Taub-NUT Black Hole Overview

• Taub-NUT black holes are a class of solutions featuring a unique NUT charge that introduces intrinsic gravitomagnetic effects and a nontrivial topological twist.
• They exhibit distinct horizon structures with analytic regularity in D=5 and weak, mildly singular behavior in D>5, ensuring finite tidal forces for free-fall observers.
• Generalizations to de Sitter spaces reveal dual horizons and modified causal structures, offering insights into higher-dimensional gravity and string theory frameworks.

The Taub-NUT black hole is a family of solutions to Einstein's field equations, and their generalizations in higher dimensions and modified gravity, characterized by the presence of a "NUT charge" that endows the spacetime with intrinsic gravitomagnetic properties. Unlike Schwarzschild or Kerr spacetimes, Taub-NUT geometries possess nontrivial topological structure, off-diagonal metric components ("twists" or Misner strings), and, in many instances, non-asymptotically flat behavior. This article details the mathematical formulation, horizon structure, extension and singularity properties, physical interpretation, generalizations, and key aspects of thermodynamics of Taub-NUT black holes, with a focus on recent research constructing extremal charged solutions with twisted extra dimensions (Tatsuoka et al., 2011).

1. Mathematical Formulation and Construction

The construction of extremal charged black holes based on generalized Taub-NUT spaces proceeds from the action for $D$-dimensional Einstein–Maxwell gravity: $$S = \frac{1}{16\pi} \int d^D x\, \sqrt{-g}\, [R - F^2]$$ where $F = dA$ is the Maxwell field strength. The ansatz utilizes a $(D-1)$-dimensional Ricci-flat base, taken as a higher-dimensional generalization of the (Euclidean) Taub–NUT metric: \begin{align*} h_{ij} dx^i dx^j ={}& r(r + 2L) d\chi^2 + F(r) (dx + \omega_n)^2 + \text{[other angular terms]} \ F(r) ={}& \frac{r}{(n+1)L} \prod_{k=1}^n \frac{1}{(n+k)(n+k-1)} \left[\text{poly}(r,L)\right] \end{align*} This base is regular and Ricci-flat for appropriate $F(r)$. The full metric in $D=2(n+1)+1$ dimensions (odd $D$) becomes: $$ds^2 = -H(r)^{-2} dt^2 + H(r)^{1/n} \left( \frac{L F(r)}{r(r+2L)} dr^2 + r(r+2L) d\Sigma^2_{2n} + L^2 F(r) (dx + \omega_n)^2 \right)$$ with $A = \pm \sqrt{(2n+1)/(4n)}\, H(r)^{-1} dt$, and $H(r)$ a harmonic function on the Taub-NUT base,

$\Delta_{h_{ij}} H(r) = 0$

A localized charge at $r=0$ yields

$$H(r) = 1 + \frac{p}{2^{2n-1}L} \frac{F(r)}{[r(r+2L)]^n}$$

where $p>0$ is arbitrary.

2. Horizon Structure and Null Hypersurfaces

A distinctive property is the appearance of a null hypersurface at $r=0$, where the expansion $\Theta_+$ of outgoing null geodesic congruence vanishes: $$\Theta_+ = h^{\alpha\beta} \nabla_\alpha k_\beta = 0 \quad \text{at}~r=0$$ With outgoing null vector $k^\mu$ constructed via orthonormal frames, this surface satisfies $\Theta_+ > 0$ for $r>0$ but $\Theta_+ = 0$ at $r=0$, signifying an apparent or trapping horizon. As in extremal black holes, this surface demarcates the boundary of causal influence for external observers.

