Dyonic Kalb-Ramond Black Hole

• Dyonic Kalb–Ramond black holes are static, spherically symmetric solutions that feature both electric and magnetic charges dynamically altered by a background Kalb–Ramond field.
• They modify the Reissner–Nordström metric by asymmetrically dressing the electric and magnetic sectors through a Lorentz-violating parameter, affecting optical and geodesic observables.
• Investigations show that the Lorentz-violating deformation influences horizon structure, thermodynamic phase transitions, and gravitational redshift, offering fresh insights into modified gravity.

A dyonic Kalb–Ramond black hole is, in the recent Lorentz-violating literature, a static, spherically symmetric black-hole solution in which a background Kalb–Ramond two-form $B_{\mu\nu}$ induces spontaneous Lorentz symmetry breaking while the spacetime carries both electric charge $Q$ and magnetic charge $p$. In the direct four-dimensional constructions, the Kalb–Ramond sector enters through a vacuum expectation value and a Lorentz-violating parameter $\ell$, whereas the dyonic charges belong to the electromagnetic sector and are dressed asymmetrically by the Kalb–Ramond background (Lin et al., 18 May 2026). This differs sharply from much of the earlier Kalb–Ramond black-hole literature, which dealt with neutral Lorentz-violating geometries, purely electric solutions, or charge-like hair without a genuine dyonic two-charge structure (Duan et al., 2023, Rahaman, 1 Jun 2025, Liu et al., 12 May 2025).

1. Historical emergence and conceptual scope

The immediate precursor of the dyonic solution class is the electrically charged black hole in gravity with a background Kalb–Ramond field, where the metric takes an RN-like form modified by the Lorentz-violating parameter $\ell$, but only a purely electric Maxwell sector is present (Duan et al., 2023). That electric branch later became the background for quasinormal-mode studies of the charged Kalb–Ramond black hole, again without magnetic charge (Gu et al., 28 Sep 2025). By contrast, the explicitly dyonic construction introduces both $Q$ and $p$ and shows that the electric and magnetic terms are renormalized differently by the Lorentz-violating background (Lin et al., 18 May 2026).

This development is best understood against a broader background in which many Kalb–Ramond black holes were not dyonic at all. Several exact or phenomenological studies treated the Kalb–Ramond field as a symmetry-breaking condensate with $H_{\mu\nu\rho}=0$, producing Schwarzschild-like or topological black holes without independent Kalb–Ramond electric and magnetic charges (Yu et al., 25 Nov 2025, Liu et al., 12 May 2025). Other works found RN-like or Kerr–Newman-like metric behavior generated by Kalb–Ramond hair or Lorentz-violating backgrounds, but still without a genuine two-charge dyonic interpretation (Lessa et al., 2019, Kumar et al., 2020). This suggests that the modern expression “dyonic Kalb–Ramond black hole” should be reserved for solutions with explicit electric and magnetic charges in a Kalb–Ramond-modified geometry, not for neutral Kalb–Ramond backgrounds that only mimic charge.

2. Lorentz-violating Kalb–Ramond framework

The underlying framework is a Lorentz-violating gravity theory in which an antisymmetric Kalb–Ramond tensor acquires a nonzero vacuum expectation value. One representative action is

$$S=\int d^4x\,\sqrt{-g}\left[\frac{1}{2\kappa}\Big(R-2\Lambda+\xi\,B^{\mu\alpha}B^{\nu}{}_{\alpha}R_{\mu\nu}\Big) -\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho} -V\!\left(B_{\mu\nu}B^{\mu\nu}\pm b^2\right) +\mathcal{L}_{\text{matter}}\right],$$

with

$H_{\mu\nu\rho}=\partial_{[\mu}B_{\nu\rho]}$

and a potential $Q$0, $Q$1, that fixes the vacuum (Lin et al., 18 May 2026). A closely related formulation includes two electromagnetic nonminimal couplings,

$Q$2

and yields the same dyonic solution branch (Filho, 27 May 2026).

For the static spherical background, the Kalb–Ramond field is taken to be purely pseudo-electric. In one explicit form,

$Q$3

so that

$Q$4

on the solution (Lin et al., 18 May 2026). The Lorentz-violating deformation is then encoded by a dimensionless parameter $Q$5, although its normalization is paper-dependent across the literature. In every case, $Q$6 is the effective parameter generated by the Kalb–Ramond vacuum background (Lin et al., 18 May 2026, Filho, 27 May 2026, Ahmed et al., 1 Jun 2026).

