Accelerating Kerr-Taub-NUT Spacetime

• Accelerating Kerr-Taub-NUT spacetime is an exact black-hole geometry combining rotation, acceleration, and NUT charge, enriched by electric charge and a cosmological constant.
• Parameter reductions recover familiar spacetimes (e.g., Kerr-Newman, Taub-NUT, and C-metric) and reveal subtle effects from conical defects and Misner strings.
• The solution exhibits complex horizon structures, anomalous precession dynamics, and thermodynamic challenges that push the limits of conventional strong-field gravity.

Accelerating Kerr-Taub-NUT spacetime denotes a class of exact black-hole geometries in which rotation, acceleration, and a NUT parameter coexist; in broader Einstein-Maxwell and string-theoretic embeddings, electric charge, a cosmological constant, and additional matter fields may also be present. In the generalized Einstein-Maxwell family with cosmological constant, the solution carries eight parameters $M,Q,a,\alpha,\lambda,S_1,S_2,S_3$, where $S_3$ is the NUT charge, $S_1$ is the rescaled parameter $C$ of the Kerr-NUT spacetime, and $S_2$ is a new parameter associated with conical defects; setting $Q\to0$, $\lambda\to0$, and appropriate values of $S_i$ reduces this family to the accelerating Kerr-Taub-NUT metric (Gao, 20 Sep 2025). In the low-energy limit of heterotic string theory, an accelerating charged and rotating black hole with NUT parameter is obtained from the accelerating Kerr-Taub-NUT seed by a Hassan-Sen transformation, adding a Maxwell field, dilaton, and Kalb-Ramond 2-form (Siahaan, 2024).

1. Exact solution families

A convenient four-dimensional Einstein-Maxwell-de Sitter representative is

$$ds^2 = \frac{1}{\Omega^2} \left\{ -\frac{\Sigma - \delta a^2 \sin^2\theta}{\Delta}dt^2 + \frac{\Delta}{\Sigma} dr^2 + \frac{\Delta}{\delta} d\theta^2 - \frac{2a \sin^2\theta [\delta (r^2+a^2) - \Sigma]}{\Delta} dt d\phi + \frac{[\delta (r^2+a^2)^2 - \Sigma a^2 \sin^2\theta]\sin^2\theta}{\Delta} d\phi^2 \right\},$$

with

$$\Omega = 1 - \alpha r \cos\theta,\qquad \Delta = r^2 + a^2 \cos^2\theta,$$

$S_3$0

$S_3$1

and Maxwell potential

$S_3$2

This family extends the accelerating Kerr-Newman-NUT-de Sitter spacetime and makes the NUT sector, the conical-defect sector, and the acceleration sector simultaneously explicit (Gao, 20 Sep 2025).

Parameter	Physical role	Note
$S_3$3	Mass	Physical mass
$S_3$4	Electric charge	Maxwell field
$S_3$5	Specific angular momentum	Rotation
$S_3$6	Acceleration	C-metric-type acceleration
$S_3$7	Cosmological constant	de Sitter/AdS sector
$S_3$8	Rescaled $S_3$9 parameter	Misner string configuration
$S_1$0	Conical defect parameter	Cosmic string tension
$S_1$1	NUT charge	Gravito-magnetic charge

In the low-energy heterotic-string construction, the dynamical fields are $S_1$2, $S_1$3, $S_1$4, and $S_1$5, governed in the string frame by

$S_1$6

where $S_1$7. The Einstein-frame metric is $S_1$8; the vector potential, dilaton, and axion are nontrivial, and the physical mass and charge depend on the Hassan-Sen parameter $S_1$9 through

$C$0

This embeds the accelerating Kerr-Taub-NUT seed into a string-theoretic sector with dilatonic and axionic backreaction (Siahaan, 2024).

2. Reductions and related geometries

Several standard spacetimes arise as parameter reductions. In the generalized Einstein-Maxwell family, $C$1 gives the accelerating Kerr-Newman-de Sitter black hole; $C$2 with $C$3 yields the Kerr-Newman-NUT-de Sitter family; $C$4 recovers the standard Plebański-Demiański structure; $C$5 removes the NUT charge; and $C$6, after transformation to planar coordinates, gives a planar black hole and further the Kasner plane metric (Gao, 20 Sep 2025).

For the accelerating NUT subclass, the metric reduces to the C-metric when $C$7, to the Taub-NUT spacetime when $C$8, to Schwarzschild when $C$9, and to Minkowski space when $S_2$0. The parameters

$S_2$1

organize these limits and the associated horizon structure (Podolsky et al., 2020).

