Self-Dual Taub-NUT Black Holes

• Self-dual Taub-NUT black holes are self-dual gravitational instantons exhibiting Ricci-flat metrics, characterized by distinctive self-duality conditions across various signatures.
• They are formulated via conditions like m=i·n, M=N, or N=-iM, thereby providing exact backgrounds for exploring scattering processes, helicity selection rules, and gauge equivalences.
• Twistor constructions and Kerr-Schild approaches reveal their equivalence to self-dual Kerr-Taub-NUT configurations, linking them to celestial holography and higher-spin gravity extensions.

Searching arXiv for recent and foundational papers on self-dual Taub-NUT black holes. Search query: self-dual Taub-NUT black holes Taub-NUT instanton Kleinian black holes scattering celestial holography Searching arXiv now. Self-dual Taub-NUT black holes are self-dual members of the Taub-NUT and, more broadly, Plebański-Demiański families, studied in several closely related but not identical senses: as complex Lorentzian self-dual limits of NUT-charged solutions, as real black-hole geometries in Kleinian signature $(2,2)$, and as Euclidean Ricci-flat self-dual metrics, i.e. hyperkähler gravitational instantons. Recent work treats them as the simplest “self-dual black holes,” as exact backgrounds for scattering and twistor constructions, and as a formal endpoint of NUT-charged black-hole dynamics in which helicity selectivity becomes exact (Adamo et al., 8 Jan 2026, Crawley et al., 2021).

1. Definitions and parameter regimes

A standard parametrization of a NUT-charged gravitational dyon uses

$$p_i^\mu = \mathcal{M}_i u_i^\mu,\qquad \mathcal{M}_i=\sqrt{m_i^2+n_i^2},\qquad e^{i\theta_i}=\frac{m_i+i n_i}{\mathcal{M}_i},$$

with $m_i$ the mass and $n_i$ the NUT charge. In the scattering literature, the self-dual limit is defined by

$m_i = i n_i,$

the gravitational analogue of the electromagnetic condition $e_i=i g_i$. Because of the explicit factors of $i$, true self-dual fields are naturally real in Euclidean or split/Kleinian signature, whereas in Lorentzian signature they are necessarily complex, so the discussion is naturally phrased in a complexified setting (Doran et al., 25 Mar 2026).

A complementary, real notion of self-duality appears in Klein space. For the analytically continued Taub-NUT family, the exact curvature condition is

$$R_{\mu\nu\rho\sigma} = \frac12\,\varepsilon_{\mu\nu\alpha\beta} R^{\alpha\beta}{}_{\rho\sigma} \qquad\text{for } M=N,$$

while $M=-N$ gives anti-self-duality. In this formulation the self-duality condition is simply

$\boxed{M=N,}$

and it holds independently of the Kerr parameter $$p_i^\mu = \mathcal{M}_i u_i^\mu,\qquad \mathcal{M}_i=\sqrt{m_i^2+n_i^2},\qquad e^{i\theta_i}=\frac{m_i+i n_i}{\mathcal{M}_i},$$0 (Crawley et al., 2021).

A third standard parametrization arises from the complexified Euclidean Taub-NUT/Plebański-Demiański family, where the self-dual limit is stated as

$$p_i^\mu = \mathcal{M}_i u_i^\mu,\qquad \mathcal{M}_i=\sqrt{m_i^2+n_i^2},\qquad e^{i\theta_i}=\frac{m_i+i n_i}{\mathcal{M}_i},$$1

After taking the Euclidean real slice, this yields the standard self-dual Taub-NUT instanton. The phrase “self-dual black hole” is used in recent work for such Euclidean-signature, Ricci-flat, self-dual metrics obtained as real slices of the complexified black-hole family (Adamo et al., 8 Jan 2026).

