Please give me some clues about the calculations for a non-rotating black hole

[park]

Homework Statement

non-rotating black holes

Relevant Equations

Kruskal coordinates

 

[TSny]

The right-hand side doesn't look dimensionally consistent to me. Could it be that the argument of the $ln$ function should be something like $|r/r^* - 1|$ instead of $|r - r^*|$?

It's a bit hard to check your work since you haven't indicated how $p$ and $q$ are defined.

 

[park]

oops! my mistake! p and q is defined ...

 

[park]

i finally got a clue with your comment about dimensions!
For consistency of dimension, i think r-r^* of (8.13) should be r/r^*-1

 

[TSny]

i finally got a clue with your comment about dimensions!
For consistency of dimension, i think r-r^* of (8.13) should be r/r^*-1

Yes, things seem to work out better with the r/r^*-1 form.

I'm not very familiar with Kruskal coordinates. Sean Carroll's notes on GR are freely available on the internet.
https://www.preposterousuniverse.com/grnotes/

He discusses Kruskal coordinates in this chapter:
https://preposterousuniverse.com/wp-content/uploads/grnotes-seven.pdf
The relevant material starts around page 182. Note equation (7.74) on page 186. I think his $\tilde u$ and $\tilde v$ are similar to your $p$ and $q$, and his $u'$ and $v'$ are essentially your $p'$ and $q'$. (But I believe Carroll is missing an overall negative sign in his expressions for $v'$ in (7.75) and (7.76). He corrects this in his published textbook.).