Einstein Weyl Research Articles

INTEGRABILITY OF EINSTEIN-WEYL EQUATIONS FOR SPATIALLY HOMOGENEOUS MODELS OF TYPE III BY BIANCHI

This paper deals with nonisotropic spatially homogeneous models for a self-consistent system of Einstein–Weyl equations with a spinor field. It is shown that for spaces of type III by Bianchi the system of Einstein–Weyl equations is integrable.

Cohomogeneity-one Einstein–Weyl structures: a local approach

We analyse in a systematic way the (non-) compact n-dimensional Einstein–Weyl spaces equipped with a cohomogeneity-one metric. In that context, with no compactness hypothesis for the manifold on which lives the Einstein–Weyl structure, we prove that, as soon as the ( n−1)-dimensional space is a homogeneous reductive Riemannian space with a unimodular group of left-acting isometries G: • there exists a Gauduchon gauge such that the Weyl-form is co-closed and its dual is a Killing vector for the metric; • in that gauge, a simple constraint on the conformal scalar curvature holds; • a non-exact Einstein–Weyl structure may exist only if the ( n−1)-dimensional homogeneous space G/ H contains a non-trivial subgroup H′ that commutes with the isotropy subgroup H; • the extra isometry due to this Killing vector corresponds to the right-action of one of the generators of the algebra of the subgroup H′. The first two results are well known when the Einstein–Weyl structure lives on a compact manifold, but our analysis gives the first hints on the enlargement of the symmetry due to the Einstein–Weyl constraint. We also prove that the subclass with G compact, a one-dimensional subgroup H′ and the ( n−2)-dimensional space G/( H× H′) being an arbitrary compact symmetric Kähler coset space, corresponds to n-dimensional Riemannian locally conformally Kähler metrics. The explicit family of structures of cohomogeneity-one under SU( n/2) being, thanks to our results, invariant under U(1)× SU( n/2), it coincides with the one first studied by Madsen; our analysis allows us to prove most of his conjectures.

Einstein metrics, hypercomplex structures and the Toda field equation

We obtain explicitly all solutions of the SU(∞) Toda field equation with the property that the associated Einstein–Weyl space admits a 2-sphere of divergence-free shear-free geodesic congruences. The solutions depend on an arbitrary holomorphic function and give rise to new hyperKähler and selfdual Einstein metrics with one-dimensional isometry group. These metrics each admit a compatible hypercomplex structure with respect to which the symmetries are triholomorphic

Applications of Riemannian and Einstein–Weyl Geometry in the Theory of Second-Order Ordinary Differential Equations

We consider some properties of four-dimensional Riemannian spaces whose metric coefficients are associated with the coefficients of second-order nonlinear differential equations, and we study the properties of three-dimensional Einstein–Weyl spaces related to the dual equations b″ = g(a,b,b′), where the function g(a,b,b′) satisfies a special partial differential equation.

Einstein–Weyl structures from hyper–Kähler metrics with conformal Killing vectors

We consider four (real or complex) dimensional hyper-Kähler metrics with a conformal symmetry K. The three-dimensional space of orbits of K is shown to have an Einstein–Weyl structure which admits a shear-free geodesics congruence for which the twist is a constant multiple of the divergence. In this case the Einstein–Weyl equations reduce down to a single second order PDE for one function. The Lax representation, Lie point symmetries, hidden symmetries and the recursion operator associated with this PDE are found, and some group invariant solutions are considered.

Einstein–Weyl geometry, the dKP equation and twistor theory

It is shown that Einstein–Weyl (EW) equations in 2+1 dimensions contain the dispersionless Kadomtsev–Petviashvili (dKP) equation as a special case: if an EW structure admits a constant-weighted vector then it is locally given by h= dy 2−4 dx dt−4u dt 2 , ν=−4u x dt , where u= u( x, y, t) satisfies the dKP equation ( u t − uu x ) x = u yy . Linearised solutions to the dKP equation are shown to give rise to four-dimensional anti-self-dual conformal structures with symmetries. All four-dimensional hyper-Kähler metrics in signature (++−−) for which the self-dual part of the derivative of a Killing vector is null arise by this construction. Two new classes of examples of EW metrics which depend on one arbitrary function of one variable are given, and characterised. A Lax representation of the EW condition is found and used to show that all EW spaces arise as symmetry reductions of hyper-Hermitian metrics in four dimensions. The EW equations are reformulated in terms of a simple and closed two-form on the CP 1 -bundle over a Weyl space. It is proved that complex solutions to the dKP equations, modulo a certain coordinate freedom, are in a one-to-one correspondence with mini-twistor spaces (two-dimensional complex manifolds Z containing a rational curve with normal bundle O(2) ) that admit a section of κ −1/4, where κ is the canonical bundle of Z . Real solutions are obtained if the mini-twistor space also admits an anti-holomorphic involution with fixed points together with a rational curve and section of κ −1/4 that are invariant under the involution.

The geometry of the Toda equation

I show that solutions of the SU(∞) Toda field equation generating a fixed Einstein–Weyl space are governed by a linear equation on the Einstein–Weyl space. From this, obstructions to the existence of Toda solutions generating a given Einstein–Weyl space are found. I also give a classification of Einstein–Weyl spaces arising from the Toda equation in more than one way. This classification coincides with a class of spaces found by Ward and hence clarifies some of their properties. I end by discussing the simplest examples.

