Kerr–Taub–NUT Black Holes

• Kerr–Taub–NUT black holes are stationary, axisymmetric vacuum solutions of Einstein’s equations that incorporate a NUT parameter acting as a gravitomagnetic monopole.
• They exhibit modified horizon structures, ergoregions, and geodesic integrability with NUT-dependent deformations influencing shadow size, accretion dynamics, and orbital stability.
• Their thermodynamics and energy extraction processes reveal novel parameter dependencies, including variations in radiative efficiency and insights into weak cosmic censorship.

Kerr–Taub–NUT black holes are stationary, axisymmetric vacuum solutions of Einstein’s equations that generalize Kerr by adding a NUT parameter $n$ or $l$, interpreted as a gravitomagnetic monopole charge or “dual mass,” alongside the mass $M$ and spin parameter $a$. In the limits $n\to 0$ and $a\to 0$, the geometry reduces to Kerr and Taub–NUT, respectively; the outer horizon is located at $r_+ = M + \sqrt{M^2 + n^2 - a^2}$, so the basic horizon condition is $a^2 \le M^2 + n^2$ (Abdujabbarov et al., 2012, Sen et al., 2024).

1. Metric structure, horizons, and singularities

A standard Boyer–Lindquist–like form of the Kerr–Taub–NUT metric is

$$ds^2= -\frac{1}{\Sigma}\left(\Delta-a^2 \sin ^2 \theta\right) dt^2 +\frac{2}{\Sigma}\left[\Delta \chi-a(\Sigma+a \chi) \sin ^2 \theta\right] dt\, d\varphi$$

$$+\frac{1}{\Sigma}\left[(\Sigma+a \chi)^2 \sin ^2 \theta-\chi^2 \Delta\right] d\varphi^2 +\frac{\Sigma}{\Delta} dr^2+\Sigma d\theta^2,$$

with

$l$0

Equivalent notation uses $l$1 במקום $l$2, with the same geometric role for the NUT parameter (Arévalo et al., 2024).

The event horizon is a root of $l$3,

$l$4

and the static limit is

$l$5

The ergoregion is the region between $l$6 and $l$7. Increasing the NUT charge increases the horizon radius, while the static limit is modified so that the relative volume of the ergoregion decreases as $l$8 grows (Abdujabbarov et al., 2011).

The curvature singularity is determined by $l$9. In one common parametrization it lies at

$M$0

and for $M$1 the spacetime is regular in the sense that $M$2 everywhere, whereas for $M$3 there is a ring singularity at $M$4, $M$5 (Sen et al., 2024, Wang et al., 9 Nov 2025).

The NUT parameter introduces Misner strings along the axis and renders the spacetime non-asymptotically flat in the usual sense. One line of analysis treats the string as an effective background defect and does not impose time periodicity; another emphasizes that imposing periodic time removes the string at the price of closed timelike curves. Recent work cited in the tilted-disk literature states that if one does not impose time periodicity, freely falling observers do not encounter CTCs and geodesics avoid causal pathologies (Sen et al., 2024).

Limit or regime	Result
$M$6	Kerr spacetime
$M$7	Taub–NUT spacetime
$M$8	Regular spacetime with no curvature singularity

2. Geodesic integrability and circular orbits

Null geodesics in Kerr–Taub–NUT remain separable under the Hamilton–Jacobi ansatz

$M$9

leading to first-order equations with a generalized Carter constant $a$0. The radial and angular potentials can be written as

$a$1

$a$2

The spacetime is therefore integrable at the geodesic level, as in Kerr, but with explicit $a$3-dependent deformations in both the radial and polar sectors (Abdujabbarov et al., 2012).

For equatorial timelike motion, the effective-potential analysis differs from Kerr in several respects. In particular, the effective potential is finite at $a$4 on the equatorial plane when $a$5, and the innermost stable circular orbit cannot always be extracted by the usual Kerr procedure. Detailed study of circular photon orbits and ISCOs shows that, for particular combinations of Kerr and NUT parameter values in KTN naked singularities, the ISCO equation does not give any positive real radius solution. In such cases the accretion efficiency generally reaches $a$6 at a particular orbit of radius $a$7, and $a$8 is then chosen as the “ISCO” for practical purposes (Chakraborty et al., 2019).

The same circular-orbit structure underlies thin-disk observables. For a general stationary, axisymmetric metric, circular equatorial geodesics are characterized by

$a$9

$n\to 0$0

with the radiative efficiency identified as $n\to 0$1. In Kerr–Taub–NUT, increasing $n\to 0$2 moves the ISCO outward and decreases $n\to 0$3 (Narzilloev et al., 2023).