Setting	Self-duality condition	Immediate consequence
Complexified scattering problem	$$p_i^\mu = \mathcal{M}_i u_i^\mu,\qquad \mathcal{M}_i=\sqrt{m_i^2+n_i^2},\qquad e^{i\theta_i}=\frac{m_i+i n_i}{\mathcal{M}_i},$$2	$$p_i^\mu = \mathcal{M}_i u_i^\mu,\qquad \mathcal{M}_i=\sqrt{m_i^2+n_i^2},\qquad e^{i\theta_i}=\frac{m_i+i n_i}{\mathcal{M}_i},$$3
Kleinian $$p_i^\mu = \mathcal{M}_i u_i^\mu,\qquad \mathcal{M}_i=\sqrt{m_i^2+n_i^2},\qquad e^{i\theta_i}=\frac{m_i+i n_i}{\mathcal{M}_i},$$4 black hole	$$p_i^\mu = \mathcal{M}_i u_i^\mu,\qquad \mathcal{M}_i=\sqrt{m_i^2+n_i^2},\qquad e^{i\theta_i}=\frac{m_i+i n_i}{\mathcal{M}_i},$$5	Real self-dual curvature
Euclidean self-dual instanton	$$p_i^\mu = \mathcal{M}_i u_i^\mu,\qquad \mathcal{M}_i=\sqrt{m_i^2+n_i^2},\qquad e^{i\theta_i}=\frac{m_i+i n_i}{\mathcal{M}_i},$$6 before taking the Euclidean slice	Standard self-dual Taub-NUT metric

2. Metric realizations and global structure

The Kleinian Taub-NUT metric obtained by analytic continuation is

$$p_i^\mu = \mathcal{M}_i u_i^\mu,\qquad \mathcal{M}_i=\sqrt{m_i^2+n_i^2},\qquad e^{i\theta_i}=\frac{m_i+i n_i}{\mathcal{M}_i},$$7

with

$$p_i^\mu = \mathcal{M}_i u_i^\mu,\qquad \mathcal{M}_i=\sqrt{m_i^2+n_i^2},\qquad e^{i\theta_i}=\frac{m_i+i n_i}{\mathcal{M}_i},$$8

In the self-dual case $$p_i^\mu = \mathcal{M}_i u_i^\mu,\qquad \mathcal{M}_i=\sqrt{m_i^2+n_i^2},\qquad e^{i\theta_i}=\frac{m_i+i n_i}{\mathcal{M}_i},$$9, this reduces to

$m_i$0

Its Kretschmann scalar is

$m_i$1

so only the singularity at $m_i$2 remains, while $m_i$3 is a coordinate horizon. In the toric Penrose-diagram description, the horizon is at $m_i$4, the singularity is at $m_i$5, and the spacetime is geodesically complete at null infinity $m_i$6 (Crawley et al., 2021).

The Euclidean self-dual Taub-NUT background is commonly written in Gibbons-Hawking form as

$m_i$7

with periodic Euclidean time

$m_i$8

Equivalent forms appearing in the recent literature include

$m_i$9

In split signature this same geometry is described as a genuine black-hole-like background with a horizon at $n_i$0, with the angular 2-spheres replaced by hyperbolic discs $n_i$1 (Adamo et al., 2023).

For self-dual Kerr-Taub-NUT in $n_i$2 signature, the striking simplification is that the rotation parameter $n_i$3 is pure gauge in a global sense. The explicit large diffeomorphism

$n_i$4

maps the self-dual Kerr-Taub-NUT metric to self-dual Taub-NUT. In rectangular coordinates adapted to the Rindler wedge this becomes

$n_i$5

so the Kerr parameter is literally a translation in the $n_i$6-direction (Crawley et al., 2021).

3. Kerr-Schild, invariant classification, and twistor constructions

A central structural development is the recognition that self-dual Taub-NUT, although long known in double Kerr-Schild and Gibbons-Hawking forms, also admits a single Kerr-Schild form. One explicit expression is

$n_i$7

with null one-form

$n_i$8

The line defect of $n_i$9 along the negative $m_i = i n_i,$0-axis is identified with the Misner string, the gravitational analogue of the Dirac string (Kim, 2024).

The equivalence between the “self-dual analog of Kerr” and the self-dual Taub-NUT instanton has also been established invariantly. In the Kerr-Schild/Newman-Penrose construction, the self-dual truncation is obtained by setting

$m_i = i n_i,$1

Applying the Cartan-Karlhede algorithm to this self-dual Kerr-Schild metric and to self-dual Taub-NUT gives the same invariant data,

$m_i = i n_i,$2

so the two geometries are locally equivalent. In this sense the self-dual analog of Kerr is precisely self-dual Taub-NUT written in a different guise, and the Kerr-like parameter becomes gauge (Desai et al., 2024).

