General Theory of Relativity or the theory of relativistic gravitation is the one which describes black holes, gravitational waves and expanding Universe. The goal of the course is to introduce you into this theory. The introduction is based on the consideration of many practical generic examples in various scopes of the General Relativity. After the completion of the course you will be able to solve basic standard problems of this theory. We assume that you are familiar with the Special Theory of Relativity and Classical Electrodynamics. However, as an aid we have recorded several complementary materials which are supposed to help you understand some of the aspects of the Special Theory of Relativity and Classical Electrodynamics and some of the calculational tools that are used in our course. Also as a complementary material we provide the written form of the lectures at the website: https://math.hse.ru/generalrelativity2015
Black hole horizon creation
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Из курса от партнера National Research University Higher School of Economics
Introduction into General Theory of Relativity
83 оценки
National Research University Higher School of Economics
83 оценки
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Collapse into black hole
We start with the derivation of the Oppenheimer-Snyder solution of the Einstein equations, which describes the collapse of a star into black hole. We derive the Penrose-Carter diagram for this solution. We end up this module with a brief description of the origin of the Hawking radiation and of the basic properties of the black hole formation.
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Emil AkhmedovAssociate Professor
Faculty of Mathematics
[MUSIC]
So, we want to complete, finish our discussion of the black hole
like solutions in general theory of relativity with a discussion of
another very complicated and subject on which there is no generally,
commonly accepted opinion in the scientific community.
So let me phrase the problem that I want to address.
We have obtained the following Penrose diagram describing the collapse process.
The following Penrose diagram which describes the collapse process.
Let me draw it clearly.
So, we have the following situation.
So, this is the region which is inside the star, in the body of the star.
So consider some observer which stays always outside of the star.
It's world line is like this.
So from the point of view of this observer, the collapse is never ending so
the surface, the last light ray which is emitted or
refracted by the star goes along the horizon, and
reaches the observer only at its future infinity.
We have seen this effect in many different situations, in many different
calculations, that body which is falling inside a black hole, it takes from
the point of view outside observer, it takes infinite time, etc., etc.
So there are different ways to see this phenomenon.
So then appears a very obvious question.
Well, of course, there is always event horizon, for
the collapsed process there is an event horizon.
This event horizon is present there always.
But the question is whether the black hole from the point of view, the matter,
the body of the black hole, the matter which creates a black hole.
The point is if from the point of view of the observer which stays
always outside of the black hole, the question is
whether this creation happens during finite period of time or not.
You see that we observe the following situation that basically,
the matter which contributes to the black hole from the point of view of
outside observer always eternally stays outside of the horizon.
It never crosses this.
So, I want to establish a few things.
First of all, I want to clearly state the approximation
in which this statement is valid, and
then I want to propose my not generally accepted opinion on what is going on.
I want to convince you that the creation of
the black hole happens within finite time even from the point of
view of those who are staying eternally outside of the black hole.
Why?
The reason is the following.
This statement, so suppose we have a star which is collapsing, and suppose
it emits some radiation or refracts some radiation
which is detected by someone staying outside of the black hole.
One may say, why address this question because this radiation experience
is exponential red shift.
The frequency experienced is exponential red shift.
So there will be no device which is capable to detect this phenomenon.
But you see this exponential factor is crucial for
many phenomena that are happening in the black hole.
For example, this exponential factor is crucial for the Hawking radiation.
So, due to this exponential factor we have the Hawking radiation.
Of course, one may say that there is no device which can detect this but
the Hawking radiation is exactly due to this factor.
So, I think that the discussion I am conducting is important to
understand deeper the Hawking radiation and the back reaction on it.
So that's the reason I am discussing this phenomenon.
Again, there is radiation which is refracted or
emitted by the body of the star and detected by this guy.
The fact that this radiation, how you call it,
if it is emitted in equal periods of time is reached by
observer but the time is increased.
So the periods, these are periods becoming bigger while these are fixed.
