Covariant differentiation(part 2)

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Del curso dictado por National Research University Higher School of Economics

Introduction into General Theory of Relativity

81 calificaciones

National Research University Higher School of Economics

Introduction into General Theory of Relativity

81 calificaciones

General Theory of Relativity or the theory of relativistic gravitation is the one which describes black holes, gravitational waves and expanding Universe. The goal of the course is to introduce you into this theory. The introduction is based on the consideration of many practical generic examples in various scopes of the General Relativity. After the completion of the course you will be able to solve basic standard problems of this theory. We assume that you are familiar with the Special Theory of Relativity and Classical Electrodynamics. However, as an aid we have recorded several complementary materials which are supposed to help you understand some of the aspects of the Special Theory of Relativity and Classical Electrodynamics and some of the calculational tools that are used in our course. Also as a complementary material we provide the written form of the lectures at the website: https://math.hse.ru/generalrelativity2015

De la lección

Covariant differential and Riemann tensor

We start with the definition of what is tensor in a general curved space-time. Then we define what is connection, parallel transport and covariant differential. We show that for Riemannian manifolds connection coincides with the Christoffel symbols and geodesic equations acquire a clear geometric meaning. We end up with the definition of the Riemann tensor and the description of its properties. We explain how Riemann tensor allows to distinguish flat space-time in curved coordinates from curved space-times. For this module we provide complementary video to help students to recall properties of tensors in flat space-time.

Covariant differentiation15:31

Parallel transport10:24

Covariant differentiation(part 2)9:11

Locally Minkowskian Reference System (LMRS)16:07

Curvature or Riemann tensor15:36

Properties of Riemann tensor13:26

Conoce a los instructores

Emil Akhmedov
Associate Professor
Faculty of Mathematics

[MUSIC]

So we have defined a rule for the parallel

transport of a vector with a upper index.

So the rule is like this.

A nu (x) d x alpha.

So the question, what is the rule for

the parallel transporting the vector with a lower index.

And to define it, we have to, it should be in agreement with this rule.

And to define it, we have to bear in mind that parallel transport is

basically a kind of transformation, kind of rotation of a vector.

Transformation of a vector.

If we move it along some paths, not necessarily geodesic,

in our space time or space, under such a transformation,

of course, scalar product doesn't change.

Scalar product under parallel transport

doesn't change as a result we have this rule.

Hence, B mu,

delta A mu is equal to,

from this relation

is minus A mu delta B mu.

Now, here we can use this rule for

this vector, as a result,

we obtain that this is A mu times gamma mu.

Gamma mu B nu of dx alpha.

Hence, from this,

that this is equivalent to this,

one can obtain that dA mu is equal

to gamma nu mu alpha (x).

A nu (x) dx alpha,

In addition to this equation.

As a result, we have the following definition of a covariant derivative.

So let me write it explicitly.

So we have the following definition of the covariant derivative.

So covariant derivative off a vector

a mu with an upper index which by

definition is the same as D alpha

of a mu is just the following, d alpha,

a mu plus gamma mu, nu alpha, A nu.

This just follows from the equation that I have been writing so far.

Similarly, A mu alpha,

which is the same as D alpha,

they mean different notation for

the same thing, is d alpha,

A mu minus gamma nu mu alpha A nu.

So this is covariant differential.

And also, one more comment is in order.

Is that apart from this gamma, we are going to use also

gamma with all lower case indices.

Beta nu alpha by definition

is just beta mu gamma mu nu alpha.

Well, natural question is to ask what is a covariant

differentiation of tensors with more indices?

Well, to show examples,

let me just write a few formulas explicitly.

Well, similarly, we have a tensor with two indices, mu and nu.

And I want to take covariant differential with respect to this

guy, which, as you may guess,

is just the same in notation as this guy.

It is equal to d alpha A mu nu,

and then I of course have to apply gamma to both indices separately.

So it's gamma mu beta alpha A beta nu,

plus another application to a different index,

gamma nu beta alpha, A mu beta.

So this is a rule for this guy.

Now if we want to apply covariant

differential to a guy which has one

upper index and one lower index,

the result is as follows,

d alpha a mu nu plus gamma mu.

I apply gamma to the upper index.

Beta alpha A beta nu and

minus, according

to this plus here and

minus here, minus gamma

beta nu alpha A mu beta.

And furthermore, if we have to

apply D alpha to something which

carries both lower case indices,

we have d alpha A mu nu minus gamma beta,

mu alpha, A beta, oops sorry,

A beta, no beta here,

A beta nu minus gamma beta nu alpha, A mu beta.

So these are the formulas, and similarly for more complicated terms,

if there's more indices, a different kind of indices in different places.

So it's not hard if you will make many exercises

with the tensors, you will get used to it.

It's not horribly complicated.

One thing one have to bear in mind at this point.

That in Minkowski space, in case if metric

is eta mu nu, so metric is eta mu nu.

Then, it follows that gamma mu nu alpha is 0.

I have to stress that for Minkowski space it is 0.

If we have flat space in different curvilinear coordinates

gamma is not necessarily 0.

It's only Minkowski space it's 0.

And, well that goes in accordance with the fact that in flat space,

the parallel transport doesn't change the vector along a curved closed contour.

But not only that, because like in principle, gamma can be non 0,

but still the vector doesn't transform if we

parallel transport it along the closed contour.

We will see this fact in a different manner at sometime during this lecture.

[MUSIC]

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