Dyonic Kerr-Newman-Kasuya-Taub-NUT Black Hole

• The dyonic Kerr-Newman-Kasuya-Taub-NUT black hole is a charged, rotating spacetime with a NUT parameter that introduces unique gravitomagnetic properties.
• It exhibits dual electromagnetic potentials and is parameterized via distinct Boyer-Lindquist forms, facilitating high-precision QPO analyses in X-ray binaries.
• Thermodynamic studies using AdS extensions and GUP corrections reveal complex phase transitions and stability regimes during black hole evaporation.

The dyonic Kerr–Newman–Kasuya–Taub–NUT black hole is a charged, rotating, NUT-parameterized four-dimensional black-hole solution specified by a mass $M$, a specific spin $a=J/M$, electric and magnetic charges, and a Taub–NUT parameter $n$ or $N$ interpreted as a gravitomagnetic-monopole charge. In the literature considered here, it appears both as a spacetime for precision timing tests using quasi-periodic oscillations (QPOs) in X-ray binaries and as a geometric object admitting a formulation in terms of self-dual and anti-self-dual Taub–NUT instantons; these perspectives also connect it to Kerr–Newman, Kerr–Taub–NUT, and AdS generalizations (Rehman et al., 27 Jul 2025, Kim, 2024).

1. Metric structure and parameter content

A Boyer–Lindquist–type form of the dyonic Kerr–Newman–Kasuya–Taub–NUT geometry is given by

$$ds^2 = - \Delta \Sigma^{-1} [dt-\Psi\,d\phi]^2 + \Sigma\,\Delta^{-1}\,dr^2 + \Sigma\,d\theta^2 + \sin^2\theta\,\Sigma^{-1}\,[a\,dt-(r^2+(a+N)^2)\,d\phi]^2,$$

with

$$\Sigma(r,\theta)\equiv r^2+(N+a\cos\theta)^2,\qquad \Delta(r)\equiv r^2-2Mr+a^2-N^2+Q_e^2+Q_m^2,$$

$\Psi(\theta)\equiv a\sin^2\theta+2N\cos\theta.$

In this notation, $M$ is the real gravitoelectric parameter, $N$ is the real gravitomagnetic parameter, $a$ is the rotation parameter, and $a=J/M$0 are the real electric and magnetic charges (Kim, 2024).

A second Boyer–Lindquist–type parameterization, used in QPO analyses, writes

$a=J/M$1

where

$a=J/M$2

$a=J/M$3

Here $a=J/M$4 is the electric charge, $a=J/M$5 the magnetic charge, and $a=J/M$6 the NUT parameter (Rehman et al., 27 Jul 2025).

These expressions encode the characteristic features of the solution: simultaneous rotation, dyonic electromagnetic charge, and a NUT sector that introduces gravitomagnetic structure beyond the Kerr–Newman family. The use of distinct notational conventions across the recent literature is itself significant, because observational, geometric, and thermodynamic analyses are often formulated in different but closely related parameterizations.

2. Electromagnetic structure and dyonic content

A convenient gauge for the electromagnetic vector potential is

$a=J/M$7

The corresponding field strength $a=J/M$8 has the explicit two-form components

$a=J/M$9

$n$0

Equivalently, $n$1 may be split into electric and magnetic parts proportional to $n$2 and $n$3 (Kim, 2024).

In the QPO-oriented formulation, the electromagnetic sector is often separated as

$n$4

with total Maxwell two-form

$n$5

The paper explicitly notes that any equivalent gauge-equivalent expression for the dyonic potential is used in the literature (Rehman et al., 27 Jul 2025).

The dyonic content also admits a complexified interpretation. In that viewpoint, the full solution arises from a self-dual and an anti-self-dual Taub–NUT constituent with complex charges

$n$6

$n$7

and the real parameters are recovered through

$n$8

$n$9

The same framework identifies electric–magnetic duality $N$0 and mass–NUT duality $N$1 as operations on the self-dual and anti-self-dual constituents (Kim, 2024).

3. Horizons, extremality, and global issues

In the asymptotically flat dyonic Kerr–Newman–Kasuya–Taub–NUT geometry, event horizons occur where

$N$2

The two real roots are

$N$3

and they coincide in the extremal limit

$N$4

This expresses the precise balance between gravitoelectric mass, gravitomagnetic charge, rotation, and dyonic electromagnetic charge (Kim, 2024).

The NUT sector introduces a Dirac–Misner singularity associated with the Misner string at $N$5. To remove the axis closed timelike curves, one must impose the periodicity

$N$6

The cited treatment characterizes this as an unavoidable NUT artifact: removing the string singularity requires periodic time (Kim, 2024).

Regularity is controlled by

$N$7

Because absence of curvature singularities demands $N$8, the case $N$9 is distinguished by the statement that there is no genuine ring singularity; the would-be ring is described as being “dissolved” into two imaginary centers in the complexified picture (Kim, 2024).

