[SOUND] So how does one observe that the Rindler metric, ds squared, which is Rho squared d tau squared minus d rho squared minus, render of space time is not singular at rho equals to 0. One just makes a transformation from this metric to the Minkovskian one. To another coordinate uuu. And in this metric, this region is absolutely regular. Similarly, if considers two dimensional line element, which is dr squared plus r squared d phi squared, how does one sees that there is nothing bad happening et r equals to zero? One makes again, coordinate transformation to dx squared dy squared where this point is not bad. So, we're going to do the same for the Schwarzschild metric. So, let us consider the Schwarzschild metric, ds squared equals to 1 minus rg divided by r, dt squared minus dr squared divided 1 minus rg divided by r minus r squared dr squared. So, let us make coordinate transformation. To do that, let us represent this metric. We continue this equality, represent this metric as (1- rg / r) (dt squared- dr squared / 1 r g / r) squared )- r squared d omega squared. And let us denote this part as dr *. So dr * is equal to dr / 1- r g / r. So this is the so-called tortoise coordinate. So, integration of this equation makes it that r * = r plus rg times log r divided by rg minus 1. And we have fixed the integration constant in this such that the relation is like this between r* and r. Totoise coordinate r*, and one can easily see that as r goes to infinity, r* is approximately equal to r, and also goes to plus plus infinity. And as r goes to rg, r* goes to minus infinity because this log goes to minus infinity. So, with this choice, this is equal to 1- rg divided by r, dt squared- dr* squared- r squared d omega squared. So, and let us introduce new coordinate, v. V, which is t plus r*. And let us make a transformation from (t,r) to (v,r) coordinates. If one makes this coordinate transformation to this v, then this metric acquires so-called in-going Eddington-Finkelstein coordinates. So these are called Eddington-Finkelstein coordinate, and the metric acquires the following form. It becomes like this, 1- rg divided by r, dv squared minus 2 dv, dr minus r squared d omega squared. So, despite the fact that, now in this coordinate, despite the fact that as r goes to rg, this guy goes to zero, this expression goes to zero. But because this metric is non-diagonal, so this part of the metric is like this. So one can perfectly go with this type of coordinate. So I can perfectly go beyond the region r equals rg. So, while there are regional coordinates, applicable only for r greater than rg. Only for this case. This coordinates are applicable for any r. For any r. For r greater than zero. But let us stress that, in this new coordinate, that invariant, r, mu, nu, Alpha, Beta, r, mu, nu, alpha beta, has the same form. And I remind you that it's proportional to some power of rg divided by r to the sixth power. And this guy blows up, goes to infinity, when r goes to zero. So, the spacetime, so while r = rg is a singularity only of this metric, but there is nothing bad happening in spacetime at r = rg. There is a blow up of the invariant at r equals to zero. So, this is a physical singularity already, physical singularity. Because tensor measures tidal forces, the strength of tidal forces, and they become enormous, enormous as r goes to zero. And that's the reason that the spacetime, that this metric, is applicable only beyond this point. Moreover, one can say that Einstein's general theory of relativity is not applicable as r goes to zero anymore. Because like higher terms and powers of curvature are becoming relevant. So the curvature becomes strong. And Einstein's theory on general grounds can be expected to be applicable only if curvature is small, sufficiently small. So now, let us draw a light like geodesics, light like geodesics for this metric. So we want to write light like geodesics for this metric. Radial like light like geodesics. So, the [INAUDIBLE] for which [INAUDIBLE] is equal to d phi is equal to zero. But, and also, ds squared is equal to zero. Then, for light like geodesics, this is the law of motion. And we obtain the following equation, that 1- rg divided by r, dv- 2dr multiplied by dv = 0. As a result, we obtain two equations. Either dv = 0, or 1- rg divided by r times dv = 2 dr. So these are two equations that are specifying light like geodesics. So this geodesic, is ingoing. Because one can notice that this is the equation t plus r* equals to constant, because V is this. As t is increasing, r goes to minus infinity. So, this means that this is ingoing geodesic, and this is outgoing geodesics. But one can easily see that these geodesics actually, so we have the falling picture, let us draw these geodesics. Basically, we have this v coordinate, and we have this r coordinate, and we encounter the following situation, that there is a line corresponding to r=rg for which case dv=0. Or dr equals, sorry, dr equals to zero. So this is this geodesic. Then we have these geodesics, ingoing. This, the ingoing geodesics. And when r is greater than rg, we encounter these geodesics, which are going like this. These are solutions of this equation. These are outgoing geodesics. And then, there is a funny thing that when r is less than rg, this equation gives us another type of geodesics, which are outgoing but at the same time they're going to r equals to zero. And only if r equals to rg, we have a vertical geodesic. So this is a picture we encounter for light like geodesics in the spacetime under consideration. Now, this is already the picture that describes a black hole. Why describes a black hole, why this is a black hole and why it's so special with this solution of Einstein equation. We're going to continue our discussion in the next lectures. [MUSIC]