Hidden Symmetries and Gyromagnetic Ratio of Kerr-Newman Black Holes in $f(R)$ Gravity

Abstract: We explore hidden symmetries in electrically charged, four-dimensional rotating Kerr-Newman black hole within $f(R)$ gravity. By deriving the Killing and Killing-Yano tensors, we establish their role in the spacetime structure. The gyromagnetic ratio is calculated and shown to retain its universal value of $g = 2$, consistent with all four-dimensional black holes. These findings show that the gyromagnetic ratio remains consistent in this modified gravity setting. Moreover, they highlight the connection between hidden symmetries and the ability to separate the Hamilton-Jacobi equation in $f(R)$ gravity. This work advances the study of black holes in modified gravity, offering implications for both theoretical frameworks and observational cosmology, such as gravitational wave analyses.

• The paper rigorously derives hidden symmetries via Killing and Killing-Yano tensors, establishing integrability in the modified Kerr-Newman metric.
• It constructs a modified black hole metric within f(R) gravity that rescales charge and preserves metric separability for geodesic motion.
• Rigorous analysis confirms the gyromagnetic ratio remains at g=2 despite f(R) modifications, aligning with general relativity and Dirac theory.

Hidden Symmetries and Gyromagnetic Ratio of Kerr-Newman Black Holes in $f(R)$ Gravity

Introduction

This essay analyzes the detailed results presented in "Hidden Symmetries and Gyromagnetic Ratio of Kerr-Newman Black Holes in $f(R)$ Gravity" (2604.14066). The paper conducts a rigorous investigation into the interplay between modified gravity theories—specifically $f(R)$ gravity—and the structural and electromagnetic properties of Kerr-Newman black holes. It presents explicit derivations for Killing tensors, Killing-Yano tensors, metric separability, and the gyromagnetic ratio in this context. The research contributes to the development of theoretical models relevant to cosmology and observational tests of fundamental physics.

Metric Construction and Modified Gravity Framework

The authors construct the metric of a charged, rotating black hole within $f(R)$ gravity, starting from the action: $$S = S_g + S_M,\qquad S_g = \frac{1}{16\pi G}\int d^4x \sqrt{-g}\left(R + f(R)\right),\qquad S_M = -\frac{1}{16\pi}\int d^4x \sqrt{-g} F_{\mu\nu}F^{\mu\nu}$$ Under the assumption of constant scalar curvature $R=R_0$, the field equations yield a spacetime with a cosmological constant-like term and a rescaled electromagnetic coupling. The resulting metric generalizes the Kerr-Newman solution by incorporating $f(R)$ terms as multiplicative factors affecting mass, charge, and angular momentum.

One notable property is the inclusion of the black hole charge in the metric, rescaled by $(1+f'(R_0))^{-1}$, distinguishing the solution from its general relativity analog. The Boyer-Lindquist-type coordinates are retained, and the metric is shown to reduce to standard Kerr-Newman (or Kerr) in appropriate limits.

Hamilton-Jacobi Equation and Metric Separability

The separability of the Hamilton-Jacobi equation is explicitly demonstrated within the derived $f(R)$-Kerr-Newman metric. The equation admits integrability via Carter's constant, confirming hidden symmetries analogous to the original Kerr solution. Variables separate cleanly, providing a quadratic constant of motion, $K$, whose existence is tied directly to the presence of a second rank Killing tensor.

Killing and Killing-Yano Tensors

The authors derive the explicit forms of the Killing tensor and Killing-Yano tensor for the $f(R)$0-Kerr-Newman metric. The Killing tensor ensures the quadratic constant in the geodesic motion, while the antisymmetric Killing-Yano tensor confirms deeper hidden symmetries, associated with integrability and separability in the spacetime.

Importantly, the Killing-Yano structure persists in this modified gravity framework—this is not a generic feature in all alternative gravity models. The metric is shown to satisfy the necessary symmetries, and the Killing-Yano tensor reduces to standard Kerr form in the $f(R)$1 limit.

Gyromagnetic Ratio Analysis

A central aspect of the paper is the rigorous computation of the gyromagnetic ratio, $f(R)$2, for Kerr-Newman black holes within $f(R)$3 gravity. The formula for the magnetic dipole moment is expressed in terms of charge, mass, and angular momentum, and using the modified expressions for these quantities, the authors prove that $f(R)$4, the same universal value as in four-dimensional general relativity and Dirac theory for electrons.

The result is remarkable in that, despite extensive parametrization from $f(R)$5 modifications, the gyromagnetic ratio remains invariant. This is not generically true for other modified gravities (e.g., Gauss-Bonnet or Kaluza-Klein in higher dimensions). The calculation hinges on rescaling factors that ultimately cancel, preserving the proportionality between charge, angular momentum, and dipole moment.

Implications and Future Directions

The preservation of key hidden symmetries (Killing and Killing-Yano tensors) and the invariance of the gyromagnetic ratio in $f(R)$6 gravity have multiple theoretical and observational implications. First, they reinforce the suitability of $f(R)$7 theories as extensions of general relativity, at least in the context of electromagnetic properties of black holes. Second, these properties are relevant for the analysis of gravitational wave signals and black hole accretion models, where deviations from GR predictions would provide constraints on modified gravity models.

The explicit confirmation of metric separability is also crucial for black hole stability analysis, geodesic motion integrability, and quantum field theory in curved spacetime. The authors suggest possible connections with holographic principles—e.g., Kerr/CFT correspondence—where hidden symmetries are instrumental in relating black hole microstates to dual quantum field theories.

Future research could extend the analysis to non-constant curvature branches, higher dimensions, or more complex topologies, where the fate of hidden symmetries and gyromagnetic invariance may differ, thereby offering new probes for modified gravity. Observational signatures, including those from the EHT or LIGO/Virgo, may further test these properties.

Conclusion

This work rigorously establishes that Kerr-Newman black holes in four-dimensional $f(R)$8 gravity retain hidden symmetries characterized by Killing and Killing-Yano tensors and exhibit an unchanged gyromagnetic ratio $f(R)$9. These results imply theoretical consistency of $f(R)$0 gravity with GR in terms of electromagnetic and symmetry properties, providing a benchmark for future studies of modified gravity models. The findings have strong implications for theoretical physics, cosmology, and high-precision observational tests of gravity.