Kundt Metrics Research Articles

Topological black holes in higher derivative gravity

We study static black holes in quadratic gravity with planar and hyperbolic symmetry and non-extremal horizons. We obtain a solution in terms of an infinite power-series expansion around the horizon, which is characterized by two independent integration constants – the black hole radius and the strength of the Bach tensor at the horizon. While in Einstein’s gravity, such black holes require a negative cosmological constant Lambda , in quadratic gravity they can exist for any sign of Lambda and also for Lambda =0. Different branches of Schwarzschild–Bach–(A)dS or purely Bachian black holes are identified which admit distinct Einstein limits. Depending on the curvature of the transverse space and the value of Lambda , these Einstein limits result in (A)dS–Schwarzschild spacetimes with a transverse space of arbitrary curvature (such as black holes and naked singularities) or in Kundt metrics of the (anti-)Nariai type (i.e., dS_2times S^2, AdS_2times H^2, and flat spacetime). In the special case of toroidal black holes with Lambda =0, we also discuss how the Bach parameter needs to be fine-tuned to ensure that the metric does not blow up near infinity and instead matches asymptotically a Ricci-flat solution.

Nil-Killing vector fields and type III deformations

This paper is concerned with deformations of Kundt metrics in the direction of type III tensors and nil-Killing vector fields whose flows give rise to such deformations. We find various characterizations within the Kundt class in terms of nil-Killing vector fields and obtain a theorem classifying algebraic stability of tensors, which has an application in finding sufficient criteria for a type III deformation of the metric to preserve spi’s. This is used in order to specify Lie algebras of nil-Killing vector fields that preserve the spi’s, for degenerate Kundt metrics. Using this we discuss the characterization of Kundt-Constant scalar invariants spacetimes in terms of nil-Killing vector fields.

Axially Symmetric Type N Space-Time with a Naked Curvature Singularity and Closed Timelike Curves

A family of type N exact solution of the Einstein's field equations, regular everywhere except on the symmetry axis where it possesses a naked curvature singularity, is present. The stress-energy tensor is of the anisotropic fluid coupled with pure radiation field satisfy the different energy conditions and the physical parameters diverge $r \rightarrow 0$. The space-time admitting a non-expanding, non-twisting, and shear-free geodesic null congruence and belongs to a special class of type N Kundt metrics. The space-time is geodesically complete along the radiation direction in the constant $z$-planes and exhibits geometrically different properties from the known pp-waves. The present family of solution admits closed time-like curves (CTC) which appear after a certain instant of time and the space-time is a four-dimensional generalization of the Misner space metric in curved space-time.

Locally homogeneous Kundt triples and CSI metrics

A pseudo-Riemannian manifold is called CSI if all scalar polynomial invariants constructed from the curvature tensor and its covariant derivatives are constant. In the Lorentzian case, the CSI spacetimes have been studied extensively due to their application to gravity theories. It is conjectured that a CSI spacetime is either locally homogeneous (l.h.) or belongs to the subclass of degenerate Kundt metrics. Independent of this conjecture, any CSI spacetime can be related to a particular l.h. degenerate Kundt metric sharing the same scalar polynomial curvature invariants. In this paper we will invariantly classify the entire subclass of l.h. CSI Kundt spacetimes which are of alignment type to all orders and show that any other CSI Kundt metric can be constructed from them.

Almost universal spacetimes in higher-order gravity theories

We study almost universal spacetimes - spacetimes for which the field equations of any generalized gravity with the Lagrangian constructed from the metric, the Riemann tensor and its covariant derivatives of arbitrary order reduce to one single differential equation and one algebraic condition for the Ricci scalar. We prove that all d-dimensional Kundt spacetimes of Weyl type III and traceless Ricci type N are almost universal. Explicit examples of Weyl type II almost universal Kundt metrics are also given. The considerable simplification of the field equations of higher-order gravity theories for almost universal spacetimes is then employed to study new Weyl type II, III, and N vacuum solutions to quadratic gravity in arbitrary dimension and six-dimensional conformal gravity. Necessary conditions for almost universal metrics are also studied.

