Kruzkal Coordinates Inside Horizon: Defining u', v

[maxverywell]

We start by defining two coordinates $u=t+r^*$ and $v=t-r^*$. Then we define another two coordinates $u'=e^{u/4GM}$ and $v'=-e^{-v/4GM}$. But from what I have understood this is true for $r>2GM$. How do we define $u'$ and $v'$ for $r<2GM$?

I think it's $u'=e^{u/4GM}$ and $v'=e^{-v/4GM}$ for $r<2GM$ so that the line element is the same for $r<2GM$ and $r>2GM$.

 

[Bill_K]

Why don't you just look up Kruskal coordinates on Wikipedia.

 

[maxverywell]

Yes, I looked at it. But I asked something specific and the wiki article doesn't answer it.
I'm reading Carroll's notes and there he defines u' and v' in the similar way, but only for r>2GM.
Same in Wald's book.

 

[WannabeNewton]

The extended Kruskal representation of the Schwarzschild metric $ds^{2} = \frac{32M^{3}e^{\frac{-r}{2M}}}{r}(-dT^{2} + dX^{2}) + r^{2}d\Omega^{2}$ is valid for all $r > 0$ which from the definition $(\frac{r}{2M} - 1)e^{\frac{r}{2M}} = X^{2} - T^{2}$ yields $-1 < X^{2} - T^{2}$. This is made clear in both the Wiki article and in Wald (page 154). Compare this with the Rindler space-time.

 

[maxverywell]

In Carroll's notes on page 186 he defines u' and v' only for r>2GM. We need also to make transformation for r<2GM and than write the line element for both regions.

Please check the page 186. I found there another mistake.

 

[WannabeNewton]

Ah ok. Yes Carroll isn't being complete. The way he wrote it down, the chart has to be defined piecewise for $r > r_g$ and $r < r_g$ (it will agree continuously at $r = r_g$ of course).