That can be seen from different ways of calculations and
is related to the fact that the time slows down as from the point of view
of this observer, every process on the star surface slows down.
But this statement is obtained in the following approximation.
In the approximation that light rays are going along light-like geodesics.
Light rays are going along light-like geodesics if they describe a probe.
If they describe such a situation that we neglect the fact that
that light ray carries energy and also curved spacetime.
So basically this spacetime is described the following observation
that it is curved by the black hole but
we neglect the fact that it is curved by the observer or by light rays.
The question is whether this neglection is meaningful
when we discuss such an exponentially suppressed phenomena.
And I'm trying to explain to you that it is in fact,
important to take into account the fact that light rays are also
bending, curving the spacetime because they're carrying energy.
And if we take into account the fact that light rays also are contributing to
the curvature of the spacetime, the process looks a bit different.
Let me clarify this point on the picture that we have described
in the lecture four.
In that lecture where we have obtained for the first time Schwarzer's solution.
In that lecture, we considered that
the light-like geodesics
in the Eddington–Finkelstein coordinates.
So there was a peculiar, so this is, let me tell,
say that this is a singularity r = 0, this is r = rg.
And then we had the full length picture.
We had outgoing light rays behaving like this.
And inside, they were actually in-going.
And there were in-going light rays, like this.
So that was the picture that we had already.
Now, this picture describes geodesics, describes light-like geodesics.
And it neglects the fact that light rays carry energy and also curved spacetime.
Of course, if a light ray carries energy, if we take into account that it
carries energy and also curved space time, it goes along the curve which is
actual light-like line, world line rather than geodesic which is for
the small energy carried by light ray is a little bit different from this guy.
So, those will be a little bit different.
Of course, this picture is not applicable anymore in that circumstances
because in this case, the picture is stationary, so
it repeats itself at every different moments of time.
It's stationary, while if we take into account that the fact that light rays also
carry energy and curved spacetime, the situation becomes nonstationary.
But approximately for short periods of time, for short period of time,
if we discuss only seeds of light rays, seeds of light rays, in that case,
we can we have a very similar picture, very similar picture.
But light rays are a bit, actual light rays are a bit different
from the geodesics, a little bit different from the geodesics.
But now what do we encounter?
In the original picture for the geodesics,
we had the following situation that this vertical line, this vertical line which
corresponds to the horizon here was also one of the light rays.
So if a light ray was standing on this line,
eternally stand on it, eternally go along it.
But if we take into account the fact that light rays also carry energy and
bent, curved spacetime, the picture will be similar.
We will also have a family of outgoing geodesics that are in this
region are going to infinity, in this region are going inside,
in the seeds we will see that.
For the seeds, this will be apparent.
But it will happen that this light ray will either belong to this family,
or to this family, but will not eternally stay on this surface,
due to the fact that it also bend to curved spacetime.
So, it will contribute to the emission of gravitational waves or
some other stuff like that.
So, let me re-phrase this what I am saying in different terms,
I'm saying the following thing.
That if this guy emits radiation which is received by
an absorber which stays always outside the black hole,
this absorber will see the very last sug,
the very last wave packet emitted by the star.
And the one which is after it will contribute to the body of the black hole
to the energy of the black hole, and will not be received by this observer.
So this observer will stop receiving light rays from the collapsing body.
Will stop receiving light rays from the collapsing body within finite
time as measured by this outside observer.
So, this moment can be considered as the moment of the creation of the black hole.
Of course, this explanation that I have gave to you,
has some drawback because the moment of the creation of the black
hole is somehow related to the energy carried by this light ray.
Stronger is the energy carried by each wave packet,
earlier the creation is happening.
This is a drawback.
And of course, I am vaguely explaining to you the picture
which demands some farther regress study, an explanation.
So I just gave you the idea, my idea of what is going on and
why the creation of the black hole happens within finite period
of time from the point of view of outside observers.
[MUSIC]
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