Kerr-Taub-NUT Black Holes

• Kerr-Taub-NUT black holes are stationary, axisymmetric solutions characterized by mass, rotation, and a NUT parameter that acts as a gravitomagnetic monopole.
• The NUT charge modifies the radial and angular structure, influencing horizon, ergosphere, and shadow geometry, while introducing Misner strings and potential closed timelike curves.
• These spacetimes serve as theoretical models for exploring strong-field dynamics, energy extraction processes, thermodynamics, and complex Euclidean geometries.

Kerr–Taub–NUT black holes are stationary, axisymmetric vacuum solutions of Einstein’s equations characterized by a mass $M$, a rotation parameter $a=J/M$, and a NUT parameter $l$ or $n$, often interpreted as a gravitomagnetic monopole charge or “dual mass.” They interpolate between Schwarzschild $(a=0,l=0)$, Kerr $(l=0)$, and Taub–NUT $(a=0)$. Relative to Kerr, the NUT parameter modifies both the radial and angular structure of the metric, changes the horizon and ergosphere, breaks ordinary asymptotic flatness, and introduces Misner strings and possible closed timelike curves unless further identifications are imposed. For that reason, Kerr–Taub–NUT is used less as a literal astrophysical model than as a controlled theoretical background for strong-field dynamics, black-hole thermodynamics, shadow formation, energy extraction, and cosmic-censorship tests [(Abdujabbarov et al., 2011); (Yang et al., 2020)].

1. Defining geometry and causal structure

A standard Boyer–Lindquist–type form used in the literature is

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

with

$$\Sigma = r^{2} + (l + a\cos\theta)^{2},\qquad \Delta = r^{2} - 2Mr - l^{2} + a^{2},\qquad \chi = a\sin^{2}\theta - 2l\cos\theta.$$

An equivalent notation replaces $l$ by $a=J/M$0, with the same horizon function $a=J/M$1 and $a=J/M$2 [(Abdujabbarov et al., 2011); (Yang et al., 2020)].

The horizon structure is determined by $a=J/M$3, giving

$a=J/M$4

or equivalently $a=J/M$5. The existence condition for a horizon is therefore

$a=J/M$6

The outer horizon radius increases with the NUT charge for fixed $a=J/M$7 and $a=J/M$8. The static limit, defined by $a=J/M$9, is

$l$0

so the ergoregion occupies $l$1 [(Abdujabbarov et al., 2011); (Yang et al., 2020)].

Several geometric peculiarities are standard. The NUT parameter introduces a gravitomagnetic monopole–like structure, Misner strings along the axis, non-asymptotically flat behavior, and possible closed timelike curves. In one formulation, the curvature singularity is tied to $l$2; for $l$3, $l$4 everywhere and the spacetime is regular, whereas for $l$5 a genuine black-hole interpretation with Misner-string singularities is adopted (Yang et al., 2020). This dual local/global character is central: many analyses focus on local geodesic and thermodynamic questions while deliberately setting aside the global regularity problem.

2. Geodesics, photon regions, and optical appearance

Kerr–Taub–NUT retains separability properties closely analogous to Kerr. For null geodesics, the Hamilton–Jacobi equation separates into radial and angular parts with conserved energy $l$6, axial angular momentum $l$7, and a Carter-like constant $l$8. In one common notation,

$l$9

with

$n$0

and a correspondingly modified $n$1 that contains explicit $n$2-dependent terms. Unstable spherical photon orbits satisfy $n$3 and $n$4, and define the shadow boundary through impact parameters $n$5 and $n$6 (1212.11463).

For an equatorial observer, the celestial coordinates simplify to

$n$7

and numerical studies show a clear NUT signature: for fixed spin, increasing the gravitomagnetic charge enlarges the shadow and reduces its deformation relative to Kerr. Thus the NUT parameter tends to make the shadow larger and more circular, while spin tends to increase the familiar Kerr-like asymmetry (1212.11463).

Near-horizon emission amplifies a second effect: the equatorial plane ceases to be a symmetry plane when $n$8. In the extremal case, photons emitted from the near-horizon region lie on a vertical NHEKTN line in the observer’s sky, with

$n$9

Because the NUT charge breaks north–south reflection symmetry, the line emission seen by observers in the Northern and Southern hemispheres is inequivalent. One analysis finds that the near-horizon electromagnetic line emission is brighter than in Kerr for observers in the equatorial plane or Southern hemisphere, but becomes fainter as the observer moves far from the equatorial plane in the Northern hemisphere (Long et al., 2018).

