far we've been talking about Newtonian physics, and how Newton's laws, force being proportional to the distance squared between objects and scaling with mass, tells us that stars moving on circular orbits have a relatively simple relationship between the star's velocity, or the planet's velocity, the star's mass, and the planet's distance from the star. And Newton and Galileo and Kepler taught us that planets move around closed ellipses, in the case of our solar system, these ellipses are fairly close to circles, closed orbits that come back to the same position. And that's what we observe basically for Earth, Venus, Mars. And Newton's laws work remarkably well In describing the solar system. But what happened as our observations began to improve late in the 19th century, is astronomers started to notice a problem. The problem occurred through the observations of Mercury. Mercury, of course, is the closest planet to the Sun. And when they studied Mercury's orbit, they found when Mercury came around, its ellipse didn't quite close. Every orbit, the position of the ellipse, precessed, slightly. As the orbit goes around, it kept precessing and precessing. Now, the precession of Mercury's orbit's quite small. It looks quite large on this figure, so you could see it. But the observed procession is only about 574 arc seconds a century. So that's a sixth of a degree every century. Or equivalently, every time it comes around, it misses closing by about six parts in a million. So it's a very small perturbation on the orbit. And it was a tremendous triumph of 19th century observational astronomy to be able to measure this very small precession. To me, an even more impressive accomplishment was they were able to calculate how much precession should occur due to the planets, because our solar system isn't just Mercury orbiting the Sun, Mercury is perturbed by Venus, by Earth, by Mars, by Jupiter, by the other planets. They included all the effects of the other planets. When you include them, you find that the predicted precession is about 531 arc seconds per century. Now I think most people would look at this tiny sensitive measurement, the reasonably good agreement between observation and theory, and think that's all there is. Newton's laws work remarkably well. Well, that's true. Newton's laws do do a pretty good job of describing what's going on in the solar system. But, there's something missing. There's something else going on here. The first thing people suspected was a new planet. Observations of Uranus' orbit had led to the discovery of Neptune. They realized that Uranus was being perturbed by another planet out there. And that's what led astronomers to look for and discover Neptune. Astronomers thought this worked the first time, they found Neptune. Maybe there's a new planet inside the orbit of Mercury. They named this new planet Vulcan. And in fact there were some erroneous observations, around the turn of the century, that led people to think they actually discovered this new planet Vulcan. We now know that it's not there, and Mercury is the innermost planet. What was needed to explain this anomaly was not a new planet, but new physics. And that was going to have to wait for Albert Einstein. Now Einstein develop two important theories that we're going to need to talk about. And they are related but they're different. One is the theory of special relativity. Theory of special relativity says light travels at a fixed speed, the speed of light, and nothing, no information can go faster than the speed of light, matter can't go faster than the speed of light. This was a switch from the Galilean view. The Galilean view of how velocities behave is velocities add. If I'm running on a train, the velocity you see, observing me from the distance, is the sum of the trains velocity and my velocity. That's what Galileo taught us. What Einstein taught us is when the velocities get very large, so that the train is moving at half the speed of light and I'm running at half the speed of light, you don't get to add velocities. And special relativity describes why you can't velocities and describes the behavior of things moving close to the speed of light. General Relativity was an idea that Einstein developed a decade later. The idea in General Relativity is that gravity behaves differently than the force of the distance view that was espoused by Newton. What Einstein suggested was that gravity consists of two simple ideas. Matter tells space how to curve and the curvature of space tells matter and radiation how to move. So the Earth curves space around it, the Sun curves space around it, and the Earth just tries to move on a straight line through the curved space created by the Sun. So you can think about a satellite orbiting the earth. It's just moving around, constant energy in this curved space. One of the remarkable things Einstein realized when he developed this theory, and he starts from very elegant mathematical principles and worked his way forward, is that his theory correctly predicted the observed precession of Mercury. And that was a good first step because it explained an existing observation. But then the remarkable thing