CCNV Space-Times as Potential Supergravity Solutions

It is of interest to study supergravity solutions preserving a nonminimal fraction of supersymmetries. A necessary condition for supersymmetry to be preserved is that the space-time admits a Killing spinor and hence a null or time-like Killing vector field. Any space-time admitting a covariantly constant null vector (CCNV) field belongs to the Kundt class of metrics and more importantly admits a null Killing vector field. We investigate the existence of additional non-space-like isometries in the class of higher-dimensional CCNV Kundt metrics in order to produce potential solutions that preserve some supersymmetries.

Kerr-Schild–Kundt metrics are universal

We define (non-Einsteinian) universal metrics as the metrics that solve the source-free covariant field equations of generic gravity theories. Here, extending the rather scarce family of universal metrics known in the literature, we show that the Kerr-Schild–Kundt class of metrics are universal. Besides being interesting on their own, these metrics can provide consistent backgrounds for quantum field theory at extremely high energies.

Kundt's Metrics with Special Properties

Kundt's Metrics with Special Properties

Pseudo-Z symmetric space-times with divergence-free Weyl tensor and pp-waves

In this paper we present some new results about [Formula: see text]-dimensional pseudo-Z symmetric space-times. First we show that if the tensor Z satisfies the Codazzi condition then its rank is one, the space-time is a quasi-Einstein manifold, and the associated 1-form results to be null and recurrent. In the case in which such covector can be rescaled to a covariantly constant we obtain a Brinkmann-wave. Anyway the metric results to be a subclass of the Kundt metric. Next we investigate pseudo-Z symmetric space-times with harmonic conformal curvature tensor: a complete classification of such spaces is obtained. They are necessarily quasi-Einstein and represent a perfect fluid space-time in the case of time-like associated covector; in the case of null associated covector they represent a pure radiation field. Further if the associated covector is locally a gradient we get a Brinkmann-wave space-time for [Formula: see text] and a pp-wave space-time in [Formula: see text]. In all cases an algebraic classification for the Weyl tensor is provided for [Formula: see text] and higher dimensions. Then conformally flat pseudo-Z symmetric space-times are investigated. In the case of null associated covector the space-time reduces to a plane wave and results to be generalized quasi-Einstein. In the case of time-like associated covector we show that under the condition of divergence-free Weyl tensor the space-time admits a proper concircular vector that can be rescaled to a time like vector of concurrent form and is a conformal Killing vector. A recent result then shows that the metric is necessarily a generalized Robertson–Walker space-time. In particular we show that a conformally flat [Formula: see text], [Formula: see text], space-time is conformal to the Robertson–Walker space-time.

I-degenerate pseudo-Riemannian metrics

In this paper we study pseudo-Riemannian spaces with a degenerate curvature structure i.e. there exists a continuous family of metrics having identical polynomial curvature invariants. We approach this problem by utilising an idea coming from invariant theory. This involves the existence of a boost which is assumed to extend to a neighbourhood. This approach proves to be very fruitful: It produces a class of metrics containing all known examples of I-degenerate metrics. To date, only Kundt and Walker metrics have been given, however, our study gives a plethora of examples showing that I-degenerate metrics extend beyond the Kundt and Walker examples. The approach also gives a useful criterion for a metric to be I-degenerate. Specifically, we use this to study the subclass of VSI and CSI metrics (i.e., spaces where polynomial curvature invariants are all vanishing or constants, respectively).

Mathematical properties of a class of four-dimensional neutral signature metrics

While the Lorenzian and Riemanian metrics for which all polynomial scalar curvature invariants vanish (the VSI property) are well-studied, less is known about the four-dimensional neutral signature metrics with the VSI property. Recently it was shown that the neutral signature metrics belong to two distinct subclasses: the Walker and Kundt metrics. In this paper we have chosen an example from each of the two subcases of the Ricci-flat VSI Walker metrics respectively. To investigate the difference between the metrics we determine the existence of a null, geodesic, expansion-free, shear-free and vorticity-free vector, and classify these spaces using their infinitesimal holonomy algebras. The geometric implications of the holonomy algebras are further studied by identifying the recurrent or covariantly constant null vectors, whose existence is required by the holonomy structure in each example. We conclude the paper with a simple example of the equivalence algorithm for these pseudo-Riemannian manifolds, which is the only approach to classification that provides all necessary information to determine equivalence.

