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.     

New Infinite Series of Einstein Metrics on Sphere Bundles from AdS Black Holes

A new infinite series of Einstein metrics is constructed explicitly on S²×S³, and the non-trivial S³-bundle over S², containing infinite numbers of inhomogeneous ones. They appear as a certain limit of 5-dimensional AdS Kerr black holes. In the special case, the metrics reduce to the homogeneous Einstein metrics studied by Wang and Ziller. We also construct an inhomogeneous Einstein metric on the non-trivial S^d–2-bundle over S² from a d-dimensional AdS Kerr black hole. Our construction is a higher dimensional version of the method of Page, which gave an inhomogeneous Einstein metric on      

Obstructions to conformally Einstein metrics in n dimensions

We construct polynomial conformal invariants, the vanishing of which is necessary and sufficient for an n-dimensional suitably generic (pseudo-)Riemannian manifold to be conformal to an Einstein manifold. We also construct invariants which give necessary and sufficient conditions for a metric to be conformally related to a metric with vanishing Cotton tensor. One set of invariants we derive generalises the set of invariants in dimension 4 obtained by Kozameh, Newman and Tod. For the conformally Einstein problem, another set of invariants we construct gives necessary and sufficient conditions for a wider class of metrics than covered by the invariants recently presented by Listing. We also show that there is an alternative characterisation of conformally Einstein metrics based on the tractor connection associated with the normal conformal Cartan bundle. This plays a key role in constructing some of the invariants. Also using this we can interpret the previously known invariants geometrically in the tractor setting and relate some of them to the curvature of the Fefferman–Graham ambient metric.     

Existence of infinitely-many smooth,static, global solutions of the Einstein/Yang-Mills equations

We prove the existence of infinitely-many globally defined singularity-free solutions, to the EYM equations withSU(2) gauge group. The solutions are indexed by a coupling constant, have distinct winding numbers, and their corresponding Einstein metrics decay at infinity to the flat Minkowski metric. Each solution has a finite (ADM) mass; these masses are derived from the solutions, and arenot arbitrary constants.Both authors supported in part by the NSF, Contract No. DMS 89-05205     

LETTER: Conformal Ricci Collineations of Space-Times

We study conformal vector fields on space-times which in addition are compatible with the Ricci tensor (so-called conformal Ricci collineations). In the case of Einstein metrics any conformal vector field is automatically a Ricci collineation as well. For Riemannian manifolds, conformal Ricci collineation were called concircular vector fields and studied in the relationship with the geometry of geodesic circles. Here we obtain a partial classification of space-times carrying proper conformal Ricci collineations. There are examples which are not Einstein metrics.     

Circularly Symmetric Static Metric in Three Dimensions and Its Killing Symmetry

In this note we consider a general three-dimensional circularly symmetric static metric and investigate possible Killing symmetries it possesses.We then summarize the work by addressing Einstein equations and briefly discuss their implications on the energy momentum content of some of the metrics that arise in the process.     

Classification of local generalized symmetries for the vacuum Einstein equations

A local generalized symmetry of a system of differential equations is an infinitesimal transformation depending locally upon the fields and their derivatives which carries solutions to solutions. We classify all local generalized symmetries of the vacuum Einstein equations in four spacetime dimensions. To begin, we analyze symmetries that can be built from the metric, curvature, and covariant derivatives of the curvature to any order; these are called natural symmetries and are globally defined on any spacetime manifold. We next classify first-order generalized symmetries, that is, symmetries that depend on the metric and its first derivatives. Finally, using results from the classification of natural symmetries, we reduce the classification of all higher-order generalized symmetries to the first-order case. In each case we find that the local generalized symmetries are infinitesimal generalized diffeomorphisms and constant metric scalings. There are no non-trivial conservation laws associated with these symmetries. A novel feature of our analysis is the use of a fundamental set of spinorial coordinates on the infinite jet space of Ricci-flat metrics, which are derived from Penrose's exact set of fields for the vacuum equations.Dedicated to the memory of H. Rund     

A New Way of Solving the Wheeler–DeWitt Equation as Applied to Point Sources

A new way of solving the Wheeler–DeWitt equation is proposed which is based on quantization over free parameters of metrics satisfying the Einstein equations. This technique is applied to two point sources described in the classical case by the Tangherlini metric (in an n-dimensional space) and the Reissner–Nordström metric (in the case of the presence of a charge). The results obtained clarify the sense of the Wheeler hypothesis about statistical weights of small dimensionalities and make possible a new approach to the problem of variation of fundamental constants.     

Velocity and velocity bounds in static spherically symmetric metrics

We find simple expressions for velocity of massless particles with dependence on the distance, r, in Schwarzschild coordinates. For massive particles these expressions give an upper bound for the velocity. Our results apply to static spherically symmetric metrics. We use these results to calculate the velocity for different cases: Schwarzschild, Schwarzschild-de Sitter and Reissner-Nordström with and without the cosmological constant. We emphasize the differences between the behavior of the velocity in the different metrics and find that in cases with naked singularity there always exists a region where the massless particle moves with a velocity greater than the velocity of light in vacuum. In the case of Reissner-Nordström-de Sitter we completely characterize the velocity and the metric in an algebraic way. We contrast the case of classical naked singularities with naked singularities emerging from metric inspired by noncommutative geometry where the radial velocity never exceeds one. Furthermore, we solve the Einstein equations for a constant and polytropic density profile and calculate the radial velocity of a photon moving in spaces with interior metric. The polytropic case of radial velocity displays an unexpected variation bounded by a local minimum and maximum.     

