 In a previous video, we saw how failure to detect the so-called luminiferous ether led Albert Einstein to propose the principle of relativity. The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. In applying this principle to inertial reference frames, we were led to the equations of special relativity as presented by Einstein in 1905. In this video, we will retrace Einstein's steps towards a general theory of relativity. The general theory itself was published in 1916, more than a decade after the special theory, but here we're going to look at some of the concepts presented in a 1911 paper titled, On the Influence of Gravitation on the Propagation of Light. Special relativity deals with the physical equivalence of inertial reference frames. Recall that an inertial reference frame is one in which objects tend to stay put or move with constant speed in a straight line. Earth's surface is approximately a two-dimensional inertial frame, but the third dimension, up and down, as we well know, is subject to gravitational acceleration. As Galileo pointed out, to the extent that air resistance can be neglected, this acceleration has a very curious property. All objects, regardless of composition or mass, fall with the same gravitational acceleration g. Einstein started thinking about the deeper meaning of this fact. In particular, what are the implications for distinguishing between different frames of reference? Well, it's a lovely spring day here, so naturally I've moved a hurl a ball of lead through the air. There it goes at one-tenth speed. Here we plot the ball's position at one-sixtieth second intervals, and what we see is that it follows a curved trajectory as is characteristic of accelerated motion. However, as mentioned previously, if we look only at the coordinate parallel to the ground, we see the constant velocity motion characteristic of an inertial reference frame. It's when we examine position in the up and down coordinate that we see the non-uniform motion due to gravitational acceleration. Yeah, man, that gravity thing, it's always pulling everything down. I mean, even if you nudge something upwards, darn gravity turns it around and pulls it right back down. And you can't ever get anything to stay still, can you? Well, look at here, Wilbur. It's a dang-nab-ball-lead of floating in the air. What the hell is going on? Well, I'll tell you what's going on, boys. I jumped up, took my feet off the ground, and that there ball just went on a floating. Now, I know you're saying, fool, that ball's falling, just you and the camera's falling with it. And indeed, you've probably seen how airplanes can be used to simulate weightlessness by putting their occupants into a freely falling reference frame. So here's a question for you. Is Dr. Hawking's apple falling or floating? More precisely, suppose we have two identical boxes containing some objects. The box on the left is floating in, quote, empty space, while the box on the right is falling in the uniform gravitational field of a massive object. Are there any mechanical experiments that we can perform that would distinguish between these two cases? The answer is no, because all objects fall with the same gravitational acceleration. There will be no relative change in position amongst the various objects in the falling box. Consequently, there will be no way to tell the difference between falling and floating. And now Einstein applies his principle of relativity. If these two frames of reference are mechanically equivalent, they must be optically equivalent also. Now imagine pulses of light sent from one side of each box towards the other side. For the floating case, of course, the pulse of light will travel at a constant speed in a straight line from one side of the box to the other. However, if light travels in a straight line relative to the ground, then the falling box will necessarily have to perceive the beam of light to be curving upwards. However, if the principle of relativity is correct, the path of light in both boxes must be identical. This will mean that people on the ground must necessarily perceive the ray of light to be bending downward. That is, the light is perceived to be falling in the gravitational field, just like a mechanical object would. This is how Einstein was able to conclude that gravitational fields bend beams of light. And moreover, he could calculate how much bending would occur. Suppose a pulse of light is traveling from left to right at the speed of light. It travels a distance L across the screen. That takes a time L over C. But during that same interval, gravitational acceleration imparts a velocity G times T in the downward direction. And since T is L over C, that's G L over C. And that results in the beam being bent at an angle which is approximately the ratio of those two velocities. That is, G L over C over C, or theta is equal to G L over C squared. A similar analysis allowed him to calculate the total bending that would occur when a beam of light passes by a star. If capital G is the gravitational constant and M is the mass of the star, Einstein's result was that theta was equal to 2 gm over the speed of light squared divided by r0, which is the minimum distance between the center of the star and the beam of light. In the final general theory of relativity, this value actually gets bumped up by a factor of 2. Nonetheless, this made the prediction that light passing near the sun would be bent by about one three thousandth of a degree, or just about at the resolution