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.).