arXiv subjects

Bernardo Araneda

Publications and source records attributed to Bernardo Araneda.

At least 19 recordsLinked to original sources

arXiv2026

Schwarzschild black holes from twistor space

Twistor theory forms the basis for many surprising advances in areas ranging from dynamical systems to quantum field theory. Yet for almost fifty years, one of the main drawbacks of twistor theory has been its inability to give non-perturbative descriptions of non-chiral (or non-self-dual) field configurations. This difficulty is known as 'the googly problem.' In this paper, we provide a resolution of the googly problem for a particular solution of the vacuum Einstein equations: the Schwarzschild metric. We start with the twistor space of the self-dual Taub-NUT Euclidean gravitational instanton, expressed in Kerr-Schild form. Within this twistor space, we then consider a quadric which corresponds to the anti-self-dual Taub-NUT metric. While the full quadric is not holomorphic with respect to the complex structure of the self-dual Taub-NUT twistor space, its holomorphic locus still has complex dimension two. This 'coincidence locus' -- points in twistor space on the holomorphic portion of the quadric -- inherits a complex structure from the twistor space and a K\"ahler form from the quadric itself. Remarkably, these structures are compatible, giving rise to a non-self-dual, four-dimensional K\"ahler metric which is conformal to Schwarzschild (in Lorentzian or Euclidean signature). This is the first instance of a non-self-dual Einstein metric constructed entirely from holomorphic data in a twistor space.

hep-th↗

arXiv2026

On Chen-Teo geometries with cosmological constant

The Chen-Teo geometry is a Riemannian, Ricci-flat ALF 4-manifold, containing an AF gravitational instanton that gives the first counterexample to the Euclidean black hole uniqueness conjecture. We investigate the problem of constructing an Einstein analogue with a non-zero cosmological constant $\lambda$. We show that the solution is either the Pleba\'nski-Demia\'nski metric with $\lambda$, or it has an anti-self-dual Weyl tensor. We study the latter case in detail: we prove that for $\lambda<0$, there is a conformal infinity separating two asymptotically hyperbolic metrics; we show that one of them is globally conformal to an ALE scalar-flat K\"ahler metric; we construct gravitational instantons with different topologies; and we show that the geometry is a 4-pole solution in the Calderbank-Pedersen classification.

math.DG↗

arXiv2026

On toric self-dual Einstein gravitational instantons

We consider the classification of toric self-dual Einstein gravitational instantons with negative cosmological constant. As is well known, any Killing vector field on a self-dual Einstein manifold defines a local conformal K\"ahler structure. We prove that if the conformal K\"ahler structure associated to one of the torus Killing fields is global and extends to an ALE manifold with no additional fixed points, then the corresponding self-dual Einstein instanton is precisely given by the infinite class of multipole solutions constructed by Calderbank, Pedersen and Singer.

math.DG↗

arXiv2026

Infinitesimal rigidity of Hermitian gravitational instantons

We prove infinitesimal rigidity and integrability of the moduli space for Hermitian gravitational instantons. Together with the recent proof by Biquard, Gauduchon, and LeBrun of local rigidity for Hermitian instantons, this completes the picture of the moduli space of Hermitian gravitational instantons, both for the compact and non-compact cases. An important step in the proof is to show that provided certain boundary conditions hold, a curve of Riemannian metrics passing through a Hermitian non-K\"ahler Einstein metric is conformally K\"ahler to second perturbative order. This uses ideas of Wu and LeBrun.

math.DG↗

arXiv2026

The dual twistor theory of self-dual black holes

The Taub-NUT and Eguchi-Hanson gravitational instantons, along with the self-dual Plebanski-Demianski metric, form a set of Euclidean metrics which can naturally be called `self-dual black holes', as they arise from self-dual slices of the most general vacuum, asymptotically flat black hole metric. These self-dual black holes are of interest for many reasons, and can famously be described through the non-linear graviton construction of twistor theory. However, the implicit nature of this twistor description obscures some features of the underlying geometry, particularly for the most general self-dual black holes. In this paper, we give a new construction of all asymptotically flat self-dual black holes based on holomorphic quadrics in flat dual twistor space, rather than the usual twistor space associated with self-duality. Remarkably, the geometry of the self-dual black holes -- including their hyperkahler structure, as well as Kerr-Schild and Gibbons-Hawking forms -- is directly encoded in the corresponding quadric. As a consequence, we obtain a previously unknown single Kerr-Schild form of the self-dual Plebanski-Demianski metric.

