[MUSIC] We finish this lecture with consideration of an anthropic but especially homogenous solutions. And that will help us to understand better the homogenous and isotropic solutions. So this so called Kushner Kasner solutions. So the Kasner solution corresponds to the folding metric d t, squared minus x, squared of t, times d x squared minus Y squared 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 of K equal to zero, I mean when the special sections are flat. 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. So because we consider the case of T mu nu equal to 0, so both rho and T equal to 0. Both rho and p = 0. So then this equation we don't need. 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. 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. Time by time shift we can put this to 0. So now, one can use this equation to find that minus a dot is minus a theta. Is equal to 0. But from the fact that theta = 1/t from here we can obtain that A = p/t. So far any p. P can be any. Similarly from this equation and from this equation we obtain that B=q over t and C=r over t. Where p, q and r so far arbitrary constants of integration following from the equations similar to this one. 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 dx squared minus t to the power 2q 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 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. Let us change r over rg square root, dr. We know that's dT. Then 2 RG over 3 one third DT we will denote as DX. 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. 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. 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. 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. 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. 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. [SOUND]