Einstein Weyl Research Articles

Conelike radiant structures

Analogues of the classical affine-projective correspondence are developed in the context of statistical manifolds compatible with a radiant vector field. These utilize a formulation of Einstein equations for special statistical structures that generalizes the usual Einstein equations for pseudo-Riemannian metrics and is of independent interest. A conelike radiant structure is a not necessarily flat affine connection equipped with a family of surfaces that behave like the intersections of the planes through the origin with a convex cone in a real vector space. A radiant structure is a torsion-free affine connection and a vector field whose covariant derivative is the identity endomorphism. A radiant structure is conelike if for every point and every two-dimensional subspace containing the radiant vector field there is a totally geodesic surface passing through the point and tangent to the subspace. Such structures exist on the total space of any principal bundle with one-dimensional fiber and on any Lie group with a quadratic structure on its Lie algebra. The affine connection of a conelike radiant structure can be normalized in a canonical way to have antisymmetric Ricci tensor. Applied to a conelike radiant structure on the total space of a principal bundle with one-dimensional fiber this yields a generalization of the classical Thomas connection of a projective structure. The compatibility of radiant and conelike structures with metrics is investigated and yields a construction of connections for which the symmetrized Ricci curvature is a constant multiple of a compatible metric that generalizes well-known constructions of Riemannian and Lorentzian Einstein–Weyl structures over Kähler–Einstein manifolds having nonzero scalar curvature. A formulation of Einstein equations for special statistical manifolds is given that generalizes the Einstein–Weyl equations and encompasses these more general examples. There are constructed left-invariant conelike radiant structures on a Lie group endowed with a left-invariant nondegenerate bilinear form, and the case of three-dimensional unimodular Lie groups is described in detail.

Analytical Approximations to Charged Black Hole Solutions in Einstein–Maxwell–Weyl Gravity

The homotopy analysis method (HAM) is a useful method to derive analytical approximate solutions of black holes in modified gravity theories. In this paper, we study the Einstein–Weyl gravity coupled with Maxwell field and obtain analytical approximation solutions for charged black holes by using the HAM. It is found that the analytical approximate solutions are sufficiently accurate in the entire spacetime outside the black hole’s event horizon and also consistent with numerical ones for charged black holes in the Einstein–Maxwell–Weyl gravity.

Towards effective actions for the continuum limit of spin foams

Spin foams arise from a quantization of classical gravity expressed via the Plebanski action. Key open questions related to the continuum limit of spin foams are whether general relativity is reproduced and what type of corrections could emerge. As a central component for spin foam dynamics, recent results on the continuum limit of the Area Regge action suggest a close relation with actions for area metrics instead of a length metric. Inspired by these results, within the framework of modified Plebanski theory we construct a family of candidate actions for area metrics. These actions are expected to describe the continuum limit of spin foams and provide a starting point to explore phenomenological aspects of the large-scale dynamics of spin foams. More generally, they set the stage for exploring consequences of an enlargement of the configuration space for gravity from length to area metrics. The actions we construct lead to an effective action for the length metric, describing a non-local and ghost-free version of Einstein–Weyl gravity.

Vaisman manifolds and transversally Kähler–Einstein metrics

We use the transverse Kähler–Ricci flow on the canonical foliation of a closed Vaisman manifold to deform the Vaisman metric into another Vaisman metric with a transverse Kähler–Einstein structure. We also study the main features of such a manifold. Among other results, using techniques from the theory of parabolic equations, we obtain a direct proof for the short-time existence of the solution for transverse Kähler–Ricci flow on Vaisman manifolds, recovering in a particular setting a result of Bedulli et al. (J Geom Anal 28:697–725, 2018), but without employing the Molino structure theorem. Moreover, we investigate Einstein–Weyl structures in the setting of Vaisman manifolds and find their relationship with quasi-Einstein metrics. Some examples are also provided to illustrate the main results.