hep-th↗

arXiv2025

All toric Hermitian ALE gravitational instantons

We prove that the only smooth, Ricci flat, ALE instanton with a toric Hermitian non-K\"ahler structure is the Eguchi-Hanson instanton. The proof is analogous to the classification of toric Hermitian ALF instantons by Biquard and Gauduchon, although we avoid the use of toric K\"ahler geometry and instead perform a direct global analysis of the Tod form of the metric in Weyl-Papapetrou coordinates. This supports a conjecture by Gibbons and Bando-Kasue-Nakajima which states that any Ricci flat ALE instanton is self-dual.

math.DG↗

arXiv2025

New asymptotically flat Einstein--Maxwell instantons

We disprove the Euclidean Einstein--Maxwell Black Hole Uniqueness Conjecture, and thus demonstrate that the semi-classical properties of coupled gravitational and electromagnetic fields are more subtle than expected from Lorentzian general relativity, where the Kerr-Newman family of metrics yields the most general stationary and asymptotically flat black holes with a single event horizon. This is achieved by an explicit construction of a new three--parameter family of asymptotically flat Einstein--Maxwell instantons. These solutions are toric, regular, and free of conical and orbifold singularities on the manifold $M=\CP^2\setminus S^1$. In the case of vanishing charge, these instantons reduce to the Chen--Teo Ricci flat instantons.

gr-qc↗

arXiv2025

Charges, complex structures, and perturbations of instantons

Hermitian non-K\"ahler Einstein 4-manifolds have a quasi-locally conserved charge associated to spin-lowering via Killing spinors, and corresponding to a parameter of the moduli space. This charge is evaluated for all explicitly known examples of gravitational instantons. Generic gravitational perturbations are shown to admit a closed 2-form that measures the perturbation to this charge, generalizing previous Lorentzian results on the linearized mass of perturbed Kerr black holes.

gr-qc↗

arXiv2024

Mode Stability of Hermitian Instantons

In this note, we prove the Riemannian analog of black hole mode stability for Hermitian, non-self-dual gravitational instantons which are either asymptotically locally flat (ALF) and Ricci-flat, or compact and Einstein with positive cosmological constant. We show that the Teukolsky equation on any such manifold is a positive definite operator. We also discuss the compatibility of the results with the existence of negative modes associated to variational instabilities.

gr-qc↗

arXiv2024

$α-$surfaces and global coordinates in black hole spacetimes

We present a discussion of the periodicity in imaginary time of maximally extended black hole spacetimes without reference to Euclidean manifolds. As motivation, we first demonstrate our approach for the Rindler geometry in flat space before then applying it for the case of Schwarzschild time in the maximally extended Kruskal geometry. One notable advantage of our approach is that it can be utilized in the Kerr case, again without reference to a Euclidean manifold, and without complexifying the angular momentum parameter, $a$. One unusual feature of our application in the Kerr case is that, even for the purely Lorentzian geometry, it is developed in terms of explicitly complex coordinates which are associated with distinct families of $α-$surfaces. Moreover, coordinatization of these gives a single set of coordinates which cover the entire geometry outside the inner horizon.

gr-qc↗

arXiv2024

Teukolsky equations, twistor functions, and conformally self-dual spaces

We prove a correspondence, for Riemannian manifolds with self-dual Weyl tensor, between twistor functions and solutions to the Teukolsky equations for any conformal and spin weights. In particular, we give a contour integral formula for solutions to the Teukolsky equations, and we find a recursion operator that generates an infinite family of solutions and leads to the construction of Cech representatives and sheaf cohomology classes on twistor space. Apart from the general conformally self-dual case, examples include self-dual black holes, scalar-flat Kähler surfaces, and quaternionic-Kähler metrics, where we map the Teukolsky equation to the conformal wave equation, establish new relations to the linearised Przanowski equation, and find new classes of quaternionic deformations.

