 Welcome to video 11 in this series on the general theory of relativity, as in previous videos we'll concentrate on concepts and predicted phenomena and leave most of the mathematical details to appendix videos. We've seen that Einstein's theory is pretty complicated mathematically. In fact, he originally doubted it would be possible to find exact solutions and expected the theory would only be applicable using approximations. Yet less than a year after he presented his equations, a fellow named Schwarzschild found the exact solution for a spherically symmetric mass. The Schwarzschild's solution allows us to calculate planetary orbits to describe the bending of light rays by stars and led to one of the most incredible predictions in all of astrophysics, the black hole. First, a couple of preliminaries. By measuring both time and distance in seconds, we've fixed the speed of light to be one, and this has simplified our expressions. We can also simplify the math by measuring mass in seconds. To do this, we consider Newton's law of gravity. At a distance r from the center of a spherical mass, m, the gravitational acceleration, a, in meters per second squared, is g times m in kilograms over r in meters squared, where g is the gravitational constant. We want to write the acceleration, a, in seconds per second squared, as the mass in seconds over the distance in seconds squared. This defines a mass in seconds as g over the speed of light in meters per second cubed times the mass in kilograms. Now, g is small and c is big, so the mass of one second is very large, about 203,000 solar masses. Another point to keep in mind is that Newton's theory is spectacularly successful in describing gravitational effects on Earth's surface and in the solar system, at least for objects traveling at much less than the speed of light. A mass bouncing on a spring and a pendulum are just two examples. Clearly then, if Einstein's theory is correct, it must reduce to and agree with Newton's theory in the limit of low velocity and small masses. In an appendix video, we show that this is indeed the case. Schwarzschild's solution uses so-called spherical coordinates. For our purpose, we only consider the two spatial dimensions of a plane represented by the screen and time. One coordinate, denoted r, is supposed to measure distance from the center of the massive body to a point of interest. The second spatial coordinate, denoted by Greek letter phi, or phi, is supposed to measure angle of the point of interest from some reference direction. r is analogous to altitude and phi to longitude. If we change these coordinates by small amounts, we create displacement, dr, and r times d phi. The metric of special relativity, the case where there is no mass and present, is ds squared equals dt squared minus dl squared. ds is an increment of proper time, that is, time is measured by a clock that freely falls between the two events. dt is the elapsed time between the events, and dl, the spatial separation between the events, is measured in our coordinate system. In spherical coordinates, dl squared is dr squared plus r squared d phi squared. In Schwarzschild's solution, the dt squared term is multiplied by a factor 1 minus 2m over r. And the dr squared term is divided by the same factor. m and r are both measured in seconds. When m is equal to 0, this reduces to the no gravity case of special relativity as it must. In an appendix video, we verify the complete solution and look at the details of the equations of motion. Let's first note that the term 2m over r is very small in our solar system. Its largest value, at sun's surface, is only about four parts in a million. So any relativistic effects in our solar system will be very small indeed. Let's now jump right into the predictions for planetary orbits and light beams. We'll start with numerically calculated orbits for both the Newton and Einstein theories. Newton's theory predicts that a particle orbiting a massive body will follow an elliptical orbit that closes on itself and repeats perpetually. Here we show the case of a slightly elliptical orbit. Einstein's theory predicts the particle will follow a path that resembles an ellipse that doesn't quite close on itself, but instead precesses or rotates through time. The orbit shown here is for the same initial position and velocity as we saw in the Newtonian case. If we make the attracting mass much larger, this effect becomes very pronounced. Again, Newton's theory would predict a perfectly elliptical orbit for this case. Although orbits in our solar system involve much smaller gravitational fields than the one we've simulated here, orbital procession can indeed be observed. Mercury is the planet closest to our sun. Its orbit has long been known to process at about 1,600th of a degree per century. Of this, 1,500th of a degree per century can be explained by the gravitational pull of the other planets. But as early as 1859, it was recognized that this left 1,100th of a degree per century unaccounted for. 1,100th of a degree per century is an incredibly small value, but astronomical observations are so precise that this is much larger than the observational uncertainty. Therefore, this was clearly a real effect and it puzzled astronomers for decades. However, using an approximate solution to his equations, Einstein was able to show that general relativity accurately explained this anomaly. This was the first significant new prediction of the theory to be verified. Indeed, in a seminal 1916 paper, the only physical evidence Einstein had for general relativity was its agreement with Newtonian