236 research outputs found
Kerr-Schild spacetimes with (A)dS background
General properties of Kerr-Schild spacetimes with (A)dS background in
arbitrary dimension are studied. It is shown that the geodetic Kerr-Schild
vector k is a multiple WAND of the spacetime. Einstein Kerr-Schild spacetimes
with non-expanding k are shown to be of Weyl type N, while the expanding
spacetimes are of type II or D. It is shown that this class of spacetimes obeys
the optical constraint. This allows us to solve Sachs equation, determine
r-dependence of boost weight zero components of the Weyl tensor and discuss
curvature singularities.Comment: 17 pages, minor change
Some remarks on degenerate hypoelliptic Ornstein-Uhlenbeck operators
37 pages, 3 figuresInternational audienceWe study degenerate hypoelliptic Ornstein-Uhlenbeck operators in spaces with respect to invariant measures. The purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators. We first show that some known results about the spectral and subelliptic properties of Ornstein-Uhlenbeck operators may be directly recovered from the general analysis of quadratic operators with zero singular spaces. We also provide new resolvent estimates for hypoelliptic Ornstein-Uhlenbeck operators. We show in particular that the spectrum of these non-selfadjoint operators may be very unstable under small perturbations and that their resolvents can blow-up in norm far away from their spectra. Furthermore, we establish sharp resolvent estimates in specific regions of the resolvent set which enable us to prove exponential return to equilibrium
Generalization of the Geroch-Held-Penrose formalism to higher dimensions
Geroch, Held and Penrose invented a formalism for studying spacetimes
admitting one or two preferred null directions. This approach is very useful
for studying algebraically special spacetimes and their perturbations. In the
present paper, the formalism is generalized to higher-dimensional spacetimes.
This new formalism leads to equations that are considerably simpler than those
of the higher-dimensional Newman-Penrose formalism employed previously. The
dynamics of p-form test fields is analyzed using the new formalism and some
results concerning algebraically special p-form fields are proved.Comment: 24 page
Newman-Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting geodetic multiple Weyl aligned null direction
Vacuum spacetimes admitting a non-twisting geodetic multiple Weyl aligned
null direction (WAND) are analyzed in arbitrary dimension using recently
developed higher-dimensional Newman-Penrose (NP) formalism. We determine
dependence of the metric and of the Weyl tensor on the affine parameter r along
null geodesics generated by the WAND for type III and N spacetimes and for a
special class of type II and D spacetimes, containing e.g.
Schwarzschild-Tangherlini black holes and black strings and branes.
For types III and N, all metric components are at most quadratic polynomials
in r while for types II and D the r-dependence of the metric as well as of the
Weyl tensor is determined by an integer m corresponding to the rank of the
expansion matrix S_{ij}. It is shown that for non-vanishing expansion, all
these spacetimes contain a curvature singularity.
As an illustrative example, a shearing expanding type N five-dimensional
vacuum solution is also re-derived using higher-dimensional NP formalism. This
solution can be, however, identified with a direct product of a known
four-dimensional type N metric with an extra dimension.Comment: 25 pages, version to be published in Class. Quantum Grav. (expanded
-background material included, 3 references added, small change in notation
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
generalised pp-waves, VSI or CSI spacetimes, and gyratons.Comment: 15 page
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.
Ricci identities in higher dimensions
We explore connections between geometrical properties of null congruences and
the algebraic structure of the Weyl tensor in n>4 spacetime dimensions. First,
we present the full set of Ricci identities on a suitable "null" frame, thus
completing the extension of the Newman-Penrose formalism to higher dimensions.
Then we specialize to geodetic null congruences and study specific consequences
of the Sachs equations. These imply, for example, that Kundt spacetimes are of
type II or more special (like for n=4) and that for odd n a twisting geodetic
WAND must also be shearing (in contrast to the case n=4).Comment: 8 pages. v2: typo corrected between Propositions 2 and 3. v3: typo in
the last term in the first line of (11f) corrected, missing term on the
r.h.s. of (11p) added, first sentence between Propositions 2 and 3 slightly
change
Perturbations of higher-dimensional spacetimes
We discuss linearized gravitational perturbations of higher dimensional
spacetimes. For algebraically special spacetimes (e.g. Myers-Perry black
holes), we show that there exist local gauge invariant quantities linear in the
metric perturbation. These are the higher dimensional generalizations of the 4d
Newman-Penrose scalars that (in an algebraically special vacuum spacetime)
satisfy decoupled equations of motion. We show that decoupling occurs in more
than four dimensions if, and only if, the spacetime admits a null geodesic
congruence with vanishing expansion, rotation and shear. Decoupling of
electromagnetic perturbations occurs under the same conditions. Although these
conditions are not satisfied in black hole spacetimes, they are satisfied in
the near-horizon geometry of an extreme black hole.Comment: 21 pages (v2:Minor corrections, accepted by CQG.
On the algebraic classification of spacetimes
We briefly overview the Petrov classification in four dimensions and its
generalization to higher dimensions.Comment: Submitted to Journal of Physics, conference series, proceedings of
4th meeting on constrained dynamics and quantum gravity, 12-16 September
2005, Sardinia, Ital
- …