Neutrino fields in a sandwich gravitational wave background

Sandwich gravitational waves are given globally in terms of step functions at the boundaries. Linearized Einstein–Weyl equations are solved exactly in this background in Rosen coordinates. Depending on the geometry and composition of the sandwich wave, the neutrino’s energy-momentum redistributes itself. At the test field level, since the background will not change, the neutrino’s energy density in particular will show variations between positive and negative extrema when crossing the sandwich wave. This may reveal facts about the weakly interacting neutrinos in cosmology.

On the lump solutions, breather waves, two-wave solutions of (2 + 1)-dimensional Pavlov equation and stability analysis

Hirota’s bilinear method (HBM) has been successfully applied to the [Formula: see text]-dimensional Pavlov equation to analyze the different wave structures in this paper. The [Formula: see text]-dimensional Pavlov equation is used for the study of integrated hydrodynamic chains and Einstein–Weyl manifolds. In our research, we find new solutions in the forms of lump solutions, breather waves, and two-wave solutions. The modulation instability (MI) of the governing model is also discussed. Moreover, a variety of 3D, 2D, and contour profiles are used to illustrate the physical behavior of the reported results. Acquired findings are useful in understanding nonlinear science and its related nonlinear higher-dimensional wave fields. Through the use of Mathematica, the obtained results are verified by inserting them into the governing equation. The strengthening of representative calculations we’ve made gives us a strong and effective mathematical framework for dealing with the most difficult nonlinear wave problems.

The quadric ansatz for the mn-dispersionless KP equation, and supersymmetric Einstein–Weyl spaces

We consider two multi-dimensional generalisations of the dispersionless Kadomtsev–Petviashvili (dKP) equation, both allowing for arbitrary dimensionality, and non-linearity. For one of these generalisations, we characterise all solutions which are constant on a central quadric. The quadric ansatz leads to a second order ODE which is equivalent to Painlevé I or II for the dKP equation, but fails to pass the Painlevé test in higher dimensions. The second generalisation of the dKP equation leads to a class of Einstein–Weyl (EW) structures in an arbitrary dimension, which is characterised by the existence of a weighted parallel vector field, together with further holonomy reduction. We construct and characterise an explicit new family of EW spaces belonging to this class, and depending on one arbitrary function of one variable.

Second-Order PDEs in 3D with Einstein–Weyl Conformal Structure

Einstein–Weyl geometry is a triple ({mathbb {D}},g,omega ) where {mathbb {D}} is a symmetric connection, [g] is a conformal structure and omega is a covector such that bullet connection {mathbb {D}} preserves the conformal class [g], that is, {mathbb {D}}g=omega g; bullet trace-free part of the symmetrised Ricci tensor of {mathbb {D}} vanishes. Three-dimensional Einstein–Weyl structures naturally arise on solutions of second-order dispersionless integrable PDEs in 3D. In this context, [g] coincides with the characteristic conformal structure and is therefore uniquely determined by the equation. On the contrary, covector omega is a somewhat more mysterious object, recovered from the Einstein–Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge–Ampère type), the covector omega is also expressible in terms of the equation, thus providing an efficient ‘dispersionless integrability test’. The knowledge of g and omega provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein–Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein–Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation.

Elements of the Metric–Affine gravity I: Aspects of F(R) theories reductions and the topologically massive gravity

Some classical aspects of Metric–Affine Gravity are reviewed in the context of the [Formula: see text] type models (polynomials of degree [Formula: see text] in the Riemann tensor) and the topologically massive gravity. At the nonperturbative level, we explore the consistency of the field equations when the [Formula: see text] models are reduced to a Riemann–Christoffel (RCh) space–time, either via a Riemann–Cartan (RC) space or via an Einstein–Weyl (EW) space. It is well known for the case [Formula: see text] that any path or reduction “classes” via RC or EW leads to the same field equations with the exception of the [Formula: see text] theories for [Formula: see text]. We verify that this discrepancy can be solved by imposing nonmetricity and torsion constraints. In particular, we explore the case [Formula: see text] for the interest in expected physical solutions as those of conformally flat class. On the other hand, the symmetries of the topologically massive gravity are reviewed, as the physical content in RC and EW scenarios. The appearance of a nonlinearly modified selfdual model in RC and existence of many nonunitary degrees of freedom in EW with the suggestion of a modified model for a massive gravity which cure the unphysical propagations shall be discussed.