Uniqueness of the Functional Determinant

The functional determinant of the conformal laplacian and the square of the Dirac operator are known to be extremized at the standard round metric of the four-sphere among all conformal metrics (up to gauge equivalence). In this article we show that this is the unique critical point, thus extending the work of Onofri and Osgood, Phillips and Sarnak for the functional determinant on S ² which characterized the constant curvature metric as the unique critical point of the determinant. In addition, we introduce a new symmetric two-tensor field which is defined on any conformally flat four-manifold and can be viewed as a fourth order generalization of the Einstein gravitational tensor. As a consequence we prove a Pohozaev identity for manifolds with boundary which admit conformal Killing vector fields. Received: 30 October 1996 / Accepted: 21 March 1997     

<Emphasis Type="Italic">D</Emphasis>-Dimensional Metrics with <Emphasis Type="Italic">D</Emphasis>−3 Symmetries

Symmetry transformations in a space of D-dimensional vacuum metrics with D?3 commuting Killing vectors are studied. We solve directly the Einstein equations in the Maison formulation under additional assumptions. We show that the Reissner-Nordström solution is related by the symmetry transformation to a particular case of the 5-dimensional Gross-Perry metric and the 5-dimensional plane wave solution is related to the Gross-Perry-Sorkin metric.     

Generalized Robertson-Walker metrics and some of their properties. II

The Robertson-Walker (RW) metrics, of dimensionality four and signature –2, are generalized to metrics of dimensionality (n+1) and of arbitrary signature,n (> 1) being an arbitrary integer. In canonical coordinates (t, x ¹,x ², ...,x ⁿ ) these generalized Robertson-Walker (GRW) metrics are functions of the coordinatet. The following statements are proved to be equivalent: The GRW metrics are (a) expressible int-independent form, (b) of constant curvature, (c) Einstein spaces. Furthermore, there are six, and only six, classes of GRW metrics satisfying these three statements. The coordinate transformations which transform these metrics to theirt-independent form are given explicitly. Two of these classes of GRW metrics reduce, in theirt-independent form, to the same flat (generalized Minkowski) metrics, three reduce to the samet-independent metrics which are generalizations of the de Sitter space-time metric, and the last class tot-independent metrics which are generalizations of the anti-de Sitter space-time metric.     

Robinson–Trautman solutions to Einstein’s equations

Solutions to Einstein’s equations in the form of a Robinson–Trautman metric are presented. In particular, we derive a pure radiation solution which is non-stationary and involves a mass m, The resulting spacetime is of Petrov Type II A special selection of parametric values throws up the feature of the particle ‘rocket’, a Type D metric. A suitable transformation of the complex coordinates allows the metrics to be expressed in real form. A modification, by setting m to zero, of the Type II metric thereby converting it to Type III, is then shown to admit a null Einstein–Maxwell electromagnetic field.     

Canonical Sasakian Metrics

Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L ²-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric. During the preparation of this work, the first two authors were partially supported by NSF grant DMS-0504367.     

Ambient Metrics for n-Dimensional pp-Waves

We provide an explicit formula for the Fefferman-Graham ambient metric of an n-dimensional conformal pp-wave in those cases where it exists. In even dimensions we calculate the obstruction explicitly. Furthermore, we describe all 4-dimensional pp-waves that are Bach-flat, and give a large class of Bach-flat examples which are conformally Cotton-flat, but not conformally Einstein. Finally, as an application, we use the obtained ambient metric to show that even-dimensional pp-waves have vanishing critical Q-curvature.     

Uniqueness and Examples of Compact Toric Sasaki-Einstein Metrics

In [11] it was proved that, given a compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on for each positive integer k.     

Higher Dimensional Cosmology with Normal Scalar Field and Tachyonic Field

We have considered N-dimensional Einstein field equations in which four-dimensional space-time is described by a FRW metric and that of extra dimensions by an Euclidean metric. We have supposed that the higher dimensional anisotropic universe is filled with only normal scalar field or tachyonic field. Here we have found the nature of potential of normal scalar field or tachyonic field. From graphical representations, we have seen that the potential is always decreases with field φ increases.     

A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein–Dirac–Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory.     

On General Solutions for Field Equations in Einstein and?Higher Dimension Gravity

We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing exact solutions in gravity. The main idea of this method is to introduce on (pseudo) Riemannian manifolds an alternative (to the Levi-Civita connection) metric compatible linear connection which is also completely defined by the same metric structure. Such a canonically distinguished connection is with nontrivial torsion which is induced by some nonholonomy frame coefficients and generic off-diagonal terms of metrics. It is possible to define certain classes of adapted frames of reference when the Einstein equations for such an alternative connection transform into a system of partial differential equations which can be integrated in very general forms. Imposing nonholonomic constraints on generalized metrics and connections and adapted frames (selecting Levi-Civita configurations), we generate exact solutions in Einstein gravity and extra dimension generalizations.     

Exact solutions of embedding the four-dimensional perfect fluid in a five- or higher-dimensional Einstein spacetime and the cosmological interpretations

We investigate an exact solution that describes the embedding of the four-dimensional (4D) perfect fluid in a five-dimensional (5D) Einstein spacetime. The effective metric of the 4D perfect fluid as a hypersurface with induced matter is equivalent to the Robertson–Walker metric of cosmology. This general solution shows interconnections among many 5D solutions, such as the solution in the braneworld scenario and the topological black hole with cosmological constant. If the 5D cosmological constant is positive, the metric periodically depends on the extra dimension. Thus we can compactify the extra dimension on S¹${\mathrm{S}}^1$ and study the phenomenological issues. We also generalize the metric ansatz to the higher-dimensional case, in which the 4D part of the Einstein equations can be reduced to a linear equation.