it also did was lead to some new predictions. Perhaps the most dramatic prediction of General Relativity is the existence of black holes. Since matter curves space, if the density of matter is large enough, if it curves space enough, it could curve space so much that nothing can escape from a black hole. And this is what we think about as a black hole, is a black hole is point where there's so much matter concentrated into such a tiny region, that the curvature of space is so large, that it traps light, it traps matter. Nothing can escape from a black hole. We have observed black holes through our universe. There are black holes that are several times the mass of the Sun, that we've detected through their effects on neighboring stars. And there are supermassive black holes whose masses range from a million up to a billion times the mass of the Sun lurking in the centers of galaxies. Our own galaxy hosts a four million solar mass black hole. These huge black holes that sit in the centers of galaxies are actually engines that power quasars and active galactic nuclei. When we think about these black holes there's a characteristic radius developed, an idea developed by Karl Schwarzschild, a German scientist who was actually working on this theory while serving in the trenches in the first world war, Carl Schwarzchild developed the solution to Einstein's theory. Sadly, died in the first world war due to disease, and in his solution, it turns out you can work out what is the characteristic radius that what we now call the Schwarzchild radius associated with the black hole. And that Schwarzchild radius depends on the strength of gravity, mass and the speed of light squared. If you think about this as a Newtonian physicist and think about light going around in an orbit around the black hole, this would be the sort of characteristic radius you'd write down. That's a pretty small radius for an object who's mass is the size of the Sun. The Schwarzchild radius is about three kilometers. So, you'd have to shrink the Sun down to the size of being much smaller than the scale of Manhattan to collapse the sun into a black hole, and make it curve space enough. But this Schwarzchild radius is a useful concept when thinking about general activity, even when we're not dealing with black holes, because it sets the characteristic size on which General Relativity is important. So, how important is General Relativity in our solar system? How big is the deflection of light, things that we expect. Well the characteristic deflection of light is going to depend on the ratio of the velocity that object move at to the speed of light, and that turns out to be that the deflection of light goes as the Schwarzschild radius divided by how far you are from the object. So if we plug in the numbers for the Earth, the general relativistic correction in the Earth's orbit is about a part in a million. So if we only use Newtonian physics, we're mostly right. We make an error of about 0.0001% in our calculation. For Mercury, which is three times closer, this number's smaller, so this correction for Mercury's about three times larger. Go through the numbers as Einstein did. You add this correction into your orbit, and you could explain the precession of the perihelion of Mercury. What Einstein's theory predicts not only is matter deflected by the curvature of space, but light itself. And Einstein recognizes [INAUDIBLE] made an important prediction, that if you were looking a distant star behind the Sun, its light would not move on a straight line. It would be deflected by the Sun. This effect was difficult to observe. You have to observe a star very close to the Sun. As we all know it is a lot easier to see the stars at night, and rather difficult to see a star right next to the Sun. Fortunately nature offers us a very nice opportunity to test this theory, because every now and then, we have a solar eclipse. And the moon passes in front of the Sun, blocks the Sun, and lets you see the stars behind it. And it was the solar eclipse expedition of 1919 that led to the confirmation the theory of General Relativity, and so it made Einstein famous. And this is a headline from The New York Times, which I love reading, from the 1919 New York Times, announcing light askew in the heavens, men of science more or less agog over results of eclipse observations. And as they noted, Einstein's theory triumphs. Stars not where they seemed, or were calculated to be, but nobody need worry. Of course this statement's not actually right, the stars actually were exactly where Einstein said they would be. They were deflected from their position by the amount that Einstein predicted. And this observation is what confirmed the theory of General Relativity. And it's this effect which we'll come back to in the next section of the lectures which we will use to study and detect extra-solar planets. So let me leave you with a question, where were going to apply General Relativity. You're going to work as a relativist and work out the characteristic size, the Schwarzschild radius of the black hole in the center of our galaxy, and compare that to the distance between the Earth and the Sun.
