Taub–NUT-like Black Holes

• Taub–NUT-like black holes are spacetimes defined by mass, spin, electromagnetic charges, and a NUT parameter that acts as a gravitomagnetic monopole, twisting time fibers.
• They exhibit distinct geometric features such as Misner strings, closed timelike curves, and non-flat asymptotics that influence horizon topology and causal structure.
• Recent analyses in modified gravity and scalar-tensor theories reveal that these black holes show unique thermodynamic behaviors, quantum emission spectra, and bifurcation phenomena.

Searching arXiv for the cited Taub–NUT-related papers to ground the article in the current literature. {"query":"Taub-NUT black hole Kerr-Taub-NUT scalarization Einstein-bumblebee arXiv (Rehman et al., 27 Jul 2025, Liu et al., 2024, Chen et al., 29 May 2025, Rodríguez et al., 2021, Ong, 2016)", "max_results": 10} {"query":"(Rehman et al., 27 Jul 2025) Constraints on extra charges in dyonic Kerr-Newman-Kasuya-Taub-NUT black hole from the observations of quasi-periodic oscillations", "max_results": 5} A Taub–NUT-like black hole is a black-hole spacetime that carries, in addition to mass and often spin and electromagnetic charges, a Taub–NUT parameter interpreted as a gravitomagnetic monopole moment. In the recent literature the label is used in several closely related senses: for the four-dimensional Lorentzian Taub–NUT and Kerr–Taub–NUT families, for dyonic Kerr–Newman–Kasuya–Taub–NUT geometries used as phenomenological deviations from Kerr, and for higher-dimensional or modified-gravity black holes built on Taub–NUT fibrations or endowed with NUT-like charges. This suggests that the term is best understood as a family resemblance rather than a single unique metric, with the common structural element being the NUT-induced twist of the time or Kaluza–Klein fiber over an angular base (Rehman et al., 27 Jul 2025, Pradhan, 2015, Nedkova et al., 2011).

1. Canonical geometries and defining structures

In four-dimensional Lorentzian Einstein gravity, the standard Taub–NUT black hole is written as

$$ds^2 = -\,B(r)\,\bigl(dt + 2n\cos\theta\,d\phi\bigr)^2 + \frac{dr^2}{B(r)} + (r^2 + n^2)\bigl(d\theta^2 + \sin^2\theta\,d\phi^2\bigr),$$

with

$B(r)=1-\frac{2(Mr+n^2)}{r^2+n^2}.$

Here $M$ is the gravito-electric mass and $n$ is the gravito-magnetic mass, or NUT charge. The horizons are at

$r_\pm=M\pm \sqrt{M^2+n^2},$

so $r_+r_-=-n^2$, a sign structure that already distinguishes Taub–NUT from Reissner–Nordström (Pradhan, 2015).

A widely used rotating and charged generalization is the dyonic Kerr–Newman–Kasuya–Taub–NUT metric,

$$ds^2 = -\frac{\Delta_r}{\Sigma}(dt-\eta\,d\varphi)^2+\frac{\Sigma}{\Delta_r}dr^2+\Sigma\,d\theta^2+\frac{\sin^2\theta}{\Sigma}(a\,dt-\rho^2\,d\varphi)^2,$$

with

$$\Sigma=r^2+(n+a\cos\theta)^2,\quad \eta=a\sin^2\theta-2n\cos\theta,$$

$$\Delta_r=\rho^2-2(Mr+n^2)+Q^2+P^2,\quad \rho^2=r^2+a^2+n^2.$$

Here $a=J/M$, $B(r)=1-\frac{2(Mr+n^2)}{r^2+n^2}.$0 is electric charge, $B(r)=1-\frac{2(Mr+n^2)}{r^2+n^2}.$1 magnetic charge, and $B(r)=1-\frac{2(Mr+n^2)}{r^2+n^2}.$2 the Taub–NUT parameter. The Kerr limit is recovered when $B(r)=1-\frac{2(Mr+n^2)}{r^2+n^2}.$3; the Kerr–Taub–NUT sector is obtained when $B(r)=1-\frac{2(Mr+n^2)}{r^2+n^2}.$4 but $B(r)=1-\frac{2(Mr+n^2)}{r^2+n^2}.$5 (Rehman et al., 27 Jul 2025).