A more recent twistor reformulation starts from flat dual twistor space $m_i = i n_i,$3 with homogeneous coordinates

$m_i = i n_i,$4

and asserts a correspondence between generic Euclidean-reality-preserving dual twistor quadrics and self-dual black holes. In the classification of quadrics, self-dual Taub-NUT is “Case A.” The construction yields directly both a self-dual Kerr-Schild perturbation and a Gibbons-Hawking form with

$m_i = i n_i,$5

The same quadric therefore encodes the hyperkähler structure, the tri-holomorphic Killing vector, the Kerr-Schild description, and the single-centre ALF Gibbons-Hawking potential of self-dual Taub-NUT (Adamo et al., 8 Jan 2026).

4. Scattering, helicity selection, and the vanishing of self-dual black-hole scattering

Self-dual Taub-NUT provides an exactly solvable background for massless-field scattering. On the self-dual dyon, charged Killing spinors generate quasi-momentum eigenstates, and these lift directly to self-dual Taub-NUT by the replacement

$m_i = i n_i,$6

For the scalar sector one obtains

$m_i = i n_i,$7

while negative-helicity gravitons are represented by

$m_i = i n_i,$8

Because of the non-trivial topology, Euclidean time periodicity implies

$m_i = i n_i,$9

and the states grow faster at infinity than flat-space plane waves (Adamo et al., 2023).

At the level of exact tree-level amplitudes on the fixed self-dual background, the integrability of the self-dual sector imposes strong helicity selection rules. The $e_i=i g_i$0 and $e_i=i g_i$1 amplitudes vanish generically, the scalar two-point amplitude vanishes, and the only non-vanishing two-point amplitudes are $e_i=i g_i$2 amplitudes for spin 1 and spin 2 (Adamo et al., 2023). This fixed-background program has been extended to arbitrary multiplicity in the MHV sector: linearised Einstein equations on self-dual Taub-NUT can be solved exactly, a twistor sigma-model description gives an explicit formula exact in the background for the tree-level MHV graviton amplitude, and the holomorphic collinear splitting function is the same as in flat space, so the celestial symmetry algebra is undeformed (Adamo et al., 24 Jul 2025).

A different question is the scattering of self-dual Taub-NUT black holes with each other. In the KMOC-based analysis of Kerr-Taub-NUT scattering, the self-dual limit is a highly constrained limit of the dyonic problem. Since

$e_i=i g_i$3

in the self-dual limit, the usual momentum $e_i=i g_i$4 degenerates, and one should instead think in terms of the velocity kick $e_i=i g_i$5. More importantly, self-duality forces the relevant amplitudes to vanish: the particle couples only to one helicity sector, so the exchanged graviton cannot mediate a nontrivial interaction. In the language of observables, the impulse vanishes, the waveform vanishes, and therefore there is no radiative memory effect. The self-dual black-hole scattering problem is accordingly described as “purely academic,” useful as a limiting consistency check in complexified gravity rather than as a realistic astrophysical process (Doran et al., 25 Mar 2026).

5. Celestial holography, Kleinian horizons, and AdS limits

Self-dual Taub-NUT has acquired a distinct role in celestial and twisted holography. One line of work places it inside the Taub-NUT-AdS$e_i=i g_i$6 family by starting from the Pedersen metric and imposing the self-duality relation

$e_i=i g_i$7

or, with $e_i=i g_i$8,

$e_i=i g_i$9

The Pedersen parameter is then

$i$0

In the limit $i$1, the metric becomes

$i$2

which is stated to be precisely self-dual Taub-NUT after the shift $i$3. In the associated twistor construction, self-dual Taub-NUT appears as the $i$4 limit of a two-parameter deformation of the celestial symmetry algebra, reducing to undeformed $i$5 (Bogna et al., 2024).

