[MUSIC] So we have obtained the following solution of the Einstein equations (1- rg 0 divided by r) dt squared- dr squared 1- rg 0 divided by r- r squared d omega squared. So, this solution is I mean, where rg is equal to kappa M and M is the mass of the gratiating center. So this solution describes very symmetric rotational field, of this sphere. And let us describe what our type of observers see this metric. Well, I already say that this metric is invariant on the time reversal and time translatial invariance, its components are time independent, and so this metric is static. And if we consider time slices of this metric, time slices of this metric are sliced as onions and spheres, the spheres have the following radii. So the radii of the sphere as, well, the distance between two points are r1 and r2 is given by dr divided by square root of 1 minus rg divided by r. And it is not equal to r2 minus r1. It's only approximately equal to this value if we are very far, if r1 and r2 are much greater than rg, then there is an approximate relation between these two quantities, so they are not equal. This is important. And, this is again the radii of the spheres are different. So this would be the radii if we were to integrate. Anyway, so the radii are not directly related to the areas of the sphere. That's the first thing. And then let us look at the observer. So consider someone who doesn't move in spatial directions. So consider this, dt d phi is equal to 0, so static observance in this metric. Static observance of this metric, so they're road lines, just t 0, 0, 0. And they're fixed at the given radius, at the given angles over the gradiating center so there is a gradiating center and these guys are fixed at the given radius and the given theta and phi. So these are non-inertial, so we say that this Schwarzchild metric is seen by non-inertial observers, which are fixed above this levitating center. This is important. And let us describe properties of this metric in greater detail. One can easily see that this metric degenerates as r goes to rg. As r goes to rg this metric degenerates. So, this component goes to 0, while this component blows up, goes to infinity. But this is not a physical singularity. This singularity is very similar to the one we have encountered in the first lecture for Rindler. For the Rindler metric there was a singularity at ro equals to 0, which is similar to the polar coordinate singularity at r equals to 0. So, this singularity of this metric at r equals to rg is similar to this sort of similarities, so it's some physical. The way to see it, to consider invariants, so for example, let us consider, we're going to show that there no invariants of the metric that blow up. I mean, invariants of the general covariant transformation invariants so we're going to build up invariants of the metric. And we're going to see that they don't blow up. So the simplest invariant that one can build is the volume form. So this invariant for the given metric, as one can see this factor and this factor and this invariant cancel each other. So this is a determinant of this metric. And it is = r squared sin theta dr d theta d phi, so it is regular, that's the first thing. Then, one can for example, study components of Riemann tensor. Components of the Riemann tensor of course have various singular properties. So for example, this component, so let me write all the non-zero components of Riemann tensor. One of them is like this. The other one, is related to this one by the sin squared of theta and equals to minus rg r minus rg two r squared. And R1212 equals to R1313 divided by sine squared theta, this is equals to rg divided by 2r minus rg. So this component of Riemann tensor is singular at r equals to rg. R2323, is equal to -rg r sin squared teta, but let us look at invariant. Invariant, for example, simplest in variant would be r, but it is zero, because r means zero for this metric. Then r is 0 also. So this is the simplest invariant. But then let us construct invariants from the Riemann tensors, the simplest invariant that one can construct like this. And it happens to be equal to 3rg squared divided by r to the sixth power. And this guy is completely regular at r equals to, as r goes to rg, this guy is regular. So this way, one can build many different invariants from these guys, and all of them will be regular at this point. That means that the singularity is as a coordinate singularity. So it's similar to this singularity for Rindler space that we have encountered in the first lecture, or this singularity in polar coordinates, two dimensional polar coordinates, that we have also discussed In the first lecture. So, there is another way to see that there is nothing horrible happens at this point, for this metric, and we're going to discuss it now. [MUSIC]