A central structural consequence follows directly from $Q$7: the four-dimensional dyonic black holes do not carry an independent propagating Kalb–Ramond flux charge. Instead, the Kalb–Ramond field acts as a Lorentz-violating background that modifies how the electric and magnetic Maxwell charges gravitate (Lin et al., 18 May 2026).

3. Exact dyonic solutions

The direct static, spherically symmetric dyonic solution uses

$Q$8

together with the gauge potential

$Q$9

so that

$p$0

and the black hole carries electric charge $p$1 and magnetic monopole charge $p$2 (Lin et al., 18 May 2026).

For the asymptotically flat case,

$p$3

while for nonzero cosmological constant,

$p$4

The electric and magnetic sectors are therefore weighted differently: $p$5 When $p$6, the usual dyonic Reissner–Nordström-(A)dS metric is recovered (Lin et al., 18 May 2026).

Later extensions preserve this basic structure while adding new matter sectors. With a Letelier cloud of strings, the lapse becomes

$p$7

where $p$8 is the string density parameter (Ahmed et al., 1 Jun 2026). In the ModMax-plus-string-cloud variant,

$p$9

with

$\ell$0

so the dyonic sector is exponentially screened by the ModMax parameter $\ell$1 (Ahmed et al., 11 Mar 2026).

4. Horizon structure, singularities, and effective charge

For the asymptotically flat dyonic Kalb–Ramond black hole, the horizon equation is

$\ell$2

and real horizons require

$\ell$3

The same paper stresses that the curvature invariants diverge at $\ell$4, so the singularity is physical rather than coordinate (Lin et al., 18 May 2026).

A useful shorthand, emphasized in perturbative and relativistic-effect analyses, is the effective dyonic charge

$\ell$5

which packages the asymmetric electric and magnetic dressing by the Lorentz-violating background (Filho, 27 May 2026). In this notation,

$\ell$6

The same combination controls the horizon radii, tidal-force sign changes, and part of the free-fall kinematics (Filho, 27 May 2026).

In the cloud-of-strings extension, it is convenient to define

$\ell$7

so that

$\ell$8

with extremality condition $\ell$9 (Ahmed et al., 1 Jun 2026). The string cloud changes the asymptotic normalization through the constant term $\ell$0, whereas the dyonic charges remain encoded in $\ell$1. The resulting spacetime is therefore RN-like in radial structure but not asymptotically identical to the Einstein–Maxwell solution.

5. Geodesics, relativistic effects, and perturbations

The direct dyonic solution admits explicit formulas for photon spheres, shadows, and timelike circular motion. The photon sphere radius $\ell$2 is fixed by

$\ell$3

which yields

$\ell$4

For a static observer at radius $\ell$5, the shadow radius is

$\ell$6

and increasing $\ell$7, $\ell$8, or $\ell$9 shrinks both the photon sphere and the shadow (Lin et al., 18 May 2026). Timelike circular orbits satisfy

$Q$0

while the ISCO follows from

$Q$1

and moves inward as $Q$2, $Q$3, or $Q$4 increase (Lin et al., 18 May 2026).

A distinct extension studies free-fall Doppler shifts, gravitational time delay, tidal forces, and perturbative spectra in the same dyonic geometry (Filho, 27 May 2026). There the dyonic sector enters through $Q$5, and larger dyonic charges reduce the gravitational redshift by shifting the frequency ratio toward unity. The radial tidal component takes the form

$Q$6

with sign-reversal radius

$Q$7

while the angular components are

$Q$8

The same work derives scalar, vector, tensor, and spinor effective potentials and computes quasinormal frequencies with the sixth-order WKB method, concluding that $Q$9 gives the dominant correction, whereas the dyonic charges produce milder shifts; time-domain profiles show damped quasinormal ringing followed by late-time power-law tails (Filho, 27 May 2026).

The string-cloud extension adds a QPO-oriented geodesic analysis. There,

$p$0

while $p$1 strongly affects the radial epicyclic sector through the lapse itself rather than its derivatives (Ahmed et al., 1 Jun 2026).

6. Thermodynamics and phase structure

In the AdS/dS dyonic Kalb–Ramond solution, the thermodynamic energy is

$p$2

the entropy is

$p$3

and the Hawking temperature is

$p$4

The electromagnetic potentials are

$p$5

and the first law and Smarr relation are

$p$6

with

$p$7

The equation of state has van der Waals form, with critical point

$p$8

$p$9

$H_{\mu\nu\rho}=0$0

Below $H_{\mu\nu\rho}=0$1, the Gibbs free energy shows a swallowtail and the system undergoes a first-order small/large black-hole transition (Lin et al., 18 May 2026).

