60 research outputs found
The type N Karlhede bound is sharp
We present a family of four-dimensional Lorentzian manifolds whose invariant
classification requires the seventh covariant derivative of the curvature
tensor. The spacetimes in questions are null radiation, type N solutions on an
anti-de Sitter background. The large order of the bound is due to the fact that
these spacetimes are properly , i.e., curvature homogeneous of order 2
but non-homogeneous. This means that tetrad components of are constant, and that essential coordinates first appear as
components of . Covariant derivatives of orders 4,5,6 yield one
additional invariant each, and is needed for invariant
classification. Thus, our class proves that the bound of 7 on the order of the
covariant derivative, first established by Karlhede, is sharp. Our finding
corrects an outstanding assertion that invariant classification of
four-dimensional Lorentzian manifolds requires at most .Comment: 7 pages, typos corrected, added citation and acknowledgemen
Maximally inhomogeneous G\"{o}del-Farnsworth-Kerr generalizations
It is pointed out that physically meaningful aligned Petrov type D perfect
fluid space-times with constant zero-order Riemann invariants are either the
homogeneous solutions found by G\"{o}del (isotropic case) and Farnsworth and
Kerr (anisotropic case), or new inhomogeneous generalizations of these with
non-constant rotation. The construction of the line element and the local
geometric properties for the latter are presented.Comment: 4 pages, conference proceeding of Spanish Relativity Meeting (ERE
2009, Bilbao
Black rings with a small electric charge: gyromagnetic ratios and algebraic alignment
We study electromagnetic test fields in the background of vacuum black rings
using Killing vectors as vector potentials. We consider both spacetimes with a
rotating S^1 and with a rotating S^2 and we demonstrate, in particular, that
the gyromagnetic ratio of slightly charged black rings takes the value g=3
(this will in fact apply to a wider class of spacetimes). We also observe that
a S^2-rotating black ring immersed in an external "aligned" magnetic field
completely expels the magnetic flux in the extremal limit. Finally, we discuss
the mutual alignment of principal null directions of the Maxwell 2-form and of
the Weyl tensor, and the algebraic type of exact charged black rings. In
contrast to spherical black holes, charged rings display new distinctive
features and provide us with an explicit example of algebraically general (type
G) spacetimes in higher dimensions. Appendix A contains some global results on
black rings with a rotating 2-sphere. Appendix C shows that g=D-2 in any D>=4
dimensions for test electromagnetic fields generated by a time translation.Comment: 22 pages, 3 figures. v2: new appendix C finds the gyromagnetic ratio
g=D-2 in any dimensions, two new references. To appear in JHE
On Spacetimes with Constant Scalar Invariants
We study Lorentzian spacetimes for which all scalar invariants constructed
from the Riemann tensor and its covariant derivatives are constant (
spacetimes). We obtain a number of general results in arbitrary dimensions. We
study and construct warped product spacetimes and higher-dimensional
Kundt spacetimes. We show how these spacetimes can be constructed from
locally homogeneous spaces and spacetimes. The results suggest a number
of conjectures. In particular, it is plausible that for spacetimes that
are not locally homogeneous the Weyl type is , , or , with any
boost weight zero components being constant. We then consider the
four-dimensional spacetimes in more detail. We show that there are severe
constraints on these spacetimes, and we argue that it is plausible that they
are either locally homogeneous or that the spacetime necessarily belongs to the
Kundt class of spacetimes, all of which are constructed. The
four-dimensional results lend support to the conjectures in higher dimensions.Comment: 25 pages, 1 figure, v2: minor changes throughou
Realizations of Real Low-Dimensional Lie Algebras
Using a new powerful technique based on the notion of megaideal, we construct
a complete set of inequivalent realizations of real Lie algebras of dimension
no greater than four in vector fields on a space of an arbitrary (finite)
number of variables. Our classification amends and essentially generalizes
earlier works on the subject.
