For a simulation, I want to compute the path that light follows near a black hole.
Non-relativistically, a massive point particle in a central newtonian gravitational field follows either an ellipse, a parabola, or a hyperbola. Is the same true relativistically for light around a black hole? A problem I see with this, is that while particles gain velocity when approaching a black hole, a photon gains energy instead. So do photons behave differently?
Best Answer
Yes, photons behave differently, which I'm sure is no surprise! Any graduate book on GR will derive the orbits for a Schwarzchild metric. In my copy of "A first course in general relativity" by Bernard F. Schutz the orbits are calculated in chapter 11. Cribbing from this, for photons the orbit is:
$$ \left(\frac {dr}{d\lambda}\right)^2 = E^2 - \left(1 - \frac {2M}{r}\right) \frac {L^2}{r^2} $$
so you get an effective central potential:
$$ V^2(r) = \left(1 - \frac{2M}{r}\right) \frac {L^2}{r^2} $$
$L$ is the $p_\phi$ component of the four momentum and is constant. For particles the first equation would give you $dr/d\tau$ but proper time is always zero for a photon hence the use of the affine parameter $\lambda$ instead, where $\lambda$ is defined by $p^r$ = $dr/d\lambda$. The $r$ and $\phi$ are the Scharwzchild co-ordinates i.e. as seen by an observer at infinity.