More general deformations appear once extra structure is included. In Kerr–Sen–Taub–NUT with the Manko–Ruiz parameter, the Hamilton–Jacobi equation remains separable, and the shadow depends not only on rotation and NUT charge but also on the Hassan–Sen charge parameter and the Manko–Ruiz parameter $(a=0,l=0)$0, which shifts and deforms the image (Siahaan, 12 Apr 2025). In thin-disk ray tracing directly on Kerr–Taub–NUT backgrounds, increasing spin deforms the critical curve into a $(a=0,l=0)$1-shape while shrinking and distorting the inner shadow; for $(a=0,l=0)$2 and $(a=0,l=0)$3, a novel “duck-cap-like” morphology with a protruding lower-right edge appears. That feature is termed the “extended inner shadow” and is partly generated by photons that are neither absorbed by the horizon nor intersect the disk plane, so they remain dark despite lying beyond the usual critical curve (Wang et al., 9 Nov 2025).

3. Energy extraction, repetitive Penrose processes, and horizon stability

The ergoregion admits negative-energy states and therefore supports Penrose-type rotational energy extraction. In a quasi-equatorial approximation appropriate to very small observationally allowed NUT charges, the radial energy equation for timelike motion can be written as

$(a=0,l=0)$4

with an effective potential containing $(a=0,l=0)$5, $(a=0,l=0)$6, $(a=0,l=0)$7, and $(a=0,l=0)$8. The main qualitative result of one single-splitting analysis is that “the presence of the parameter $(a=0,l=0)$9 slightly shifts the shape of the effective potential down,” and that the relative extracted energy grows with the dimensionless NUT parameter $(l=0)$0 (Abdujabbarov et al., 2011).

A later analysis of the repetitive Penrose process reaches a different conclusion for a different notion of efficiency. Using the horizon area

$(l=0)$1

it defines the extractable energy as

$(l=0)$2

Since $(l=0)$3 increases with $(l=0)$4, the irreducible mass increases and the extractable energy decreases with the gravitomagnetic charge. In that framework the horizon and ergosphere both enlarge, but the maximum energy return on investment of the repetitive Penrose process is higher for smaller $(l=0)$5, and the iteration terminates after finitely many steps with a substantial residual extractable energy left unextracted (Alloqulov et al., 26 Mar 2026). This suggests that local access to negative-energy trajectories and global reversible-energy bounds do not vary identically with $(l=0)$6.

Kerr–Taub–NUT also provides a sharp arena for weak cosmic censorship. For the horizon indicator

$(l=0)$7

over-spinning would require $(l=0)$8 after absorbing matter or fields. In a scalar-field gedanken experiment, both extremal and near-extremal Kerr–Taub–NUT black holes remain protected: keeping the thermodynamic Misner charge $(l=0)$9 fixed yields $(a=0)$0, and neither extremal nor near-extremal black holes can be over-spun by classical scalar-field scattering in an infinitesimal-time process. By contrast, in the test-particle approximation, extremal holes remain protected but near-extremal ones admit a parameter window $(a=0)$1 and can be over-spun at first order (Yang et al., 2020).

That near-extremal loophole is closed by second-order analysis. For the subset of near-extremal Kerr–Taub–NUT black holes that can be over-spun at first order by test fields satisfying the null energy condition, evaluation of the Sorce–Wald second-order inequality shows that the second-order terms drive the final state away from extremality. In the notation

$(a=0)$2

the corrected final state satisfies

$(a=0)$3

so the horizon is preserved and the spacetime is not over-spun once second-order variations are included (Düztaş, 14 Jul 2025).

4. Thermodynamics, evaporation, and higher-curvature deformations

Thermodynamic behavior is unusually subtle in Kerr–Taub–NUT because the NUT charge changes both asymptotics and the role of axial singularities. In Lorentzian signature, the inner and outer horizon areas are

$(a=0)$4

so the area product becomes

$(a=0)$5

Unlike Kerr or Reissner–Nordström, this product depends explicitly on the mass parameter and is therefore neither universal nor naturally quantized. The same mass dependence afflicts the entropy and irreducible-mass products, and the standard first law and Smarr–Gibbs–Duhem relations do not hold for Lorentzian Taub–NUT and Kerr–Taub–NUT. The stated reasons are the nontrivial NUT charge, asymptotic non-flatness, and the presence of Dirac–Misner type singularities (Pradhan, 2014).