A related geometric perspective comes from double-root limits of Kerr-NUT-(A)dS spacetimes. In the NUT-like limit, degenerate angular directions become $S_2$2 fibers with enhanced $S_2$3 symmetry, while the extreme near-horizon limit produces an $S_2$4 sector with enhanced $S_2$5 symmetry. In four dimensions these limits recover Taub-NUT-(A)dS and the familiar extremal near-horizon geometries, supplying a systematic limit-based relation between rotating NUT geometries and their high-symmetry degenerations (Kolar et al., 2017).

3. Horizons, algebraic type, and global geometry

In the generalized accelerating Kerr-Newman-NUT-de Sitter family, horizons are determined by the quartic equation

$S_2$6

A typical configuration has four real roots, interpreted as two black-hole horizons and two cosmic horizons. The existence of four physical horizons is identified as a novel feature of this solution class (Gao, 20 Sep 2025).

The nonrotating accelerating NUT solution provides the clearest global picture. Its Weyl tensor is algebraically general type I with four distinct principal null directions, which explains why it is not contained in the Plebański-Demiański family of type D spacetimes. In this subclass there are four Killing horizons, there are asymptotically flat regions related to conformal infinities, and the boost-rotation form shows that the spacetime contains a pair of black holes uniformly accelerating in opposite directions due to rotating cosmic strings or struts along the two axes. When the NUT parameter is nonzero, scalar invariants show that there are no curvature singularities (Podolsky et al., 2020).

The string-theoretic accelerating Kerr-Sen-Taub-NUT solution has two black-hole horizons,

$S_2$7

and two acceleration horizons,

$S_2$8

Its ergoregion is fixed by $S_2$9, equivalently

$Q\to0$0

and extremality occurs when

$Q\to0$1

Violation of this condition yields a naked singularity (Siahaan, 2024).

4. Axis structure, conical defects, and closed timelike curves

The axis sector separates three distinct effects that are often conflated. $Q\to0$2 is the NUT charge and is associated with the Misner string; $Q\to0$3 is the rescaled $Q\to0$4 parameter controlling how the Misner string is distributed between the poles; and $Q\to0$5 is a new conical-defect parameter. In the static limit, $Q\to0$6 gives

$Q\to0$7

with deficit angle

$Q\to0$8

and cosmic string tension

$Q\to0$9

For $\lambda\to0$0 there is a deficit, while $\lambda\to0$1 can produce an excess (Gao, 20 Sep 2025).

In the string-theoretic accelerating Kerr-Taub-NUT solution, conical singularities are controlled by

$\lambda\to0$2

with

$\lambda\to0$3

Regularity can be imposed at one pole only by a rescaling of $\lambda\to0$4, but not both; global removal is impossible without introducing other pathologies (Siahaan, 2024).

Closed timelike curves are a persistent feature of NUT sectors. In the accelerating NUT subclass they occur near the rotating defects associated with the axes, while in the accelerating Kerr-Sen-Taub-NUT spacetime they are present where $\lambda\to0$5. The string analysis further notes that increasing the acceleration parameter $\lambda\to0$6 can eliminate CTCs in certain regions (Podolsky et al., 2020, Siahaan, 2024).

5. Geodesic structure and precessional dynamics

Analyses of the Kerr-Taub-NUT spacetime isolate the dynamical role of the NUT parameter and thereby clarify the local orbital effects expected whenever NUT charge is present. The metric is not invariant under $\lambda\to0$7 when $\lambda\to0$8, so the $\lambda\to0$9 equatorial reflection symmetry is broken. Circular orbits are no longer equatorial, spherical-orbit motion is not reflection-symmetric, and the deviation from the equator increases with $S_i$0. In the Hamiltonian formulation,

$S_i$1

with conserved quantities $S_i$2 and $S_i$3; the ISCO is determined by

$S_i$4

For spherical orbits, the precession angular velocity is

$S_i$5

At $S_i$6, spherical orbits degenerate into tilted circular orbits without precession; for nonzero spin, the precession angular velocity increases with the absolute value of the NUT charge, and jet-precession measurements cannot distinguish the sign of $S_i$7 (Meng et al., 14 Aug 2025).