3. Curvature Behavior and Mild Singularities

The spacetime's regularity at the horizon depends sharply on the dimensionality. For $D=5$ (i.e., $n=2$), the Riemann tensor components in a parallelly transported frame remain finite at $r=0$. However, for $D>5$, certain components (e.g., $R^{(0)}{}_{(2)(0)(2)}$) diverge as $r\to0$, scaling as a fractional power of $r$. Despite these divergences:

• The Kretschmann scalar $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ remains finite at $r=0$.
• A free-fall observer crossing $r=0$ experiences only finite tidal forces. These are thus "mild" singularities: the metric is only $C^0$-extendable across the horizon for $D>5$, but is analytic for $D=5$. The non-divergent invariants sharply distinguish these from curvature singularities (e.g., $r=0$ in Schwarzschild), which cannot be crossed by free-fall geodesics.

4. Extension Properties and Coordinate Transformations

In $D=5$, coordinate charts (e.g., using an advanced null coordinate $v = t + \ldots dr$) can be employed to show the analyticity of the metric across the horizon at $r=0$. For $D > 5$, even after chart extension, metric components involve fractional powers such as $p^{1/n}$, precluding a $C^1$ extension. Nevertheless, a $C^0$-extension—the metric and gauge fields remain continuous though not all derivatives do—can always be constructed.

This horizon is thus regarded as a "weakly singular" surface: the inability to extend analytically results from dimensionality-dependent power-law behavior in $r \to 0$ expansions.

5. Generalization to de Sitter and Cosmological Horizons

With a positive cosmological constant $\Lambda > 0$, the construction generalizes. The metric ansatz becomes: $$ds^2 = -H(t,r)^{-2}dt^2 + H(t,r)^{1/n}\left( e^{-2At/n} \left[ \frac{L F(r)}{r(r+2L)} dr^2 + r(r+2L) d\Sigma_{2n}^2 \right] + L^2 F(r) (dx + \omega_n)^2 \right)$$ where

$$H(t,r) = 1 + e^{-At} \frac{p}{2^{2n}L}\frac{F(r)}{[r(r+2L)]^n}$$

The presence of de Sitter expansion modifies the causal structure: the expansion analysis shows two roots for $\Theta_+=0$—a smaller $r$ root (black hole horizon) and a larger $r$ root (cosmological horizon). As $t \to -\infty$, the geometry locally approaches that of an extremal Reissner–Nordström–de Sitter black hole. These solutions retain regularity (for sufficiently early times) in the vicinity of both horizons.

6. Physical Interpretation and Implications

These extremal charged black holes with twisted extra dimensions manifest distinct features determined by the higher-dimensional topology:

• Their base is a generalization of Euclidean Taub-NUT space, incorporating a nontrivial $S^1$ fibration (the "twisted" extra dimension).
• The horizon at $r=0$ is not an isolated singularity but a smooth (for $D=5$) or mildly singular ($D > 5$) null surface separating causally disconnected regions.
• The presence or absence of analytic extension, and the mildness of singularities, is intimately tied to the dimension of the theory.
• De Sitter generalization results in two apparent horizons, modifying the global causal and thermodynamic properties.

These properties illuminate the possible range of black hole structures in higher-dimensional and string-inspired theories where nontrivial base geometries and extra dimensions are generic. The existence of black holes with only $C^0$ horizons but finite tidal forces underscores the variety of possible horizon regularities beyond four-dimensional general relativity.

7. Table: Dimensional Dependence of Horizon Regularity

Number of Dimensions	Extension Across r=0	Riemann Components	Physical Effect for Free-fall Observer
$D=5$	Analytic ($C^\infty$)	Finite	Horizon regular
$D > 5$	$C^0$ only	Diverge as $r \to 0$ (fractional powers)	Finite tidal force, horizon is weak singularity

The physical tidal experience of free fall across $r=0$ does not render the black hole horizon impassable, even when the metric is only $C^0$, further clarifying the difference between geometric and physical singularities in this class.

---

In summary, these extremal charged black holes constructed with a generalized Taub–NUT base highlight the impact of base geometry, dimensionality, and cosmological constant on horizon structure, regularity, and physical traversability. Their paper provides a framework for evaluating the global and local properties of black hole solutions with topological or "twisted" features, with relevance for higher-dimensional gravity, string theory compactifications, and analytic studies of horizon formation and singularity strength (Tatsuoka et al., 2011).