 Welcome to the second video on differential geometry in this series on the theory of relativity. As in the last video, we're talking about an area of mathematics, so there will be naturally quite a bit of math. However, it's still the case that the equations are simply ways to precisely describe physical concepts, and you don't need to follow the math to follow the concepts. In the last video, we saw how Gauss pioneered the study of curved two-dimensional surfaces, what we now call differential geometry. This is primarily based on the idea of being able to make length measurements and relating those to surface coordinates using a generalized Pythagorean theorem over small regions of the surface. Two-dimensional surface geometry is easy to visualize and has many important practical applications. Yet, inevitably, in any field of mathematics, one will eventually hear the command, go forth and generalize. One of Gauss's students, Riemann, did just that. Indeed, the type of differential geometry used in general relativity is often referred to as Riemannian geometry. During the 19th century, other mathematicians made further contributions. Among the names will come across are Christophe and Rishi. So, by the time Einstein went looking for a mathematical formalism for his ideas of curved spacetime, they had already put together an ideal package and had it waiting for them. Here are the relevant ideas they came up with. In place of the two-dimensional surface, they imagined an abstract, n-dimensional space. Since it's not easy to draw, say, a four-dimensional space, I'll continue to use simple two-dimensional surfaces for illustrative purposes. But the dimension of the space is just the number of coordinates you need to locate a point. The coordinates themselves do not need to have any physical significance. You can imagine, for example, giving one group of people cans of red spray paint and telling them to go running through the countryside spraying lines. Another group can do this with blue paint. You don't care how they do this, as long as no two lines are the same color cross, because then each red line represents a constant value of a coordinate x1, say, and each blue line represents a constant value of a coordinate x2. Every point in space now has a unique set of n coordinates, and these serve as the address or the phone number of the point. Again, the coordinate values don't have to have any physical significance. We just need to know that if two points have the same coordinates, they're indeed the same point. And that the coordinates vary smoothly, so we can use the generalized Pythagorean theorem trick that we developed in the last video. In general relativity, n is four, three dimensions of space and one of time. But soon people were working on five-dimensional theories, and today we have string theory where n is equal to ten dimensions. If somebody in the future wants to make a 37-dimensional theory, all they have to do is change the number of coordinates n. Then we have the idea of the metric tensor. In principle, we can measure or calculate this at every point in the space. For instance, on the bottom blue line, the first coordinate varies between x1 and x1 plus dx1, where dx1 is a differential that is very, very tiny amount. Using some sort of measurement system, we find the length ds1 of this segment. Relating these by ds1 squared equals a metric coefficient g11 times dx1 squared. We can solve for the coefficient g11 because we know ds1 and dx1. Then we can do the same thing on the left red line for coordinate x2 and get the coefficient g22. Finally, measuring the diagonal distance ds12, which corresponds to simultaneous changes dx1 and dx2, and writing out the corresponding terms of the generalized Pythagorean theorem and realizing that we now know every value in that expression, except for g12, we can solve for that coefficient also. Then we collect these values into the array that we call the metric tensor. For n equals four dimensions, as we have in spacetime, we'd also have steps dx3 and dx4 and end up with a 4 by 4 array. Anyway, if we do this at every point, then we can calculate any distance and hence do any kind of geometry we want to in the space. Although we talk about ds as a distance, abstractly it's just a quote invariant, meaning it's an intrinsic property of space independent of the coordinate system. For the same two points, we'd have to get the same ds, regardless of how we assign coordinate values. And this is what anchors the subjective math to objective reality. Now that we can locate points, we need to be able to describe arbitrary paths through space. Suppose we want to describe this green path. We'll do so by specifying the coordinates as a function of distance s along the path. The analogy with physics is to specify a particle's position as a function of time. We place our invariant yardstick at the start of the path, read the coordinates of the two ends, and plot those on the graph corresponding to s equals zero and say s equals one. And we move the yardstick end to end and plot the coordinates of the new endpoint at s equals two. And we continue on this way until we have a graph that describes the entire path. In higher dimensional spaces, we'd simply have more curves on the graph. Now we have all end coordinates on