Exploring black hole shadows in axisymmetric spacetimes with coordinate-independent methods and neural networks

TL;DR

This work develops a coordinate-independent framework to study black hole shadows in axisymmetric spacetimes, applying Fourier and Legendre descriptors to quantify shadow morphology in Kerr, γ, Taub–NUT, and Kaluza–Klein metrics. It demonstrates that rotation-invariant Fourier coefficients provide robust descriptors for cross-model comparison and parameter inference, and couples this with a two-stage neural-network pipeline trained on extensive ray-traced data to perform fast shadow generation and inverse parameter recovery from shadow features. Using observational constraints from EHT, Keck, and VLTI, the study derives bounds on shadow size and deformation, and on metric-specific parameters, illustrating the potential and limits of shadow-based tests of general relativity. The results highlight the value of coordinate-independent techniques and machine-learning-assisted inference for strong-field gravity, while noting degeneracies that necessitate complementary observables such as polarization or multi-wavelength data to fully break parameter degeneracies.

Abstract

The study of black hole shadows provides a powerful tool for testing the predictions of general relativity and exploring deviations from the standard Kerr metric in the strong gravitational field regime. Here, we investigate the shadow properties of axisymmetric gravitational compact objects using a coordinate-independent formalism. We analyze black hole shadows in various spacetime geometries, including the Kerr, Taub-NUT, $γ$, and Kaluza-Klein metrics, to identify distinctive features that can be used to constrain black hole parameters. To achieve a more robust characterization, we employ both Legendre and Fourier expansions, demonstrating that the Fourier approach may offer better coordinate independence and facilitate cross-model comparisons. Finally, we develop a machine learning framework based on neural networks trained on synthetic shadow data, enabling precise parameter estimation from observational results. Using data from observational astronomical facilities such as the Event Horizon Telescope (EHT), Keck, and the Very Large Telescope Interferometer (VLTI), we provide constraints on black hole parameters derived from shadow observations. Our findings highlight the potential of coordinate-independent techniques and machine learning for advancing black hole astrophysics and testing fundamental physics beyond general relativity.

Figures (14)

• Figure 1: Ray-tracing setup: photons are launched from the observer’s screen at distance $d$, inclined by angle $i$ relative to the spin axis. The screen is parametrized by coordinates $(\alpha,\beta)$, which map to the photon impact parameters used in the geodesic integration.
• Figure 2: Shadow areal radius $r_{\rm sh}$ (left) and Schwarzschild deviation parameter $\delta$ (right) of a Kerr black hole as functions of observer inclination angle $i$ for spin parameters $a_* = 0.00,\,0.10,\,0.20,\,\dots,\,0.99$. Curves are colour-coded from blue ($a_*=0$) to dark red ($a_*=0.99$) and use alternating solid/dashed styles for clarity.
• Figure 3: Same as Fig. \ref{['kerr_sh']} for the $\gamma$-metric ($\gamma=0.50$--$1.50$).
• Figure 4: Sgr A* Kerr shadow constraints: $r_{\rm sh}$ (left) and Schwarzschild deviation parameter $\delta$ (right) vs. spin $a_*$ for $i=0^\circ$ (red solid) and $90^\circ$ (blue dash--dotted). Shaded pink and light-blue bands indicate VLTI and Keck limits.
• Figure 5: Same as Fig. \ref{['kerr_const']} for the $\gamma$ metric at $i=0^\circ,\,45^\circ,\,90^\circ$.