# How to retrace the paths of light back in time, for light around a black hole?

I have been thinking about black-hole raytracing, and have found this writeup which talks about deriving the shape of a photon's orbit around a Schwarzchild BH, enabling raytracing around a BH from a stationary observer's perspective, and with only stationary light sources (such as an accretion disk in the examples). The author uses this equation:

$$u′′(\phi)+u = \frac{3}{2}u^3$$

I have also found the following equation for null geodesics around a Schwarzchild BH (from https://s3.cern.ch/inspire-prod-files-4/4125fb620def30c9536ff6eb62ff583d page 24), which also does not feature time:

$$\left(\frac{du}{d\phi}\right)^{2} = 2mu^3 - u^2 + \frac{E^2}{L^2}$$

($$u=\frac{1}{r}$$ above)

The limitation of the above is that the equations do not feature time, so moving light sources aren't supported. In the interests of creating a rudimentary but accurate numerical simulation, what's the best way to go about solving the equations for a Schwarzchild black hole to get the path of a photon, parametrised by time? Something that would enable tracing the photon as it travels around the hole.

The other limitation is lack of support for moving observers. I assume incorporating this would be the easier challenge though, as one would just need to account for Doppler shift, gravitational red/blue shift, and relativistic aberration?

I'm also a bit confused about light paths forward in time being different from paths backwards in time. When you're outside the event horizon, it's fine. But if you're inside the event horizon and you see a photon falling in towards you, and you emit a photon back in the same direction, the photon won't follow the path of the first photon in the reverse direction, but will fall back because it can't leave the event horizon. Illustration below. Would this need to be a consideration? Is this right or am I missing something?

Outside: Inside:

I am aware this topic might be a slightly obscure area of research, but any help/hints/resources would be greatly appreciated.

• Relativistic raytracing is a tricky subject to say the least, but it exists, so I guess your concens are unwarranted ;) If you want a really detailed explanation, with working software to back it up (public domain c sources), this guy has done a lot of hard work: madore.org/~david/math/kerr.html BTW Starless is mostly concerned with the exterior view of a black hole, Madore's software lets you see the guts! Mar 28, 2021 at 17:17
• @m4r35n357 Thank you, this looks like a great resource! I'll definitely have a read Mar 28, 2021 at 19:24