---
title: Taub-NUT Black Hole Overview
url: https://www.emergentmind.com/topics/taub-nut-black-hole
type: topic
---

# Taub-NUT Black Hole Overview

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 [1110.6731].

## 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.

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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 study 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 [1110.6731].

Source: https://www.emergentmind.com/topics/taub-nut-black-hole