0:13

Now we are ready to define the last and

Â most important element of the Riemannian geometry.

Â That quantity which distinguishes flat space from curved space,

Â flat space time from curved space time.

Â That is curvature, or Riemannian tensor.

Â To do that, let us do parallel transport.

Â So we have a vector, V mu, at some point, V mu.

Â And we want to make at some point A, and

Â we want to make it parallel transport to a different point A,

Â B, C along this path and along this path.

Â 1:06

Here we have delta 1 of x, this is delta 2 of x and

Â this is delta 2 of x, delta 1 of x.

Â We assume that these guys are very small.

Â And we do not distinguish this guy from this guy, and this guy from this guy.

Â Just because distinction appears in the calculations

Â that we're going to do of the higher order of smallness.

Â So we make two parallel transports of this guy, V mu to this point, we obtain

Â two different vectors after parallel transporting like that and like that.

Â And then we want to see the difference between these two guys.

Â And the difference between these two guys defines for

Â us the curvature, the Riemannian tensor.

Â Let us see how it works.

Â So we have vector V mu here and we want to obtain vector V mu here.

Â So V mu at the point after the parallel

Â transporting along A B is the following saying.

Â 2:17

It's approximately V mu minus gamma,

Â mu, nu alpha given at the point A,

Â at this point A, V nu delta one x alpha.

Â So this is according to just the law of parallel transformation,

Â parallel transport that we have introduced before.

Â Now, we want to make parallel transport of this quantity along this path.

Â To do that, we have to bear in mind that gamma mu

Â nu alpha at the point (B) is not the same as gamma at the point A.

Â It is approximately,

Â after Taylor expansion of this guy,

Â is gamma mu nu alpha at A plus

Â first d beta gamma mu nu

Â alpha A delta 1 x beta.

Â So, we keep here and here only linear terms in these guys.

Â Because the difference these two reactors will

Â apparently happen to be of the second order in delta x.

Â And to obtain the precise expression for this difference,

Â we have to keep these terms only linear order terms, in terms of this guy.

Â So now, after parallel transporting V mu,

Â along the path A B C, we obtain the following.

Â It's just approximately V mu A B, so

Â we are parallel transporting this guy from here to here.

Â Now V mu a mu minus gamma

Â mu nu alpha at the point

Â B V nu A B delta 2 now x alpha.

Â 4:35

So now I plug here this quantity, and

Â here, and here, this quantity.

Â Then I get the following expression,

Â approximately, V mu minus

Â gamma mu nu alpha at the point A,

Â V nu delta 1 x alpha, minus,

Â 5:12

Gamma mu nu alpha at the point A,

Â plus d beta gamma mu nu

Â alpha at the point

Â A times delta 1 x beta,

Â multiplied by V nu minus gamma nu,

Â beta delta at the point A V beta,

Â delta 1 x delta.

Â And all that multiplied by delta 2 x alpha.

Â So now we open up the brackets and obtain approximately to the second

Â order in delta 1 and delta 2, to the second order in this quantity.

Â We keep not all terms, we keep only part of the terms.

Â We obtain the following expression.

Â Minus gamma mu nu alpha,

Â V nu, delta 1 x alpha

Â minus gamma mu nu alpha,

Â v nu delta 2 x alpha minus

Â d beta gamma mu nu alpha.

Â V nu delta one x beta delta

Â two x alpha plus gamma mu nu

Â alpha gamma nu beta gamma.

Â Multiplied by, that's

Â the continuation of this formula,

Â v beta delta 1 x gamma delta 2 x alpha.

Â So this is the expression for

Â the vector V after parallel transported along this line.

Â Similarly, one can write the expression for

Â the parallel transporting along this line.

Â The result is as follows, so this is just V mu A B C.

Â Now let me write V mu

Â 9:04

These are two vectors, two results of the parallel transport,

Â this guy to here, through this pass, and through this pass.

Â Now we want to see the difference between these two,

Â to obtain what means Riemannian tensor.

Â So the difference between these two

Â ways of parallel transporting V mu,

Â V A B C minus V mu A D C is the following.

Â Approximately, of course,

Â minus d alpha gamma mu

Â nu beta minus d beta

Â gamma mu nu alpha.

Â Plus gamma mu gamma alpha,

Â gamma gamma, nu beta,

Â minus gamma mu gamma,

Â beta, gamma, gamma,

Â nu, alpha multiplied

Â by v nu delta 1 x alpha

Â delta 2 x beta.

Â 10:38

Delta 2 x beta.

Â Now we want to express this quantity in a bit different way.

Â Let us introduce, notice that this quantity is antisymmetric,

Â and the exchange of alpha and beta in this is antisymmetric.

Â So it changes its sign and the change of alpha and beta [INAUDIBLE].

Â And let me introduce this guy,

Â delta S of beta which is the following,

Â it's delta 1 x alpha delta 2 x beta

Â minus delta 1 x beta delta 2 x alpha.

Â This quantity defines the area of that

Â parallelogram that we have been considering,

Â A B C, A D C, this quantity.

Â So, if we introduce this quantity, we can write this expression.

Â So, we basically can use the antisymmetry of this quantity, we can,

Â instead of this guy, we can use this guy.

Â But, we have to, so

Â this is equal then to 1 minus

Â one half R mu nu alpha beta

Â V nu delta S alpha beta.

Â So, the angle of the rotation of these guys, the result of the rotation of

Â this vector, after parallel transporting along two different guys.

Â Is proportional to the area of this parallelogram,

Â proportional to the vector itself, and

Â it is proportional to this quantity.

Â Which has the following form, by definition as follows from this formula.

Â R r mu nu alpha beta is just

Â d alpha gamma mu, mu,

Â beta minus d beta gamma mu,

Â nu alpha plus gamma mu,

Â gamma alpha,

Â gamma gamma nu beta.

Â Minus gamma mu gamma beta,

Â gamma gamma nu alpha.

Â So, this quantity is nothing but the Riemann tensor.

Â This is exactly Riemann tensor.

Â One frequently uses also a different expression for

Â it when we use all lower case indices.

Â So this is just g mu gamma R gamma nu alpha beta.

Â So lower case indices.

Â This guy is nothing but curvature for this connection.

Â That is another interpretation of this guy.

Â So in flat spacetime,

Â well first of all, this is a tensor.

Â So it transforms as a tensor on the coordinate transformation, so

Â it means multiplicatively.

Â In flat spacetime, one can choose everywhere Minkowskian metric.

Â In Minkowskian metric, this is zero so all is zero.

Â So this guy is zero.

Â In Minkowskian metric that is obviously zero.

Â But because it's zero in one coordinate system

Â which globally covers whole space time.

Â It is zero in any other coordinate system.

Â So in flat space, independently on the coordinate system that we use,

Â this guy is zero, flat space.

Â