Magnetic field effects on spherical orbit in Kerr-Bertotti-Robinson spacetime: constraints from jet precession of M87*

The recently reported precession period of about $11.24$ years of the M87* jet provides a sensitive probe of strong field gravity and the electromagnetic environment in the immediate vicinity of supermassive black holes. In this work, we study the precession of the spherical orbit in the Kerr-Bertotti-Robinson geometry describing a rotating black hole immersed in a uniform electromagnetic field. Although the timelike geodesics is non-separable, we develop a Hamiltonian approach to investigate the spherical orbits. For sufficiently strong magnetic fields, the study shows that the spherical orbits can only exist within a finite radial range for given orbital inclination. Requiring the existence of the spherical orbits, we obtain an upper bound of the magnetic field, i.e., $B<0.33 M^{-1}$ for prograde and $B<0.0165 M^{-1}$ for retrograde motion. Furthermore, imposing the observed jet precession period, we obtain a significantly tighter constraint, $B\lesssim 0.0145 M^{-1}$, providing a new constrain on the magnetic field of M87* independent of the shadow. Our results provide unified constraints on the parameters of the KBR black hole and demonstrate that the jet precession offers a robust and complementary probe of magnetized black holes in the strong gravity regime.

💡 Research Summary

**
The paper investigates the influence of a uniform magnetic field on spherical (constant‑radius) timelike geodesics in the Kerr‑Bertotti‑Robinson (KBR) spacetime, a solution of the Einstein‑Maxwell equations describing a rotating black hole immersed in a large‑scale electromagnetic field. Because the Hamilton‑Jacobi equation for massive particles is not separable in this geometry, the authors adopt a Hamiltonian formalism. Starting from the Lagrangian (L=\frac12 g_{\mu\nu}\dot x^\mu\dot x^\nu), they define canonical momenta, construct the Hamiltonian (H=\frac12 g^{\mu\nu}p_\mu p_\nu), and exploit the stationarity and axisymmetry to identify conserved energy (E) and axial angular momentum (L). The effective potential (V_{\rm eff}(r,\theta)) follows from the normalization condition, and spherical orbits are defined by the simultaneous conditions (V_{\rm eff}=0) and (\partial_r V_{\rm eff}=0) at a fixed radius (r_0) and polar turning points (\theta_t=\pi/2\pm\zeta). Solving these algebraic equations yields expressions for the specific energy (E_s) and angular momentum (L_s) that depend on the metric functions and a ratio (\chi) determined by a quadratic equation involving radial derivatives of the metric components. The sign of the square‑root distinguishes prograde (upper sign) from retrograde (lower sign) orbits.

Stability analysis requires (\partial_r^2 V_{\rm eff}>0) and (\partial_\theta^2 V_{\rm eff}>0). Numerical scans show that as the magnetic field strength (B) increases, the region of stable spherical orbits shrinks dramatically; retrograde orbits disappear at much lower (B) than prograde ones. This leads to existence limits: (B<0.33,M^{-1}) for prograde and (B<0.0165,M^{-1}) for retrograde motion.

The authors then compute the Lense‑Thirring (LT) precession frequency for a spherical orbit. Using the Hamiltonian equations for (\dot t) and (\dot\phi), they obtain the orbital angular velocity (\Omega_{\rm orb}=\langle\dot\phi\rangle/\langle\dot t\rangle) and define the LT precession as (\Omega_{\rm LT}=\Omega_{\rm orb}-\Omega_{\rm spin}), where (\Omega_{\rm spin}=a/(r_+^2+a^2)) is the frame‑dragging rate of the black hole. The observed precession period of the M87* jet, (T_{\rm obs}=11.24\pm0.47) yr, corresponds to a frequency (\Omega_{\rm obs}=2\pi/T_{\rm obs}). By scanning the parameter space ((a/M,,B M)) and solving (\Omega_{\rm LT}=\Omega_{\rm obs}) while enforcing the spherical‑orbit existence conditions, the authors identify the allowed region that reproduces the observed jet precession.

The combined constraints are significantly tighter than those obtained from the existence condition alone. The final bound on the magnetic field strength is (B\lesssim0.0145,M^{-1}), i.e. roughly an order of magnitude below the previous upper limits derived from black‑hole shadow measurements. This result demonstrates that jet precession provides an independent and powerful probe of magnetized black holes in the strong‑gravity regime.

In conclusion, the study showcases (i) the KBR spacetime as a rare analytic model that fully incorporates back‑reaction of a uniform magnetic field, (ii) the utility of Hamiltonian methods for non‑separable geodesic problems, and (iii) the potential of long‑term VLBI jet‑precession observations to place stringent constraints on the magnetic environment of supermassive black holes. The authors suggest that future work should incorporate plasma effects, non‑uniform magnetic fields, and full GRMHD simulations to refine the connection between spherical‑orbit dynamics and observable jet behavior.