Kundt Class Research Articles

On the metrizability ofm-Kropina spaces with closed null one-form

We investigate the local metrizability of Finsler spaces with m-Kropina metric F = α1+mβ−m, where β is a closed null one-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric α and one-form β have a very specific form in certain coordinates. In particular, when the signature of α is Lorentzian, α belongs to a certain subclass of the Kundt class and β generates the corresponding null congruence, and this generalizes in a natural way to arbitrary signature. We use this result to prove that the affine connection on such an m-Kropina space is locally metrizable by a (pseudo-)Riemannian metric if and only if the Ricci tensor constructed from the affine connection is symmetric. In particular, we construct all counterexamples of this type to Szabo’s metrization theorem, which has only been proven for positive definite Finsler metrics that are regular on all of the slit tangent bundle.

All solutions of Einstein-Maxwell equations with a cosmological constant in 2+1 dimensions

We present a general solution of the coupled Einstein-Maxwell field equations (without the source charges and currents) in three spacetime dimensions. We also admit any value of the cosmological constant. The whole family of such $\mathrm{\ensuremath{\Lambda}}$-electrovacuum local solutions splits into two distinct subclasses, namely the nonexpanding Kundt class and the expanding Robinson-Trautman class. While the Kundt class only admits electromagnetic fields which are aligned along the geometrically privileged null congruence, the Robinson-Trautman class admits both aligned and also more complex nonaligned Maxwell fields. We derive all the metric and Maxwell field components, together with explicit constraints imposed by the field equations. We also identify the most important special spacetimes of this type, namely the coupled gravitational-electromagnetic waves and charged black holes.

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.

Einstein-Maxwell fields with vanishing higher-order corrections

We obtain a full characterization of Einstein-Maxwell $p$-form solutions $(\boldsymbol{g},\boldsymbol{F})$ in $D$-dimensions for which all higher-order corrections vanish identically. These thus simultaneously solve a large class of Lagrangian theories including both modified gravities and (possibly non-minimally coupled) modified electrodynamics. Specifically, both $\boldsymbol{g}$ and $\boldsymbol{F}$ are fields with vanishing scalar invariants and further satisfy two simple tensorial conditions. They describe a family of gravitational and electromagnetic plane-fronted waves of the Kundt class and of Weyl type III (or more special). The local form of $(\boldsymbol{g},\boldsymbol{F})$ and a few examples are also provided.

All solutions of Einstein’s equations in 2+1 dimensions: Λ-vacuum, pure radiation, or gyratons

Under a weak assumption of the existence of a geodesic null congruence, we present the general solution of the Einstein field equations in three dimensions with any value of the cosmological constant, admitting an aligned null matter field, and also gyratons (a matter field in the form of a null dust with an additional internal spin). The general local solution consists of the expanding Robinson–Trautman class and the non-expanding Kundt class. The gyratonic solutions reduce to spacetimes with a pure radiation matter field when the spin is set to zero. Without matter fields, we obtain new forms of the maximally symmetric vacuum solutions. We discuss these complete classes of solutions and their various subclasses. In particular, we identify the gravitational field of an arbitrarily accelerating source (the Kinnersley photon rocket, which reduces to a Vaidya-type non-moving object) in the Robinson–Trautman class, and pp-waves, vanishing scalar invariants (VSI) spacetimes, and constant scalar invariants (CSI) spacetimes in the Kundt class.

Ultrarelativistic boost of a black hole in the magnetic universe of Levi-Civita–Bertotti–Robinson

We consider an exact Einstein-Maxwell solution constructed by Alekseev and Garcia which describes a Schwarzschild black hole immersed in the magnetic universe of Levi-Civita, Bertotti and Robinson (LCBR). After reviewing the basic properties of this spacetime, we study the ultrarelativistic limit in which the black hole is boosted to the speed of light, while sending its mass to zero. This results in a non-expanding impulsive wave traveling in the LCBR universe. The wave front is a 2-sphere carrying two null point particles at its poles -- a remnant of the structure of the original static spacetime. It is also shown that the obtained line-element belongs to the Kundt class of spacetimes, and the relation with a known family of exact gravitational waves of finite duration propagating in the LCBR background is clarified. In the limit of a vanishing electromagnetic field, one point particle is pushed away to infinity and the single-particle Aichelburg-Sexl pp-wave propagating in Minkowski space is recovered.

