Self-dual Weyl Research Articles

Self-dual cosmology

We construct cosmological spacetimes with a self-dual Weyl tensor whose dynamics are described by conformally coupled scalars with only cubic self-interactions. Similar to the previously discovered cases in flat and (Anti) de Sitter backgrounds, the interactions are characterized by a bracket that encodes a kinematic algebra. We discuss how the color-kinematics duality and double copy are realized in these cosmological backgrounds. If we further impose that the Ricci scalar is that of an FLRW spacetime, we find two new self-dual metrics corresponding to radiation-dominated and coasting (non-accelerating) FLRW backgrounds. Relaxing this requirement, we find an infinite family of solutions given by three different conformal classes of cosmological self-dual metrics. These solutions approximate those of FLRW as long as we impose a simple additional constraint on the scalar theory.

The anti-self-dual deformation complex and a conjecture of Singer

Abstract Let ( M 4 , g ) (M^{4},g) be a smooth, closed, oriented anti-self-dual (ASD) four-manifold. ( M 4 , g ) (M^{4},g) is said to be unobstructed if the cokernel of the linearisation of the self-dual Weyl tensor is trivial. This condition can also be characterised as the vanishing of the second cohomology group of the ASD deformation complex, and is central to understanding the local structure of the moduli space of ASD conformal structures. It also arises in construction of ASD manifolds by twistor and gluing methods. In this article, we give conformally invariant conditions which imply an ASD manifold of positive Yamabe type is unobstructed.

Almost-Kähler four-manifolds with harmonic self-dual Weyl curvature

We show that a compact almost-Kähler four-manifold (M,g,ω) with harmonic self-dual Weyl curvature and constant scalar curvature is Kähler if c1⋅[ω]≥0. We also prove an integral curvature inequality for compact almost-Kähler four-manifolds with harmonic self-dual Weyl curvature.

Hyperheavenly spaces and their application in Walker and para-Kähler geometries: Part II

4-dimensional spaces equipped with congruences of null strings are considered. It is assumed that a space admits a congruence of expanding self-dual null strings and its self-dual part of the Weyl tensor is algebraically degenerate. Different Petrov-Penrose types of such spaces are analyzed. A special attention is paid to para-Kähler Einstein spaces. All para-Kähler Einstein metrics of spaces with algebraically degenerate self-dual Weyl spinor are found in all the generality.

An alternative to the Simon tensor

The Simon tensor gives rise to a local characterization of the Kerr-NUT family in the stationary class of vacuum spacetimes. We find that a symmetric and traceless tensor in the quotient space of the stationary Killing trajectory offers a useful alternative to the Simon tensor. Our tensor is distinct from the spatial dual of the Simon tensor and illustrates the geometric property of the three dimensional quotient space more manifest. The reconstruction procedure of the metric for which the generalized Simon tensor vanishes is spelled out in detail. We give a four dimensional description of this tensor in terms of the Coulomb part of the imaginary selfdual Weyl tensor, which corresponds to the generalization of the three-index tensor defined by Mars. This allows us to establish a new and simple criterion for the Kerr-NUT family: the gradient of the Ernst potential becomes the non-null eigenvector of the Coulomb part of the imaginary selfdual Weyl tensor. We also discuss the SU(1, 2) covariant extension of the obstruction tensor into the Einstein–Maxwell system as an intrinsic characterization of the Kerr–Newman-NUT family.

Einstein manifolds, self-dual Weyl curvature, and conformally Kähler geometry

Einstein manifolds, self-dual Weyl curvature, and conformally Kähler geometry

Einstein Four-Manifolds With Self-Dual Weyl Curvature of Nonnegative Determinant

Abstract We prove that simply connected Einstein four-manifolds of positive scalar curvature are conformally Kähler if and only if the determinant of the self-dual Weyl curvature is positive.