Einstein–Weyl spaces and third-order differential equations

The three-dimensional null-surface formalism of Tanimoto [M. Tanimoto, “On the null surface formalism,” Report No. gr-qc/9703003 (1997)] and Forni et al. [Forni et al., “Null surfaces formation in 3D,” J. Math Phys. (submitted)] are extended to describe Einstein–Weyl spaces, following Cartan [E. Cartan, “Les espaces généralisées et l’integration de certaines classes d’equations différentielles,” C. R. Acad. Sci. 206, 1425–1429 (1938); “La geometria de las ecuaciones diferenciales de tercer order,” Rev. Mat. Hispano-Am. 4, 1–31 (1941)]. In the resulting formalism, Einstein–Weyl spaces are obtained from a particular class of third-order differential equations. Some examples of the construction which include some new Einstein–Weyl spaces are given.

The Painlevé III, V and VI transcendents as solutions of the Einstein–Weyl equations

We demonstrate that the integrable Ernst–Weyl equation governing neutrino and gravitational fields in axially symmetric space–times of general relativity admits symmetry reductions to the Painlevé III, V and VI equations with arbitrary constants. In particular, the matrix form of PVI is shown to be a canonical symmetry reduction of the integrable Loewner–Konopelchenko–Rogers (LKR) system.

Local Constraints on Einstein-Weyl Geometries: The 3-Dimensional Case

Following an earlier study [3], we consider the Einstein–Weyl equations on a fixed (complex) background metric as an equation for a 1-form and its first few derivatives. If the background is flat then we conclude that the only solutions are conformal rescalings of constant curvature metrics. If the background is a homogeneous 3-geometry in Bianchi class A (i.e., with unimodular isometry group), we find necessary and sufficient conditions on the 3-geometry for solutions of the Einstein–Weyl equations to exist. The solutions we find are complexifications of known ones. In particular, we find that the general left-invariant metric on S3 and the metric 'Sol' admit no local solutions of the Einstein–Weyl equations.

Cosmological solution of the Einstein—Weyl equation

The accurate integration of the Einstein—Weyl field equations is considered for the case when the spinor field depends only on the time, while the metric specifies a uniform space—time of type I in the Bianci classification, i.e., a particular case of a Steckel space of type (3.0).

The Self-Dual Einstein–Weyl Metric and Classical Solution of Painlevé VI

In the natural correspondence between the self-dual Bianchi type IX metrics and solutions of Painleve VI, the self-dual Ricci-flat metrics or the nontrivial self-dual Einstein–Weyl metrics correspond to the classical solutions of Painleve VI that determine isomonodromic deformations with reducible monodromy.

Homogeneous Einstein–Weyl Structures on Symmetric Spaces

In this paper we examine homogeneous Einstein–Weyl structures and classify them on compact irreducible symmetric spaces. We find that the invariant Einstein–Weyl equation is very restrictive: Einstein–Weyl structures occur only on those spaces for which the isotropy representation has a trivial component, for example, the total space of a circle bundle.

Integration of the Einstein–Dirac equations

The problem of exact integration of the Einstein–Dirac equation is studied. To solve this set of equations metrics admitting complete separation of variables in the Dirac equation are considered. At the first stage Stäckel spaces of types (3.1) and (3.0) are studied. For the first case obtained solutions contain arbitrary functions depending on one variable only (five real functions for the Einstein–Weyl equations and four real functions for the Einstein–Dirac ones). For the second one all solutions belonging to the diagonal metrics of Bianchi type 1 are found.

Exact solutions of the Einstein equations with sources from linearized solutions

It is shown that if (gμν,Fμν) is an exact solution of the Einstein–Maxwell equations and lμ is a principal null direction of Fμν then (gμν+2Hlμlν,Fμν) is also an exact solution of the Einstein–Maxwell equations if and only if (2Hlμlν,0) satisfies the Einstein–Maxwell equations linearized about (gμν,Fμν). It is also shown that an analogous result applies in the case of the Einstein–Weyl equations if lμ is parallel to the flux vector of the neutrino field.

The Einstein–Weyl equations in complex and quaternionic geometry

Einstein–Weyl manifolds with compatible complex structures are shown to be given as torus bundles on Kähler–Einstein manifolds, extending known results on locally conformal Kähler manifolds. The Weyl structure is derived from a Ricci-flat metric constructed by Calabi on the canonical bundle of the Kähler–Einstein manifold. Similar questions are addressed when the Weyl geometry admits compatible hypercomplex or quaternionic structures.

Intrinsic wave mechanical behaviour in a Weyl-like geometry which conserves lengths

Einstein's lambda transformation and Weyl's gauge transformation are coupled to produce an asymmetric Weyl-like geometry. In the models of interest, lengths do not change under parallel transport. Automatic internal constraints of such geometries very closely resemble the equations of wave mechanics. Allowing an asymmetric metric tensor appears to add phenomena suggesting spin, although the equations are still scalar equations, not spinor ones. However, correspondence with Dirac theory exists for non-null, constant, uniform, electromagnetic fields in the limit that the symmetric part of the metric is Lorentzian.

Local symmetries in extended conformal gravity and conformal supergravity in superspace

The extended conformal field theory of gravity is discussed, and its local symmetries are compared to those of Einstein and standard Weyl theories. It is then shown that the local symmetries of the extended conformal geometry are also natural for the construction of SU(N)-extended conformal supergravity theories in superspace. The natural superspace for the formulation of such theories is one with two sets of four-component fermionic variables which transform oppositely under scale transformations.

Null strings and Bianchi identities

The results of the theory of complexified V4’s which admit a null string are reexamined by using systematically the Bianchi identities. The formal reasons which permit the integration of the Gμν=0 equations to one differential constraint in the case of the ℋℋ structures are exhibited. Some new results concerning complexified Einstein–Weyl equations in the presence of a null string are given as well.

On the nonexistence of static solutions of the Einstein–Weyl equations in general relativity

It is proved that the combined gravitational-neutrino equations in general relativity admit no nontrivial solutions in a static space-time provided that the energy density of the neutrino field is nonnull for all observers.