The exact Lense-Thirring precession in Kerr-Taub-NUT spacetime shows an anomalous strong-field behavior: instead of obeying the inverse-cube law near the horizon, it becomes maximum just near the horizon, falls sharply and becomes zero near the horizon, increases again, and only then settles into the usual $S_i$8 decay. The anomaly is maximum at the polar region and disappears after crossing a critical angle toward the equator. In the special case $S_i$9, the outer horizon is at $$ds^2 = \frac{1}{\Omega^2} \left\{ -\frac{\Sigma - \delta a^2 \sin^2\theta}{\Delta}dt^2 + \frac{\Delta}{\Sigma} dr^2 + \frac{\Delta}{\delta} d\theta^2 - \frac{2a \sin^2\theta [\delta (r^2+a^2) - \Sigma]}{\Delta} dt d\phi + \frac{[\delta (r^2+a^2)^2 - \Sigma a^2 \sin^2\theta]\sin^2\theta}{\Delta} d\phi^2 \right\},$$0 and the inner horizon is at $$ds^2 = \frac{1}{\Omega^2} \left\{ -\frac{\Sigma - \delta a^2 \sin^2\theta}{\Delta}dt^2 + \frac{\Delta}{\Sigma} dr^2 + \frac{\Delta}{\delta} d\theta^2 - \frac{2a \sin^2\theta [\delta (r^2+a^2) - \Sigma]}{\Delta} dt d\phi + \frac{[\delta (r^2+a^2)^2 - \Sigma a^2 \sin^2\theta]\sin^2\theta}{\Delta} d\phi^2 \right\},$$1 (Chakraborty, 2014).

6. High-energy processes and astrophysical constraints

For unbound high-energy particles moving along the rotation axis in Kerr-Taub-NUT spacetime, the repulsive effect of gravity depends on the Carter constant, the position, and the particle velocity. On the axis, only particles with $$ds^2 = \frac{1}{\Omega^2} \left\{ -\frac{\Sigma - \delta a^2 \sin^2\theta}{\Delta}dt^2 + \frac{\Delta}{\Sigma} dr^2 + \frac{\Delta}{\delta} d\theta^2 - \frac{2a \sin^2\theta [\delta (r^2+a^2) - \Sigma]}{\Delta} dt d\phi + \frac{[\delta (r^2+a^2)^2 - \Sigma a^2 \sin^2\theta]\sin^2\theta}{\Delta} d\phi^2 \right\},$$2 can leave the ergosphere along geodesic trajectories. The NUT charge modifies the critical Carter constant and can enlarge the range of $$ds^2 = \frac{1}{\Omega^2} \left\{ -\frac{\Sigma - \delta a^2 \sin^2\theta}{\Delta}dt^2 + \frac{\Delta}{\Sigma} dr^2 + \frac{\Delta}{\delta} d\theta^2 - \frac{2a \sin^2\theta [\delta (r^2+a^2) - \Sigma]}{\Delta} dt d\phi + \frac{[\delta (r^2+a^2)^2 - \Sigma a^2 \sin^2\theta]\sin^2\theta}{\Delta} d\phi^2 \right\},$$3 for which $$ds^2 = \frac{1}{\Omega^2} \left\{ -\frac{\Sigma - \delta a^2 \sin^2\theta}{\Delta}dt^2 + \frac{\Delta}{\Sigma} dr^2 + \frac{\Delta}{\delta} d\theta^2 - \frac{2a \sin^2\theta [\delta (r^2+a^2) - \Sigma]}{\Delta} dt d\phi + \frac{[\delta (r^2+a^2)^2 - \Sigma a^2 \sin^2\theta]\sin^2\theta}{\Delta} d\phi^2 \right\},$$4. Numerically, the observable velocity $$ds^2 = \frac{1}{\Omega^2} \left\{ -\frac{\Sigma - \delta a^2 \sin^2\theta}{\Delta}dt^2 + \frac{\Delta}{\Sigma} dr^2 + \frac{\Delta}{\delta} d\theta^2 - \frac{2a \sin^2\theta [\delta (r^2+a^2) - \Sigma]}{\Delta} dt d\phi + \frac{[\delta (r^2+a^2)^2 - \Sigma a^2 \sin^2\theta]\sin^2\theta}{\Delta} d\phi^2 \right\},$$5 increases with $$ds^2 = \frac{1}{\Omega^2} \left\{ -\frac{\Sigma - \delta a^2 \sin^2\theta}{\Delta}dt^2 + \frac{\Delta}{\Sigma} dr^2 + \frac{\Delta}{\delta} d\theta^2 - \frac{2a \sin^2\theta [\delta (r^2+a^2) - \Sigma]}{\Delta} dt d\phi + \frac{[\delta (r^2+a^2)^2 - \Sigma a^2 \sin^2\theta]\sin^2\theta}{\Delta} d\phi^2 \right\},$$6, while the asymptotic transverse scale $$ds^2 = \frac{1}{\Omega^2} \left\{ -\frac{\Sigma - \delta a^2 \sin^2\theta}{\Delta}dt^2 + \frac{\Delta}{\Sigma} dr^2 + \frac{\Delta}{\delta} d\theta^2 - \frac{2a \sin^2\theta [\delta (r^2+a^2) - \Sigma]}{\Delta} dt d\phi + \frac{[\delta (r^2+a^2)^2 - \Sigma a^2 \sin^2\theta]\sin^2\theta}{\Delta} d\phi^2 \right\},$$7 also increases, indicating less collimated jets for larger NUT charge (Zhang et al., 2017).

