[SOUND] Now let us discuss the properties of the Riemann tensor and also, well, this probably will follow by product. Also, we will see the power of the trick that one uses in the use of locally Minkowskian reference system that was introduced before in this lecture. So, locally Minkowskian reference system where I remind you, g mu nu = to eta mu nu plus subletting corrections and gamma mu nu alpha = to 0 plus subletting corrections. So let us fix this reference frame. In around the navigator point then Reimann tensor from it's definition, because gamma is zero, but derivatives are not necessary zero, then this guy is d alpha gamma new new beta minus d beta Mu alpha. Or, which is the same, after plugging in expressions for gamma is one half of d squared mu alpha, g mu beta minus d squared nu beta g mu alpha minus d squared mu alpha g mu beta plus d squared mu beta g mu alpha, where we have introduced a notion that d squared alpha deta is just second derivative d alpha d beta. So this is how a Reimann tensor looks in local systems. As you see, if the space is curved, it means that the first derivative of gamma doesn't vanish. It means that the correction to these guys is of order of psi. And the second derivative of the metric doesn't vanish, which means that the correction to this is psi squared. So that's exactly the point. That's exactly the reason why we have these corrections in case of this space. So, these guys, the metric is equal to this at the mu nu, only if the space is exactly flat around that point, or is equal to zero. So that's an important command. But now, from this form of the metric of the Riemann tensor, it is not hard to see its properties. We already notice that R, that Riemann tensor, changes its sign under the exchange of this indices that already have been used before in the definition. Also, now from this form one can easily see that it is also anti-semetric under the change of the first couple of indices. But, also one can also easily see that if one exchanges the pair of indices, so I mean this couple with this couple, it doesn't change its sign. And the last equality that one can observe for this expression for the Riemann tensor is as follows, that r mu nu alpha beta plus r mu alpha beta mu plus R mu beta mu alpha is 0. Differentiating this expression with covalently differentiating this expression. One can obtain the Riemann which is as follows, mu nu alpha beta covariant derivative with respect to gamma plus R mu mu gamma alpha covariance data with respect to beta plus r mu mu beta gamma covariant with respect to alpha is equal to zero. Now we have obtained all this relations from this form of the Riemann tensor means we have obtained it in locally Minkowskian reference system. But these relations are tensor relations, they relate tensor quantities, those quantities, which transform multiplicatively under coordinate transformations. It means that these relations will remain the same in any other reference system although they have been obtained only in this reference system. It means that these relations are always valid, always in any reference system. And this relation is referred to as Bianchi Identity. Now, using Riemann tensor, one can define a new quantity. First of all, if one will contract the first couple of indices, say, nu mu, contract means to make them equivalent, and sum over, then we'll get 0 due to anti-symmetry of this guy. But we can contract the first and the third index. Then this will be mu new mu beta, and we will obtain this stanza which is not 0. This is called Ricci tensor. And this quantity is actually symmetric under the exchange of its indices as it can sequence of this relation. So Ricci tensor is symmetric. Furthermore, we can contract indices of Ricci tensor. With the use of the that's the usual way of contraction of indices. And we will obtain the Ricci scalar. So this is a Ricci scalar, and this is Ricci tensor. Now, if one will contract indices here in this expression, this index and this, index mu and alpha, he will obtain the following consequence of the Bianchi identity, that R Mu nu mu covariant derivative with respect to nu. It's equal to 1/2 d mu r. This is the consequence of Bianchi identity that we have for the Ricci tensor and Ricci scale. Notice that this is a covariant derivative, because it acts on the scalar. Covariant derivative, when acting on the scalar, is equivalent to the regular derivative. Now we are in a position to say a few things about the number of the components of the Riemann tensor. One, we have these relations, we can say that. First of all like we will do that in dimensions although, so far we have been considering four dimensional space then. Let us consider d dimensional space then. Basically in any manipulations that we have been making during this lecture, we never use the number of dimensions in space-time. So what I ever been saying is valid for any D, and in fact for any signature of the metric, not only in Minkowski and also in flat space, etcetera, etcetera so far. Now let us find number of the independent components of Riemann tensor. Due to this anti symmetry properties, anti symmetry property we have d times d minus one half. Number of combinations of new mu. Mu new and for beta, for each of them. At the same time, due to this symmetry property, we have the falling number of whole combination of this. And it's one half of this times [D(D- 1) / 2 + 1]. So this is the total number of independent components of Riemann tensor according to these symmetry properties. But we have this relation. To find the number of these relations, so let us consider this quantity, let's consider this, let's denote this as B nu nu alpha beta. This sum as B nu nu alpha beta. And it is not very hard to observe that this quantity is actually totally antisymmetric, and they exchange of all of any couple of it's indices. Let us see that immediately from here using these properties. So it's not hard to see that B mu nu alpha beta, which is, by definition, this quantity. It is equal after exchange of the indices. It is equal to r nu mu alpha beta minus r nu alpha. Beta mu minus R mu beta mu alpha, and it is equal to minus B nu mu alpha beta. Similarly, one can check that this anti-symmetric under exchange of alpha beta nu mu, etc, etc. So, it means that this is totally anti-symmetric tensor, it means that this relation establishes the following number of conditions. So, it reduces the number of quantities, independent quantities in terms by which the following amount. We have the following number of conditions here d times d minus one, d minus two times d minus three over four factorial. This is a number of relations of a sort so we have to subtract from this quantity, this quantity. As a result we obtain D squared minus D squared minus 1 over 12. That leads In four dimensions this is 20, in four dimensions, 4D. In three dimensions this is 6, in 3D. And in two dimensions this is 2, in two dimensions. But we can use in principle in local Minkowskian reference system in a around any point. We can reduce this amount, even by bigger, using the symmetries, using extra symmetries D(D-1)/2 number of lawrencium and Lorentzian, and Lorentzian information in the rotations, the total number is this, which is six in four dimensions. This we have already been counting before. So using this number of symmetries, we can even reduce this number, so we have smaller number of independent components of the Riemann tensor. In is [INAUDIBLE] smaller than these given numbers. So those are the properties of the Riemann tensor. In the upcoming lectures, we are going to use Riemann tensor to define, under the other tensorial quantities. To define equations of motions and general relativity, and to use these equations in starting physics. So far that was just geometry of curved space times. [MUSIC] [NOISE]