A different picture emerges in AdS. For Kerr–Taub–NUT AdS, a generalized Komar construction yields a consistent thermodynamic framework with entropy

$(a=0)$6

physical mass $(a=0)$7, black-hole angular momentum $(a=0)$8, total angular momentum $(a=0)$9, Misner charges $$\begin{aligned} ds^{2} &= -\frac{1}{\Sigma} \left(\Delta - a^{2}\sin^{2}\theta\right)dt^{2} + \frac{\Sigma}{\Delta}dr^{2} + \Sigma\, d\theta^{2} \ &\quad + \frac{2}{\Sigma}\left[\Delta\chi - a(\Sigma+a\chi)\sin^{2}\theta\right]dt\,d\varphi \ &\quad + \frac{1}{\Sigma}\left[(\Sigma+a\chi)^{2}\sin^{2}\theta - \chi^{2}\Delta\right]d\varphi^{2}, \end{aligned}$$0, and thermodynamic volume

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

The corresponding first law is

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

or equivalently

$$\begin{aligned} ds^{2} &= -\frac{1}{\Sigma} \left(\Delta - a^{2}\sin^{2}\theta\right)dt^{2} + \frac{\Sigma}{\Delta}dr^{2} + \Sigma\, d\theta^{2} \ &\quad + \frac{2}{\Sigma}\left[\Delta\chi - a(\Sigma+a\chi)\sin^{2}\theta\right]dt\,d\varphi \ &\quad + \frac{1}{\Sigma}\left[(\Sigma+a\chi)^{2}\sin^{2}\theta - \chi^{2}\Delta\right]d\varphi^{2}, \end{aligned}$$3

Its key novelty is the independent variation of black-hole and Misner-string angular momenta (Rodríguez et al., 2021).

Higher-curvature extensions preserve this thermodynamic program. In Einstein gravity corrected by a cubic curvature invariant, first-order perturbative Taub–NUT solutions can be constructed explicitly, and the Reall–Santos method remains applicable even though the metrics are no longer asymptotic to Minkowski spacetime. That method then yields leading corrections to the thermodynamics of the more complicated Kerr–Taub–NUT black holes, again satisfying the first law and Smarr relation at the perturbative level (Chen et al., 2024).

Evaporation adds an information-theoretic layer. For a Kerr–Taub–NUT black hole radiating massless scalar particles, the Hawking temperature can be written as

$$\begin{aligned} ds^{2} &= -\frac{1}{\Sigma} \left(\Delta - a^{2}\sin^{2}\theta\right)dt^{2} + \frac{\Sigma}{\Delta}dr^{2} + \Sigma\, d\theta^{2} \ &\quad + \frac{2}{\Sigma}\left[\Delta\chi - a(\Sigma+a\chi)\sin^{2}\theta\right]dt\,d\varphi \ &\quad + \frac{1}{\Sigma}\left[(\Sigma+a\chi)^{2}\sin^{2}\theta - \chi^{2}\Delta\right]d\varphi^{2}, \end{aligned}$$4

and the Bekenstein–Hawking entropy as

$$\begin{aligned} ds^{2} &= -\frac{1}{\Sigma} \left(\Delta - a^{2}\sin^{2}\theta\right)dt^{2} + \frac{\Sigma}{\Delta}dr^{2} + \Sigma\, d\theta^{2} \ &\quad + \frac{2}{\Sigma}\left[\Delta\chi - a(\Sigma+a\chi)\sin^{2}\theta\right]dt\,d\varphi \ &\quad + \frac{1}{\Sigma}\left[(\Sigma+a\chi)^{2}\sin^{2}\theta - \chi^{2}\Delta\right]d\varphi^{2}, \end{aligned}$$5