Connections between Weyl geometry, quantum potential and quantum entanglement

The Weyl geometry promises potential applications in gravity and quantum mechanics. We study the relationships between the Weyl geometry, quantum entropy and quantum entanglement based on the Weyl geometry endowing the Euclidean metric. We give the formulation of the Weyl Ricci curvature and Weyl scalar curvature in the n-dimensional system. The Weyl scalar field plays a bridge role to connect the Weyl scalar curvature, quantum potential and quantum entanglement. We also give the Einstein–Weyl tensor and the generalized field equation in 3D vacuum case, which reveals the relationship between Weyl geometry and quantum potential. Particularly, we find that the correspondence between the Weyl scalar curvature and quantum potential is dimension-dependent and works only for the 3D space, which reveals a clue to quantize gravity and an understanding why our space must be 3D if quantum gravity is compatible with quantum mechanics. We analyze numerically a typical example of two orthogonal oscillators to reveal the relationships between the Weyl scalar curvature, quantum potential and quantum entanglement based on this formulation. We find that the Weyl scalar curvature shows a negative dip peak for separate state but becomes a positive peak for the entangled state near original point region, which can be regarded as a geometric signal to detect quantum entanglement.

Dispersionless BKP Equation, the Manakov–Santini System and Einstein–Weyl Structures

We construct a map from solutions of the dispersionless BKP (dBKP) equation to solutions of the Manakov–Santini (MS) system. This map defines an Einstein–Weyl structure corresponding to the dBKP equation through the general Lorentzian Einstein–Weyl structure corresponding to the MS system. We give a spectral characterisation of reduction in the MS system, which singles out the image of the dBKP equation solution, and also consider more general reductions of this class. We define the BMS system and extend the map defined above to the map (Miura transformation) of solutions of the BMS system to solutions of the MS system, thus obtaining an Einstein–Weyl structure for the BMS system.

Gravitational Decoupling in Higher Order Theories

Gravitational decoupling via the Minimal Geometric Deformation (MGD) approach has been used extensively in General Relativity (GR), mainly as a simple method for generating exact anisotropic solutions from perfect fluid seed solutions. Recently this method has also been used to generate exact spherically symmetric solutions of the Einstein-scalar system from the Schwarzschild vacuum metric. This was then used to investigate the effect of scalar fields on the Schwarzschild black hole solution. We show that this method can be extended to higher order theories. In particular, we consider fourth order Einstein–Weyl gravity, and in this case by using the Schwarzschild metric as a seed solution to the associated vacuum field equations, we apply the MGD method to generate a solution to the Einstein–Weyl scalar theory representing a hairy black hole solution. This solution is expressed in terms of a series using the Homotopy Analysis Method (HAM).

Isospectral deformation of the reduced quasi-classical self-dual Yang–Mills equation

We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang–Mills equation and applying the technique of twisted extensions to the obtained Lie algebra. Then we find a recursion operator for symmetries of the new equation and construct a Bäcklund transformation between this equation and the four-dimensional Martínez Alonso–Shabat equation. Finally, we construct extensions of the integrable hierarchies associated to the hyper-CR equation for Einstein–Weyl structures, the reduced quasi-classical self-dual Yang–Mills equation, the four-dimensional universal hierarchy equation, and the four-dimensional Martínez Alonso–Shabat equation.

Nonlinear nonisospectral differential coverings for the hyper-CR equation of Einstein–Weyl structures and the Gibbons–Tsarev equation

We apply the technique based on twisted extensions of symmetry algebras to construct new nonlinear four-dimensional differential coverings for the hyper-CR equation of Einstein–Weyl structures and for the associated integrable hierarchy. We expose related multi-component three-dimensional covering. By the symmetry reduction of the hyper-CR equation of Einstein–Weyl structures we derive nonlinear three-dimensional differential covering for the Gibbons-Tsarev equation.

