[MUSIC] Now we are ready to define the last and most important element of the Riemannian geometry. That quantity which distinguishes flat space from curved space, flat space time from curved space time. That is curvature, or Riemannian tensor. To do that, let us do parallel transport. So we have a vector, V mu, at some point, V mu. And we want to make at some point A, and we want to make it parallel transport to a different point A, B, C along this path and along this path. Here we have delta 1 of x, this is delta 2 of x and this is delta 2 of x, delta 1 of x. We assume that these guys are very small. And we do not distinguish this guy from this guy, and this guy from this guy. Just because distinction appears in the calculations that we're going to do of the higher order of smallness. So we make two parallel transports of this guy, V mu to this point, we obtain two different vectors after parallel transporting like that and like that. And then we want to see the difference between these two guys. And the difference between these two guys defines for us the curvature, the Riemannian tensor. Let us see how it works. So we have vector V mu here and we want to obtain vector V mu here. So V mu at the point after the parallel transporting along A B is the following saying. It's approximately V mu minus gamma, mu, nu alpha given at the point A, at this point A, V nu delta one x alpha. So this is according to just the law of parallel transformation, parallel transport that we have introduced before. Now, we want to make parallel transport of this quantity along this path. To do that, we have to bear in mind that gamma mu nu alpha at the point (B) is not the same as gamma at the point A. It is approximately, after Taylor expansion of this guy, is gamma mu nu alpha at A plus first d beta gamma mu nu alpha A delta 1 x beta. So, we keep here and here only linear terms in these guys. Because the difference these two reactors will apparently happen to be of the second order in delta x. And to obtain the precise expression for this difference, we have to keep these terms only linear order terms, in terms of this guy. So now, after parallel transporting V mu, along the path A B C, we obtain the following. It's just approximately V mu A B, so we are parallel transporting this guy from here to here. Now V mu a mu minus gamma mu nu alpha at the point B V nu A B delta 2 now x alpha. So now I plug here this quantity, and here, and here, this quantity. Then I get the following expression, approximately, V mu minus gamma mu nu alpha at the point A, V nu delta 1 x alpha, minus, Gamma mu nu alpha at the point A, plus d beta gamma mu nu alpha at the point A times delta 1 x beta, multiplied by V nu minus gamma nu, beta delta at the point A V beta, delta 1 x delta. And all that multiplied by delta 2 x alpha. So now we open up the brackets and obtain approximately to the second order in delta 1 and delta 2, to the second order in this quantity. We keep not all terms, we keep only part of the terms. We obtain the following expression. Minus gamma mu nu alpha, V nu, delta 1 x alpha minus gamma mu nu alpha, v nu delta 2 x alpha minus d beta gamma mu nu alpha. V nu delta one x beta delta two x alpha plus gamma mu nu alpha gamma nu beta gamma. Multiplied by, that's the continuation of this formula, v beta delta 1 x gamma delta 2 x alpha. So this is the expression for the vector V after parallel transported along this line. Similarly, one can write the expression for the parallel transporting along this line. The result is as follows, so this is just V mu A B C. Now let me write V mu A D C, just the expression for it, it's all derivation, the derivation is very similar. V mu minus gamma mu nu alpha V nu delta 2 x alpha minus gamma mu nu alpha V nu delta 1 x alpha. Minus d beta gamma mu nu alpha V nu delta 1 x alpha delta 2 x beta. Plus gamma mu nu alpha gamma nu beta gamma V beta delta 1 x alpha delta 2 x gamma. Now you can see how hard one should work to work out the details of General Theory of Relativity. So, we have these two quantities. These are two vectors, two results of the parallel transport, this guy to here, through this pass, and through this pass. Now we want to see the difference between these two, to obtain what means Riemannian tensor. So the difference between these two ways of parallel transporting V mu, V A B C minus V mu A D C is the following. Approximately, of course, minus d alpha gamma mu nu beta minus d beta gamma mu nu alpha. Plus gamma mu gamma alpha, gamma gamma, nu beta, minus gamma mu gamma, beta, gamma, gamma, nu, alpha multiplied by v nu delta 1 x alpha delta 2 x beta. Delta 2 x beta. Now we want to express this quantity in a bit different way. Let us introduce, notice that this quantity is antisymmetric, and the exchange of alpha and beta in this is antisymmetric. So it changes its sign and the change of alpha and beta [INAUDIBLE]. And let me introduce this guy, delta S of beta which is the following, it's delta 1 x alpha delta 2 x beta minus delta 1 x beta delta 2 x alpha. This quantity defines the area of that parallelogram that we have been considering, A B C, A D C, this quantity. So, if we introduce this quantity, we can write this expression. So, we basically can use the antisymmetry of this quantity, we can, instead of this guy, we can use this guy. But, we have to, so this is equal then to 1 minus one half R mu nu alpha beta V nu delta S alpha beta. So, the angle of the rotation of these guys, the result of the rotation of this vector, after parallel transporting along two different guys. Is proportional to the area of this parallelogram, proportional to the vector itself, and it is proportional to this quantity. Which has the following form, by definition as follows from this formula. R r mu nu alpha beta is just d alpha gamma mu, mu, beta minus d beta gamma mu, nu alpha plus gamma mu, gamma alpha, gamma gamma nu beta. Minus gamma mu gamma beta, gamma gamma nu alpha. So, this quantity is nothing but the Riemann tensor. This is exactly Riemann tensor. One frequently uses also a different expression for it when we use all lower case indices. So this is just g mu gamma R gamma nu alpha beta. So lower case indices. This guy is nothing but curvature for this connection. That is another interpretation of this guy. So in flat spacetime, well first of all, this is a tensor. So it transforms as a tensor on the coordinate transformation, so it means multiplicatively. In flat spacetime, one can choose everywhere Minkowskian metric. In Minkowskian metric, this is zero so all is zero. So this guy is zero. In Minkowskian metric that is obviously zero. But because it's zero in one coordinate system which globally covers whole space time. It is zero in any other coordinate system. So in flat space, independently on the coordinate system that we use, this guy is zero, flat space. So in flat space, this guy is zero in all reference frames everywhere. So the difference of the curved space is that this guy is not zero. At least in some places of the space where the space is different from flat one. So that is exactly the quantity which distinguishes flat spaces form curved ones. And that is the main quantity that we are going to use in our lectures follow. [MUSIC]