These features separate the KNKTN family from the Kerr and Kerr–Newman cases in two ways. First, the horizon bound depends explicitly on the NUT charge. Second, the global structure inherits Misner-string issues that are not reducible to ordinary rotation or electromagnetic charging.

4. Geodesic dynamics and the relativistic-precession QPO model

For QPO applications, one considers a neutral test particle with Lagrangian

$$ds^2 = - \Delta \Sigma^{-1} [dt-\Psi\,d\phi]^2 + \Sigma\,\Delta^{-1}\,dr^2 + \Sigma\,d\theta^2 + \sin^2\theta\,\Sigma^{-1}\,[a\,dt-(r^2+(a+N)^2)\,d\phi]^2,$$0

and conserved energy and angular momentum

$$ds^2 = - \Delta \Sigma^{-1} [dt-\Psi\,d\phi]^2 + \Sigma\,\Delta^{-1}\,dr^2 + \Sigma\,d\theta^2 + \sin^2\theta\,\Sigma^{-1}\,[a\,dt-(r^2+(a+N)^2)\,d\phi]^2,$$1

Restricting to equatorial circular orbits $$ds^2 = - \Delta \Sigma^{-1} [dt-\Psi\,d\phi]^2 + \Sigma\,\Delta^{-1}\,dr^2 + \Sigma\,d\theta^2 + \sin^2\theta\,\Sigma^{-1}\,[a\,dt-(r^2+(a+N)^2)\,d\phi]^2,$$2, the orbital motion gives rise to the Keplerian frequency $$ds^2 = - \Delta \Sigma^{-1} [dt-\Psi\,d\phi]^2 + \Sigma\,\Delta^{-1}\,dr^2 + \Sigma\,d\theta^2 + \sin^2\theta\,\Sigma^{-1}\,[a\,dt-(r^2+(a+N)^2)\,d\phi]^2,$$3 and the radial and vertical epicyclic frequencies $$ds^2 = - \Delta \Sigma^{-1} [dt-\Psi\,d\phi]^2 + \Sigma\,\Delta^{-1}\,dr^2 + \Sigma\,d\theta^2 + \sin^2\theta\,\Sigma^{-1}\,[a\,dt-(r^2+(a+N)^2)\,d\phi]^2,$$4 and $$ds^2 = - \Delta \Sigma^{-1} [dt-\Psi\,d\phi]^2 + \Sigma\,\Delta^{-1}\,dr^2 + \Sigma\,d\theta^2 + \sin^2\theta\,\Sigma^{-1}\,[a\,dt-(r^2+(a+N)^2)\,d\phi]^2,$$5, measured at infinity: $$ds^2 = - \Delta \Sigma^{-1} [dt-\Psi\,d\phi]^2 + \Sigma\,\Delta^{-1}\,dr^2 + \Sigma\,d\theta^2 + \sin^2\theta\,\Sigma^{-1}\,[a\,dt-(r^2+(a+N)^2)\,d\phi]^2,$$6 The squared epicyclic frequencies are

$$ds^2 = - \Delta \Sigma^{-1} [dt-\Psi\,d\phi]^2 + \Sigma\,\Delta^{-1}\,dr^2 + \Sigma\,d\theta^2 + \sin^2\theta\,\Sigma^{-1}\,[a\,dt-(r^2+(a+N)^2)\,d\phi]^2,$$7

with explicit lengthy expressions for $$ds^2 = - \Delta \Sigma^{-1} [dt-\Psi\,d\phi]^2 + \Sigma\,\Delta^{-1}\,dr^2 + \Sigma\,d\theta^2 + \sin^2\theta\,\Sigma^{-1}\,[a\,dt-(r^2+(a+N)^2)\,d\phi]^2,$$8 given in Appendix A of the QPO study (Rehman et al., 27 Jul 2025).

In the relativistic-precession model, the observed QPOs are identified as

$$ds^2 = - \Delta \Sigma^{-1} [dt-\Psi\,d\phi]^2 + \Sigma\,\Delta^{-1}\,dr^2 + \Sigma\,d\theta^2 + \sin^2\theta\,\Sigma^{-1}\,[a\,dt-(r^2+(a+N)^2)\,d\phi]^2,$$9

Thus the upper high-frequency QPO is mapped to the orbital frequency, the lower high-frequency QPO to the periastron-precession frequency, and the type-C low-frequency QPO to the nodal-precession frequency (Rehman et al., 27 Jul 2025).

