Covariant differentiation

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From the course by National Research University Higher School of Economics

Introduction into General Theory of Relativity

94 ratings

National Research University Higher School of Economics

Introduction into General Theory of Relativity

94 ratings

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

From the lesson

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

Meet the Instructors

Emil Akhmedov
Associate Professor
Faculty of Mathematics

[SOUND]

[MUSIC]

So, let me clarify my point.

Well, first of all.

And with the use of the metric tensor and its inverse tensor,

I can also lower and higher the indices of any tensor.

For example, if I have a tensor like this mu nu, say,

alpha beta.

Then I can higher the index of alpha like this, and define corresponding

with the corresponding number of indices, like this.

And here, I stress that the order of indices is important.

That's the first thing.

So now, I'm ready to explain why transfers are important.

For example, if I have such a quantity, say like this,

T mu of mu Gamma and

I multiply it by another tensor, S,

say carrying indices alpha, nu, gamma.

This quantity should transform also as

a tensor because it carries one index, tensor index.

So it should transform as a vector.

And it does transform as a vector

under the conditions that these guys transform appropriately.

Because transformation of every such index is compensated by a transformation of

every such index and the same is true for

this, so the only index which remains is this.

As a result, this quantity is transformed as a vector.

So manipulations with are self consistent and

have obvious properties under the coordinate transformations.

And that's what we are seeking for.

So, and also, it is obvious that a scale or product of two vectors,

scalar product of two vectors is invariant under

the coordinate transformations like this.

That is this quantity.

And one can write it like this.

Sorry, this is all the same.

This is a different ways of writing of the same quantity.

So, tanzers as a tool to use to write equations in special and

general theory of relativity, so metric is, of course, is a tanza.

Because as we discussed during the previous lecture for

the metric, we have the following consideration.

That integral in one coroney system looks like this.

And if we transform it to the other coordinate

systems, and then it does like this.

G bar of beta of x bar

d x bar alpha d x bar beta.

Because of these and

after coordinate interval doesn't change,

we have the falling law of transformation for

the metric that g mu nu affects,

is equal to g bar alpha beta x bar d x bar alpha d x mu d x bar beta d x nu.

And this transformation law we have been using during the previous lecture

at the very beginning, so metric is just a concrete example of a tenant.

So, but then, after all these definitions, we encounter the following problem.

That differentials, differentials are not transforming appropriately, suppose.

Suppose we consider a differential of a tensor.

I mean, suppose we consider a difference of A mu at the point (x+dx) and

subtract from it A mu (x).

To linear order in dx,

this is equal to d

nu A mu of x d x nu.

And now I'm going to show that despite the fact that this quantity carries

two indices, carries two indices, so nu and

mu, but it doesn't transform appropriately as.

Now I'm going to show that.

I'm going to show that this quantity, this quantity d nu a mu,

dx nu which is just to linear order by definition

of the difference of these two factors.

It is not a tensor with two indices, this by the fact that it carries two linear.

Well, by the way, I'm going to use several different notations for

the ordinary differential.

Namely dAmu/dxnu is the same as dnu

A mu and the same as A mu

comma nu.

So, why this quantity doesn't transform appropriately.

Consider this thing.

Consider A bar alpha comma beta function of x bar.

Well, this is d over d x bar beta

d of a bar alpha, by definition,

by definition, but this guy,

let us express this guy through

that unbarred quantity.

Well, this is a function of x bar, through that unbarred quantity, so this becomes

e over ex bar dX bar

alpha over

dX mu, A mu of X.

So this, I just used the expression for the transformed form of this stanza.

And now, I have to understand that, X.

X is a function of x bar.

A mu is a complicated function of x bar, it's a function of x which is and so

on, write as a function of x bar.

So, now I use Leibniz rule to differentiate this.

First, I differentiate this quantity with x bar.

But this is the same as differentiating complicated functions,

meaning that I first take the derivative.

Well, let me write it and then explain.

So x.

Of course, I write this DX Alpha, DX Mu.

And then, I differentiate.

So, what I do here, I explain in a second.

A mu, comma.

Nu x plus,

before writing the remaining quantity let me just stress what I did here.

So I applied this differential to this function, so

I differentiated it, this guy, with respect to x nu.

And here, and it's written and

then differentiated x nu as a function of x bar with respect to x bar.

So if we would have had only this term, this term.

Then this quantity would transform as a tender.

But the problem is that we also have to apply the differential,

this differential to this guy.

And that is a problem, in fact, because, what we get is a falling stage.

So, we differentiate this guy with respect to twice

x bar alpha d x mu d x nu.

And then, so we assume that this is a function of x unbarred.

And then we differentiate unbarred x nu with respect to the bar beta.

So, I mean, we assume that this differential,

this derivative here is a function of x, unbarred x.

Times a mu (x)

exactly because of this term, this quantity

with that simple standard differentiation, doesn't transform as intended.

Notice that if we would have had only linear transformations,

linear coordinate transformations, this term would be absent and

then this quantity would be a quantity.

That's the situations we encounter in special relativity, in flat space-time.

But in curved space-time, and for generic coordinate transformations,

this guy doesn't vanish.

As a result,

this quantity doesn't transform as a tensor, that complicates things.

My goal here is to define co-variant differential,

which does transform as an appropriate tensor quantity.

Let me define it.

The problem with this saying,

that we subtract a vector at the point.

At this point, from the vector at this point.

To make appropriate quantity, we have to parallel transport

this quantity to the point x plus [INAUDIBLE].

How it is done, I'm going to explain right now.

Now the co-variant derivative of

a mu is the following thing.

It's a difference between this quantity, And

the following thing.

[A mu(x) + delta

A mu(x)].

And this guy is exactly result of the parallel transport of this

quantity to the point (x+dx).

Who is this guy, and what it means parallel transport?

I will explain a bit later, but at this moment I am going to define this quantity,

this quantity.

So I'm going to explain this, the meaning of parallel transport on some

simple example But at this moment, let me just specify.

So I have a point x, I have a nearby point x plus dx,

I have some vector field here, so I have,

say, for vector a mu at this point, and

I have a full vector a mu at this point.

So this is A(x+dx),

and this is A(x).

So, this guy doesn't coincide with the part of the transport of this quantity.

It's some value of this field at this point, but then I parallel transport.

According to some rule how parallel transport is done,

I paratransport this quantity here, and I obtain some different vector,

which is A plus delta A, at this.

And then I subtract from this is and that's how I get this quantity.

So, what is this thing?

What is this thing?

It is as follows, it is as follows.

First of all, I assume that this interval is very short, so

the dx is very small despite the fact that I draw it big.

So this delta a should be delta a should

be a result of some transformation, so

there is some matrix M, which does some

transformation on our vector A at this point.

So who is this matrix?

So it's some rotation of this vector.

The result of this result is one plus, so

unit matrix plus this short rotation given by this M.

So, who is this guy?

For small dx, it should be proportional to dx.

So I define it.

This is just the definition

with the minus sign with

some quantity gamma mu,

nu alpha of x, dx alpha.

A mew of x.

So, this quantity is referred to as a connection for the parallel transport.

And I'm going to explain it geometric meaning in a second.

[MUSIC]

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