New identities for linearized gravity on the Kerr spacetime

In this paper we derive a differential identity for linearized gravity on the Kerr spacetime and more generally on vacuum spacetimes of Petrov type D. We show that a linear combination of second derivatives of the linearized Weyl tensor can be formed into a complex symmetric 2-tensor $\mathcal{M}_{ab}$ which solves the linearized Einstein equations. The identity makes this manifest by relating $\mathcal{M}_{ab}$ to two terms solving the linearized Einstein equations by construction. The self-dual Weyl curvature of $\mathcal{M}_{ab}$ gives a covariant version of the Teukolsky-Starobinsky identities for linearized gravity which, in addition to the two classical identities for linearized Weyl scalars with extreme spin weights, includes three additional equations. In particular, they are not consequences of the classical Teukolsky-Starobinsky identities, but are additional integrability conditions for linearized gravity. The result has direct application in the construction of symmetry operators and also yields a set of non-trivial gauge invariants for linearized gravity.

Euclidean supersymmetric solutions with the self-dual Weyl tensor

We explore the Euclidean supersymmetric solutions admitting the self-dual gauge field in the framework of N=2 minimal gauged supergravity in four dimensions. According to the classification scheme utilizing the spinorial geometry or the bilinears of Killing spinors, the general solution preserves one quarter of supersymmetry and is described by the Przanowski–Tod class with the self-dual Weyl tensor. We demonstrate that there exists an additional Killing spinor, provided the Przanowski–Tod metric admits a Killing vector that commutes with the principal one. The proof proceeds by recasting the metric into another Przanowski–Tod form. This formalism enables us to show that the self-dual Reissner–Nordström–Taub–NUT–AdS metric possesses a second Killing spinor, which has been missed over many years. We also address the supersymmetry when the Przanowski–Tod space is conformal to each of the self-dual ambi-toric Kähler metrics. It turns out that three classes of solutions are all reduced to the self-dual Carter family, by virtue of the nondegenerate Killing–Yano tensor.

Limits of Riemannian 4-manifolds and the symplectic geometry of their twistor spaces

The twistor space of a Riemannian 4-manifold carries two almost complex structures, $J_+$ and $J_-$, and a natural closed 2-form $\omega$. This article studies limits of manifolds for which $\omega$ tames either $J_+$ or $J_-$. This amounts to a curvature inequality involving self-dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti-self-dual Einstein manifolds with non-zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperk\"ahler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorphic 2-sphere (for any of its hyperk\"ahler complex structures). In particular, this rules out the formation of bubbles modelled on ALE gravitational instantons in such families of metrics.

Non-diagonal four-dimensional cohomogeneity-one Einstein metrics in various signatures

Most known four-dimensional cohomogeneity-one Einstein metrics are diagonal in a basis defined by the left-invariant one-forms, though some essentially non-diagonal ones are known. We consider the problem of explicitly seeking non-diagonal Einstein metrics, and we find solutions which in some cases exhaust the possibilities. In particular we construct new examples of neutral signature non-diagonal Bianchi type VIII Einstein metrics with self-dual Weyl tensor.

Spacetime characterizations of Λ-vacuum metrics with a null Killing 2-form

An exhaustive list of four-dimensional Λ-vacuum spacetimes admitting a Killing vector ξ whose self-dual Killing two-form is null is obtained assuming that the self-dual Weyl tensor is proportional to the tensor product of by itself. Our analysis complements previous results (Mars 1999 Class Quantum Grav. 16 2507–23; Mars and Senovilla 2015 Ann. H. Poincaré 16 1509–50) concerning the case with non-null . We analyze both cases with Λ vanishing or not. In the latter case we prove that must hold necessarily, and we find a characterization of the Einstein spacetimes conformal to pp-waves. In the former case we obtain spacetime characterizations of vacuum plane waves and of the stationary vacuum Brinkmann spacetimes. At the light of the full set of results, old and new, we reformulate the case with non-null and . We finally present a table collecting the results for both null, and non-null, .