A note on Kundt spacetimes of type N with a cosmological constant

In recent literature there appeared conflicting claims about whether the Ozsváth–Robinson–Rózga family of type N electrovac spacetimes of the Kundt class with Λ is complete. We show that indeed it is.

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.

Kundt spacetimes minimally coupled to scalar field

We derive an exact solution belonging to the Kundt class of spacetimes both with and without a cosmological constant that are minimally coupled to a free massless scalar field. We show the algebraic type of these solutions and give interpretation of the results. Subsequently, we look for solutions additionally containing an electromagnetic field satisfying nonlinear field equations.

Weyl type N solutions with null electromagnetic fields in the Einstein–Maxwell p-form theory

We consider $d$-dimensional solutions to the electrovacuum Einstein-Maxwell equations with the Weyl tensor of type N and a null Maxwell $(p+1)$-form field. We prove that such spacetimes are necessarily aligned, i.e. the Weyl tensor of the corresponding spacetime and the electromagnetic field share the same \textit{aligned null direction} (AND). Moreover, this AND is geodetic, shear-free, non-expanding and non-twisting and hence Einstein-Maxwell equations imply that Weyl type N spacetimes with a null Maxwell $(p+1)$-form field belong to the Kundt class. Moreover, these Kundt spacetimes are necessarily $CSI$ and the $(p+1)$ form is $VSI$. Finally, a general coordinate form of solutions and a reduction of the field equations are discussed.

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.

Extended Kerr—Schild spacetimes of any dimension

We study geometric and algebraic properties of extended Kerr-Schild space times(xKS), i.e. an extension of the Kerr-Schild(KS) ansatz where, in addition to the null KS vector, a spacelike vector field appears in the metric. In contrast to the KS case, it turns out that xKS space times with a geodetic KS vector are not necessarily algebraically special and we obtain, in general, only a necessary condition under which the KS vector is geodetic. However, it is shown that this condition becomes sufficient if we appropriately restrict the geometry of the null and spacelike vector fields. Examples of xKS space times belonging to the Kundt class and also expanding xKS space times, namely the CCLP black hole, are provided and briefly discussed.

Absence of gyratons in the Robinson-Trautman class

We present the Riemann and Ricci tensors for a fully general nontwisting and shear-free geometry in arbitrary dimension $D$. This includes both the nonexpanding Kundt and expanding Robinson--Trautman family of spacetimes. As an interesting application of these explicit expressions, we then integrate the Einstein equations and prove a surprising fact that in any $D$ the Robinson--Trautman class does not admit solutions representing gyratonic sources, i.e., a matter field in the form of a null fluid (or particles propagating with the speed of light) with an additional internal spin. Contrary to the closely related Kundt class and $pp$-waves, the corresponding off-diagonal metric components thus do not encode the angular momentum of some gyraton. Instead, we demonstrate that in standard $D=4$ general relativity they directly determine two independent amplitudes of the Robinson--Trautman exact gravitational waves.