Known results on classification of low-dimensional real Lie algebras, their
automorphisms, differentiations, ideals, subalgebras and realizations are
reviewed.Comment: LaTeX2e, 39 pages. Essentially exetended version. Misprints in
Appendix are correcte
Invariant classification and the generalised invariant formalism: conformally flat pure radiation metrics, with zero cosmological constant
Metrics obtained by integrating within the generalised invariant formalism
are structured around their intrinsic coordinates, and this considerably
simplifies their invariant classification and symmetry analysis. We illustrate
this by presenting a simple and transparent complete invariant classification
of the conformally flat pure radiation metrics (except plane waves) in such
intrinsic coordinates; in particular we confirm that the three apparently
non-redundant functions of one variable are genuinely non-redundant, and easily
identify the subclasses which admit a Killing and/or a homothetic Killing
vector. Most of our results agree with the earlier classification carried out
by Skea in the different Koutras-McIntosh coordinates, which required much more
involved calculations; but there are some subtle differences. Therefore, we
also rework the classification in the Koutras-McIntosh coordinates, and by
paying attention to some of the subtleties involving arbitrary functions, we
are able to obtain complete agreement with the results obtained in intrinsic
coordinates. In particular, we have corrected and completed statements and
results by Edgar and Vickers, and by Skea, about the orders of Cartan
invariants at which particular information becomes available.Comment: Extended version of GRG publication, with some typos etc correcte
Generalizations of pp-wave spacetimes in higher dimensions
We shall investigate -dimensional Lorentzian spacetimes in which all of
the scalar invariants constructed from the Riemann tensor and its covariant
derivatives are zero. These spacetimes are higher-dimensional generalizations
of -dimensional pp-wave spacetimes, which have been of interest recently in
the context of string theory in curved backgrounds in higher dimensions.Comment: 5 pages, RevTex, to appear in Physical Review
Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension
We consider time reversal transformations to obtain twofold orthogonal
splittings of any tensor on a Lorentzian space of arbitrary dimension n.
Applied to the Weyl tensor of a spacetime, this leads to a definition of its
electric and magnetic parts relative to an observer (i.e., a unit timelike
vector field u), in any n. We study the cases where one of these parts vanishes
in particular, i.e., purely electric (PE) or magnetic (PM) spacetimes. We
generalize several results from four to higher dimensions and discuss new
features of higher dimensions. We prove that the only permitted Weyl types are
G, I_i and D, and discuss the possible relation of u with the WANDs; we provide
invariant conditions that characterize PE/PM spacetimes, such as Bel-Debever
criteria, or constraints on scalar invariants, and connect the PE/PM parts to
the kinematic quantities of u; we present conditions under which direct product
spacetimes (and certain warps) are PE/PM, which enables us to construct
explicit examples. In particular, it is also shown that all static spacetimes
are necessarily PE, while stationary spacetimes (e.g., spinning black holes)
are in general neither PE nor PM. Ample classes of PE spacetimes exist, but PM
solutions are elusive, and we prove that PM Einstein spacetimes of type D do
not exist, for any n. Finally, we derive corresponding results for the
electric/magnetic parts of the Riemann tensor. This also leads to first
examples of PM spacetimes in higher dimensions. We also note in passing that
PE/PM Weyl tensors provide examples of minimal tensors, and we make the
connection hereof with the recently proved alignment theorem. This in turn
sheds new light on classification of the Weyl tensors based on null alignment,
providing a further invariant characterization that distinguishes the types
G/I/D from the types II/III/N.Comment: 43 pages. v2: new proposition 4.10; some text reshuffled (former sec.
2 is now an appendix); references added; some footnotes cancelled, others
incorporated into the main text; some typos fixed and a few more minor
changes mad
From Navier-Stokes To Einstein
We show by explicit construction that for every solution of the
incompressible Navier-Stokes equation in dimensions, there is a uniquely
associated "dual" solution of the vacuum Einstein equations in
dimensions. The dual geometry has an intrinsically flat timelike boundary
segment whose extrinsic curvature is given by the stress tensor of
the Navier-Stokes fluid. We consider a "near-horizon" limit in which
becomes highly accelerated. The near-horizon expansion in gravity is shown to
be mathematically equivalent to the hydrodynamic expansion in fluid dynamics,
and the Einstein equation reduces to the incompressible Navier-Stokes equation.
For , we show that the full dual geometry is algebraically special Petrov
type II. The construction is a mathematically precise realization of
suggestions of a holographic duality relating fluids and horizons which began
with the membrane paradigm in the 70's and resurfaced recently in studies of
the AdS/CFT correspondence.Comment: 15 pages, 2 figures, typos correcte
- …