Self-Dual Taub-NUT Black Hole

• The self-dual Taub-NUT black hole is a four-dimensional Ricci-flat gravitational instanton defined by an (anti-)self-dual curvature, with its mass and NUT charge intertwined by strong algebraic constraints.
• This solution exhibits smooth, ALF geometry with a globally complete, non-singular manifold, where the NUT parameter governs both topological charge and physical mass, ensuring regularity at the bolt.
• It serves as a fundamental building block in quantum gravity and integrable field theories, underpinning constructions in twistor theory, celestial holography, and the formation of more complex rotating black holes.

The self-dual Taub-NUT black hole refers to a four-dimensional Riemannian (Euclidean-signature) solution to Einstein’s equations characterized by a metric that is Ricci-flat and has (anti-)self-dual Riemann curvature. This geometry, historically termed a gravitational instanton, has become central in mathematical physics and quantum gravity, including integrable models, supersymmetric gauge theory, twistor constructions, and celestial holography. The “self-dual” condition imposes strong algebraic and geometric constraints that relate the black hole’s mass, NUT (“Newman–Unti–Tamburino”) charge, and—when included—a cosmological constant, leading to distinctive global and physical properties. The self-dual Taub-NUT black hole arises as a limit of the broader family of Taub-NUT–AdS spacetimes, admits a complete regular instanton structure, and participates in both classical and quantum aspects of gauge and gravity theories.

1. Geometric Structure and Self-Duality

The self-dual Taub-NUT metric is expressed, in “Gibbons–Hawking” form, as

$$ds^2 = V(r)^{-1}(d\tau + 2n\cos\theta\,d\varphi)^2 + V(r)\,(dr^2 + r^2\,d\Omega_2^2)$$

with harmonic function $V(r) = 1 + \frac{2n}{r}$, Euclidean coordinate $\tau$ periodic with period $8\pi n$, and NUT parameter $n$ serving both as mass and “magnetic” charge. The manifold is smooth for $r \geq n$, with a bolt (totally geodesic 2-sphere) of radius $2n$ at $r = n$ and no curvature singularity. The regularity at the bolt requires the periodic identification of $\tau$ to avoid Misner string singularities.

The Riemann curvature two-forms in an orthonormal frame $e^a$ ($a=0,1,2,3$) satisfy the (anti-)self-duality condition: $$R^{ab} = \pm \frac{1}{2} \varepsilon^{ab}{}_{cd} R^{cd}$$ ensuring the Weyl tensor is algebraically special of Petrov type D, with all self-dual (or anti-self-dual) curvature components vanishing.

2. Origin from the Pedersen Metric and Self-Duality Constraints

The self-dual Taub-NUT arises as a particular limit of the two-parameter Pedersen metric on the 4-ball: $$ds^2 = f^2(r)\Bigl[ h_r(r)\,dr^2 + h_{12}(r)(\sigma_1^2+\sigma_2^2) + h_3(r)\sigma_3^2 \Bigr]$$ with parameters related to cosmological constant $\Lambda=-3/l^2$ and squashing parameter $\nu^2$. The self-duality condition on the Weyl tensor sets the relation: $$n = \pm iM,\qquad m = M\bigl(1 - \frac{4M^2}{l^2}\bigr)$$ and, in the $\Lambda\to 0$ (“asymptotically flat”) limit, yields the classic self-dual Taub-NUT: $$ds^2 = \left(1+\frac{2M}{r}\right)^{-1}(dt+2M\cos\theta\,d\phi)^2 + \left(1+\frac{2M}{r}\right)(dr^2 + r^2\,d\Omega_{2}^2)$$ with the $S^1$ fiber degenerating smoothly as $r\to0$ (the “nut”) and no conical singularity, ensured by the appropriate period of $t$.

Other limits of the Pedersen family produce related gravitational instantons: the Eguchi–Hanson double cover, Euclidean $\mathrm{AdS}_4$, and noncompact $\widetilde{\mathbb{CP}}^2$ conformal to Burns space (Bogna et al., 26 Aug 2024).

3. Physical and Topological Features

The self-dual Taub-NUT solution is globally complete, non-singular, and ALF (“asymptotically locally flat”), with spatial infinity exhibiting an $S^1$ fibration over $S^2$ and nontrivial first Chern class $2n$: $$ds^2 \sim (d\tau + 2n\cos\theta\,d\varphi)^2 + dr^2 + r^2\,d\Omega_2^2$$ as $r\to\infty$. The metric topology is $\mathbb{R}^4$. The NUT parameter enters as both Komar mass and a topological (magnetic) charge of the $S^1$ fibration.

Thermodynamic quantities in the self-dual case satisfy: $$I_E = \beta m - S = 4\pi n^2, \quad S = 4\pi n^2, \quad \beta = 8\pi n, \quad T = 1/(8\pi n), \quad F = n/2$$ with the Euclidean action and entropy coinciding, and the first law $F = E - TS$, $E = m = n$ (Skvortsov et al., 26 Aug 2025). This underlines the topological and modular properties of the instanton.