theory and the low-velocity small mass limit and the explanation of the orbit of Mercury. Einstein wrote, these equations give us to a first approximation, Newton's law of attraction, and to a second approximation, the explanation of the motion of the perihelion of the planet Mercury. These facts must, in my opinion, be taken as a convincing proof of the correctness of the theory. In previous videos, we've seen that in general relativity, no single coordinate system can provide an undistorted view of all spacetime, but instead provides only a particular perspective. What perspective do our coordinates represent? Consider our metric as we move far away from the mass, that is, the coordinate r increases towards infinity. The 2m over r term shrinks to zero, and we are left with the metric of flat spacetime. That is, far from the mass, the coordinates have their normal or flat meaning. These are sometimes referred to as observer coordinates, or far away coordinates. But what about near the mass? This is where things get interesting. Schwarzschild's solution forces us to confront the concept of a singularity. A singularity is where a mathematical expression breaks down, typically, because some term becomes infinite. Assuming no real physical property can be infinite, this is usually taken to indicate a failure of the corresponding theory. In the Newtonian expression for acceleration, a equals m over r squared. There is seemingly a singularity at r equals zero, because 1 over 0 blows up to infinity. But the formula only applies outside the body, so this would only be a problem for an infinitely dense body, that is, 1 compressed all the way down to zero radius. The theory for a finite sized body looks like this plot. Outside the body's radius, gravity varies as m over r squared, but inside it actually decreases down to zero at r equals zero. If you could drill a shaft to Earth's center, you'd find no gravity there. In Newtonian theory, there are no singularities for objects of finite density. The Schwarzschild's solution has a singularity at r equals zero two, which we might dismiss by the same reasoning. However, it also has a singularity at r equals 2m, because there, 2m over r equals 1. 1 minus 1 equals zero, and 1 over zero is, quote, infinity. This singularity only requires the mass m to be compressed into a sphere of radius 2m, not zero, thus resulting in a finite density. So it seems possible to have an object of finite density produce a physical singularity. One physical implication of the Schwarzschild's singularity is that time stops there, at least as measured by a distant observer. Here's the metric again. For a clock at rest, the dr and d phi increments are zero, and we're left with ds equals a factor times dt. Turning this around, dt equals ds over the same factor. Recall that ds corresponds to the tick of a clock at rest, and dt to the tick of the distant observer's clock. As r approaches 2m, this factor approaches zero. So even though an observer near the singularity would experience a normal flow of time, measured by ds, his clock will appear to a distant observer to stop ticking, measured by dt. This is gravitational time dilation in the extreme. Another phenomenon is that light appears to freeze at the singularity. Recall that the increment of proper time, ds, is zero for a light beam. Consider light moving along the altitude or r direction so that there's no angle increment. With ds and d phi both zero, our metric looks like this. We can solve for the increment of r over the increment of t, and we interpret this as the speed of light in the r direction as seen by a distant observer. When r equals 2m, this expression is zero, that is, light freezes. Doing the same thing in the longitude direction, we find that the speed of light is the square root of that in the r direction. It's also zero at the singularity. We see that in general, the speed of light varies with position and direction, again, as seen by a distant observer, and the speed is zero at the singularity. Apparently, if light were generated at the singularity, it could not propagate and be seen by anyone above the singularity. No light means darkness, and appropriately, such an object is called a black hole. The singularity that separates the inside and outside of the black hole is usually called the event horizon. Let's see how this works by looking at numerical solutions of the equations of motion for light. First, let's shoot a light pulse across the screen. Every blue dot represents the position of the light pulse at one second intervals of coordinate time. The coordinate axes are labeled in light seconds. There's no mass, so the pulse takes 20 seconds to travel 20 light seconds in a straight line. Now, we'll shoot the same pulse from the presence of increasing black hole masses placed near the center of our plot. The light paths are progressively more curved. Eventually, they crash into the black hole. Looking at one of these crashing paths, we see that the dots get closer together as they approach the black hole. This is the perceived decreasing speed of light. Even though the simulation continues to run, the light pulse stops moving. It freezes at the horizon. Now imagine running the simulation in reverse. In that case, the light pulse would start off frozen and never move. No light, in fact, no thing can move outward from the black hole horizon. In between curving and crashing, we can actually get light to go into orbit around the black hole for a while. The orbits are unstable, so they eventually either crash or take off toward infinity, as in the case shown here.