Particle collisions in Kerr-Taub-NUT spacetime show a related sensitivity to $$ds^2 = \frac{1}{\Omega^2} \left\{ -\frac{\Sigma - \delta a^2 \sin^2\theta}{\Delta}dt^2 + \frac{\Delta}{\Sigma} dr^2 + \frac{\Delta}{\delta} d\theta^2 - \frac{2a \sin^2\theta [\delta (r^2+a^2) - \Sigma]}{\Delta} dt d\phi + \frac{[\delta (r^2+a^2)^2 - \Sigma a^2 \sin^2\theta]\sin^2\theta}{\Delta} d\phi^2 \right\},$$8 and $$ds^2 = \frac{1}{\Omega^2} \left\{ -\frac{\Sigma - \delta a^2 \sin^2\theta}{\Delta}dt^2 + \frac{\Delta}{\Sigma} dr^2 + \frac{\Delta}{\delta} d\theta^2 - \frac{2a \sin^2\theta [\delta (r^2+a^2) - \Sigma]}{\Delta} dt d\phi + \frac{[\delta (r^2+a^2)^2 - \Sigma a^2 \sin^2\theta]\sin^2\theta}{\Delta} d\phi^2 \right\},$$9. The center-of-mass energy depends on both parameters, and in the extremal case an unlimited center-of-mass energy can be approached if

$$\Omega = 1 - \alpha r \cos\theta,\qquad \Delta = r^2 + a^2 \cos^2\theta,$$0

which differs from Kerr and Kerr-Newman. The critical angular momentum is $$\Omega = 1 - \alpha r \cos\theta,\qquad \Delta = r^2 + a^2 \cos^2\theta,$$1, and the NUT charge enlarges the spin range permitting arbitrarily high center-of-mass energy (Liu et al., 2010).

An observational use of NUT-induced precession has been developed through spherical orbits in Kerr-Taub-NUT spacetime applied to jet precession in M87*. Using the observed rate

$$\Omega = 1 - \alpha r \cos\theta,\qquad \Delta = r^2 + a^2 \cos^2\theta,$$2

and tilt angle $$\Omega = 1 - \alpha r \cos\theta,\qquad \Delta = r^2 + a^2 \cos^2\theta,$$3, the model excludes low-spin and large-$$\Omega = 1 - \alpha r \cos\theta,\qquad \Delta = r^2 + a^2 \cos^2\theta,$$4 regions because they would place the warp radius inside the ISSO/ISCO. The excluded region is larger for retrograde disks than for prograde ones, the sign of the NUT charge cannot be distinguished, and jet precession observations alone do not allow a clear distinction between black holes and naked singularities. The empirical exclusion boundary is

$$\Omega = 1 - \alpha r \cos\theta,\qquad \Delta = r^2 + a^2 \cos^2\theta,$$5

with $$\Omega = 1 - \alpha r \cos\theta,\qquad \Delta = r^2 + a^2 \cos^2\theta,$$6 and $$\Omega = 1 - \alpha r \cos\theta,\qquad \Delta = r^2 + a^2 \cos^2\theta,$$7 in units of $$\Omega = 1 - \alpha r \cos\theta,\qquad \Delta = r^2 + a^2 \cos^2\theta,$$8 (Meng et al., 14 Aug 2025).

7. Thermodynamic quantities and unresolved issues

In the generalized accelerating Kerr-Newman-NUT-de Sitter family, the horizon angular velocity, temperature, entropy, and electric potential at $$\Omega = 1 - \alpha r \cos\theta,\qquad \Delta = r^2 + a^2 \cos^2\theta,$$9 are

$S_3$00

$S_3$01

$S_3$02

However, the asymptotic angular velocity is angle-dependent, and the first law and Smarr formula are obtained only in the specific limit $S_3$03, $S_3$04, $S_3$05 (Gao, 20 Sep 2025).

The heterotic-string solution modifies these expressions through the Hassan-Sen parameter. The horizon area is

$S_3$06

the entropy is $S_3$07, and the Hawking temperature is

$S_3$08

The area-temperature product satisfies

$S_3$09

while the area product $S_3$10 is not independent of the mass (Siahaan, 2024).

Two recurring open issues follow directly from these results. First, observables based on precession do not determine the sign of the NUT charge and do not by themselves separate black holes from naked singularities. Second, once acceleration and NUT charge are both present, the asymptotic rotational structure and axis defects obstruct a straightforward global thermodynamic formulation. These features place the accelerating Kerr-Taub-NUT spacetime at the intersection of exact-solution theory, strong-field orbital dynamics, and the geometry of cosmic strings, Misner strings, and nontrivial asymptotics.