The cloud-of-strings extension changes the thermodynamic dictionary. There,

$H_{\mu\nu\rho}=0$2

and the first law becomes

$H_{\mu\nu\rho}=0$3

with

$H_{\mu\nu\rho}=0$4

The heat capacity,

$H_{\mu\nu\rho}=0$5

changes sign at $H_{\mu\nu\rho}=0$6, and the shadow-controlled geometric-optics emission rate is

$H_{\mu\nu\rho}=0$7

(Ahmed et al., 1 Jun 2026). The ModMax-plus-string-cloud variant likewise preserves a first law and generalized Smarr relation, while exhibiting a Hawking-Page-type phase transition in the specific heat and a spectral energy emission rate governed by the shadow radius (Ahmed et al., 11 Mar 2026).

7. Relation to the broader Kalb–Ramond black-hole literature

The dyonic Kalb–Ramond black hole is not synonymous with every charged or charge-like Kalb–Ramond geometry. A recurring source of confusion is that much of the earlier literature involved either neutral Lorentz-violating Kalb–Ramond backgrounds or purely electric solutions. The atom-infall study of a Lorentz-violating Kalb–Ramond black hole analyzes detector response, Hawking temperature, and HBAR entropy in a neutral Schwarzschild-like background and explicitly states that it is not dyonic (Rahaman, 1 Jun 2025). The revisitation of Einstein–Kalb–Ramond gravity constructs exact topological black holes with $H_{\mu\nu\rho}=0$8 and no independent Kalb–Ramond electric or magnetic charges, so it is again non-dyonic (Yu et al., 25 Nov 2025). The exact background-field solutions derived with a purely pseudo-electric Kalb–Ramond condensate likewise remain non-dyonic even though they modify thermodynamics and asymptotics (Liu et al., 12 May 2025).

Other works produced RN-like or Kerr–Newman-like metric behavior without genuine dyonic content. The “modified black hole solution with a background Kalb-Ramond field” has an RN-like term $H_{\mu\nu\rho}=0$9 for a special Lorentz-violating parameter value, but this occurs “despite the absence of charge” (Lessa et al., 2019). The rotating Kalb–Ramond hairy black hole interpolates between Kerr and Kerr–Newman at the metric level, yet the paper is explicit that it does not introduce independent electric and magnetic charges and therefore is not dyonic (Kumar et al., 2020). Even the electrically charged Kalb–Ramond black hole and its quasinormal-mode analysis remain purely electric precursors rather than full dyonic solutions (Duan et al., 2023, Gu et al., 28 Sep 2025).

The same caution applies to adjacent variants. The magnetized Kalb–Ramond black hole used in QPO studies is a neutral Kalb–Ramond geometry immersed in an external magnetic field, not a black hole carrying intrinsic electric and magnetic charges (Rodrigues et al., 2 Jun 2026). In $$S=\int d^4x\,\sqrt{-g}\left[\frac{1}{2\kappa}\Big(R-2\Lambda+\xi\,B^{\mu\alpha}B^{\nu}{}_{\alpha}R_{\mu\nu}\Big) -\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho} -V\!\left(B_{\mu\nu}B^{\mu\nu}\pm b^2\right) +\mathcal{L}_{\text{matter}}\right],$$0 dimensions, Kalb–Ramond-supported black strings and their dual black holes carry an axion-like charge rather than an electric–magnetic pair, so they are also not dyonic in the usual four-dimensional sense (Asrat, 2024). Against that background, the recent four-dimensional dyonic solutions mark a sharper notion: the black hole carries electric and magnetic charges in a geometry deformed by a Lorentz-violating Kalb–Ramond condensate, while the Kalb–Ramond field itself remains a background sector with $$S=\int d^4x\,\sqrt{-g}\left[\frac{1}{2\kappa}\Big(R-2\Lambda+\xi\,B^{\mu\alpha}B^{\nu}{}_{\alpha}R_{\mu\nu}\Big) -\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho} -V\!\left(B_{\mu\nu}B^{\mu\nu}\pm b^2\right) +\mathcal{L}_{\text{matter}}\right],$$1 on the solution (Lin et al., 18 May 2026).

In that sense, the present subject occupies a specific niche within Kalb–Ramond black-hole theory. It is neither a standard Einstein–Maxwell dyon dressed by a spectator tensor, nor a black hole with conserved Kalb–Ramond flux charges. Rather, it is a dyonic Einstein–Maxwell-type black hole whose asymptotics, charge weights, optical observables, free-fall kinematics, and thermodynamic response are all controlled by spontaneous Lorentz violation induced by a background Kalb–Ramond field (Lin et al., 18 May 2026, Filho, 27 May 2026, Ahmed et al., 1 Jun 2026).