159 research outputs found
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 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, is a
multiple of the metric. A Ricci flat classical solution is called {\it strongly
universal} if, when evaluated on that Ricci flat metric,
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 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 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
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