the path to find as end functions of distance along the path. As a shorthand, we'll refer to all coordinates as a group as x with no subscript, something we call a vector. The individual coordinates are then the components of this vector. Another way to think about a path is in terms of the direction of travel, what a pilot or sailor would call the heading. We think of direction as represented by an arrow called the tangent vector u. The end components of u tell us how the coordinates change when we take a step ds along the path. The change dx1 is u1 times ds, dx2 is u2 times ds, and so on. Dividing everywhere by ds, we see that each component of u is the change in the corresponding coordinate divided by the distance traveled. The analogy to physics is that if s was time, then these would look like the components of velocity. If we take our length formula and divide both sides by ds squared, we find that the vector u has a length one. And we say that u is a unit vector. Later, we'll see that this means that in some sense, we're always moving through space time with the same speed. If we don't move through space, we have to move through time. And if we go faster in space, we move slower in time. Now, suppose our path has some property, such as being the shortest path between two points, what we've called a geodesic. We need to know how to change direction as we move along the path. The tangent vector tells us how to change coordinate values at each step, and something has to tell us how to change the components of the tangent vector to keep us on the right path. This is what the laws of physics do. They tell each particle how to change direction so as to stay on its proper path. The geodesic concept is central to differential geometry and relativity. As we've seen in flat space, this is simply the directions for drawing a straight line. In curved space, it's not a line, but it still gives you the shortest path between two points. We find that the change in the i-th component of u, dui on the left, is the output of an equation for which the inputs are the current components of u, the uj on the right, and the coordinate changes dxk. Our summation convention applies so we sum over both j and k from one to n since they appear twice in the same term, and the capital Greek letter gamma with one output superscript i and two input subscripts, j and k, is called the Christoffel symbol and represents the coefficients of the equation. These coefficients are derived from the metric coefficients. The details are fairly technical, so if you're interested, they're given in an appendix video link in the description box. Now we have a recipe that tells us the change in direction for any change in position. If we divide both sides by our step size ds, realize that dxk over ds is just uk, and think of s as time and u as velocity, we end up with an equation for the change in velocity over change in time. But that's what we call acceleration. So as Einstein points out in his 1916 paper, this means that the right side of the equation is nothing other than the gravitational field. What we call gravity is nature's way of telling particles how to follow a geodesic through spacetime. The summation over two subscripts proceeds as it does for the metric tensor. Take the n by n array of coefficients, write the dx values above and the u values to the left, then for each gamma coefficient, multiply by the corresponding u and dx values, add all these terms, and the negative of that is the change in the u component. The new wrinkle is that we have to do this for each of the n components of u, and that's what the superscript on the gamma represents. So Einstein's geometric picture of physics is that dynamics is the process of tracing a geodesic through spacetime. We started a point specified by coordinates x1 and x2 in two dimensions, in four dimensions you'd also have x3 and x4. We head off in some direction specified by a spacetime velocity u, which has length 1. We take a step of length ds in direction u and get new coordinate values, then we let spacetime tell us how to change the components of u. We continue doing this as time goes on. So far so good. Now we start thinking about curvature. As we've seen in previous videos, Einstein came to realize that matter curves spacetime. So how do we tell if our space is curved? If our space is flat, it might appear to be curved because we're using curved coordinates. But in that case, we can lay a piece of graph paper flat on the surface. Then using those rectangular coordinates, we have Euclidean geometry and special relativity. In three dimensions, the squares would be cubes, and in four dimensions, they would be hyper cubes, whatever those are. But if space is intrinsically curved, we can't lay the graph paper flat over the entire surface. So a test for flatness would be to simply try and draw squares, or cubes or whatever. Starting at any point, draw two lines of the same length s at right angles. On both lines, turn 90 degrees and then draw another line of length s. If space is flat, the two endpoints will be the same and we'll have drawn a square. This draw-square test leads to a new object called the Riemann tensor. Again, the details are technical and there's an appendix video if you're interested. If this vanishes at every point