In that model, angular momentum is lost faster than the NUT parameter and the mass, and the von Neumann entropy follows approximately a Page curve. Larger values of the radiation parameter $$\begin{aligned} ds^{2} &= -\frac{1}{\Sigma} \left(\Delta - a^{2}\sin^{2}\theta\right)dt^{2} + \frac{\Sigma}{\Delta}dr^{2} + \Sigma\, d\theta^{2} \ &\quad + \frac{2}{\Sigma}\left[\Delta\chi - a(\Sigma+a\chi)\sin^{2}\theta\right]dt\,d\varphi \ &\quad + \frac{1}{\Sigma}\left[(\Sigma+a\chi)^{2}\sin^{2}\theta - \chi^{2}\Delta\right]d\varphi^{2}, \end{aligned}$$6 accelerate evaporation and move the Page time earlier as a fraction of the total decay time (Arévalo et al., 2024).

5. Euclidean structures, complex geometry, and soft infrared data

The Euclidean continuation of the Taub–NUT and Kerr sectors reveals an additional layer of structure. Euclidean Taub–NUT is hyper-Kähler: it admits three integrable complex structures $$\begin{aligned} ds^{2} &= -\frac{1}{\Sigma} \left(\Delta - a^{2}\sin^{2}\theta\right)dt^{2} + \frac{\Sigma}{\Delta}dr^{2} + \Sigma\, d\theta^{2} \ &\quad + \frac{2}{\Sigma}\left[\Delta\chi - a(\Sigma+a\chi)\sin^{2}\theta\right]dt\,d\varphi \ &\quad + \frac{1}{\Sigma}\left[(\Sigma+a\chi)^{2}\sin^{2}\theta - \chi^{2}\Delta\right]d\varphi^{2}, \end{aligned}$$7 satisfying quaternionic relations, with closed Kähler forms

$$\begin{aligned} ds^{2} &= -\frac{1}{\Sigma} \left(\Delta - a^{2}\sin^{2}\theta\right)dt^{2} + \frac{\Sigma}{\Delta}dr^{2} + \Sigma\, d\theta^{2} \ &\quad + \frac{2}{\Sigma}\left[\Delta\chi - a(\Sigma+a\chi)\sin^{2}\theta\right]dt\,d\varphi \ &\quad + \frac{1}{\Sigma}\left[(\Sigma+a\chi)^{2}\sin^{2}\theta - \chi^{2}\Delta\right]d\varphi^{2}, \end{aligned}$$8

Euclidean Kerr is not Kähler, but it is globally conformally Kähler: the Lee form is exact,

$$\begin{aligned} ds^{2} &= -\frac{1}{\Sigma} \left(\Delta - a^{2}\sin^{2}\theta\right)dt^{2} + \frac{\Sigma}{\Delta}dr^{2} + \Sigma\, d\theta^{2} \ &\quad + \frac{2}{\Sigma}\left[\Delta\chi - a(\Sigma+a\chi)\sin^{2}\theta\right]dt\,d\varphi \ &\quad + \frac{1}{\Sigma}\left[(\Sigma+a\chi)^{2}\sin^{2}\theta - \chi^{2}\Delta\right]d\varphi^{2}, \end{aligned}$$9

so the conformal factor

$$\Sigma = r^{2} + (l + a\cos\theta)^{2},\qquad \Delta = r^{2} - 2Mr - l^{2} + a^{2},\qquad \chi = a\sin^{2}\theta - 2l\cos\theta.$$0

turns the metric into a genuine Kähler metric. The same conformal factor also follows from the self-dual Weyl tensor $$\Sigma = r^{2} + (l + a\cos\theta)^{2},\qquad \Delta = r^{2} - 2Mr - l^{2} + a^{2},\qquad \chi = a\sin^{2}\theta - 2l\cos\theta.$$1 via Derdziński’s theorem (Kelekçi, 2022). A plausible implication is that Euclidean Kerr–Taub–NUT geometries interpolate between a hyper-Kähler Taub–NUT limit and a conformally Kähler Kerr limit, although that full statement is presented in the source as a suggestion rather than a proved classification (Kelekçi, 2022).

