I am reading the book by Birrell and Davies on QFT on curved spacetime.
On page 40 they write the following metric:

and on the following page they provide the Kruskal Coordiantes and the changed metric:

Now, I get something a little bit different than theirs.
According to my computations, if we change ##2M/r## to ##(1-2M/r)## and ##e^{-r/2M}## to ##e^{-r^*/2m}##, then I get that by Kruskal transformation that (3.20) is equivalent to (3.18).
Am I right?

If I understand you correctly: wouldn't that mean that your version of the Kruskal metric still has a coordinate singularity at r=2M?

So no, that doesn't seem right.

mad mathematician said:

Now, I get something a little bit different than theirs.

If you'll post your calculations, we'll be able to comment (and if you do so, please use LaTeX (which enables quoting)).

Shouldn't it be
##dr^*=dr+dr/(r/2M-1)=dr/(1-2M/r)##
?

I didn't read beyond that. I'll do, if necessary, after your reply.

JimWhoKnew said:

Shouldn't it be
##dr^*=dr+dr/(r/2M-1)=dr/(1-2M/r)##
?

I didn't read beyond that. I'll do, if necessary, after your reply.

good catch, I'll change it. you are right.

mad mathematician said:

good catch, I'll change it. you are right.

I lost track.
Can you see that after the correction, you have$$\frac{2M}r e^{-r/2M}d\bar{u}d\bar{v}=\frac{2M}r e^{-r/2M}e^{r^*/2M}(dt^2-dr^2/(1-2M/r)^2)$$which correctly yields the corresponding terms in Schwarzschild metric?
(when ##~r>2M~##)

JimWhoKnew said:

I lost track.
Can you see that after the correction, you have$$\frac{2M}r e^{-r/2M}d\bar{u}d\bar{v}=\frac{2M}r e^{-r/2M}e^{r^*/2M}(dt^2-dr^2/(1-2M/r)^2)$$which correctly yields the corresponding terms in Schwarzschild metric?
(when ##~r>2M~##)

There should be a difference between ##r## and ##r^*##, they aren't the same.

Thanks that cleared my perplexion.