 Welcome to this ongoing series on the theory of relativity. In this video we want to investigate the topic of differential geometry, a subject in mathematics that plays a central role in our theory. I'll try to take a visual approach as much as possible. Still, math is the language of physics and no explanation of relativity is complete without it. There's no need to follow the math in any detail, but if you are interested, more details are given in the appendix videos that are listed in the description box. Our story begins with Carl Friedrich Goss, a fellow who was immensely influential in several areas of math and science. In 1818 he was involved in an extensive survey of the German state of Hanover, and this contributed to his interest in the study of curved surfaces, what we now call differential geometry. Suppose that you're the royal road builder and there are a couple of villages that you've been told to build a road between, and the road is supposed to be as short as possible. Where do you put it? I mean, it's not as simple as drawing a straight line because you've got hills and valleys. Do you use intuition, trial and error? Well a scientific approach is to develop a mathematical model of the terrain that describes it in terms of coordinates, numbers, functions, and so on. Then you formulate the path problem as a mathematical equation that you solve to get a mathematical description of the best path. Although our terrain exists in three-dimensional space, it forms a two-dimensional surface. What are the surfaces represented by a topographic contour map? Position on the map is uniquely specified by two coordinates, say y1 and y2, typically longitude and latitude. The third coordinate, y3, is represented by labeled constant elevation contours. Where the contours are close together, the terrain is steep. Where they are far apart, the ground is relatively flat. We say this curved two-dimensional surface is, quote, embedded in flat three-dimensional space. It's two-dimensional because each point on the surface is uniquely specified by two numbers, y1 and y2. Of course the y3 coordinate will have important physical consequences back in the real world. Consider the road building problem again. It might be tempting to say that the shortest path between two points is a straight line, so there's our road. But, one look at the elevation contours and we see that this would take us over a peak. Probably not the shortest path. No, the shortest path will probably be a curve that avoids the peak. This is analogous to the picture that general relativity paints a particle motion in the presence of a gravitational field. In an inertial frame, flat spacetime, particles move along straight lines at constant velocity. But when gravity is present, curved spacetime, particles move along curves as dictated by what we might think of as a type of spacetime equivalent to elevation contours. In both cases, we're just taking the shortest path between two points. We're going to develop our surface model using length measurements and relating those to position on the surface as denoted by coordinate values. To get started, imagine we have a rectangular graph paper type coordinate system with one yard spacing between coordinate lines. An arbitrary displacement can be thought of as a displacement in the y1 direction followed by a displacement in the y2 direction. Then the net displacement s forms the hypotenuse of a right triangle and we can use the Pythagorean theorem to calculate it. s squared is equal to y1 squared plus y2 squared. So the Pythagorean theorem is the basic tool that allows us to interpret coordinate changes as the distance between two points on the surface. Now we've seen in previous videos that a graph paper coordinate system won't fit on a curved surface. So we've got to consider more general types of coordinate systems. To start, assume we have rectangular coordinates but the coordinates are not measured in yards. We still want to calculate the displacement s in yards. So say the coordinate x1 is measured in fathoms. The Pythagorean theorem only works if the sides of the triangle are all measured in the same units. So we're going to need a scale factor a1 equals two yards per fathom. Then a coordinate change x1 corresponds to a distance a1 x1 yards. And suppose the other coordinate x2 is measured in feet, then we'll need a scale factor a2 is equal to one third yard per foot. And a coordinate change x2 will correspond to a distance a2 x2 yards. Now we can use the Pythagorean theorem to calculate f squared in yards squared is equal to a1 squared x1 squared plus a2 squared x2 squared. These scale factors we have to employ are called metric coefficients and they will play a central role in everything that follows. Another thing that can happen is that the coordinate axes are not at right angles. These are called non-orthogonal coordinates. Suppose the angle between the axes is theta and we've used the scale factors a1 and a2 as before so that the sides of the triangle are all measured in yards. The Pythagorean theorem only works for right triangles. So if theta is not 90 degrees, we have to make a right triangle with the green lines shown here. Now a2 x2 is the hypotenuse of a right triangle and we can use trigonometry to get the lengths of the green sides. a2 x2 times the sine and cosine of theta. Finally, we have a right triangle with s is the hypotenuse and we can apply the Pythagorean theorem to find s. Using and simplifying the expression, we end up with our previous a1 squared x1 squared plus a2 squared x2 squared terms plus a new term that contains both scale factors and coordinates in the cosine of the angle theta. When theta equals 90 degrees, that term goes away, cosine