Relativistic disc spectra

The simplest (but not necessarily realistic!) model of a relativistic accretion disc is an optically thick (i.e., opaque), geometrically thin (i.e., $\theta = \pi/2$) disc around the black hole. The disc has an inner radius at the Innermost Stable Circular Orbit (ISCO) around the black hole, which depends on the black hole spin, and extends out to large radii. This disc is illuminated by X-rays and produces a back-scattered "reflection" spectrum with prominent iron $\text{K}\alpha$ fluorescence line. The observed energy and intensity of this line is modified by a combination of Doppler, transverse-Doppler, gravitational red-shifting, and light bending effects. The overall shape of the iron line can be calculated by taking into account all of these effects.

Testing general relativity

X-ray spectral fitting can also be used to test if the Kerr metric, used to describe spinning black holes, provides a good description of the spacetime close to a black hole. A review article describing this approach is provided by Bambi (2022).

Numerical models of relativistic spectral distortions

Gradus

You can use the Gradus.jl relativistic ray tracing software to model the effects of General Relativity on the emission spectra from the accretion disc. This is written in Julia with accompanying instructions. Gradus is quiet general and can work with many different spacetimes, disc geometries, and corona geometries. More details in Baker & Young (2026).

Note: if you run into an ERROR: cannot find name corresponding to UUID please see this issue, as you need to fix the General Julia registry.

kerrz

Fergus Baker has also written a Kerr metric specific ray tracer (perhaps a ray solver might be a more appropriate description) that is incredibly fast and increasingly flexible. Well worth checking out the kerrz package.

relxill

One of the most well established models of relativistically blurred reflection is relxill. This includes the relativistic ray tracing routines Thomas Dauser combined with Javier García's XILLVER rest-frame reflection code. There are many XILLVER papers, several of which are listed in the Papers section below.

Papers

• Baker & Young (2026), "GRADUS.JL: spacetime-agnostic general relativistic ray-tracing for X-ray spectral modelling"
• Bambi (2022), "Testing Gravity with Black Hole X-Ray Data"
• Garcia & Kallman (2010), "X-ray Reflected Spectra from Accretion Disk Models. I. Constant Density Atmospheres"
• Garcia et al. (2013), "X-Ray Reflected Spectra from Accretion Disk Models. III. A Complete Grid of Ionized Reflection Calculations"
• Garcia et al. (2014), "Improved Reflection Models of Black Hole Accretion Disks: Treating the Angular Distribution of X-Rays"
• Tanaka et al. (1995), "Gravitationally redshifted emission implying an accretion disk and massive black hole in the active galaxy MCG-6-30-15". A very convincing detection of a broad iron line in an active galactic nucleus using ASCA X-ray observations.