3. Shadows, photon rings, and optical appearance

The shadow boundary is determined by unstable spherical photon orbits. In terms of impact parameters

$n\to 0$4

the boundary satisfies

$n\to 0$5

which yields

$n\to 0$6

and the corresponding $n\to 0$7 given explicitly in the shadow analysis. For an observer at infinity,

$n\to 0$8

For an equatorial observer, these reduce to $n\to 0$9 and $a\to 0$0 (Abdujabbarov et al., 2012).

The basic shadow result is that, for fixed spin, increasing the gravitomagnetic charge enlarges the shadow and reduces its deformation. In the numerical analysis, $a\to 0$1 is an increasing function of $a\to 0$2, while the distortion parameter $a\to 0$3 is a decreasing function of $a\to 0$4. Spin and NUT charge therefore act in opposite directions: larger $a\to 0$5 yields smaller and more distorted shadows, whereas larger $a\to 0$6 yields larger and more circular shadows (Abdujabbarov et al., 2012).

Thin-disk ray tracing reveals a distinct horizon-scale morphology beyond the classical shadow. In disk-illuminated Kerr–Taub–NUT images, increasing spin deforms the critical curve into a characteristic “D-shape,” but increasing $a\to 0$7 enlarges the critical curve and reduces that distortion. More strikingly, for $a\to 0$8, $a\to 0$9, and $r_+ = M + \sqrt{M^2 + n^2 - a^2}$0, the dark region develops an “extended inner shadow” with a “duck-cap-like” morphology: part of this region is formed by photons that do not hit the horizon and do not intersect the equatorial disk plane, so they remain dark despite not being captured (Wang et al., 9 Nov 2025).

Near-horizon line emission in the extremal regime is likewise NUT-sensitive. Because the equatorial plane is no more a symmetry plane of the KTN spacetime, the dependence of electromagnetic line emission on the NUT charge differs between the Northern and Southern hemispheres. The line emission is brighter than in Kerr for observers in the equatorial plane or the Southern Hemisphere, but becomes more faint as the observer’s position deviates far from the equatorial plane in the Northern one (Long et al., 2018).

4. Accretion disks, frame dragging, and jet phenomenology

A tilted thin inner accretion disk around a KTN black hole can be described in the diffusive warp regime by the steady-state Pringle–Scheuer–Feiler equation,

$r_+ = M + \sqrt{M^2 + n^2 - a^2}$1

For KTN, the low-order expansion of the Lense–Thirring frequency is

$r_+ = M + \sqrt{M^2 + n^2 - a^2}$2

The NUT contribution enters at order $r_+ = M + \sqrt{M^2 + n^2 - a^2}$3, and there can be a radius $r_+ = M + \sqrt{M^2 + n^2 - a^2}$4 where the total Lense–Thirring precession vanishes and reverses sign. This “negative LT precession” does not occur for a Kerr black hole. Depending on the numerical values of the viscosity of the accreting material and Kerr parameter, the gravitomagnetic monopole tends the angular momentum of the disk to align along the black hole’s spin axis, or to make it more tilted (Sen et al., 2024).

For equatorial thin disks, the radiative efficiency is still set by the ISCO binding energy, $r_+ = M + \sqrt{M^2 + n^2 - a^2}$5, but Kerr–Taub–NUT introduces an $r_+ = M + \sqrt{M^2 + n^2 - a^2}$6–$r_+ = M + \sqrt{M^2 + n^2 - a^2}$7 degeneracy: increasing $r_+ = M + \sqrt{M^2 + n^2 - a^2}$8 decreases $r_+ = M + \sqrt{M^2 + n^2 - a^2}$9, so the same observed thermal spectrum can be fitted by different combinations of spin and NUT charge. This degeneracy was applied to the X-ray binaries GRS 1915+105, GRO J1655–40, XTE J1550–564, A0620–00, H1743–322, and GRS 1124–683, where KTN was shown to explain the radiative efficiency inferred from the continuum fitting method (Narzilloev et al., 2023).

Jet power was modeled through the Blandford–Znajek scaling

$a^2 \le M^2 + n^2$0

with the KTN horizon angular velocity

$a^2 \le M^2 + n^2$1

Joint fits of radiative efficiency and jet power yielded more stringent constraints on $a^2 \le M^2 + n^2$2. The resulting source-by-source analysis found that Kerr–Taub–NUT can reproduce both observables for several systems, but, as in the Kerr case, it cannot simultaneously explain the observed jet power and radiative efficiency of GRS 1915+105 (Narzilloev et al., 2023).