gr-qc↗

arXiv2024

Generalized Siklos space-times

Motivated by supersymmetry methods in general relativity, we study four-dimensional Lorentzian space-times with a complex Dirac spinor field satisfying a Killing-spinor-like equation where the Killing constant is promoted to a complex function. We call the resulting geometry a generalized Siklos space-time. After deriving a number of identities for complex spaces, we specialize to Lorentz signature, where we show that the Killing function must be real and that the corresponding Dirac spinor is Majorana (as long as the space-time is not conformally flat), and we obtain the local form of the metric. We show that the purely gravitational degrees of freedom correspond to waves, whereas the matter sources generically correspond, via Einstein's field equations, to a sum of pure radiation and a space-like perfect fluid. Consequently, we conclude that the physically relevant case is obtained when the Killing function is homogeneous on the wave surfaces.

gr-qc↗

arXiv2023

Hyper-Kähler instantons, symmetries, and flat spaces

We find all hyper-Kähler 4-manifolds admitting conformal Kähler structures with respect to either orientation, and we show that these structures can be expressed as a combination of twistor elementary states (and possibly a self-dual dyon) in locally flat spaces. The complex structures of different flat pieces are not compatible however, reflecting that the global geometry is not a linear superposition. For either orientation the space must be Gibbons-Hawking (thus excluding the Atiyah-Hitchin metric), and, if the orientations are opposite, it must also be toric and have an irreducible Killing tensor. We also show that the only hyper-Kähler 4-metric with a non-constant Killing-Yano tensor is the half-flat Taub-NUT instanton.

math.DG↗

arXiv2023

Hidden symmetries of generalised gravitational instantons

For conformally K\"ahler Riemannian four-manifolds with a Killing field, we present a framework to solve the field equations for generalised gravitational instantons corresponding to conformal self-duality and to cosmological Einstein-Maxwell. After deriving generic identities for the curvature of such manifolds without assuming field equations, we obtain $SU(\infty)$ Toda formulations for the Page-Pope, Plebanski-Demianski, and Chen-Teo classes, we show how to solve the (modified) Toda equation, and we use this to find conformally self-dual and Einstein-Maxwell generalisations of these geometries.

gr-qc↗

arXiv2023

Complex conformal transformations and zero-rest-mass fields

We give a simple prescription for relating different solutions to the zero-rest-mass field equations in conformally flat space-time via complex conformal transformations and changes in reality conditions. We give several examples including linearized black holes. In particular, we show that the linearized Plebanski-Demianski and Schwarzschild fields are related by a complex translation and a complex special conformal transformation. Similar results hold for the linearized Kerr and C-metric fields, and for a peculiar toroidal singularity.

gr-qc↗

arXiv2022

Twistor quadrics and black holes

A simple procedure is given to construct curved, non-self-dual (complexified) Kaehler metrics on space-time in terms of deformations of holomorphic quadric surfaces in flat twistor space. Imposing Lorentzian reality conditions, the Schwarzschild, Kerr, and Plebanski-Demianski space-times (among others) are derived as examples of the construction.

gr-qc↗

arXiv2022

Kaehler geometry of black holes and gravitational instantons

We obtain a closed formula for the Kaehler potential of a broad class of four-dimensional Lorentzian or Euclidean conformal "Kaehler" geometries, including the Plebanski-Demianski class and various gravitational instantons such as Fubini-Study and Chen-Teo. We show that the Kaehler potentials of Schwarzschild and Kerr are related by a Newman-Janis shift. Our method also shows that a class of supergravity black holes, including the Kerr-Sen spacetime, is Hermitian (but not conformal Kaehler). We finally show that the integrability conditions of complex structures lead naturally to the (non-linear) Weyl double copy, and we give new vacuum and non-vacuum examples of this relation.

gr-qc↗

arXiv2022

Parallel spinors, pp-waves, and gravitational perturbations

We prove that any real, vacuum gravitational perturbation of a 4-dimensional vacuum pp-wave space-time can be locally expressed, modulo gauge transformations, as the real part of a Hertz/Debye potential, where the scalar potential satisfies the wave equation. We discuss relations with complex perturbations, complex space-times, non-linear structures, and real spaces with split (ultra-hyperbolic/Kleinian) signature. Motivated by generalized notions of parallel spinors, we also discuss generalizations of the result to other space-times.

gr-qc↗