Astrofiend
Oct21-09, 09:59 PM
1. The problem statement, all variables and given/known data
I'm having problems seeing how the transformation to Eddington-Finkelstein in the Schwarzschild geometry works. Any help would be great!
2. Relevant equations
So we have the Schwarzschild Geometry given by:
ds^2 = -(1-2M/r)dt^2 + (1-2M/r)^-^1 dr^2 + r^2(d\theta^2+sin^2\theta d\phi^2)
and the Edd-Fink transformation assigns
t = v - r -2Mlog\mid r/2M-1 \mid
The textbook says it is straight-forward to simply sub this into the S.G line element to get the transformed geometry, but I can't seem to get it.
3. The attempt at a solution
OK, so differentiating the expression for the new t coordinates above, I get:
dt = dv - dr -2M. \frac{d}{dr} [log(r-2m)-log(2m)]
dt = dv - dr -2M. \left(\frac{1}{r-2m}\right)
dt = dv - dr - \left(\frac{2m}{r}-1\right)
...but the book says the answer is
dt = -\frac{dr}{1-\frac{2M}{r}} + dv
>where am I going wrong here? Given this last expression, it is fairly easy to sub it into the SG line element to get the transformed coordinates. Problem is, I can't seem to get that far! Any help much appreciated...
I'm having problems seeing how the transformation to Eddington-Finkelstein in the Schwarzschild geometry works. Any help would be great!
2. Relevant equations
So we have the Schwarzschild Geometry given by:
ds^2 = -(1-2M/r)dt^2 + (1-2M/r)^-^1 dr^2 + r^2(d\theta^2+sin^2\theta d\phi^2)
and the Edd-Fink transformation assigns
t = v - r -2Mlog\mid r/2M-1 \mid
The textbook says it is straight-forward to simply sub this into the S.G line element to get the transformed geometry, but I can't seem to get it.
3. The attempt at a solution
OK, so differentiating the expression for the new t coordinates above, I get:
dt = dv - dr -2M. \frac{d}{dr} [log(r-2m)-log(2m)]
dt = dv - dr -2M. \left(\frac{1}{r-2m}\right)
dt = dv - dr - \left(\frac{2m}{r}-1\right)
...but the book says the answer is
dt = -\frac{dr}{1-\frac{2M}{r}} + dv
>where am I going wrong here? Given this last expression, it is fairly easy to sub it into the SG line element to get the transformed coordinates. Problem is, I can't seem to get that far! Any help much appreciated...