And this is r0(theta).

So this is Ergo region.

One can escape from this part, one cannot escape from here, but

one can escape from here.

So, to see that one cannot escape from here one

has to consider normal vector to this surface and find that there is like light.

And to see that one can escape from this region is to consider one has to

consider normal vector to this surface and again see that it is not like light.

So one doesn't have to have speed of light to escape from this region.

Then, that is a way to make these observations.

Now, one can notice that,

so when r equals to this,

g 00 = 0.

Now, one can see that not always

this has real solutions.

One has real solutions for this case and for this case, if for this case specially.

If kappa m is greater than a otherwise, there is no real solution for this guy.

And also one can observe that this metric has closed time like curves.

So this is unphysical situation, and we have a mate singularity,

because there is no solution for this equation.

There is no real R+, it means that no horizon.

But at the same time, we have a singularity.

So, this metric, this spacetime has a singularity

which is not surrounded by the horizon.

The explanation why goes beyond this lecture.

I do not give explanation for that, but it is un-physical

to have such a situation that there is a singularity which can

be reached with infinite time from the point of view of an observer

which stays always outside of the body, of the black hole.

So there is so-called cosmic censorship.

Physicists old school study black holes, believe in so-called cosmic censorship.

That in real situations during collapse one cannot create such an object.

So this guy by this metric was cut by M greater than zero.

When cut by M is equal to a it's called critical solution,

but one cannot create such a metric which corresponds to this situation.

Or more generally, it states that, during real physical situations,

one cannot create spacetimes with the singularities which

are not surrounded by horizons, event horizon.

Why?

Well, one believes that in real physical situations,

starting from generic conditions which are smooth.

We have at our disposal something smooth initial metric,

smooth initial matter distribution, smooth forces, etc..

And then we're using various devices or tools.

want to create from this matter some black hole like object.

And physicists believe that using all possible tools we cannot create

something which will have a singularity, which is not surrounded by event horizon.

Well, this is just generic belief of sounding like possible.

Although, as far as I understand,

there is no rigorous proof of this statement from first principles.

And the proof should rely on the fact that we shouldn't go beyond

the general theory of relativity.