A useful organizational summary is:

Class	Defining feature	Representative source
Lorentzian Taub–NUT	$B(r)=1-\frac{2(Mr+n^2)}{r^2+n^2}.$6 fiber and $B(r)=1-\frac{2(Mr+n^2)}{r^2+n^2}.$7	(Pradhan, 2015)
Kerr–Taub–NUT / KNKTN	Rotation, charge, and NUT charge in a stationary axisymmetric metric	(Rehman et al., 27 Jul 2025)
Einstein-bumblebee Taub–NUT-like	Same NUT structure, but $B(r)=1-\frac{2(Mr+n^2)}{r^2+n^2}.$8 rescaled by Lorentz-breaking parameter	(Chen et al., 29 May 2025)
5D Taub–NUT-instanton black hole	Squashed $B(r)=1-\frac{2(Mr+n^2)}{r^2+n^2}.$9 horizon on a Taub–NUT instanton with ALF asymptotics	(Nedkova et al., 2011)

The same structural motif also appears in generalized Euclidean Taub–NUT bases with twisted $M$0 fibers in odd-dimensional Kaluza–Klein black holes, where the NUT-like parameter controls the size and twist of the extra dimension (Tatsuoka et al., 2011).

2. Gravitomagnetic monopole moment and geodesic dynamics

The NUT parameter is treated in the recent astrophysical literature as a gravitomagnetic monopole moment. In the KNKTN metric it appears in $M$1, $M$2, and especially in

$M$3

so even when $M$4, the $M$5 term generates an effective $M$6. In this sense the NUT charge contributes a “magnetic” part to the gravitational field and induces a frame-dragging-like twist independent of ordinary spin (Rehman et al., 27 Jul 2025).

For equatorial circular motion, the orbital angular velocity is

$M$7

and the corresponding frequencies are

$M$8

Within the relativistic precession model one identifies

$M$9

and

$n$0

The KNKTN analysis reports that all three frequencies increase with increasing Taub-NUT charge, while increasing magnetic charge $n$1 lowers them (Rehman et al., 27 Jul 2025).

Using QPO data from GRO J1655–40, XTE J1550–564, XTE J1859+226, GRS 1915+105, and H1743–322, the Markov Chain Monte Carlo analysis finds no significant evidence for nonzero $n$2 or $n$3, and no compelling indication of nonzero $n$4 for four of the five sources. The notable exception is GRS 1915+105, for which the Kerr–Taub–NUT fit gives

$n$5

and the full KNKTN fit yields

$n$6

so that the posterior is bounded away from zero (Rehman et al., 27 Jul 2025).

A related Kerr–Taub–NUT phenomenology based on continuum fitting and Blandford–Znajek jet power finds that the NUT parameter decreases radiative efficiency for fixed spin and modifies the horizon angular velocity entering jet power. Several X-ray binaries can be accommodated in this framework, but GRS 1915+105 remains problematic: the Kerr–Taub–NUT solution, like Kerr itself, cannot simultaneously explain its observed jet power and radiative efficiency (Narzilloev et al., 2023).

3. Global structure, Misner strings, and causal pathologies

The characteristic global feature of Taub–NUT geometry is the Misner-string structure associated with the one-form $n$7. In the Lorentzian Taub–NUT literature these string-like singularities lie at $n$8, and Misner’s classic cure is to identify $n$9 modulo $r_\pm=M\pm \sqrt{M^2+n^2},$0, which removes the string at the price of making time periodic (Pradhan, 2015).

A major clarification in the modern literature is that the supposedly novel “twisted black hole” is exactly Taub–NUT after the coordinate transformation

$r_\pm=M\pm \sqrt{M^2+n^2},$1

Its unusual opposite frame dragging in the two hemispheres, vanishing ADM angular momentum, and absence of curvature singularity are therefore not signs of a new asymptotically flat rotating black hole, but of Taub–NUT structure (Ong, 2016).

