Ichimaru
- 9
- 0
Hi,
I've found geodesic equations for the metric:
\begin{equation}
ds^{2} = -c^{2} \alpha dt^{2} + \frac{1}{ \alpha } dr^{2} + d \omega ^{2}
\end{equation}
where
\begin{equation}
\alpha = 1 - \frac{r^{2}}{r_{s}^{2}}
\end{equation}
I have found that for a light ray:
\begin{equation}
\alpha \frac{dt}{d \lambda} = 1 ;
\frac{dr}{d \lambda} = c
\end{equation}
Where lambda is an affine parameter and i have appliedthe conditions:
\begin{equation} \frac{dt}{d \lambda} = 1 , r = 0
\end{equation}
I am then told to integrate these equations to get:
\begin{equation}
r = r_{S}tanh(\frac{ct}{r_{s}})
\end{equation}
However when I try to integrate I get:
\begin{equation}
\int_{0}^{r} \frac{1}{1-\frac{r^2}{r_{s}^{2}}}dr = ct
\end{equation}
goes to:
\begin{equation}
\frac{r_{s}}{2} ln \frac{r+r_{s}}{r-r_{s}} = ct
\end{equation}
Which rearranges to:
\begin{equation}
r = r_{s}coth( \frac{ct}{r_s})
\end{equation}
Any help would be appreciated, thanks!
I've found geodesic equations for the metric:
\begin{equation}
ds^{2} = -c^{2} \alpha dt^{2} + \frac{1}{ \alpha } dr^{2} + d \omega ^{2}
\end{equation}
where
\begin{equation}
\alpha = 1 - \frac{r^{2}}{r_{s}^{2}}
\end{equation}
I have found that for a light ray:
\begin{equation}
\alpha \frac{dt}{d \lambda} = 1 ;
\frac{dr}{d \lambda} = c
\end{equation}
Where lambda is an affine parameter and i have appliedthe conditions:
\begin{equation} \frac{dt}{d \lambda} = 1 , r = 0
\end{equation}
I am then told to integrate these equations to get:
\begin{equation}
r = r_{S}tanh(\frac{ct}{r_{s}})
\end{equation}
However when I try to integrate I get:
\begin{equation}
\int_{0}^{r} \frac{1}{1-\frac{r^2}{r_{s}^{2}}}dr = ct
\end{equation}
goes to:
\begin{equation}
\frac{r_{s}}{2} ln \frac{r+r_{s}}{r-r_{s}} = ct
\end{equation}
Which rearranges to:
\begin{equation}
r = r_{s}coth( \frac{ct}{r_s})
\end{equation}
Any help would be appreciated, thanks!