The Weyl tensor of gradient Ricci solitons

This paper derives new identities for the Weyl tensor on a gradient Ricci soliton, particularly in dimension four. First, we prove a Bochner-Weitzenb\"ock type formula for the norm of the self-dual Weyl tensor and discuss its applications, including connections between geometry and topology. In the second part, we are concerned with the interaction of different components of Riemannian curvature and (gradient and Hessian of) the soliton potential function. The Weyl tensor arises naturally in these investigations. Applications here are rigidity results.

Einstein metrics, harmonic forms, and symplectic four-manifolds

If $M$ is the underlying smooth oriented $4$-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics $h$ on $M$ such that $W^+(\omega , \omega )> 0$, where $W^+$ is the self-dual Weyl curvature of $h$, and $\omega$ is a non-trivial self-dual harmonic $2$-form on $(M,h)$. While this open region in the space of Riemannian metrics contains all the known Einstein metrics on $M$, we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on $M$.

A Spacetime Characterization of the Kerr-NUT-(A)de Sitter and Related Metrics

A characterization of the Kerr-NUT-(A)de Sitter metric among four dimensional Λ-vacuum spacetimes admitting a Killing vector ξ is obtained in terms of the proportionality of the self-dual Weyl tensor and a natural self-dual double two-form constructed from the Killing vector. This result recovers and extends a previous characterization of the Kerr and Kerr-NUT metrics (Mars, Class Quant Grav 16:2507–2523, 1999). The method of proof is based on (i) the presence of a second Killing vector field which is built in terms of geometric information arising from the Killing vector ξ exclusively, and (ii) the existence of an interesting underlying geometric structure involving a Riemannian submersion of a conformally related metric, both of which may be of independent interest. Other related metrics can also be similarly characterized, in particular the Λ < 0 “black branes” recently used in AdS/CFT correspondence to describe, via holography, the physics of Quark–Gluon plasma.

Convergence of metrics under self-dual Weyl tensor and scalar curvature bounds

Abstract We establish a C 1,α-compactness theorem for metrics with bounded self-dual Weyl tensor and scalar curvature. The main ingredient is the C 1,α-harmonic radius estimate, where a blow up analysis as in [Invent. Math. 102 (1990), 429–445] is used. Our result is motivated by, and may be applied to, the Calabi flow on complex surfaces.

Geometric models of matter

Inspired by soliton models, we propose a description of static particles in terms of Riemannian 4-manifolds with self-dual Weyl tensor. For electrically charged particles, the 4-manifolds are non-compact and asymptotically fibred by circles over physical 3-space. This is akin to the Kaluza–Klein description of electromagnetism, except that we exchange the roles of magnetic and electric fields, and only assume the bundle structure asymptotically, away from the core of the particle in question. We identify the Chern class of the circle bundle at infinity with minus the electric charge and, at least provisionally, the signature of the 4-manifold with the baryon number. Electrically neutral particles are described by compact 4-manifolds. We illustrate our approach by studying the Taub–Newman, Unti, Tamburino (Taub–NUT) manifold as a model for the electron, the Atiyah–Hitchin manifold as a model for the proton, with the Fubini–Study metric as a model for the neutron and S 4 with its standard metric as a model for the neutrino.

Algebraically special, real alpha-geometries

We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of α -planes ( algebraically special real α - geometry). In particular, we determine the behaviour of Walker geometry under conformal rescaling and provide a derivation of the hyperheavenly equation from conformal rescaling formulae.

Four-dimensional manifolds with degenerate self-dual Weyl curvature operator

It is shown that any four-dimensional Walker metric of nowhere zero scalar curvature has a natural almost para-Hermitian structure. In contrast to the Goldberg–Sachs theorem, if this structure is self-dual and *-Einstein, it is symplectic but not necessarily integrable. This is due to the non-diagonalizability of the self-dual Weyl conformal curvature tensor.

Einstein–Weyl from Kaluza–Klein

We discuss the Kaluza–Klein reduction of spaces with (anti-)self-dual Weyl tensor and point out the emergence of the Einstein–Weyl equations for the reduction from four to three dimensions. As a byproduct we get a simple expression for the gravitational instanton density in terms of the Kaluza–Klein functions.