0:59

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

Â dz squared.

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

Â homogenous because this is a function independent of special coordinates.

Â So they functions of time only.

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

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

Â 1:37

So we are going to discuss the vacuum solution.

Â So when energy momentum tensor is 0.

Â So for this case one can calculate Richie denser.

Â The not real components of the Ritchie tensor as follows.

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

Â A squared + B squared + C squared and

Â R11 = -A dot- theta A R22

Â = minus B dot

Â minus theta B.

Â R33 = minus C dot minus theta C and A and

Â B and Ca = X dot / X.

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

Â So A = X dot / X.

Â B = Y dot / Y, and

Â C = Z dot / Z, this Z.

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

Â So these are the expressions for

Â the non trivial components of Ricci Tensor for this metric.

Â Now the energy momentum conservation condition for

Â this case is rho dot plus theta

Â times p plus rho equals to 0.

Â In case if p and rho are both non zero.

Â 3:54

Just for general discussion we listed it here.

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

Â That -AB- BC- CA = 0.

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

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

Â is just A squared Plus b squared plus c squared.

Â 4:34

Hence again using this using

Â this one vacuum Einstein

Â equation Where equation is R00 = 0.

Â This is not the same.

Â This is R00 minus 1/2 g00R = 0.

Â But from R00 = 0, vacuum Einstein equation,

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

Â The solution to this equation obviously is theta

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

Â 6:29

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

Â specially homogeneous.

Â And we have introduced a notation that there is a theta

Â which is x dot over x +

Â y dot over y + z dot over z.

Â And at the same time we have obtained the condition.

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

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

Â and from the Einstein equation we have obtained

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

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

Â that this is one over T and

Â X dot over X is equal to P/t,

Â Y dot/Y = q/t,

Â z dot/z = r/t.

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

Â arbitrary constants of integration.

Â But from here and here and these relations

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

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

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

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

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

Â and this coordinate,

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

Â Nd to obtain the Kasner matrix is as follows.

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

Â 8:47

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

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

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

Â not related to each other, expanding with different, or

Â shrinking, it depends on where time goes.

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

Â So let us explain why one physical

Â situation where the Krasner solution Appears, naturally appears.

Â Consider schwarshil metric under there, under the horizon.

Â So it means that we consider the following metric.

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

Â Minus 1 minus r g

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

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

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

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

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

Â related to the coordinates, Schwarzschild standard, Schwarzschild coordinates.

Â Anyway this solution describes space time under the horizon for

Â these values, and it is not time independent, because

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

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

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

Â 10:57

the following metric, that ds squares becomes approximately r

Â over rg dr squared minus Rg

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

Â Remember that here, we have angles.

Â D theta squared, etc.

Â Now, let us make the following coordinate change.

Â 11:46

Then 3 over 2 square root of RG

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

Â We do know that dY and finally 3/2

Â Square root of rg to the power

Â 2/3 d phi we denote as dz.

Â After this changes this metric acquires the following approximate form.

Â 12:23

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

Â explain their approximation.

Â It's t to the power minus t to three

Â dx squared minus t to the power

Â four-third dy squared minus to

Â the power four-third, dz squared.

Â 12:53

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

Â As t goes to 0 the angles theta

Â and phi, big difference between two values of theta and

Â two value of phi are causally separated from each other.

Â 13:14

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

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

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

Â theta is here.

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

Â So we don't consider big differences in theta and

Â phi because they are causally separated.

Â And now one can see that this metric is

Â exactly of this type, exactly of this type.

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

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

Â And they do, indeed, obey this condition.

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

Â we encounter like and homogeneous.

Â 14:18

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

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

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

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

Â 14:35

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

Â that as we approach some singularities in space time.

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

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

Â But it's not necessary have to be vacuum

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

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

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

Â non vacuum type.

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

Â And next lecture we will discuss cosmological solutions with

Â non 0 cosmological solutions with 0 cosmological constant.

Â [SOUND]

Â