- #1
binbagsss
- 1,259
- 11
So I have the metric as ##ds^{2}=-(1-\frac{2m}{r})dt^{2}+(1-\frac{2m}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}##*
I have transformed to coordinate system ##u,r,\phi, \theta ##, where ##u=t-r*##(2),
where ##r*=r+2m In(\frac{r}{2m}-1)##
and to the coordinate system ##v,r,\phi, \theta ##,
where ##v=t+r*##,(1)
From (1) and (2) I see that ##dt=dv-\frac{dr}{(1-\frac{2m}{r})}## and ##dt=du+\frac{dr}{(1-\frac{2m}{r})}##
(On a side note, what is the proper name of these types of derivative expressions?)
Substituting these into * in turn it is easy enough to get the metrics:(which I believe are correct?).
Question:
I am now want to get the metric using both \(v\) and \(u\) in favour of \(r\) and \(t\).
To do this I make use of:
##\frac{1}{2}(v-u)=r+2M In(\frac{r}{2M}-1) ##
therefore ##\frac{1}{2}(dv-du)(1-\frac{2m}{r})=dr##
and I sub this into either (1) or (2),
say (1) , I then get:
##ds^{2}= - (1-\frac{2M}{r}) dudv+r^{2}d\Omega^{2}##
And the first term is a minus sign out.
(in accord to source sean m carroll lecture notes on general relativity eq.7.73.)
I have no idea why I am a sign out,
Thanks,your assistance is greatly appreciated !
I have transformed to coordinate system ##u,r,\phi, \theta ##, where ##u=t-r*##(2),
where ##r*=r+2m In(\frac{r}{2m}-1)##
and to the coordinate system ##v,r,\phi, \theta ##,
where ##v=t+r*##,(1)
From (1) and (2) I see that ##dt=dv-\frac{dr}{(1-\frac{2m}{r})}## and ##dt=du+\frac{dr}{(1-\frac{2m}{r})}##
(On a side note, what is the proper name of these types of derivative expressions?)
Substituting these into * in turn it is easy enough to get the metrics:(which I believe are correct?).
Question:
I am now want to get the metric using both \(v\) and \(u\) in favour of \(r\) and \(t\).
To do this I make use of:
##\frac{1}{2}(v-u)=r+2M In(\frac{r}{2M}-1) ##
therefore ##\frac{1}{2}(dv-du)(1-\frac{2m}{r})=dr##
and I sub this into either (1) or (2),
say (1) , I then get:
##ds^{2}= - (1-\frac{2M}{r}) dudv+r^{2}d\Omega^{2}##
And the first term is a minus sign out.
(in accord to source sean m carroll lecture notes on general relativity eq.7.73.)
I have no idea why I am a sign out,
Thanks,your assistance is greatly appreciated !