The same analysis stresses that the geometry is not physically innocuous. Asymptotically,

$r_\pm=M\pm \sqrt{M^2+n^2},$2

so the spacetime is not asymptotically flat. The ADM angular momentum vanishes, but the Komar angular momentum does not. Constant-$r_\pm=M\pm \sqrt{M^2+n^2},$3, constant-$r_\pm=M\pm \sqrt{M^2+n^2},$4 surfaces are not globally Riemannian because $r_\pm=M\pm \sqrt{M^2+n^2},$5 changes sign near the poles, producing closed timelike curves outside the horizon. In addition, the axis has a severe conical or wire singularity, even though the Kretschmann scalar is finite everywhere and independent of angle (Ong, 2016).

This distinction between curvature regularity and global pathology is central. A Taub–NUT-like black hole can be free of curvature blow-up at $r_\pm=M\pm \sqrt{M^2+n^2},$6 yet still have Misner strings, non-flat asymptotics, and causality violations. A related charged Taub–NUT regular black hole studied in Einstein–Maxwell theory has finite curvature invariants and can be interpreted as a one-way hidden wormhole, but it still inherits the Taub–NUT-type Misner-tube structure and the associated thermodynamic subtleties (Feng et al., 2020).

The causal status of higher-dimensional Taub–NUT-like constructions is more varied. A 5D static black hole on a Taub–NUT instanton has a spacelike compact fiber and is described as free of Misner string singularities and closed timelike curves in the intended physical region, whereas odd-dimensional extremal charged black holes on generalized Euclidean Taub–NUT spaces admit only a $r_\pm=M\pm \sqrt{M^2+n^2},$7 horizon extension and develop divergent frame components of the Riemann tensor for dimensions higher than five (Nedkova et al., 2011, Tatsuoka et al., 2011).

4. Thermodynamics and conserved charges

For the four-dimensional Lorentzian Taub–NUT black hole, the horizon areas and entropies of the outer and inner horizons are

$r_\pm=M\pm \sqrt{M^2+n^2},$8

$r_\pm=M\pm \sqrt{M^2+n^2},$9

The area and entropy products depend explicitly on $r_+r_-=-n^2$0,

$r_+r_-=-n^2$1

so they are not universal. By contrast, the temperature product is universal,

$r_+r_-=-n^2$2

and one has the mass-independent relation

$r_+r_-=-n^2$3

Both horizons have

$r_+r_-=-n^2$4

so the specific heats are negative and the Lorentzian Taub–NUT black hole is thermodynamically unstable in this treatment (Pradhan, 2015).

For rotating AdS Taub–NUT black holes, the thermodynamic structure is richer because the Misner strings themselves carry charges and angular momentum. The Kerr–Taub–NUT–AdS first law includes variations of the cosmological constant, NUT charges, and two angular momenta: the black-hole angular momentum and the Misner-string angular momentum. The generalized Komar construction yields charges satisfying the Smarr formula by construction, and the Brown–York quasilocal definitions reproduce the same mass and black-hole spin (Rodríguez et al., 2021).

Modified-gravity Taub–NUT-like solutions preserve the first-law structure but can alter the entropy conceptually. In Einstein–bumblebee gravity, one exact Taub–NUT-like black hole has

$r_+r_-=-n^2$5

with the standard Taub–NUT radial function $r_+r_-=-n^2$6, is not Ricci-flat for $r_+r_-=-n^2$7, and obeys

$r_+r_-=-n^2$8

while the Wald entropy

$r_+r_-=-n^2$9

differs from the thermodynamic entropy

$$ds^2 = -\frac{\Delta_r}{\Sigma}(dt-\eta\,d\varphi)^2+\frac{\Sigma}{\Delta_r}dr^2+\Sigma\,d\theta^2+\frac{\sin^2\theta}{\Sigma}(a\,dt-\rho^2\,d\varphi)^2,$$0

The same theory admits a Taub–NUT–AdS-like extension with effective cosmological constant

$$ds^2 = -\frac{\Delta_r}{\Sigma}(dt-\eta\,d\varphi)^2+\frac{\Sigma}{\Delta_r}dr^2+\Sigma\,d\theta^2+\frac{\sin^2\theta}{\Sigma}(a\,dt-\rho^2\,d\varphi)^2,$$1

and an extended first law with $$ds^2 = -\frac{\Delta_r}{\Sigma}(dt-\eta\,d\varphi)^2+\frac{\Sigma}{\Delta_r}dr^2+\Sigma\,d\theta^2+\frac{\sin^2\theta}{\Sigma}(a\,dt-\rho^2\,d\varphi)^2,$$2 (Chen et al., 29 May 2025).