5. Thermodynamics, evaporation, and conserved charges

In the AdS generalization, Kerr Taub–NUT AdS black holes admit a first law that includes the cosmological constant, NUT charges, and both black-hole and Misner-string angular momenta. The central result is

$a^2 \le M^2 + n^2$3

with a corresponding generalized Smarr relation. A key new ingredient is the independent variation of both angular momenta, the black hole and Misner string angular momenta; Brown–York quasilocal charges coincide with the generalized Komar expressions for the mass and black-hole spin (Rodríguez et al., 2021).

Semi-classical evaporation of a Kerr–Taub–NUT black hole emitting massless scalar particles yields a characteristic hierarchy of parameter loss rates,

$a^2 \le M^2 + n^2$4

so angular momentum disappears faster than the NUT parameter and mass. In the same model, the Bekenstein–Hawking entropy

$a^2 \le M^2 + n^2$5

decreases monotonically, while the radiation entropy grows, and the von Neumann entropy follows approximately the so-called Page curve. Larger $a^2 \le M^2 + n^2$6 values accelerate the evaporation process (Arévalo et al., 2024).

Higher-derivative corrections have also been studied perturbatively. In Einstein gravity extended with a cubic curvature invariant, first-order perturbative Taub–NUT solutions satisfy the first law and Smarr relation, and the Reall–Santos method remains applicable even though the metrics are no longer asymptotic to Minkowski spacetime. This framework was then used to obtain the leading correction to the thermodynamics of the complicated Kerr–Taub–NUT black holes (Chen et al., 2024).

6. Energy extraction and weak cosmic censorship

The Penrose process in Kerr–Taub–NUT spacetime retains the standard ergoregion mechanism but with NUT-dependent kinematics. The NUT parameter slightly shifts the shape of the effective potential down, and numerical analysis shows that the extracted energy increases with the dimensionless NUT parameter $a^2 \le M^2 + n^2$7. The extractable energy is therefore enhanced even though the relative volume of the ergoregion decreases as $a^2 \le M^2 + n^2$8 grows (Abdujabbarov et al., 2011).

Weak cosmic censorship has been tested in several gedanken experiments. For a Kerr–Taub–NUT black hole probed by a test massive scalar field and a test particle, extremal and near-extremal black holes cannot be over-spun by the scalar field, while an extremal black hole cannot be over-spun by a test particle but a near-extremal black hole can be over-spun in the test-particle approximation. The comparison suggests that the time interval for particles crossing the black hole horizon might be important for consideration of the weak cosmic censorship conjecture (Yang et al., 2020).

For Kerr–Newman Taub–NUT black holes, a consistent first law with NUT thermodynamics leads to the corresponding charged result: an extremal black hole cannot be destroyed by a charged test particle and a charged complex scalar field, whereas a near-extremal black hole with small NUT parameter can be destroyed by a charged test particle but cannot be destroyed by a complex scalar field (Yang et al., 2023).

A second-order analysis changes the overspinning conclusion for classical fields satisfying the null energy condition. Incorporating the Sorce–Wald condition for Kerr–Taub–NUT black holes shows that non-generic first-order overspinning is repaired by the second-order variations: the final state is driven away from extremality, the event horizon is preserved, and the result accords with the cosmic censorship conjecture and the laws of black hole dynamics (Düztaş, 14 Jul 2025).

7. Generalizations, mathematical structures, and conceptual status

Kerr–Taub–NUT geometry appears in several broader settings. In the low-energy limit of heterotic string theory, an accelerating Kerr–Sen–Taub–NUT spacetime extends the vacuum KTN solution by adding acceleration, stringy gauge fields, a dilaton, and a Kalb–Ramond field. Its analysis includes locations of horizons, ergoregions, conic singularity, closed timelike curves, and the area temperature product, and several properties resemble those of accelerating Kerr–Newman–Taub–NUT spacetime (Siahaan, 2024).

On the Euclidean side, Taub–NUT and Kerr exhibit distinguished complex-geometric structures. Euclidean Taub–NUT is hyper-Kähler with respect to the usual almost complex structures, while Euclidean Kerr is globally conformally Kähler. The conformally scaled Euclidean Kerr space admits a Kähler structure by a conformal factor derived either from the Lee form or from the self-dual part of the Weyl tensor. This suggests a mathematically structured relation between the pure NUT and pure rotation sectors that is relevant to Kerr–Taub–NUT instanton geometry (Kelekçi, 2022).

Across these developments, the main conceptual tension remains the status of the NUT parameter itself. In phenomenological applications it is treated as a gravitomagnetic monopole or dual mass, and in several contexts—shadows, tilted disks, jet power, and evaporation—it behaves as a measurable deformation parameter beyond Kerr. At the same time, Misner strings, asymptotic non-flatness, and possible CTCs continue to make the physical interpretation of Lorentzian Kerr–Taub–NUT black holes contingent on the global prescription adopted for the spacetime (Sen et al., 2024).