At null infinity and in scattering theory, NUT charge reappears as a dual gravitational charge. In the amplitude-based treatment of Kerr–Taub–NUT scattering, each object carries an effective mass

$$\Sigma = r^{2} + (l + a\cos\theta)^{2},\qquad \Delta = r^{2} - 2Mr - l^{2} + a^{2},\qquad \chi = a\sin^{2}\theta - 2l\cos\theta.$$2

and a duality phase

$$\Sigma = r^{2} + (l + a\cos\theta)^{2},\qquad \Delta = r^{2} - 2Mr - l^{2} + a^{2},\qquad \chi = a\sin^{2}\theta - 2l\cos\theta.$$3

The leading soft-graviton factor must be modified by a term proportional to $$\Sigma = r^{2} + (l + a\cos\theta)^{2},\qquad \Delta = r^{2} - 2Mr - l^{2} + a^{2},\qquad \chi = a\sin^{2}\theta - 2l\cos\theta.$$4 in order to remain gauge invariant, and the resulting memory tensor takes the form

$$\Sigma = r^{2} + (l + a\cos\theta)^{2},\qquad \Delta = r^{2} - 2Mr - l^{2} + a^{2},\qquad \chi = a\sin^{2}\theta - 2l\cos\theta.$$5

The $$\Sigma = r^{2} + (l + a\cos\theta)^{2},\qquad \Delta = r^{2} - 2Mr - l^{2} + a^{2},\qquad \chi = a\sin^{2}\theta - 2l\cos\theta.$$6 term is the ordinary electric part of the memory, while the $$\Sigma = r^{2} + (l + a\cos\theta)^{2},\qquad \Delta = r^{2} - 2Mr - l^{2} + a^{2},\qquad \chi = a\sin^{2}\theta - 2l\cos\theta.$$7 term is a genuinely magnetic contribution sourced by NUT charge. In the self-dual Taub–NUT limit, the leading scattering and the associated memory become trivial (Doran et al., 25 Mar 2026).

6. Astrophysical modeling, observational constraints, and interpretive limits

Kerr–Taub–NUT has been used as an effective phenomenological spacetime to test whether a gravitomagnetic charge could help reconcile multiple observables. In one survey of six X-ray binaries—A0620-00, H1743-322, XTE J1550-564, GRS 1124-683, GRO J1655-40, and GRS 1915+105—the radiative efficiency of thin Novikov–Thorne disks and the transient jet power expected from the Blandford–Znajek mechanism were both evaluated in the Kerr–Taub–NUT metric. The horizon angular velocity becomes

$$\Sigma = r^{2} + (l + a\cos\theta)^{2},\qquad \Delta = r^{2} - 2Mr - l^{2} + a^{2},\qquad \chi = a\sin^{2}\theta - 2l\cos\theta.$$8

and both the ISCO location and the radiative efficiency depend on the pair $$\Sigma = r^{2} + (l + a\cos\theta)^{2},\qquad \Delta = r^{2} - 2Mr - l^{2} + a^{2},\qquad \chi = a\sin^{2}\theta - 2l\cos\theta.$$9, leading to a degeneracy absent in pure Kerr. Within the adopted assumptions, simultaneous fits to radiative efficiency and jet power are possible for several sources, but not for GRS 1915+105; moreover, pure Kerr remains allowed in the systems that can be fit, so a nonzero NUT charge is not required by the data (Narzilloev et al., 2023).

The same study emphasizes the central limitation of all such phenomenology: NUT charge is allowed but not uniquely preferred, and the interpretation is clouded by global issues—Misner strings, closed timelike curves, and the nonstandard asymptotics of NUT spacetimes—as well as by ordinary astrophysical degeneracies in disk and jet modeling (Narzilloev et al., 2023). A separate dynamical analysis also notes that observational bounds on the NUT parameter are extremely small, with representative estimates $l$0 from microlensing and $l$1 from matter-wave interferometry and similar experiments. That scale supports the widespread use of quasi-equatorial and small-$l$2 expansions, but it also implies that any realistic astrophysical NUT effect, if present at all, is likely to be subtle (Abdujabbarov et al., 2011).

Taken together, the literature presents Kerr–Taub–NUT black holes as mathematically rich and physically instructive rather than observationally established. Their importance lies in the way a single extra parameter—the gravitomagnetic monopole charge—simultaneously reshapes horizons, photon regions, Penrose energetics, overspinning bounds, Euclidean complex structure, soft gravitational memory, and accretion-flow images. That breadth is precisely why Kerr–Taub–NUT continues to function as a benchmark geometry in classical and semiclassical gravity.