5. Quantum emission, evaporation, and horizon stability

A useful quantum probe of Taub–NUT-like black holes is the moving-mirror correspondence. For general classes of Taub–NUT black holes, the Bogoliubov coefficients of a relativistically accelerated mirror in $$ds^2 = -\frac{\Delta_r}{\Sigma}(dt-\eta\,d\varphi)^2+\frac{\Sigma}{\Delta_r}dr^2+\Sigma\,d\theta^2+\frac{\sin^2\theta}{\Sigma}(a\,dt-\rho^2\,d\varphi)^2,$$3 dimensions reproduce the Hawking spectrum of the black-hole spacetime. At late times the energy flux approaches

$$ds^2 = -\frac{\Delta_r}{\Sigma}(dt-\eta\,d\varphi)^2+\frac{\Sigma}{\Delta_r}dr^2+\Sigma\,d\theta^2+\frac{\sin^2\theta}{\Sigma}(a\,dt-\rho^2\,d\varphi)^2,$$4

and the particle number spectrum becomes Planckian in the high-frequency limit,

$$ds^2 = -\frac{\Delta_r}{\Sigma}(dt-\eta\,d\varphi)^2+\frac{\Sigma}{\Delta_r}dr^2+\Sigma\,d\theta^2+\frac{\sin^2\theta}{\Sigma}(a\,dt-\rho^2\,d\varphi)^2,$$5

The paper emphasizes that particle production is suppressed by the NUT parameter, and extends the same analysis to Kerr–Newman–Taub–NUT and extremal Taub–NUT-like cases, where the extremal spectrum is non-thermal (Foo et al., 2020).

A complementary semiclassical picture is obtained from an evaporating Kerr–Taub–NUT black hole emitting massless scalar particles. In that model, scalar emission increases the angular-momentum loss rate while decreasing the loss rates of both the NUT parameter and the mass. The resulting hierarchy is

$$ds^2 = -\frac{\Delta_r}{\Sigma}(dt-\eta\,d\varphi)^2+\frac{\Sigma}{\Delta_r}dr^2+\Sigma\,d\theta^2+\frac{\sin^2\theta}{\Sigma}(a\,dt-\rho^2\,d\varphi)^2,$$6

so the black hole first loses angular momentum, then NUT charge, and finally mass. The corresponding von Neumann entropy approximately follows the Page curve, and larger values of the parameter $$ds^2 = -\frac{\Delta_r}{\Sigma}(dt-\eta\,d\varphi)^2+\frac{\Sigma}{\Delta_r}dr^2+\Sigma\,d\theta^2+\frac{\sin^2\theta}{\Sigma}(a\,dt-\rho^2\,d\varphi)^2,$$7 accelerate evaporation and move the Page time earlier in units of the total decay time (Arévalo et al., 2024).

Horizon stability has also been investigated directly. For a regular Reissner–Nordström Taub–NUT black hole, charged scalar-field scattering cannot overcharge either near-extremal or extremal configurations, and test charged particles cannot overcharge the extremal case. A near-extremal black hole can be overcharged in the idealized discrete test-particle approximation, but the paper argues that if charge and energy cross the horizon in a continuous path, the near-extremal black hole might not be overcharged (Feng et al., 2020).

These results collectively indicate that the NUT parameter affects not only classical orbits and thermodynamics but also Hawking fluxes, information-theoretic evolution, and gedanken tests of cosmic censorship.

6. Modified gravity, scalarization, and higher-dimensional generalizations

Taub–NUT-like black holes are especially prominent in modified-gravity constructions. In quadratic gravity with Weyl-squared and $$ds^2 = -\frac{\Delta_r}{\Sigma}(dt-\eta\,d\varphi)^2+\frac{\Sigma}{\Delta_r}dr^2+\Sigma\,d\theta^2+\frac{\sin^2\theta}{\Sigma}(a\,dt-\rho^2\,d\varphi)^2,$$8 terms, the Ricci-flat Taub–NUT black hole remains a solution, but unstable Lichnerowicz modes signal bifurcations to new Taub–NUT black holes with massive spin-2 hair. These non-Einstein solutions satisfy $$ds^2 = -\frac{\Delta_r}{\Sigma}(dt-\eta\,d\varphi)^2+\frac{\Sigma}{\Delta_r}dr^2+\Sigma\,d\theta^2+\frac{\sin^2\theta}{\Sigma}(a\,dt-\rho^2\,d\varphi)^2,$$9, come in two branches connected at a minimal horizon radius, and can yield up to three black holes at the same temperature when the Ricci-flat branch is included (Chen et al., 2024).

