0:59

of t, times dy squared minus z squared of t,

dz squared.

So this is exactly well first of all it's partially

homogenous because this is a function independent of special coordinates.

So they functions of time only.

But these are neither tropic because every direction scales was different scale.

Only when these are equal to each other, we'll reduce to the previous case

1:37

So we are going to discuss the vacuum solution.

So when energy momentum tensor is 0.

So for this case one can calculate Richie denser.

The not real components of the Ritchie tensor as follows.

It's theta dot, where the notations I will explain in a moment,

A squared + B squared + C squared and

R11 = -A dot- theta A R22

= minus B dot

minus theta B.

R33 = minus C dot minus theta C and A and

B and Ca = X dot / X.

Dot means differential with respect to time, here and here.

So A = X dot / X.

B = Y dot / Y, and

C = Z dot / Z, this Z.

And theta Standing here is actually A + B + C.

So these are the expressions for

the non trivial components of Ricci Tensor for this metric.

Now the energy momentum conservation condition for

this case is rho dot plus theta

times p plus rho equals to 0.

In case if p and rho are both non zero.

3:54

Just for general discussion we listed it here.

So the condition, the (00) component of Einstein equation imposes this condition.

That -AB- BC- CA = 0.

This is 0, 0 component of Einstein's equation.

As a result because this is 0 from this one sees the set squared

is just A squared Plus b squared plus c squared.

4:34

Hence again using this using

this one vacuum Einstein

equation Where equation is R00 = 0.

This is not the same.

This is R00 minus 1/2 g00R = 0.

But from R00 = 0, vacuum Einstein equation,

we obtain for here that theta dot plus theta squared equal to 0.

The solution to this equation obviously is theta

equal to 1 minus t 1 over t minus t0 and adjusting.

6:29

We are discussing Kasner-like solution which is an isotropic but

specially homogeneous.

And we have introduced a notation that there is a theta

which is x dot over x +

y dot over y + z dot over z.

And at the same time we have obtained the condition.

Addition that teta squared it actually from the Einstein equation it folds.

We had the notation that this is A this is B this is C,

and from the Einstein equation we have obtained

that this is X squared plus B squared plus C squared.

That what we have obtained and also from the Einstein equation we have obtained

that this is one over T and

X dot over X is equal to P/t,

Y dot/Y = q/t,

z dot/z = r/t.

Where so far p, q and r are constants of an integration,

arbitrary constants of integration.

But from here and here and these relations

we obtained that p + y + r is equal to 1 and

p squared + q squared + r squared is equal to 1.

This is their conditions to which p,q, and r are subjective.

Now, one can solve this equation for X, for Y and for Zed and plug them there.

After the proper rescaling of this coordinate, and this coordinate,

and this coordinate,

one can get rid of the constants of integration which appear here.

Nd to obtain the Kasner matrix is as follows.

The dx squared is dt squared minus t to the power 2p

8:47

dy squared- t to the power 2r dz squared.

So you see Kasner's solution describes the following situations.

As time goes by we have that these three directions are independently,

not related to each other, expanding with different, or

shrinking, it depends on where time goes.

That they're, in different manner, expand with different powers.

So let us explain why one physical

situation where the Krasner solution Appears, naturally appears.

Consider schwarshil metric under there, under the horizon.

So it means that we consider the following metric.

DR squared, one minus, sorry, RG over R.

Minus 1 minus r g

over r 1 minus 1 dt squared minus r squared d omega squared.

And here r is less than rg so this is not a Schwarzschild metric, not quite that.

But it does solve Weinstein equations for this value of r and

these r and t are not directly related to r and t in Schwarzschild solution.

But they are related to the Crusco coordinates and that's how they can be

related to the coordinates, Schwarzschild standard, Schwarzschild coordinates.

Anyway this solution describes space time under the horizon for

these values, and it is not time independent, because

r is now playing the role of time and t is playing the role of spatial coordinate.

So now let us consider the situation that r is going to 0.

In this limit, one can neglect this and this and obtain

10:57

the following metric, that ds squares becomes approximately r

over rg dr squared minus Rg

over r dt squared minus r squared d omega squared.

Remember that here, we have angles.

D theta squared, etc.

Now, let us make the following coordinate change.

11:46

Then 3 over 2 square root of RG

to the power of 2/3 d theta which is here.

We do know that dY and finally 3/2

Square root of rg to the power

2/3 d phi we denote as dz.

After this changes this metric acquires the following approximate form.

12:23

For small values of angles, let me first write the metric and

explain their approximation.

It's t to the power minus t to three

dx squared minus t to the power

four-third dy squared minus to

the power four-third, dz squared.

12:53

And here, t goes to 0, t goes to 0.

As t goes to 0 the angles theta

and phi, big difference between two values of theta and

two value of phi are causally separated from each other.

13:14

Not as T very small, the points, which are big

distance away from each other by big angles, are causally separated.

That's the reason we can consider physically meaningful situation that

theta is here.

We assume that theta is of order of 0 very close to 0, and 5 is very close to 0.

So we don't consider big differences in theta and

phi because they are causally separated.

And now one can see that this metric is

exactly of this type, exactly of this type.

With the only difference that T's going to 0.

And we have concrete values of p, q and r, as follows from here.

And they do, indeed, obey this condition.

So, in fact, as we are approaching the singularity of the metric,

we encounter like and homogeneous.

14:18

Homogeneous, but anisotropic, Kasner-like solution, which is vacuum-like.

In fact, which is a vacuum solution, because in fact,

the solution is a vacuum solution, and this is a vacuum Kasner solution.

This is the condition for p, q and r for the vacuum solutions.

14:35

And it's important that in fact it's a generic phenomenon

that as we approach some singularities in space time.

We encounter in the vicinity of the singularities this kind of behavior for

the metric, for the different values of x, y and z.

But it's not necessary have to be vacuum

casual solution, because it depends on the energy momentum tensor.

In case of energy momentum tensor it's not 0 in 1.

And counts as this solution in the vicinity of the singularity but

non vacuum type.

So that's the end of the story for the standard cosmological solutions.

And next lecture we will discuss cosmological solutions with

non 0 cosmological solutions with 0 cosmological constant.

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