in space, then space is flat. And this will be true regardless of the system of coordinates we use. The Riemann tensor is a four-index object. It has one superscript i that corresponds to a single output and three subscripts, j, k, and l, that correspond to inputs. It tells you everything there is to know about the curvature of your space. The Riemann tensor is derived from the Christoffel symbols and those, in turn, are derived from the metric coefficients. So we see that the metric coefficients tell us everything we need to know. In two dimensions, the Riemann tensor can be arranged as a two-by-two array of two-by-two arrays. In four dimensions, you get a four-by-four array of four-by-four arrays. Each component is an expression that tells you about some aspect of the curvature of your space. If all the components vanish, then your space is perfectly flat. If spacetime is flat, then, as we've seen, that would mean the inertial frame physics of special relativity would describe everything. We know that's not the case, but we have to find out how spacetime curves. What limits does nature put on the curvature? Those limits, presumably, will serve as our law of gravity. For a clue, let's turn to Newtonian physics. In Newtonian physics, we say that the gravitational field gets stronger the closer we are to a massive object. So if we drop balls from two different heights, both will accelerate toward the massive object. But the closer ball will do so more rapidly. Suppose we surround a planet with a dust cloud lying at rest between two spheres. We can calculate the volume V of the dust cloud in terms of the inner radius R1 and the outer radius R2. The gravitational field will be stronger at this smaller radius, so when the dust cloud starts falling, the inner radius will shrink more rapidly than the outer radius. Due to Newton's 1 over R squared law, as this happens, the volume V does not change at least for short periods of time. That is, starting from rest, the two spheres will collapse and the entire cloud will move towards the planet, but in a way that maintains its volume. In fact, this is a general statement, regardless of the gravitational field and the shape of the dust cloud. It's a geometric way to state Newton's law of gravity in empty space. That is, when there is no source of gravity inside the dust cloud, in empty space, a dust cloud falling from rest can change shape, but not volume, at least for a short period of time. If there's a source of gravity inside the dust, then the cloud will tend to collapse and the volume will decrease. But that's a different case, and we'll come back to that in a future video. We now ask, does differential geometry have a way to describe the conservation of volume? The answer is yes. There's an object called the Ricci tensor. It's derived by adding together some of the components of the Riemann tensor. It's a two-index object, a four-by-four array of expressions, and if the Ricci tensor vanishes, even though the space may be curved, then a small region, the points of which move a short distance along parallel geodesics, may change in shape, but not in volume. We then say that the space is, quote, Ricci-flat. In four dimensions, the Riemann tensor is a four-by-four array of four-by-four arrays, and it has 256 components. Fortunately, it has a lot of symmetry, then this reduces the number of unique components to only 20. Of the 16 components of the Ricci tensor, 10 are unique. So in a sense, the Ricci tensor captures those parts of the Riemann tensor that have to do with volume changes. Since the Ricci tensor is the same size as the metric tensor, it seems that requiring Ricci flatness would result in 10 equations and 10 unknowns. Just what we need to figure out the metric tensor, which, recall, tells us everything we need to know. For illustration, in two dimensions, the Riemann tensor is a two-by-two array of two-by-two arrays, and the Ricci tensor is the sum of the two diagonal two-by-two arrays. In four dimensions, we'd have a sum of four diagonal four-by-four arrays. So differential geometry, this branch of mathematics motivated by practical two-dimensional problems such as land surveying, and later generalized to abstract n-dimensional spaces, we've tailor-made for describing general relativity. Choose any system of four coordinates, the first three representing space and the fourth representing time. Find a corresponding metric that gives the Ricci flatness property. Then derive and solve the equations of motion. Beautiful. A perfect fit. A match made in heaven. You can see why Einstein was very excited. But there's a problem, and it's a showstopper. This entire subject requires the existence of a, quote, yardstick. Our generalized Pythagorean theorem is supposed to allow us to convert coordinate changes into invariant yardstick measurements. And this invariant has to be the same for all possible coordinate systems, all possible frames of reference. But special relativity has already shown us that different coordinate systems will, in general, not agree on time and space measurements. Think of the famous twin paradox. So we're left wondering, what is the yardstick of spacetime? We just need that one foundational piece and everything else will drop into place. And that is what the next video will be about.