In extended scalar-tensor–Gauss–Bonnet theory, Ricci-flat Taub–NUT is again the scalar-free branch. The scalarized black holes are constructed numerically from a probe zero-mode analysis and display two smoothly connected branches. Their entropy is larger than that of the scalar-free Taub–NUT black holes, and a novel feature emerges: the entropy at the bifurcation point is constant for positive mass parameter, leading to a conjectured maximal entropy bound for all scalarized solutions whose bifurcation mass is positive (Liu et al., 2024).

The charged Einstein–Maxwell–scalar–Gauss–Bonnet extension exhibits an analogous phenomenon in a two-dimensional parameter space spanned by electric charge and NUT parameter. The scalarized charged Taub–NUT black holes have entropy strictly greater than their scalar-free counterparts, the entropy reaches a local maximum at the bifurcation point, and for fixed electric charge this maximum entropy becomes universal over a specific mass range, independent of the NUT parameter (Zhang et al., 19 Mar 2026).

Einstein–bumblebee gravity supplies another family of exact Taub–NUT-like geometries. One construction gives a non-rotating metric

$$\Sigma=r^2+(n+a\cos\theta)^2,\quad \eta=a\sin^2\theta-2n\cos\theta,$$0

with

$$\Sigma=r^2+(n+a\cos\theta)^2,\quad \eta=a\sin^2\theta-2n\cos\theta,$$1

The Lorentz symmetry breaking alters the mass and temperature but not the thermodynamic topological class: the solution remains in Class I with topological charge $$\Sigma=r^2+(n+a\cos\theta)^2,\quad \eta=a\sin^2\theta-2n\cos\theta,$$2 (Azreg-Aïnou et al., 22 Sep 2025). A dyonic Einstein–bumblebee generalization further extends these results to Taub–NUT-like black holes with independent electric and magnetic charges, with mass and entropy derived from the Wald formalism and a consistent first law (Li et al., 6 Oct 2025).

Higher-dimensional variants broaden the concept further. In five-dimensional Einstein–Maxwell–dilaton gravity, a static black hole on a Taub–NUT instanton has a squashed $$\Sigma=r^2+(n+a\cos\theta)^2,\quad \eta=a\sin^2\theta-2n\cos\theta,$$3 horizon, ALF asymptotics, nut charge

$$\Sigma=r^2+(n+a\cos\theta)^2,\quad \eta=a\sin^2\theta-2n\cos\theta,$$4

and Smarr-like relations involving tension, nut potential, magnetic flux, and effective current (Nedkova et al., 2011). In odd-dimensional Einstein–Maxwell theory, extremal charged Kaluza–Klein black holes built on generalized Euclidean Taub–NUT spaces have twisted $$\Sigma=r^2+(n+a\cos\theta)^2,\quad \eta=a\sin^2\theta-2n\cos\theta,$$5 fibers, a null hypersurface with vanishing outgoing expansion, and a $$\Sigma=r^2+(n+a\cos\theta)^2,\quad \eta=a\sin^2\theta-2n\cos\theta,$$6 extension across the horizon, but for dimensions higher than five some frame components of the Riemann tensor diverge there while remaining weak enough for free-fall traversal (Tatsuoka et al., 2011).

Taken together, these developments show that Taub–NUT-like black holes occupy a technically diverse but conceptually unified niche: they encode gravitomagnetic charge through a NUT parameter or Taub–NUT fibration, they modify orbit structure and thermodynamics in ways that are often observationally testable, and they repeatedly serve as bifurcation backgrounds for new hairy or Lorentz-violating black holes. At the same time, the classical four-dimensional Lorentzian examples remain entangled with Misner strings, closed timelike curves, and non-flat asymptotics, so their role as astrophysical black holes remains more controversial than their role as exact solutions and theoretical laboratories (Ong, 2016, Rehman et al., 27 Jul 2025).