The observational sample comprises five stellar-mass black-hole X-ray binaries:

Source	Mass	QPO data
GRO J1655–40	$$\Sigma(r,\theta)\equiv r^2+(N+a\cos\theta)^2,\qquad \Delta(r)\equiv r^2-2Mr+a^2-N^2+Q_e^2+Q_m^2,$$0	$$\Sigma(r,\theta)\equiv r^2+(N+a\cos\theta)^2,\qquad \Delta(r)\equiv r^2-2Mr+a^2-N^2+Q_e^2+Q_m^2,$$1, $$\Sigma(r,\theta)\equiv r^2+(N+a\cos\theta)^2,\qquad \Delta(r)\equiv r^2-2Mr+a^2-N^2+Q_e^2+Q_m^2,$$2, $$\Sigma(r,\theta)\equiv r^2+(N+a\cos\theta)^2,\qquad \Delta(r)\equiv r^2-2Mr+a^2-N^2+Q_e^2+Q_m^2,$$3 Hz
XTE J1550–564	$$\Sigma(r,\theta)\equiv r^2+(N+a\cos\theta)^2,\qquad \Delta(r)\equiv r^2-2Mr+a^2-N^2+Q_e^2+Q_m^2,$$4	$$\Sigma(r,\theta)\equiv r^2+(N+a\cos\theta)^2,\qquad \Delta(r)\equiv r^2-2Mr+a^2-N^2+Q_e^2+Q_m^2,$$5, $$\Sigma(r,\theta)\equiv r^2+(N+a\cos\theta)^2,\qquad \Delta(r)\equiv r^2-2Mr+a^2-N^2+Q_e^2+Q_m^2,$$6 Hz
XTE J1859+226	$$\Sigma(r,\theta)\equiv r^2+(N+a\cos\theta)^2,\qquad \Delta(r)\equiv r^2-2Mr+a^2-N^2+Q_e^2+Q_m^2,$$7	$$\Sigma(r,\theta)\equiv r^2+(N+a\cos\theta)^2,\qquad \Delta(r)\equiv r^2-2Mr+a^2-N^2+Q_e^2+Q_m^2,$$8, $$\Sigma(r,\theta)\equiv r^2+(N+a\cos\theta)^2,\qquad \Delta(r)\equiv r^2-2Mr+a^2-N^2+Q_e^2+Q_m^2,$$9, $\Psi(\theta)\equiv a\sin^2\theta+2N\cos\theta.$0 Hz
GRS 1915+105	$\Psi(\theta)\equiv a\sin^2\theta+2N\cos\theta.$1	$\Psi(\theta)\equiv a\sin^2\theta+2N\cos\theta.$2, $\Psi(\theta)\equiv a\sin^2\theta+2N\cos\theta.$3 Hz
H1743–322	$\Psi(\theta)\equiv a\sin^2\theta+2N\cos\theta.$4	$\Psi(\theta)\equiv a\sin^2\theta+2N\cos\theta.$5, $\Psi(\theta)\equiv a\sin^2\theta+2N\cos\theta.$6, $\Psi(\theta)\equiv a\sin^2\theta+2N\cos\theta.$7 Hz

Parameter estimation is performed with MCMC sampling using emcee. The priors are truncated Gaussians for $\Psi(\theta)\equiv a\sin^2\theta+2N\cos\theta.$8, $\Psi(\theta)\equiv a\sin^2\theta+2N\cos\theta.$9, and $M$0, centered on published means and $M$1 errors, and uniform priors

$M$2

The total log-likelihood is the sum of three Gaussian contributions, one for each observed frequency. Four separate runs isolate $M$3, $M$4, and $M$5 individually, and a final run samples the full parameter set $M$6 (Rehman et al., 27 Jul 2025).

5. Observational constraints on electric, magnetic, and NUT charges

The QPO analysis finds no significant evidence for nonzero electric charge in any of the five sources. The 90% upper limits on $M$7 are

$M$8

$M$9

No significant detection of $N$0 is reported (Rehman et al., 27 Jul 2025).

The same conclusion holds for the magnetic charge. The 90% upper limits on $N$1 are

$N$2

for GRO J1655–40, XTE J1550–564, XTE J1859+226, GRS 1915+105, and H1743–322, respectively. These results are stated to be consistent with zero magnetic charge for all sources (Rehman et al., 27 Jul 2025).

The Taub–NUT sector behaves differently. With $N$3, the 90% limits on $N$4 are

$N$5

$N$6

The study therefore reports no compelling indication of a nonzero Taub–NUT parameter for GRO J1655–40, XTE J1859+226, XTE J1550–564, and H1743–322, but a posterior distribution for GRS 1915+105 that peaks at a nonzero value, interpreted as a possible gravitomagnetic monopole moment (Rehman et al., 27 Jul 2025).

In the full six-parameter dyonic run, the 90% limits are weaker for some sources: $N$7, $N$8, $N$9 for GRO J1655–40; $a$0, $a$1, with $a$2 effectively unconstrained for XTE J1550–564; $a$3, $a$4, $a$5 for XTE J1859+226; $a$6, $a$7, $a$8 for GRS 1915+105; and $a$9, $a=J/M$00, $a=J/M$01 for H1743–322 (Rehman et al., 27 Jul 2025).