New exact solutions of quadratic curvature gravity

It is a known fact that the Kerr-Schild type solutions in general relativity satisfy both exact and linearized Einstein field equations. We show that this property remains valid also for a special class of the Kerr-Schild metrics in arbitrary dimensions in generic quadratic curvature theory. In addition to the anti-de Sitter (AdS) wave (or Siklos) metric which represents plane waves in an AdS background, we present here a new exact solution, in this class, to the quadratic gravity in $D$ dimensions which represents a spherical wave in an AdS background. The solution is a special case of the Kundt metrics belonging to spacetimes with constant curvature invariants.

Einstein–Weyl structures and de Sitter supergravity

The geometric structure of the null solutions of de Sitter D = 5 gauged supergravity coupled to vector multiplets is investigated. These solutions are Kundt metrics, constructed from a one-parameter family of three-dimensional Gauduchon–Tod base spaces. We give examples, including the near-horizon geometries previously found in Gutowski and Sabra (2011 J. High Energy Phys. JHEP05(2011)020). In addition, two special cases are considered in detail. In the first case, we consider solutions for which the Gauduchon–Tod base space is the Berger sphere. In the second case, we take the null 1-form Killing spinor bilinear to be recurrent, so that the holonomy of the Lévi-Cività connection is contained in Sim(3).

Pseudo-Riemannian VSI spaces II

In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of the polynomial curvature invariants vanish (VSI spaces). Using an algebraic classification of pseudo-Riemannian spaces in terms of the boost-weight decomposition, we first show more generally that a space which is not characterized by its invariants must possess the SG1-property. As a corollary, we then show that a VSI space must possess the NG-property (these results are the analogues of the alignment theorem, including corollaries, for Lorentzian spacetimes). As an application we classify all 4D neutral VSI spaces and show that these belong to one of two classes: (1) those that possess a geodesic, expansion-free, shear-free, and twist-free null congruence (Kundt metrics), or (2) those that possess an invariant null plane (Walker metrics). By explicit construction we show that the latter class contains a set of VSI metrics which have not previously been considered in the literature.

ISOMETRIES IN HIGHER-DIMENSIONAL CCNV SPACETIMES

We study the class of higher-dimensional Kundt metrics admitting a covariantly constant null vector, known as CCNV spacetimes. We pay particular attention to those CCNV spacetimes with constant (polynomial) curvature invariants (CSI). We investigate the existence of an additional isometry in CCNV spacetimes, by studying the Killing equations for the general form of the CCNV metric. In particular, we list all CCNV spacetimes allowing an additional non-spacelike isometry for all values of the light-cone coordinate v, which are of interest due to the invariance of the metric under a translation in v. As an application we use our results to find all CSICCNV spacetimes with an additional isometry as well as the subset of these spacetimes in which the isometry is non-spacelike for all values v.

On the existence of Kundt's metrics and degenerate (or extremal) Killing horizons

It is shown that Kundt's metric for vacuum cannot be constructed when two-dimensional spacelike sections of null hypersurfaces are compact, connected manifolds with no boundary unless they are tori or spheres, i.e. higher genus g ⩾ 2 is excluded by vacuum Einstein equations. The so-called basic equation (resulting from Einstein equations) is examined. This is a nonlinear PDE for an unknown covector field and an unknown Riemannian structure on the two-dimensional manifold. It implies several important results derived in this paper. It arises not only for Kundt's class but also for degenerate Killing horizons and vacuum degenerate isolated horizons.

Spacetimes characterized by their scalar curvature invariants

In this paper, we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an -non-degenerate spacetime metric, which implies that the spacetime metric is locally determined by its curvature invariants. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, we are able to prove a number of results (both in the algebraically general and in algebraically special cases) of when a spacetime metric is -non-degenerate. This enables us to prove our main theorem that a spacetime metric is either -non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of degenerate Kundt form. We then discuss the inverse question of what properties of the underlying spacetime can be determined from a given a set of scalar polynomial invariants, and some partial results are presented. We also discuss the notions of strong and weak non-degeneracy.