Exploiting Rotational Symmetry to Solve the Photon Orbit Equation Around a Black Hole

Physics-informed neural networks (PINNs) are a type of Machine Learning algorithm that include physics knowledge directly in their optimization process. In other words: the quantity to minimize, which measures how well the model fits the data, receives extra terms that enforce known constraints and established results. PINNs are mostly used to solve partial differential equations, which are the mathematical translation of how physical systems evolve in space and time.

In this project, we developed a method to compute the orbit of a photon, a particle of light, approaching a black hole. A black hole warps spacetime so strongly that its gravity bends even the path of a massless photon. Depending on its starting distance and angle, the photon is deflected away, or it spirals in and is swallowed. The one exception is a perfectly circular orbit, possible only at a special distance and angle. The orbit equation comes from Albert Einstein’s General Relativity.

But while classical solvers would need to be recomputed each time the initial conditions are new (distance to the black hole and incident angle), PINN-Slicer is trained once for all possible initial conditions, and can then give any orbit with an RMS deviation below 10⁻³.

The trick is to exploit a feature of the photon orbit differential equation: it is an autonomous differential equation, that is to say invariant under translations of the independent variable (azimuthal angle $\phi$). In other words, the coordinate system can be rotated to a different starting azimuthal angle. Therefore, it is possible to train the PINN only over a thinner range of the angle $\phi$ (a $\phi$-slice) and deploy it recursively across the desired angular domain.

Technical details

We solve the null geodesic equation in the Schwarzschild metric, which describes a non-rotating, uncharged, spherically symmetric black hole of Schwarzschild radius $r_s$. The initial conditions are the distance $r_0$ from the centre and the incident angle $\delta_0$, converted into $u_0 = r_s/r_0$ and $v_0 = du/d\phi$, where $\phi$ is the azimuthal angle in spherical polar coordinates. Our method accurately reproduces a custom fourth-order Taylor-series solver, and it generalizes to any second-order autonomous ODE of the form $\ddot{u} = G(u)$, where $G(u)$ does not depend explicitly on the independent variable $\phi$.

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