capability of Earth-based astronomical observations. For a very large mass, say a cluster of galaxies, this bending could be enough to produce what's called gravitational lensing, in which a single object appears in multiple ghost images. Here's a beautiful example from the Hubble Space Telescope. In the center you have a cluster of galaxies, and surrounding that you have these ghost images of a single object. Now, what is the physical mechanism by which a beam of light is bent? Light is an electromagnetic wave. The bending of a beam corresponds to the bending of wave fronts. Wave fronts bend or refract as they pass through regions in which the speed of light is different. Suppose the blue lines here represent wave fronts, and the red curves represent light paths. In order for the wave fronts to be tilted relative to each other, for there to be bending of the light rays, those two paths must be different in length. Suppose the difference in height between the two paths is a little amount dH, and suppose the length of the lower path is L. Then the upper path must have a length of L, plus a little bit more delta, which is theta dH, where theta is the angle through which the light beam has bent, that same theta that we derived previously. The only way to travel a greater distance in the same time is to travel at a greater velocity, and therefore the speed of light must be larger on the upper path than on the lower path. So let's call the speed of light C on the lower path. On the upper path, it will be C plus some small change dC. The ratio of those two velocities, C plus dC over C, is equal to the ratio of the path lengths of the distances covered in that same amount of time, L plus delta over L. Now, delta, as we said, is theta dH, and plugging in our previous result for theta and cancelling the common factor of L throughout, we end up with this. The ratio of velocities is 1 plus G over C squared times dH. So the ratios of the speed of light on the two paths is 1 plus G over C squared dH. dH is the change in height between the two paths. You can solve this to get directly dC over dH, the change in the speed of light per change in height, and the result is simply G over C, the gravitational acceleration over the speed of light. For Earth, if you plug in 9.8 meters per second per second for G and 300 million meters per second for C, you get 33 nanometers per second per meter. 33 nanometers is about the size of a virus, and so the speed of light increases for every meter you go above the Earth by about one virus width per second. Now that's a very small number, but if you had a much larger mass, and if you looked at the change over a very large distance or height, you might get very significant changes in the speed of light. Let's take a look at this. So we have this basic result that the change in speed of light with change in height is G over C. G is the gravitational acceleration. Newton's law of gravitation says that that value G falls off as 1 over the distance from the center of the object squared. That is, little G is equal to big G, the gravitational constant, times the mass of the object M over H squared. H is the height measured from the center of the object. If you take that expression and plug it into the equation on the left, and if you assume that as H goes to infinity as you get very, very far from the object, so that the gravitational field is essentially zero, that the speed of light takes on some value C0, which would be the speed of light in empty space 300 million meters per second, then you can solve for the speed of light as a function of height from the center of the object. The result is that the speed of light as a fraction of the speed of light in empty space is the square root of 1 minus some constant R s over the height H. And the constant R s is 2 GM over C0 squared. If you plot the speed of light versus height H, you see that as H decreases down to be equal to this constant R s, so that H over R s is equal to 1, the speed of light decreases down to zero. Apparently, under these conditions, light would stop propagating, would freeze in some sense, and we would certainly be justified in calling such a thing a black hole. Interestingly, this constant R s turns out to be precisely the so-called Schwarzschild radius that comes out of the rigorous general theory of relativity. Or we could call it the radius of a black hole, or typically the radius of the black hole's event horizon. It's important to emphasize that we just got lucky this is not a rigorous derivation, but it certainly is showing some of the concepts that will have to go into a general theory of relativity, and some of the bizarre things that will come out of this application of this very simple concept of the principle of relativity. Now, just to get some orders of magnitude to see what kinds of conditions might pertain to this frozen light phenomenon, if you plug in the mass of the sun, you get a radius of about 3 kilometers. If you plug in the mass of the earth, you get a radius of about 9 millimeters. So in the case of the earth, you'd be talking about a ball about 2 centimeters in diameter, about the size of your thumb, containing the entire mass of the earth. Or in the case of the sun, a ball about 6 kilometers across with the entire mass of the sun. That's an incredible prediction, and it's come out of nothing more than Galileo's observation that all things fall with the same gravitational acceleration and Einstein's insight into this fundamental principle of relativity.