- #1
Confused Physicist
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Hi, I'm the given the following line element:
ds^2=\Big(1-\frac{2m}{r}\large)d\tau ^2+\Big(1-\frac{2m}{r}\large)^{-1}dr^2+r^2(d\theta ^2+\sin ^2 (\theta)d\phi ^2)
And I'm asked to calculate the null geodesics.
I know that in order to do that I have to solve the Euler-Lagrange equations. For this I always do the following. First I calculate the Lagrangian squared in terms of the proper time ##\tau##. In this case first I have written the line element as:
ds^2=-\Big(1-\frac{2m}{r}\Big)c^2dt ^2+\Big(1-\frac{2m}{r}\Big)^{-1}dr^2+r^2(d\theta ^2+\sin ^2 (\theta)d\phi ^2)
And then the Lagrangian squared:
\mathcal{L}^2=-\Big(1-\frac{2m}{r}\Big)c^2\dot{t}^2+\Big(1-\frac{2m}{r}\Big)^{-1}\dot{r}^2+r^2(\dot{\theta}^2+\sin^2(\theta)\dot{\phi}^2)
Where ##\dot{ }## denotes derivative with respect to proper time: ##d/d\tau##.
When I solve the E-L equation for ##t## and ##r## I get:
\dot{t}=\frac{k_t}{1-2m/r}
\dot{r}=k_r\Big(1-2m/r\Big)
I'm doing this because I'm looking for a change of variable which I believe has to be:
\frac{dr}{dt}=\pm \Big(\frac{1}{1-2m/r}\Big)
What am I doing wrong?
Thansk!
ds^2=\Big(1-\frac{2m}{r}\large)d\tau ^2+\Big(1-\frac{2m}{r}\large)^{-1}dr^2+r^2(d\theta ^2+\sin ^2 (\theta)d\phi ^2)
And I'm asked to calculate the null geodesics.
I know that in order to do that I have to solve the Euler-Lagrange equations. For this I always do the following. First I calculate the Lagrangian squared in terms of the proper time ##\tau##. In this case first I have written the line element as:
ds^2=-\Big(1-\frac{2m}{r}\Big)c^2dt ^2+\Big(1-\frac{2m}{r}\Big)^{-1}dr^2+r^2(d\theta ^2+\sin ^2 (\theta)d\phi ^2)
And then the Lagrangian squared:
\mathcal{L}^2=-\Big(1-\frac{2m}{r}\Big)c^2\dot{t}^2+\Big(1-\frac{2m}{r}\Big)^{-1}\dot{r}^2+r^2(\dot{\theta}^2+\sin^2(\theta)\dot{\phi}^2)
Where ##\dot{ }## denotes derivative with respect to proper time: ##d/d\tau##.
When I solve the E-L equation for ##t## and ##r## I get:
\dot{t}=\frac{k_t}{1-2m/r}
\dot{r}=k_r\Big(1-2m/r\Big)
I'm doing this because I'm looking for a change of variable which I believe has to be:
\frac{dr}{dt}=\pm \Big(\frac{1}{1-2m/r}\Big)
What am I doing wrong?
Thansk!
Last edited: