I am trying to integrate

$\displaystyle \frac{d\tau}{dp} = \left(\frac{2\ r_g}{3(p-\tau)}\right)^{1/3} $

where $\displaystyle r_g$ is a constant. I am trying to obtain an expression that can be plotted on a $\displaystyle \tau$ versus p chart.

Attempt at a solution:

My naive attempt was to simply integrate wrt p and I obtained

$\displaystyle \tau = \int \left(\frac{2\ r_g}{3(p-\tau)}\right)^{1/3} dp \ = \ \left(\frac{3(p-\tau)}{2}\right)^{2/3} r_g^{1/3} +C $

which then solves to

$\displaystyle p = \tau \pm \frac{2 \ (\tau-C)^{3/2}}{3\sqrt{r_g}} $

Unfortunately when plotted it seems unphysical, and I think I may have done the integration wrong. I suspect the problem is that the two variables p and $\displaystyle \tau$ are not independent and multivariable integration is required, but I have no experience of that and no idea where to start. Can anyone help?

Background:

References http://www.phys.au.dk/~fedorov/GTR/09/note11.pdf and Lemaitre metric - Wikipedia, the free encyclopedia

I am using the simplified version of the Lemaitre metric

$\displaystyle ds^{2} = d\tau^{2} - \frac{r_{g}}{r} dp^{2} $

which ignores angular motion and I am trying to obtain an expression for light paths which always have ds=0, so that for light:

$\displaystyle dp/d\tau = \sqrt{ \frac{r}{r_g}} $

and from the references

$\displaystyle r = \left(\frac{3}{2}(p-\tau)\right)^{2/3}\ r_g^{1/3} $

which I substituted into the dp/d$\displaystyle \tau$ equation to obtain the expression that I am trying to integrate. If it helps any, the value 2 can be substituted for $\displaystyle r_g$. Any help would be greatly appreciated.