Black holes are difficult to study using the usual tranche of telescopes because definitionally, black holes are black. There are no photons which come off the surface of a black hole. Some fraction of black holes, however, have matter falling into them, and this matter does emit photons. In fact, it emits very hot photons because black holes are tiny, miniscule things on stellar scales, and getting close to them naturally produces copious energy.
Due to this dynamic, we usually observe this hot matter with X-ray observatories such as Chandra, NuStar, XRISM et al. These are satellites orbiting Earth whose primary job is to detect sometimes as few as hundreds of individual X-rays coming from a black hole. When we get enough of these X-rays, we can plot them in the following way:
This figure has energy on the horizontal axis and flux (number of photons per unit area, per unit time, per unit energy) on the vertical axis. This shows us the energies at which there are more or less photons. For instance, at E = 6.5 keV, there are more photons relative to the nearby energies (this can be seen on the bottom panel too). This is an iron emission line, and the exact shape of this line is an important tool for studying black holes.
How can we predict this shape?
The short answer is that it is not easy. The above schematic shows a workflow developed over the course of the last fifteen years in order to answer this question.
It first involves running general relativistic magnetohydrodynamical (GRMHD) simulations of black hole accretion. Then a single snapshot is taken from the simulation which will be post-processed using radiation solvers. We require two different solvers, one for the hot, diffuse region around the black hole, one for the less hot, more dense region. We take into account all relativistic effects and all atomic radiation, two things which are not easy to do.
The good news is that this all works, and we can now predict black hole spectra for black holes of all masses and the spectra of multiple black holes orbiting each other! Stay tuned for upcoming results!