Dyonic Black Holes in Lorentz-Violating Gravity with a Background Kalb--Ramond Field

Abstract: By introducing a nonminimal coupling between the Kalb--Ramond field and the electromagnetic field, we construct an exact four-dimensional static, spherically symmetric dyonic black hole solution in Lorentz-violating gravity with a background Kalb--Ramond field. The curvature invariants show that the spacetime retains a genuine curvature singularity at $r=0$. We then analyze the geodesic motion of null and timelike particles and obtain the photon-sphere radius, the shadow radius, and the innermost stable circular orbit, demonstrating that both the Lorentz-violating parameter and the dyonic charges can appreciably modify the shadow size and the domain of stable circular motion. In the extended phase space, we derive the thermodynamic quantities and verify the first law of black hole thermodynamics together with the Smarr relation. The system also exhibits a first-order phase transition between small and large black holes, and its phase structure is strongly influenced by the Lorentz-violating parameter and the dyonic charges.

• The paper demonstrates dyonic black hole solutions in a Lorentz-violating model where the Kalb–Ramond field modifies Einstein and Maxwell equations.
• Analytical results reveal how the combined effects of electric/magnetic charges and the Lorentz violation parameter ℓ alter photon spheres, ISCOs, and shadow sizes.
• Thermodynamic analysis uncovers a rich phase structure with swallowtail Gibbs free energy behavior and corrections to the Bekenstein–Hawking area law.

Dyonic Black Holes in Lorentz-Violating Gravity with a Background Kalb–Ramond Field

Theoretical Framework: Lorentz-Violating Kalb–Ramond Gravity

The paper develops a Lorentz-violating gravity model by coupling the Kalb–Ramond (KR) antisymmetric tensor field nonminimally to gravity and electromagnetism. This approach is rooted in string theory, where the KR field and Lorentz symmetry breaking are natural features. The action incorporates tensorial and scalar-trace couplings between the KR field and the electromagnetic field, extending beyond previously studied models (e.g., the Bumblebee scenario).

The KR field acquires a vacuum expectation value through an appropriate potential, leading to spontaneous Lorentz violation. The field equations governing the metric, electromagnetic potential, and KR field are derived, revealing modifications to both the Einstein and Maxwell equations, with additional terms encoding the Lorentz-violating background dynamics.

Analytical Dyonic Black Hole Solutions

The authors construct exact static, spherically symmetric solutions describing black holes with both electric and magnetic charges (dyonic). The analysis is performed for two classes of potentials:

• Quadratic KR potential with vanishing cosmological constant: The solution generalizes the Reissner–Nordström (RN) black hole, with explicit dependence on the Lorentz-violating parameter $\ell$ set by the KR vacuum expectation value and coupling.
• Linear KR potential with nonvanishing cosmological constant: Analytical solutions are obtained even in AdS spacetime, with the cosmological term also rescaled by the Lorentz-violating background.

Key features of the metric include nontrivial $1/r^2$ corrections (with different renormalization for electric and magnetic charges due to $\ell$) and additional higher-order terms in the curvature invariants. All cases retain a genuine curvature singularity at $r=0$, indicating that the singularity is robust against this type of Lorentz symmetry breaking.

Particle and Photon Dynamics

Geodesic analysis encompasses both null and timelike orbits, yielding:

• Photon sphere and black hole shadow: The photon sphere radius is altered by both $\ell$ and dyonic charges, which monotonically reduce the shadow size. This results in a broader parameter region where a photon sphere exists compared to the region with an event horizon, implying possible naked singularities with observable shadows.
• Innermost stable circular orbit (ISCO): The ISCO radius for massive particles is compressed as either the dyonic charges or Lorentz violation increase, with the magnetic charge contribution amplified relative to electric charge at large $\ell$ values.

The distinctions between photon sphere and horizon formation conditions have observational significance for strong gravity lensing and accretion disk phenomenology, especially in contexts with significant Lorentz violation.

Modified Black Hole Thermodynamics and Phase Structure

Employing the Iyer–Wald formalism, the black hole's mass and entropy receive explicit $\ell$-dependent corrections, leading to deviations from the Bekenstein–Hawking area law. In the extended phase space (with the cosmological constant interpreted as pressure), an enriched thermodynamic structure emerges:

• Critical behavior: A first-order phase transition occurs between small and large black holes, characterized by the presence of a swallowtail in the Gibbs free energy for $P<P_c$. The transition terminates at a second-order critical point as $P\to P_c$.
• Parameter dependence: The critical point and phase structure are strongly controlled by $\ell$, $1/r^2$0, and $1/r^2$1; increasing any of these reduces the coexistence region and shifts critical parameters. The heat capacity analysis reveals that all such parameters modulate thermodynamic stability, with notable shrinking or reshaping of stable regions as they increase.
• Smarr relation and generalized first law: The modified energy and entropy are shown to satisfy generalized thermodynamic identities, confirming a coherent extension of black hole thermodynamics to Lorentz-violating backgrounds.

Implications and Prospects

This work demonstrates that nonminimal couplings of the KR field induce Lorentz violation, which in turn produces measurable modifications across black hole geometry, orbital dynamics, and thermodynamics. Notably:

• The existence and size of photon spheres, black hole shadows, and ISCOs are sensitive to both $1/r^2$2 and dyonic charges, providing a plausible astrophysical channel for constraining spontaneous Lorentz symmetry breaking via shadow imaging, strong lensing, and disk emission profiles.
• The thermodynamic phase structure is fundamentally distinct from RN-AdS black holes, with richer phase diagrams and critical behavior accessible through the combined effect of dyonic charges and Lorentz-violating terms.
• Despite significant modifications far from the singularity, $1/r^2$3-induced Lorentz violation does not resolve the core singularity, indicating the necessity for complementary mechanisms (e.g., higher-curvature or quantum gravity corrections) for singularity regularization.

Future expansions are anticipated to include the analysis of rotating solutions, higher-order interaction terms in the action, and incorporation of observational data from EHT and related instruments to put direct bounds on the Lorentz-violating sector. Additionally, gravitational wave and quasinormal mode studies in these backgrounds could further probe the phenomenology of the KR mechanism.

Conclusion

The exact analytical dyonic black hole solutions in Lorentz-violating gravity with a background Kalb–Ramond field offer a robust theoretical laboratory for investigating spontaneous Lorentz symmetry breaking in gravitational systems. The results establish clear signatures in both dynamics and thermodynamics that are, in principle, accessible to high-precision astrophysical and experimental tests, laying the groundwork for future explorations of Lorentz-violating extensions of general relativity and the search for low-energy quantum gravity signatures (2605.18371).