4. Twistor Theory, Integrability, and Celestial Algebras

The self-dual Taub-NUT geometry admits a manifestly integrable structure in twistor theory. The twistor space is constructed as a deformation of flat twistor space via a line defect operator wrapping a “twistor line” at infinity, yielding a curved twistor space. Holomorphic coordinates and patching yield a quadric relation

$$Y^{\alpha\beta}Y_{\alpha\beta} = -\Lambda\nu^2\,\eta^2$$

with deformation parameters $(\Lambda,\nu^2)$, and the twistor space is a double cover bundle over $\mathcal{O}(2)$ (Bogna et al., 26 Aug 2024).

In the context of celestial holography, this twistor construction provides a two-parameter deformation of the celestial chiral algebra $Lw_\wedge$, governed by generators with quadratic relations and a deformed Poisson bracket. Notably, at the self-dual Taub-NUT point ($\Lambda = 0,\, \nu^2 = 1/(4M^2)$) the algebra is undeformed, whereas nonzero cosmological constant or squashing parameter interpolates to AdS$_4$ or Eguchi–Hanson algebras.

The self-dual Taub-NUT is also hyperkähler, with SU(2) holonomy and covariantly constant anti-self-dual two-forms. Its twistor data underlies the non-linear graviton construction and explicitly realizes integrable linear and non-linear field equations on the background (Adamo et al., 24 Jul 2025).

5. Horizon Structure, Signature Continuations, and Symmetries

In Euclidean signature, the “nut” is a zero-size fixed point; there is no horizon. In Kleinian (2,2) signature, the analytic continuation leads to metrics

$$ds^2 = \frac{r - M}{r + M}(dt - 2M\,\cosh\theta\,d\phi)^2 + \frac{r + M}{r - M}dr^2 - (r^2 - M^2)(d\theta^2 + \sinh^2\theta\,d\phi^2)$$

with a true null horizon at $r = M$, and allows for exploration of near-horizon symmetry algebras.

Imposing generalized near-horizon boundary conditions, the asymptotic symmetry algebra of the self-dual Taub-NUT “Kleinian horizon” is the semi-direct sum of supertranslations (arbitrary functions of the advanced time $v$ and horizon coordinates) with the Witt algebra (superrotations) acting on the transverse torus. The corresponding Noether charges are integrable and depend only on data at the horizon. Importantly, in Kleinian signature, the addition of an angular momentum parameter $a$ is pure gauge in the self-dual sector—it can be undone by a large diffeomorphism, and does not break the near-horizon symmetry (Giribet et al., 16 May 2025).

6. Self-Duality as the Building Block of Four-Dimensional Black Holes

The self-dual Taub-NUT instanton functions as a fundamental “building block” in the construction of rotating and charged four-dimensional black holes. The Kerr and Kerr–Taub–NUT metrics can be realized as nonlinear superpositions of a self-dual and an anti-self-dual Taub-NUT instanton, each localized at complex-conjugate points in a complexified coordinate space. The Newman–Janis algorithm is interpreted as the holomorphic displacement of these chiral centers, giving rise to the physical angular momentum and NUT charge parameters in real Lorentzian spacetime (Kim, 27 Dec 2024).

7. Generalizations, Scattering, and Physical Implications

The self-dual Taub-NUT metric admits higher-spin generalizations, in which higher-spin fields solve self-dual Yang-Mills or gravity equations on the instanton background, leading to a tractable perturbation theory with closed, non-associative algebras governing interactions (Skvortsov et al., 26 Aug 2025). In non-linear electrodynamics (ModMax theory), the self-duality condition yields vanishing stress tensor and a “test” gauge field decoupled from the geometry—preserving the self-dual Taub-NUT solution for arbitrary nonlinearity parameter, with charges fixed by $SO(2)$ duality (Bordo et al., 2020).

Gravitational wave (graviton) scattering on the self-dual Taub-NUT background is completely integrable: linearized field equations admit exact quasi-momentum eigenstates labeled by both null 4-momentum and an integer topological charge. The maximal helicity-violating (MHV) graviton scattering amplitudes can be computed exactly to all orders in the background using twistor sigma models; these amplitudes retain the flat-space holomorphic splitting functions and celestial $w_{1+\infty}$ symmetry algebra, i.e., the celestial algebra is undeformed by the nontrivial geometry (Adamo et al., 24 Jul 2025).

---

In summary, the self-dual Taub-NUT black hole encompasses a uniquely regular, integrable, and algebraically constrained gravitational instanton whose geometric, topological, and symmetry properties provide a foundational framework for four-dimensional quantum gravity, celestial holography, and integrable field theories. It forms the archetypal member of a family of self-dual metrics interpolating between flat space, AdS$_4$, and Eguchi–Hanson-type instantons, and continues to be vital in both mathematical and physical investigations of gauge/gravity correspondence, higher-spin theory, and exact gravitational amplitudes.