Explicit algebraic classification of Kundt geometries in any dimension

We present an algebraic classification, based on the null alignment properties of the Weyl tensor, of the general Kundt class of spacetimes in arbitrary dimension D for which the non-expanding, non-twisting, shear-free null direction k is a (multiple) Weyl aligned null direction (WAND). No field equations are used, so that the results apply not only to Einstein’s gravity and its direct extension to higher dimensions, but also to any metric theory of gravity which admits the Kundt spacetimes. By an explicit evaluation of the Weyl tensor in a natural null frame we demonstrate that all Kundt geometries are of type I(b) or more special, and we derive simple necessary and sufficient conditions under which k becomes a double, triple or quadruple WAND. All possible algebraically special types, including the refinement to subtypes, are identified, namely II(a), II(b), II(c), II(d), III(a), III(b), N, O, IIi, IIIi, D(a), D(b), D(c) and D(d). The corresponding conditions are surprisingly clear and expressed in an invariant geometric form. Some of them are always satisfied in four dimensions. To illustrate our classification scheme, we apply it to the most important subfamilies of the Kundt class, namely the pp-waves, the VSI spacetimes, and the generalization of the Bertotti–Robinson, Nariai, and Plebański–Hacyan direct-product spacetimes of any dimension.

Types III andNsolutions to quadratic gravity

We study exact vacuum solutions to quadratic gravity (QG) of the Weyl types N and III. We show that in an arbitrary dimension all Einstein spacetimes of the Weyl type N with an appropriately chosen effective cosmological constant $\Lambda$ are exact solutions to QG and we refer to explicitly known metrics within this class. For type III Einstein spacetimes, an additional constraint follows from the field equations of QG and examples of spacetimes obeying such constraint are given. However, type III pp-waves do not satisfy this constraint and thus do not solve QG. For type N, we also study a wider class of spacetimes admitting a pure radiation term in the Ricci tensor. In contrast to the Einstein case, the field equations of generic QG determine optical properties of the geometry and restrict such exact solutions to the Kundt class. We provide examples of these metrics.

General Kundt spacetimes in higher dimensions

We investigate a general metric of the Kundt class of spacetimes in higher dimensions. Geometrically, it admits a non-twisting, non-shearing and non-expanding geodesic null congruence. We calculate all components of the curvature and Ricci tensors, without assuming any specific matter content, and discuss algebraic types and main geometric constraints imposed by general Einstein's field equations. We explicitly derive Einstein–Maxwell equations, including an arbitrary cosmological constant, in the case of vacuum or possibly an aligned electromagnetic field. Finally, we introduce canonical subclasses of the Kundt family and we identify the most important special cases, namely generalized pp-waves, VSI or CSI spacetimes, and gyratons.

SUPERGRAVITY SOLUTIONS WITH CONSTANT SCALAR INVARIANTS

We study a class of constant scalar invariant (CSI) space–times which belong to the higher-dimensional Kundt class and which are solutions of supergravity. We review the known CSI supergravity solutions in this class and we explicitly present a number of new exact CSI supergravity solutions, some of which are Einstein.

Holonomy, decomposability, and relativity1This paper was presented at the Theory CANADA 4 conference, held at Centre de recherches mathématiques, Montréal, Québec, Canada on 4–7 June 2008.

We discuss the geometrical properties of spacetimes in the context of higher dimensional theories of gravity. If the spacetime admits a covariantly constant time-like vector, the spacetime is static and (1+10)-decomposable, where the 10-dimensional transverse space is Riemannian. The second class of solutions consists of spacetimes that admit a covariantly constant light-like vector and belong to the higher dimensional Kundt class of spacetimes. These spacetimes are genuinely Lorentzian and have many mathematical properties quite different from their Riemannian counterparts, which can lead to interesting and novel physics. This paper is a review of results published elsewhere (J. Brannlund, A. Coley, and S. Hervik. Class. Quantum Grav. 25, 195007 (2008)).

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.

Supersymmetry, holonomy and Kundt spacetimes

Supersymmetric solutions of supergravity theories, and consequently metrics with special holonomy, have played an important role in the development of string theory. We describe how a Lorentzian manifold is either completely reducible, and thus essentially known, or not completely reducible so that there exists a degenerate holonomy invariant lightlike subspace and consequently admits a covariantly constant or a recurrent null vector and belongs to the higher-dimensional Kundt class of spacetimes. These Kundt spacetimes (which contain the vanishing and constant curvature invariant spacetimes as special cases) are genuinely Lorentzian and have a number of interesting and unusual properties, which may lead to novel and fundamental physics.