0:13

Now let us discuss the properties of

Â the Riemann tensor and also, well, this probably will follow by product.

Â Also, we will see the power of the trick that one uses in the use of

Â locally Minkowskian reference system that was introduced before in this lecture.

Â So, locally Minkowskian reference system where I remind you,

Â g mu nu = to eta mu nu plus subletting corrections and

Â gamma mu nu alpha = to 0 plus subletting corrections.

Â So let us fix this reference frame.

Â In around the navigator point then

Â Reimann tensor from it's definition,

Â because gamma is zero, but

Â derivatives are not necessary zero,

Â then this guy is d alpha gamma new new beta minus d beta Mu alpha.

Â Or, which is the same,

Â after plugging in expressions for

Â gamma is one half of d squared mu alpha,

Â g mu beta minus d squared nu beta g

Â mu alpha minus d squared mu alpha

Â g mu beta plus d squared mu beta g mu alpha,

Â where we have introduced a notion

Â that d squared alpha deta is just

Â second derivative d alpha d beta.

Â So this is how a Reimann tensor looks in local systems.

Â As you see, if the space is curved,

Â 2:31

And the second derivative of the metric doesn't vanish,

Â which means that the correction to this is psi squared.

Â So that's exactly the point.

Â That's exactly the reason why we have these corrections in case of this space.

Â So, these guys, the metric is equal to this at the mu nu,

Â only if the space is exactly flat around that point, or is equal to zero.

Â So that's an important command.

Â But now, from this form of the metric of the Riemann tensor,

Â it is not hard to see its properties.

Â We already notice that R, that Riemann tensor,

Â changes its sign under the exchange of this indices

Â that already have been used before in the definition.

Â Also, now from this form one can easily see that it is also anti-semetric

Â under the change of the first couple of indices.

Â But, also one can also easily see that if one exchanges the pair of indices,

Â so I mean this couple with this couple, it doesn't change its sign.

Â And the last equality that one can

Â observe for this expression for

Â the Riemann tensor is as follows,

Â that r mu nu alpha beta plus r mu alpha

Â beta mu plus R mu beta mu alpha is 0.

Â Differentiating this expression with covalently

Â differentiating this expression.

Â One can obtain the Riemann which is as follows,

Â mu nu alpha beta covariant derivative

Â with respect to gamma plus R mu mu gamma

Â alpha covariance data with respect to

Â beta plus r mu mu beta gamma covariant

Â with respect to alpha is equal to zero.

Â 4:59

Now we have obtained all this relations from this form of the Riemann

Â tensor means we have obtained it in locally Minkowskian reference system.

Â But these relations are tensor relations,

Â they relate tensor quantities, those quantities,

Â which transform multiplicatively under coordinate transformations.

Â It means that these relations will remain the same in any other

Â reference system although they have been obtained only in this reference system.

Â It means that these relations are always valid, always in any reference system.

Â And this relation is referred to as Bianchi Identity.

Â 5:44

Now, using Riemann tensor,

Â one can define a new quantity.

Â First of all, if one will contract the first couple of indices, say,

Â nu mu, contract means to make them equivalent, and

Â sum over, then we'll get 0 due to anti-symmetry of this guy.

Â But we can contract the first and the third index.

Â Then this will be mu new mu beta, and we will obtain this stanza which is not 0.

Â This is called Ricci tensor.

Â And this quantity is actually symmetric under the exchange

Â of its indices as it can sequence of this relation.

Â So Ricci tensor is symmetric.

Â Furthermore, we can contract indices of Ricci tensor.

Â 7:01

Now, if one will contract indices here in this expression,

Â this index and this, index mu and

Â alpha, he will obtain the following consequence of the Bianchi identity,

Â that R Mu nu mu covariant derivative with respect to nu.

Â 7:23

It's equal to 1/2 d mu r.

Â This is the consequence of Bianchi identity that we have for

Â the Ricci tensor and Ricci scale.

Â Notice that this is a covariant derivative, because it acts on the scalar.

Â Covariant derivative, when acting on the scalar,

Â is equivalent to the regular derivative.

Â Now we are in a position to say a few things about

Â the number of the components of the Riemann tensor.

Â One, we have these relations, we can say that.

Â First of all like we will do that in dimensions although, so

Â far we have been considering four dimensional space then.

Â Let us consider d dimensional space then.

Â Basically in any manipulations that we have been making during this lecture,

Â we never use the number of dimensions in space-time.

Â So what I ever been saying is valid for any D, and

Â in fact for any signature of the metric, not only in Minkowski and

Â also in flat space, etcetera, etcetera so far.

Â Now let us find number of the independent components of Riemann tensor.

Â Due to this anti symmetry properties,

Â anti symmetry property we have d times d minus one half.

Â Number of combinations of new mu.

Â 8:48

Mu new and for beta, for each of them.

Â At the same time, due to this symmetry property,

Â we have the falling number of whole combination of this.

Â And it's one half of this times [D(D- 1) / 2 + 1].

Â So this is the total number of independent components

Â of Riemann tensor according to these symmetry properties.

Â But we have this relation.

Â To find the number of these relations, so

Â let us consider this quantity, let's consider this,

Â let's denote this as B nu nu alpha beta.

Â This sum as B nu nu alpha beta.

Â And it is not very hard to observe that this quantity is actually

Â totally antisymmetric, and they exchange of all of any couple of it's indices.

Â Let us see that immediately from here using these properties.

Â So it's not hard to see that B mu nu alpha beta,

Â which is, by definition, this quantity.

Â It is equal after exchange of the indices.

Â It is equal to r nu mu alpha

Â beta minus r nu alpha.

Â Beta mu minus R mu beta mu

Â alpha, and it is equal to

Â minus B nu mu alpha beta.

Â Similarly, one can check that this anti-symmetric under exchange

Â of alpha beta nu mu, etc, etc.

Â So, it means that this is totally anti-symmetric tensor,

Â it means that this relation establishes the following number of conditions.

Â So, it reduces the number of quantities,

Â independent quantities in terms by which the following amount.

Â We have the following number of conditions here d times d minus one,

Â d minus two times d minus three over four factorial.

Â This is a number of relations of a sort so

Â we have to subtract from this quantity, this quantity.

Â 11:41

In three dimensions this is 6, in 3D.

Â And in two dimensions this is 2, in two dimensions.

Â But we can use in principle

Â in local Minkowskian reference system in a around any point.

Â We can reduce this amount, even by bigger,

Â using the symmetries, using extra

Â symmetries D(D-1)/2 number of lawrencium and

Â 12:19

Lorentzian, and Lorentzian information in the rotations,

Â the total number is this, which is six in four dimensions.

Â This we have already been counting before.

Â So using this number of symmetries, we can even reduce this number, so

Â we have smaller number of independent components of the Riemann tensor.

Â In is [INAUDIBLE] smaller than these given numbers.

Â 12:44

So those are the properties of the Riemann tensor.

Â In the upcoming lectures, we are going to use Riemann tensor to define,

Â under the other tensorial quantities.

Â To define equations of motions and general relativity,

Â and to use these equations in starting physics.

Â So far that was just geometry of curved space times.

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Â