Note on Invariants of the Weyl Tensor

Algebraically special gravitational fields are described using algebraic and differential invariants of the Weyl tensor. A type III invariant is also given and calculated for Robinson-Trautman spaces.Comment: 3 pages, no figures, corrected expression (12

On higher dimensional Einstein spacetimes with a warped extra dimension

We study a class of higher dimensional warped Einstein spacetimes with one extra dimension. These were originally identified by Brinkmann as those Einstein spacetimes that can be mapped conformally on other Einstein spacetimes, and have subsequently appeared in various contexts to describe, e.g., different braneworld models or warped black strings. After clarifying the relation between the general Brinkmann metric and other more specific coordinate systems, we analyze the algebraic type of the Weyl tensor of the solutions. In particular, we describe the relation between Weyl aligned null directions (WANDs) of the lower dimensional Einstein slices and of the full spacetime, which in some cases can be algebraically more special. Possible spacetime singularities introduced by the warp factor are determined via a study of scalar curvature invariants and of Weyl components measured by geodetic observers. Finally, we illustrate how Brinkmann's metric can be employed to generate new solutions by presenting the metric of spinning and accelerating black strings in five dimensional anti-de Sitter space.Comment: 14 pages, minor changes in the text, mainly in Section 2.

Vanishing Scalar Invariant Spacetimes in Higher Dimensions

We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher-dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.Comment: final versio

Alignment and algebraically special tensors in Lorentzian geometry

We develop a dimension-independent theory of alignment in Lorentzian geometry, and apply it to the tensor classification problem for the Weyl and Ricci tensors. First, we show that the alignment condition is equivalent to the PND equation. In 4D, this recovers the usual Petrov types. For higher dimensions, we prove that, in general, a Weyl tensor does not possess aligned directions. We then go on to describe a number of additional algebraic types for the various alignment configurations. For the case of second-order symmetric (Ricci) tensors, we perform the classification by considering the geometric properties of the corresponding alignment variety.Comment: 19 pages. Revised presentatio

Spinor classification of the Weyl tensor in five dimensions

We investigate the spinor classification of the Weyl tensor in five dimensions due to De Smet. We show that a previously overlooked reality condition reduces the number of possible types in the classification. We classify all vacuum solutions belonging to the most special algebraic type. The connection between this spinor and the tensor classification due to Coley, Milson, Pravda and Pravdov\'a is investigated and the relation between most of the types in each of the classifications is given. We show that the black ring is algebraically general in the spinor classification.Comment: 40 page

Curvature invariants in type N spacetimes

Scalar curvature invariants are studied in type N solutions of vacuum Einstein's equations with in general non-vanishing cosmological constant Lambda. Zero-order invariants which include only the metric and Weyl (Riemann) tensor either vanish, or are constants depending on Lambda. Even all higher-order invariants containing covariant derivatives of the Weyl (Riemann) tensor are shown to be trivial if a type N spacetime admits a non-expanding and non-twisting null geodesic congruence. However, in the case of expanding type N spacetimes we discover a non-vanishing scalar invariant which is quartic in the second derivatives of the Riemann tensor. We use this invariant to demonstrate that both linearized and the third order type N twisting solutions recently discussed in literature contain singularities at large distances and thus cannot describe radiation fields outside bounded sources.Comment: 17 pages, to appear in Class. Quantum Gra

Higher dimensional VSI spacetimes

We present the explicit metric forms for higher dimensional vanishing scalar invariant (VSI) Lorentzian spacetimes. We note that all of the VSI spacetimes belong to the higher dimensional Kundt class. We determine all of the VSI spacetimes which admit a covariantly constant null vector, and we note that in general in higher dimensions these spacetimes are of Ricci type III and Weyl type III. The Ricci type N subclass is related to the chiral null models and includes the relativistic gyratons and the higher dimensional pp-wave spacetimes. The spacetimes under investigation are of particular interest since they are solutions of supergravity or superstring theory.Comment: 14 pages, changes in second paragraph of the discussio

Robinson-Trautman spacetimes in higher dimensions

As an extension of the Robinson-Trautman solutions of D=4 general relativity, we investigate higher dimensional spacetimes which admit a hypersurface orthogonal, non-shearing and expanding geodesic null congruence. Einstein's field equations with an arbitrary cosmological constant and possibly an aligned pure radiation are fully integrated, so that the complete family is presented in closed explicit form. As a distinctive feature of higher dimensions, the transverse spatial part of the general line element must be a Riemannian Einstein space, but it is otherwise arbitrary. On the other hand, the remaining part of the metric is - perhaps surprisingly - not so rich as in the standard D=4 case, and the corresponding Weyl tensor is necessarily of algebraic type D. While the general family contains (generalized) static Schwarzschild-Kottler-Tangherlini black holes and extensions of the Vaidya metric, there is no analogue of important solutions such as the C-metric.Comment: 11 page

Metrics With Vanishing Quantum Corrections

We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor $T_{\mu \nu}$ constructed from sums of terms the involving contractions of the metric and powers of arbitrary covariant derivatives of the curvature tensor. A classical solution, such as an Einstein metric, is called {\it universal} if, when evaluated on that Einstein metric, $T_{\mu \nu}$ is a multiple of the metric. A Ricci flat classical solution is called {\it strongly universal} if, when evaluated on that Ricci flat metric, $T_{\mu \nu}$ vanishes. It is well known that pp-waves in four spacetime dimensions are strongly universal. We focus attention on a natural generalisation; Einstein metrics with holonomy ${\rm Sim} (n-2)$ in which all scalar invariants are zero or constant. In four dimensions we demonstrate that the generalised Ghanam-Thompson metric is weakly universal and that the Goldberg-Kerr metric is strongly universal; indeed, we show that universality extends to all 4-dimensional ${\rm Sim}(2)$ Einstein metrics. We also discuss generalizations to higher dimensions.Comment: 23 page

Classification of Higher Dimensional Spacetimes

We algebraically classify some higher dimensional spacetimes, including a number of vacuum solutions of the Einstein field equations which can represent higher dimensional black holes. We discuss some consequences of this work.Comment: 16 pages, 1 Tabl