A complementary development concerns null-surface symmetry in Klein space. For the self-dual Schwarzschild-Taub-NUT solution with $i$6, the horizon at $i$7 is a Kleinian horizon. Near this horizon, after adopting boundary conditions more general than the standard horizon gauge,

$i$8

$i$9

the asymptotic symmetry algebra is generated by supertranslations and superrotations. The resulting Noether charges are integrable but generally $$R_{\mu\nu\rho\sigma} = \frac12\,\varepsilon_{\mu\nu\alpha\beta} R^{\alpha\beta}{}_{\rho\sigma} \qquad\text{for } M=N,$$0-dependent because the induced horizon metric depends explicitly on the advanced time $$R_{\mu\nu\rho\sigma} = \frac12\,\varepsilon_{\mu\nu\alpha\beta} R^{\alpha\beta}{}_{\rho\sigma} \qquad\text{for } M=N,$$1. The same work stresses that the global diffeomorphism relating static and stationary self-dual solutions is not among the regular near-horizon asymptotic symmetries, because its radial component is singular at the horizon (Giribet et al., 16 May 2025).

6. Relation to Kerr, chiral factorization, and extensions

Self-dual Taub-NUT has also been recast as a basic chiral constituent of rotating black holes. In a holomorphic complexification of gravity, the extremal self-dual and anti-self-dual Taub-NUT instantons satisfy

• SD: mass $$R_{\mu\nu\rho\sigma} = \frac12\,\varepsilon_{\mu\nu\alpha\beta} R^{\alpha\beta}{}_{\rho\sigma} \qquad\text{for } M=N,$$2 times the NUT charge,
• ASD: mass $$R_{\mu\nu\rho\sigma} = \frac12\,\varepsilon_{\mu\nu\alpha\beta} R^{\alpha\beta}{}_{\rho\sigma} \qquad\text{for } M=N,$$3 times the NUT charge.

Within this framework, Kerr is interpreted as a nonlinear superposition of one SD and one ASD Taub-NUT instanton. After Wick rotation, the Kerr ring singularity splits into two point singularities,

$$R_{\mu\nu\rho\sigma} = \frac12\,\varepsilon_{\mu\nu\alpha\beta} R^{\alpha\beta}{}_{\rho\sigma} \qquad\text{for } M=N,$$4

connected by a finite Misner string, and the Newman-Janis algorithm is interpreted as the operation of separating the SD and ASD constituents in complexified space. The same framework extends to Kerr-Taub-NUT, with the SD limit reducing to an SD Taub-NUT instanton (Kim, 2024).

Self-duality also controls more recent generalizations. In higher-spin extensions of self-dual Yang-Mills and self-dual gravity, Taub-NUT is used as the canonical self-dual gravitational instanton background. The Gibbons-Hawking potential is

$$R_{\mu\nu\rho\sigma} = \frac12\,\varepsilon_{\mu\nu\alpha\beta} R^{\alpha\beta}{}_{\rho\sigma} \qquad\text{for } M=N,$$5

or more generally

$$R_{\mu\nu\rho\sigma} = \frac12\,\varepsilon_{\mu\nu\alpha\beta} R^{\alpha\beta}{}_{\rho\sigma} \qquad\text{for } M=N,$$6

with self-duality encoded by

$$R_{\mu\nu\rho\sigma} = \frac12\,\varepsilon_{\mu\nu\alpha\beta} R^{\alpha\beta}{}_{\rho\sigma} \qquad\text{for } M=N,$$7

On such backgrounds, massless higher-spin fields propagate consistently; in HS-SDYM the positive-helicity solutions remain exact, while in HS-SDGR one obtains a genuine higher-spin Taub-NUT through a perturbation theory that converges for any fixed spin and is governed by a commutative non-associative algebra (Skvortsov et al., 26 Aug 2025).

Taken together, these developments give self-dual Taub-NUT black holes a sharply defined but specialized status. They are exact self-dual geometries that can be real in Euclidean or Kleinian signature, degenerate in the complexified Lorentzian scattering limit, diffeomorphic to their self-dual Kerr-Taub-NUT counterparts, and exceptionally tractable in twistor and amplitude formalisms. Their direct phenomenological role is correspondingly narrow: for realistic Kerr-Taub-NUT scattering, NUT-dependent soft effects and memory are nontrivial, whereas the self-dual sector collapses to a chiral limit with no impulse, no waveform, and no memory (Doran et al., 25 Mar 2026).