The physical interpretation stated in the paper is twofold. First, the charge limits reinforce the expectation that stellar-mass astrophysical black holes carry negligible net electric or magnetic charge. Second, if the GRS 1915+105 result were confirmed by future high-precision QPO measurements, it would signal a genuine deviation from the Kerr solution and could provide astrophysical evidence for a gravitomagnetic monopole moment. For the other four sources, the absence of a NUT signature strongly favors the standard Kerr metric (Rehman et al., 27 Jul 2025).

6. Instanton interpretation and the Newman–Janis algorithm

A distinct line of work interprets four-dimensional black holes as systems of chiral Taub–NUT instantons. In this framework, the Kerr metric is described as a pair of self-dual and anti-self-dual gravitational dyons joined by a finite Misner string carrying gravitomagnetic flux, and this program extends to Kerr–Newman and Kerr–Taub–NUT solutions as well (Kim, 2024).

For the complexified construction, the two instantons are placed at

$a=J/M$02

with complexified gravitational charges

$a=J/M$03

so that

$a=J/M$04

Each Taub–NUT instanton thereby carries both gravitoelectric and gravitomagnetic charge and is described literally as a gravitational dyon (Kim, 2024).

Within the same picture, the Newman–Janis algorithm is reinterpreted as a sequence of splitting, complex shifting, and recombination. One begins from a complexified Taub–NUT seed in Kerr–Schild form, promotes the null congruences and scalar potentials by separate complex shifts to $a=J/M$05 and $a=J/M$06, and then reassembles the two sectors through a nonlinear “double-copy” superposition theorem. The result is stated to reproduce the Kerr metric in Kerr–Schild coordinates and, more generally, to provide a first-principles derivation of the most general charged, rotating, NUT-parameterized solution in four dimensions (Kim, 2024).

This interpretation does not replace the standard coordinate presentation, but it changes the conceptual status of the solution. The dyonic Kerr–Newman–Kasuya–Taub–NUT black hole becomes not merely a deformation of Kerr–Newman by extra charges, but a nonlinear composition of self-dual and anti-self-dual constituents carrying half the total mass/NUT charge and half the total electric/magnetic charge.

7. AdS extension, Hawking temperature, and thermodynamic stability

A related generalization is the hot NUT Kerr Newman Kasuya Anti de Sitter black hole, formulated in Boyer–Lindquist–like coordinates with AdS length $a=J/M$07, $a=J/M$08, NUT parameter $a=J/M$09, charges $a=J/M$10, and $a=J/M$11. In this case

$a=J/M$12

$a=J/M$13

$a=J/M$14

In the limits $a=J/M$15, $a=J/M$16, and $a=J/M$17, one recovers Kerr–Newman; further limits yield Kerr, Reissner–Nordström, or Schwarzschild (Singh et al., 28 Jan 2026).

The horizons are the real positive roots of $a=J/M$18. For the Kerr–Newman–NUT limit $a=J/M$19,

$a=J/M$20

whereas for $a=J/M$21 the roots are more involved and one orders the four roots of $a=J/M$22, identifying the outermost two as $a=J/M$23 and $a=J/M$24 (Singh et al., 28 Jan 2026).

The classical Hawking temperature is

$a=J/M$25

To include minimal-length effects, the analysis employs the GUP-modified Klein–Gordon equation for a charged scalar, based on the Nozari–Karami Dirac Hamiltonian squared up to $a=J/M$26. Using a WKB ansatz and tunnelling formalism, it obtains the corrected temperature

$a=J/M$27

or, to first order,

$a=J/M$28

The paper states that the modified Hawking temperature is affected by the cosmological constant, magnetic mass, and electric and magnetic charges (Singh et al., 28 Jan 2026).

The same study gives the horizon area and entropy,

$a=J/M$29

and a specific heat $a=J/M$30 whose denominator $a=J/M$31 controls vertical asymptotes. Zeros of $a=J/M$32 signal second-order phase transitions. The reported result is that a significant number of discontinuities occur in the heat capacities, with alternating stable $a=J/M$33 and unstable $a=J/M$34 phases as one varies $a=J/M$35, $a=J/M$36, $a=J/M$37, and $a=J/M$38; the system becomes unstable as the black-hole size decreases (Singh et al., 28 Jan 2026).

As a thermodynamic extension of the asymptotically flat KNKTN spacetime, this AdS/GUP analysis suggests a broader role for the same parameter set. Rotation, NUT charge, dual electromagnetic charges, cosmological constant, and quantum-gravity corrections jointly determine evaporation and stability, while the asymptotically flat QPO constraints indicate that, in currently studied stellar-mass systems, only the NUT sector shows any possible departure from the Kerr baseline.