0:13

So let me again clarify what we have made.

We have obtained the falling sin at the beginning of this lecture,

we have introduced up sell the tanzer t mu nu, energy momentum tensor for

the gray, such that we have the following conservation law.

0:38

And this guy has a tedious expression

through the components of the metric tanzer.

At the same time after that we have considered linearized perturbations

in flat space time.

So we have considered flat space time plus perturbations on top of it.

1:14

exactly the same T mu nu plus t mu nu where

this guy expresses linearized perturbations, so psi mu nu Is

just traceless part of h so it's just h

mu nu minus one half at the mu nu

2:01

This equation, in a sense, is very similar to the one we have in electromagnetism,

where we have the following situation.

4pi/c U r two mu in the appropriate gauge,

and we want to consider free gravitational waves.

Free electromagnetic waves are obtained when the right hand side is 0 here,

and free gravitational waves are obtained when this part is 0.

So no sources for the gravitational waves.

And we first going to solve the equation with the right hand side equal to 0 and

then we're going to consider this quantity for the gradational waves.

That's we going to do now.

3:21

For this, to solve this equation, here epsilon and K are some constant.

They are each independent.

K is real.

Epsilon is complex.

So, for quantity to solve this equation, K should- if we just flag it here,

we obtain that K alpha times K alpha should be 0 well, this is just

the consequence of the application of two derivatives on the exponent.

And also,

K mu epsilon mu nu, minus

half K nu epsilon mu.

4:04

Mu should be 0.

This is a consequence.

This equation is a consequence of gauge condition.

I remind you that the gauge condition,

the condition in which we have obtained this equation is as follows.

It's d mu times psi mu nu should be 0.

4:27

So from this equation we have this.

If we flag this here, we obtain this equation.

Now we're going to solve these two equations in some particular frame.

Let us choose This equation just says that we have a light like.

4:44

This K is light like, which describes waves propagating with the speed of light.

And so we want to choose such a frame, call it a frame in space,

such that K alpha is just K,0,0,K.

If we plug this vector into the exponent we obtain the following situation,

the exponent just acquires the following form, it's k z minus t,

t minus z which describes

a wave propagating along the cert direction with the speed of light.

5:27

Now using this vector k which solves this equation, we want to solve this equation.

This equation we have 4 for each mu we have equation.

Nu 0, 1, 2, 3.

So, for new 1 and 2, we have the following

from this equation for new 1 and 2.

We have the following situation,

that Epsilon 01 is equal to epsilon 31 and

epsilon 02 is equal to epsilon 32.

6:32

using these equations already obtained relations between components of

epsilon tanzer which is a polarization tanzer for the gravitational wave.

Mu = 0 component of this equation implies that

epsilon 0 3 is = to 1/2 epsilon 0 0 plus epsilon 3 3.

6:57

Now let us perform, remember that this equation was obtained in this gauge.

But there is a remaining gauge freedom that we can transform.

We can, and transform h mu nu according to

the law h mu nu plus d mu epsilon 0

mu such that epsilon 0.

7:36

So now, then the solution of this operator can be represented like this,

at the solution of this equation can be represented like this,

if S minus i Epsilon new.

7:56

real path of course, over this quantity minus i.

k alpha x alpha, where k alpha solves this equation.

So represented like this, it does solve this equation,

if k alpha obeys this quantity, obeys this relation.

9:14

So if we fix this epsilon through this, components of this term like this.

As a result we obtain, we achieve the following

goal, that epsilon13 bar is

9:32

equal to epsilon bar 2 3 is equal to bar epsilon 0 0 and

equal to epsilon bar 3 3 and all of them are equal to 0.

As a result of these all relations

we have the following situation that polizaration tanzer.

Which is this quantity.

Has the only none 0 components as follows.

That's epsilon 1 1 bar is equal to minus epsilon 22 bar.

And epsilon bar 12 is equal to epsilon bar 21.

These are the only none 0 components of this tanzer.

And they have these relations.

10:29

traceless part of h, basically.

And we have been looking for the solution of this together with the gauge condition,

the gauge condition is dim sin is equal to 0.

And we have been looking for the solution of this equation in the form

real part of epsilon mu where this is Tanzer,

complex tanzer minus i k alpha x alpha and

k and absolute r constant x independent, k is real absolute is complex.

11:09

So, what we have found that After fixing this gauge and

using remnant gauge transformation,

you do harmonic solutions of right there.

We have found that this tanzer has

no 0 components epsilon 1 1 we meet,

we do not use bar on top of epsilon so

we consider already the concrete gauge and the need of the bars.

11:48

21, as a result we have the following situation that the metric

tensor the metric for this gravitational waves as the following form.

So it's a linearized solution of Einstein equation.

This is linearized approximation to the Einstein equations in the vacuum.

12:10

(1+h11)dx^2-

2h12dxdy-

(1-h11 d

y squared- d z squared.

So this form follows from the fact that due to this,

these are the only known 0 components of this tenzor.

So the only known 0 components of this guy are 1 1, 2 2, 1 2, 2 1.

As a result Curvature of the metric is as follows.

So we have the following picture basically so

in space, in space coordinates so

if this is z direction this is y and this is x.

We have a gravitational wave which propagates along this direction.

13:32

keep in mind that this guy is complex.

So, h11 is equal to Modulus

of absolute 11 cosine of k

times Zet minus t plus pi 0 And

h 1 2 is models of epsilon 1 2

times cosine of k times z minus t si

0 where this are the initial phases

which are hidden in this complex.

Quantities.

14:14

So this sense here, so

this guy solves the linearized approximation to Einstein equations, and

what it does, it describes a propagation of the wave which affinely transforms,

so it's like compresses one direction for appropriately chosen, if we choose phi 0.

Equal to 0 and psi 0 equal to pi over 2, this is a picture we obtained.

They're like half a period,

like goes along this direction, compresses in one direction and expand in the other.

And then compresses in this direction and

expands in which was previously compressed.

And so this sit he situation we have for the gravitational leaves.

For the gravitational leaves.

Now let us see what happens with T nu.

So we want to plug this quantity obtain quantity

into so we want to consider the formation on the right hand side.

We want to consider T mu nu.

Which is as we remember is of the order of H square.

So to find it one has to use that tedious

complicated expression for t mu nu through the components of the metric stanza.

And I remind you that there was g mu nu, this kind of quantities.

And in our linearized approximation,

it's approximately Y value equal to g mu nu comma alpha

which is approximately minus h mu nu comma alpha.

So the only non-zero contribution

to T mu new as far as Rho is relations is obtained

follows from the term like this, g mu alpha one has to look for

the expression for this t mu nu through the components of the terms.

G mu alpha, g mu beta, g gamma.

18:03

So.

What we have is that along the direction of propagation of the gravitational wave,

along the said direction, the reason energy flux,

that's exactly the point we obtain for the gravitational wave,

gravitational wave carries energy on top of the fixed background.

So, we have a background metric, flat metric and we have perturbation of this,

this perturbation curved space-time, and propagates with the speed of light,

and carries energy gravitational energy along this direction.

That's the reason we call this as gravitational waves,

and that's what they do.

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