Parameter estimation of Kerr-Bertotti-Robinson black holes using their shadows

Technical Summary: Parameter Estimation of Kerr-Bertotti-Robinson Black Holes Using Their Shadows

Problem Statement
While the Kerr metric serves as the standard model for astrophysical black holes in General Relativity, it describes vacuum solutions that neglect the significant influence of magnetic fields often present in accretion environments. Previous models, such as Kerr-Melvin spacetimes, suffer from pathological features like unbounded ergoregions or lack asymptotic flatness in a physically natural way. The Kerr-Bertotti-Robinson black hole (KBRBH) offers an exact solution to the Einstein-Maxwell equations representing a rotating black hole confined in a uniform magnetic field. Unlike test-field approximations where the background geometry remains unaffected, the KBRBH incorporates the backreaction of the magnetic field into the spacetime geometry itself via a deviation parameter $B$. This paper addresses the need to characterize the observational signatures of KBRBHs, specifically their shadows, to determine if they can be distinguished from standard Kerr black holes and to estimate their physical parameters ($a$, $B$) using Event Horizon Telescope (EHT) constraints.

Methodology
The authors investigate the KBRBH metric in Boyer-Lindquist coordinates, which is of Petrov type D and asymptotically non-flat. The study proceeds through the following steps:

1. Geometric Analysis: The horizon structure, stationary limit surface (SLS), surface gravity, Hawking temperature, and horizon area are derived. The existence of horizons is determined by the roots of the metric function $\Delta(r)$, establishing the parameter space $(a, B)$ where black holes exist versus naked singularities.
2. Null Geodesics: Utilizing the separability of the Hamilton-Jacobi equation in Petrov type D spacetimes, the authors derive the equations of motion for photons. They identify the critical impact parameters $(\xi_c, \eta_c)$ corresponding to unstable spherical photon orbits, which define the boundary of the black hole shadow.
3. Shadow Simulation: Ray-tracing procedures are employed to compute shadow silhouettes. Unlike standard Kerr studies that assume observers at spatial infinity, the authors place static observers at a large but finite radial distance ($r_O$) due to the non-asymptotically flat nature of KBRBH spacetime. The shadow is mapped to celestial coordinates $(X, Y)$ using an orthonormal tetrad.
4. Parameter Estimation: To overcome limitations of circularity-based observables in asymmetric shadows, the authors employ the Kumar-Ghosh method. They calculate two shadow observables: the total shadow area ($A$) and the oblateness ($D$). Contour plots of constant $A$ and $D$ are generated in the $(a, B)$ parameter space to facilitate parameter estimation.
5. Thermodynamics: The energy emission rate is analyzed by linking the high-frequency absorption cross-section to the shadow radius and calculating the Hawking radiation spectrum modified by the magnetic field.

Key Results

• Shadow Morphology: The magnetic field parameter $B$ significantly alters the shadow. Increasing $B$ enlarges the shadow size and introduces additional distortions, particularly at high spin values. The shadow shifts in the direction of rotation, and the asymmetry is amplified by both the spin $a$ and the magnetic field $B$.
• Observer Dependence: For finite-distance observers, the shadow exhibits substantial deviations compared to the asymptotic shape. Decreasing the observer's radial distance $r_O$ increases the apparent angular size and asymmetry, an effect that compounds with the distortion caused by $B$.
• Parameter Estimation: Contour plots in the $(a, B)$ plane demonstrate that the intersection of constant area ($A$) and constant oblateness ($D$) curves can uniquely determine the spin and magnetic field strength. The study provides specific estimated values for $a$ and $B$ for various observational inputs, showing that the method is effective even for asymmetric shadows.
• Thermodynamic Effects: The magnetic field suppresses the Hawking temperature and the energy emission rate. As $B$ increases, the backreaction effect lowers the effective thermal radiation output, a trend consistent with other modified gravity scenarios.
• Horizon Structure: The presence of a uniform magnetic field restricts the maximum allowed spin parameter $a$ for a given mass, tightening the conditions for black hole formation and shifting the event horizons outward.

Significance and Claims
The paper claims that KBRBH shadows encode distinct imprints of magnetic deviations, offering a potential avenue to distinguish Kerr spacetimes from non-Kerr spacetimes in the strong-gravity regime. The authors assert that their work provides a robust theoretical framework and a "ready-to-use methodology" for constraining the magnetic field parameter $B$ using future high-precision observations.

The study emphasizes that while current EHT observations of M87* and Sgr A* are compatible with Kerr shadows within $\sim 10\%$ uncertainties (hiding small $B$ deviations), the proposed parameter estimation technique using shadow observables ($A$ and $D$) is physically significant for observers relatively close to the black hole (within $\sim 100r_+$). The authors conclude that the next-generation EHT (ngEHT) and future space-VLBI missions, aiming for percent-level precision, may be able to detect these magnetic backreaction effects, thereby probing the role of strong magnetic fields in shaping the spacetime of astrophysical black holes. The work serves as a preparatory step for moving beyond the test-field approximation in black hole physics.