The eigenvalue problem of the general Einstein–Weyl metric equation and exact self-similar and multi-traveling waves solutions

The integrability of the general Einstein–Weyl Metric equation is studied . Here, the approach proposed is based on using the extended and generalized unified methods. These methods inspect the integrability conditions, whenever exist. The exact solutions are thus obtained. They provide the existence and uniqueness of solution for initial value problem where initial value is taken from the exact solutions obtained. Similarity variables are introduced, and the extended unified method is applied to find a class of explicit self-similar wave solutions to the GEWM equation. Numerical evaluations of the solutions are done via symbolic computation. These results show periodic solitons and multi-lumps. It is found that the generalized heavenly equation admits a class of infinite solutions, among them the chaotic ones. The generalized unified method is used to find multi-traveling solitary waves solutions. We think that the presented methods generalize the known ones in the literature. This will be illustrated later on.

Symmetries of Einstein-Weyl manifolds with boundary

Starting from a real analytic surface ℳ with a real analytic conformal Cartan connection A. Bor´owka constructed a minitwistor space of an asymptotically hyperbolic Einstein–Weyl manifold with ℳ being the boundary. In this article, starting from a symmetry of conformal Cartan connection, we prove that symmetries of conformal Cartan connection on ℳ can be extended to symmetries of the obtained Einstein–Weyl manifold.

Einstein–Weyl structures on real hypersurfaces of complex two-plane Grassmannians

We study real Hopf hypersurfaces with Einstein–Weyl structures in the complex two-plane Grassmannian $G_2(\mathbb {C}^{m+2})$, $m\geq 3$. First we prove that a real Hopf hypersurface with a closed Einstein–Weyl structure $W=(g,\theta )$ is of type (B) if

Self-dual Einstein spaces and the general heavenly equation. Eigenfunctions as coordinates

Eigenfunctions are shown to constitute privileged coordinates of self-dual Einstein spaces with the underlying governing equation being revealed as the general heavenly equation. The formalism developed here may be used to link algorithmically a variety of known heavenly equations. In particular, the classical connection between Plebański’s first and second heavenly equations is retrieved and interpreted in terms of eigenfunctions. In addition, connections with travelling wave reductions of the recently introduced TED equation which constitutes a 4 + 4-dimensional integrable generalisation of the general heavenly equation are found. These are obtained by means of (partial) Legendre transformations. As a particular application, we prove that a large class of self-dual Einstein spaces governed by a compatible system of dispersionless Hirota equations is genuinely four-dimensional in that the (generic) metrics do not admit any (proper or non-proper) conformal Killing vectors. This generalises the known link between a particular class of self-dual Einstein spaces and the dispersionless Hirota equation encoding three-dimensional Einstein–Weyl geometries.

Dispersionless integrable systems and the Bogomolny equations on an Einstein–Weyl geometry background

We derive a dispersionless integrable system describing a local form of a general three-dimensional Einstein-Weyl geometry with an Euclidean (positive) signature, construct its matrix extension and demonstrate that it leads to the Bogomolny equations for a non-abelian monopole on an Einstein-Weyl geometry background. The corresponding dispersionless integrable hierarchy, its matrix extension and the dressing scheme are also considered.

An action principle for the Einstein–Weyl equations

A longstanding open problem in mathematical physics has been that of finding an action principle for the Einstein–Weyl (EW) equations. In this paper, we present for the first time such an action principle in three dimensions in which the Weyl vector is not exact. More precisely, our model contains, in addition to the Weyl nonmetricity, a traceless part. If the latter is (consistently) set to zero, the equations of motion boil down to the EW equations. In particular, we consider a metric affine f(R) gravity action plus additional terms involving Lagrange multipliers and gravitational Chern–Simons contributions. In our framework, the metric and the connection are considered as independent objects, and no a priori assumptions on the nonmetricity and the torsion of the connection are made. The dynamics of the Weyl vector turns out to be governed by a special case of the generalized monopole equation, which represents a conformal self-duality condition in three dimensions.