1. Mar 31, 2014

### Ichimaru

Hi,

I've found geodesic equations for the metric:

ds^{2} = -c^{2} \alpha dt^{2} + \frac{1}{ \alpha } dr^{2} + d \omega ^{2}

where

\alpha = 1 - \frac{r^{2}}{r_{s}^{2}}

I have found that for a light ray:

\alpha \frac{dt}{d \lambda} = 1 ;
\frac{dr}{d \lambda} = c

Where lambda is an affine parameter and i have appliedthe conditions:

\frac{dt}{d \lambda} = 1 , r = 0

I am then told to integrate these equations to get:

r = r_{S}tanh(\frac{ct}{r_{s}})

However when I try to integrate I get:

\int_{0}^{r} \frac{1}{1-\frac{r^2}{r_{s}^{2}}}dr = ct

goes to:

\frac{r_{s}}{2} ln \frac{r+r_{s}}{r-r_{s}} = ct

Which rearranges to:

r = r_{s}coth( \frac{ct}{r_s})

Any help would be appreciated, thanks!

2. Mar 31, 2014

### Bill_K

The same integral can have several different forms. You need to think about whether r > rs or r < rs.