of 90 degrees is equal to zero and this reduces to our previous result. Now for bookkeeping convenience, we collect the various factors we've developed into an array called the metric tensor. The metric tensor is absolutely at the core of general relativity. For example, one form of metric tensor describes a black hole and other describes big band cosmology and so on. For our two-dimensional surface, the metric tensor is a 2 x 2 array or matrix and we use subscripts that denote the elements as g11, g12, g21, and g22. And these represent those a1 squared a2 squared and a1 a2 cosine theta factors we saw previously. It's convenient to break the 2 times a1 a2 cosine theta factor into 2, 1 times a1 a2 cosine theta factors because it'll make all the math more symmetric later on. So with this labeling, our previous result for s squared can be written as g11 x1 squared plus g22 x2 squared plus g12 x1 x2 plus g21 x2 x1. This is the Pythagorean theorem generalized to scaled and or tilted coordinates. A more compact notation is to use the capital Greek letter sigma to represent a sum and then we have the sum from i equals 1 to 2, sum from j equals 1 to 2 of gij xi xj. We sequentially substitute 1 and 2 for i and j to get the above expression. Einstein said let's just drop the sigmas and write s squared is equal to gij xi xj with the understanding that anytime you see a subscript, say i, appear twice you're supposed to sum over all possible values. So now we have a very nice compact notation for the generalized Pythagorean theorem. Finally, we have to consider what happens when our coordinate system is curved. Even the generalized Pythagorean theorem only works on triangles. So what we have to do is limit consideration, at least at any one time, to a very small patch of the surface over which the coordinate curves are essentially lines. Rigorously, this is only valid for vanishingly small displacements that we'll call differential displacements. That's the differential and differential geometry. And we'll denote these starting with a d as in dx1 and dx2. Coordinate changes dx1 and dx2 correspond to distances that are represented here by little arrows. Notice that the same dx2 change, that is from one green line to the next, can represent a different length depending on where it occurs on the surface. So when we write our generalized Pythagorean theorem, now the metric coefficients, the scale factors, have to be functions of position. And notice also that the distance ds here is a differential distance, a very, very small distance instead of the s squared we had before. Now we have all the tools we need to figure out distances in terms of any coordinate system on any surface. One more visual aid for thinking about our distance formula. You can take the 2 by 2 metric tensor and write the differential displacements, dx1 and dx2, above and to the left. And then for each metric coefficient, multiply by the dx value in that row and the dx value in that column and do that for all four coefficients. Add the results together and you get the generalized Pythagorean theorem, what's called the first fundamental form of differential geometry. Now let's apply all this to our path problem on a curved landscape. Suppose this surface is a mathematical representation of the landscape. In an aerial view, we lose the 3D perspective on elevation, although we can represent it by elevation contours. We want to be able to relate displacement on the map to displacement on the ground. Suppose we're interested in the black path shown here. Let's take a cross-sectional slice of the surface and view it in profile. We end up with a plotted elevation vertically versus the x1 coordinate from the map horizontally. Now suppose we go to x1 equals 10 on the map and ask what ground displacement ds results from a map displacement dx1. In this case, the ground is essentially level, so ds and dx1 would be essentially the same. However, if we do the same thing at x1 equals 30, because the surface is sloped there, we'll get a larger ds for the same dx1. These effects are what the metric coefficients account for. At x1 equals 10, g1, 1 is essentially 1. While at x1 equals 30, g1, 1 is something larger than 1. And we can see that as the slope of the ground gets larger, the g1, 1 coefficient will have to also. In the extreme case that the slope is infinity, that would be a sheer cliff, g1, 1 would approach infinity. In other words, if you fall off a cliff, you don't change your longitude or latitude. So your position on the map won't be changing, but you'll still be doing a whole lot of moving. It'll just all be in the downward direction. In fact, we'll see that this is one way to view what happens at the event horizon of a black hole. There, one of the space-time metric coefficients goes to infinity. And the interpretation is that someone approaching the horizon would perceive herself to be moving rapidly, while a distant observer would not see her moving at all. So now if we know the metric coefficients at all points on the map, we can take any path, say the black curve shown here, and go along it, adding up all the ds displacements on the ground, corresponding to our dx1 and dx2 displacements on the map, and arrive at the total on the ground length of the path. It's then possible in principle to solve for the shortest possible path between any two points. So we've looked at some examples of the way differential geometry was developed and how it can be applied to analyze surfaces. In the next video, we'll see how other mathematicians extended Gauss's work from two-dimensional surfaces into abstract n-dimensional spaces, and how this provided an ideal mathematical framework for the physical concept Einstein had developed.