0:13

So before drawing Penrose-Carter diagram,

we have to recall a few points, a few properties of the Schwarzschild

metric that we have discussed during the previous lecture.

So we have ended up writing one of the final formulas that we have

been using during the previous lecture has the following form.

So we have rewritten Schwarzschild metric in this form,

(dt squared- dr star squared)- r squared dOmega squared.

Where r star is a tortoise coordinate,

which is equal to r + rg log r divided by rg- 1.

And it is important that these r and

t coordinates do not cover entire space time.

They are valid only for r > rg.

So they are similar in this spirit, they are similar to a Rindler

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coordinates that have been discussed during the first lecture and

which cover only some part of the entire Minkovski spacetime.

So to draw the Penrose-Carter diagram, to do that honestly

one has to find such coordinates which cover this spacetime entirely.

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And how one does that, one should embed this spacetime in a flat,

well, one of the ways to do that is to embed such as a space time,

four-dimensional spacetime,

into a big enough flat spacetime as some hyperplane.

So we basically consider some big enough

Minkowski spacetime which has bigger than four dimensions and

embed this thing is hyperplane into some sort of hyperplane which is curved and

then find some coordinates system which covers this spacetime entirely.

2:26

Well there is mathematical theorem stating that with the use of map which

map has appropriate mathematical properties, any four dimensional

spacetime can be mapped into a flat ten-dimensional space.

Intutively this theorem can be understood on general grounds.

So a 4 by 4 matrix, symmetric 4 by 4

matrix which is g mu nu has (4 + 1)/ 2,

so 10 parameters, but

we can use general covariance.

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General covariance, which is four functions, with the use of four functions

we can fix four of those, so that reduces the number of parameters to six.

So every four-dimensional metric basically has six parameters.

So to fix these six parameters one needs six extra dimensions,

that is the reason the embedding can be done into ten-dimensional spacetime.

But so we are not going to go into great detail in this theorem.

Also we in our concise course on general relativity we

will not describe this kind of embedding in great detail.

But you will have to believe me that the coordinates that I'm going to

derive right now which are called Kruskal-Szekeres coordinates

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after those who have invented them.

So I'm going

to propose a way to derive, well, a standard way,

to derive Kruskal-Szekeres coordinates which is not using embedding.

But I will just declare that these coordinates cover

the Schwarzschild spacetime entirely.

And you basically have to believe me,

because I'm not in a position to prove it rigorously.

So how do we do that?

We want to get rid of the singularity of this metric at r.

So this metric is singular at r = rg.

We want to get rid of this.

How do we do that?

We use the following coordinates.

We propose coordinates light-like coordinates.

4:54

So, u and v, u, which is,

so we propose u = t- r

star, v = t + r star.

After such a change of the coordinates,

this metric can be represented as 1 -rg/r,

which is a function of u and v, times du dv- r squared,

which is a function of u and v, dOmega squared.

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And, so here, r is basically

understood as some implicit function of u and

v and the form of this function is fixed by this and

this relation because from here we can find r star.

Which is equal to 1/2 (v- u) and

basically I rewrite this expression like this,

and this is a function of u and

v given by this relation, log r(,v)/rg- 1.

So this is a relation specifying this function.

6:25

So the singularity of this metric which appears again when this

guy goes to rg, when this guy to rg, this goes to minus infinity,

so the singularity's basically moved to minus infinity.

So but let us see what happens as we approach this point,

in terms of this coordinate.

One can see from this formula that this term is limited as we approach rg.

But this becomes huge, so

approximately we can drop this term off and keep on with this.

Then in this limit, in this approximation, we can write that 1- rg/r

is approximately equal to v-u/2 rg.

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So this is the approximation, and now we can use this here.

So in the vicinity of this point or of this point,

we have that the metric is approximated by like this.

That is, exponent of

(v- u)/2rg du dv-

rg squared dOmega squared.

7:54

And this is equal to exponent of

-u/2 rg du exponent of v/2rg dv,

this is just the rewriting of this metric,

-rg squared dOmega squared.

Remember that we are around the point where r approximately equal to r g.

So now one can make the following

change of the coordinates,

one can use U coordinates,

which is U =- 2 rg times exponent

of -u/rg, and V, big V,

which is 2rg times exponent of v/2rg.

8:50

So using these coordinates,

this metric can be represented as follows,

dV dU- rg squared dOmega squared.

So, amazingly enough, in these new coordinates,

this plot becomes flat in the vicinity of the point r goes to rg and not singular.

And so that is the goal of the transformation that we have been making.

So let us make this transformation situation from the very beginning.

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So, if in this metric, in this metric,

ds squared = (1- rg/r) [dt squared-

dr star squared]- r squared dOmega squared.

From the very beginning,

we have made the following coordinate change so

that's what we basically found,

-2 rg exponent -t- r star which is u.

10:32

Which is, this is v just to remind you.

So this is u, this is v.

So in original r coordinate

instead r star this coordinate

transformation would look like this,

-t-r/2rg multiplied by r/ rg-1, 1/2.

And this would be 2rg times

exponent t + r divided by 2rg, r/rg- 1.

This expression I have obtained just by putting into here and

here the expression for r star [INAUDIBLE] r, so then one would get this.

So if one would make this transformation in this metric

one would obtain the metric in this form.

ds squared = rg/r,

U, which is an implicit

function of U and V,

times exponent of minus

implicit function r of U and

V divided by rg times dU

times dV minus r squared

(U,V) times dOmega squared.

Where this function can be found from here, from these expressions.

This function is easy to see to be equal to,

so if we just multiply this to this,

one will get rid of t, and then the function r of U and

V divided by rg minus 1 multiplied by,

so this is expression defining this

implicit function of r, U and V,

divided by rg = -U times v divided by 2rg squared.

So this metric is called black hole metric in Kruskal–Szekeres coordinates,

U and V are Kruskal-Szekeres coordinates.

And this metric is regular when r goes to rg.

This term is regular and everything is perfectly good, etc, etc.

So this metric doesn't have singularity.

And moreover as I promised, well, without explanation,

this metric covers entire black hole spacetime.

So, in these coordinates now we can do a conformal map to

the Penrose-Carter diagram of the black hole.

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