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.