321 research outputs found
Hidden Symmetries of Higher Dimensional Black Holes and Uniqueness of the Kerr-NUT-(A)dS spacetime
We prove that the most general solution of the Einstein equations with the
cosmological constant which admits a principal conformal Killing-Yano tensor is
the Kerr-NUT-(A)dS metric. Even when the Einstein equations are not imposed,
any spacetime admitting such hidden symmetry can be written in a canonical form
which guarantees the following properties: it is of the Petrov type D, it
allows the separation of variables for the Hamilton-Jacobi, Klein-Gordon, and
Dirac equations, the geodesic motion in such a spacetime is completely
integrable. These results naturally generalize the results obtained earlier in
four dimensions.Comment: 5 pages, no figure
Parallel-propagated frame along null geodesics in higher-dimensional black hole spacetimes
In [arXiv:0803.3259] the equations describing the parallel transport of
orthonormal frames along timelike (spacelike) geodesics in a spacetime
admitting a non-degenerate principal conformal Killing-Yano 2-form h were
solved. The construction employed is based on studying the Darboux subspaces of
the 2-form F obtained as a projection of h along the geodesic trajectory. In
this paper we demonstrate that, although slightly modified, a similar
construction is possible also in the case of null geodesics. In particular, we
explicitly construct the parallel-transported frames along null geodesics in
D=4,5,6 Kerr-NUT-(A)dS spacetimes. We further discuss the parallel transport
along principal null directions in these spacetimes. Such directions coincide
with the eigenvectors of the principal conformal Killing-Yano tensor. Finally,
we show how to obtain a parallel-transported frame along null geodesics in the
background of the 4D Plebanski-Demianski metric which admits only a conformal
generalization of the Killing-Yano tensor.Comment: 17 pages, no figure
Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets
In this paper we derive the most general first-order symmetry operator
commuting with the Dirac operator in all dimensions and signatures. Such an
operator splits into Clifford even and Clifford odd parts which are given in
terms of odd Killing-Yano and even closed conformal Killing-Yano inhomogeneous
forms respectively. We study commutators of these symmetry operators and give
necessary and sufficient conditions under which they remain of the first-order.
In this specific setting we can introduce a Killing-Yano bracket, a bilinear
operation acting on odd Killing-Yano and even closed conformal Killing-Yano
forms, and demonstrate that it is closely related to the Schouten-Nijenhuis
bracket. An important non-trivial example of vanishing Killing-Yano brackets is
given by Dirac symmetry operators generated from the principal conformal
Killing-Yano tensor [hep-th/0612029]. We show that among these operators one
can find a complete subset of mutually commuting operators. These operators
underlie separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes in all
dimensions [arXiv:0711.0078].Comment: 37 pages, no figure
Bi-Legendrian manifolds and paracontact geometry
We study the interplays between paracontact geometry and the theory of
bi-Legendrian manifolds. We interpret the bi-Legendrian connection of a
bi-Legendrian manifold M as the paracontact connection of a canonical
paracontact structure induced on M and then we discuss many consequences of
this result both for bi-Legendrian and for paracontact manifolds. Finally new
classes of examples of paracontact manifolds are presented.Comment: to appear in Int. J. Geom. Meth. Mod. Phy
Do Killing-Yano tensors form a Lie Algebra?
Killing-Yano tensors are natural generalizations of Killing vectors. We
investigate whether Killing-Yano tensors form a graded Lie algebra with respect
to the Schouten-Nijenhuis bracket. We find that this proposition does not hold
in general, but that it does hold for constant curvature spacetimes. We also
show that Minkowski and (anti)-deSitter spacetimes have the maximal number of
Killing-Yano tensors of each rank and that the algebras of these tensors under
the SN bracket are relatively simple extensions of the Poincare and (A)dS
symmetry algebras.Comment: 17 page
3-quasi-Sasakian manifolds
In the present paper we carry on a systematic study of 3-quasi-Sasakian
manifolds. In particular we prove that the three Reeb vector fields generate an
involutive distribution determining a canonical totally geodesic and Riemannian
foliation. Locally, the leaves of this foliation turn out to be Lie groups:
either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian
manifolds have a well-defined rank, obtaining a rank-based classification.
Furthermore, we prove a splitting theorem for these manifolds assuming the
integrability of one of the almost product structures. Finally, we show that
the vertical distribution is a minimum of the corrected energy.Comment: 17 pages, minor modifications, references update
A Deformation of Sasakian Structure in the Presence of Torsion and Supergravity Solutions
We discuss a deformation of Sasakian structure in the presence of totally
skew-symmetric torsion by introducing odd dimensional manifolds whose metric
cones are K\"ahler with torsion. It is shown that such a geometry inherits
similar properties to those of Sasakian geometry. As an example of them, we
present an explicit expression of local metrics and see how Sasakian structure
is deformed by the presence of torsion. We also demonstrate that our example of
the metrics admits the existence of hidden symmetries described by non-trivial
odd-rank generalized closed conformal Killing-Yano tensors. Furthermore, using
these metrics as an {\it ansatz}, we construct exact solutions in five
dimensional minimal (un-)gauged supergravity and eleven dimensional
supergravity. Finally, we discuss the global structures of the solutions and
obtain regular metrics on compact manifolds in five dimensions, which give
natural generalizations of Sasaki--Einstein manifolds and
. We also discuss regular metrics on non-compact manifolds in eleven
dimensions.Comment: 38 pages, 1 table, v2: version to appear in Class. Quant. Gra
Conformal Yano-Killing tensor for the Kerr metric and conserved quantities
Properties of (skew-symmetric) conformal Yano--Killing tensors are reviewed.
Explicit forms of three symmetric conformal Killing tensors in Kerr spacetime
are obtained from the Yano--Killing tensor. The relation between spin-2 fields
and solutions to the Maxwell equations is used in the construction of a new
conserved quantity which is quadratic in terms of the Weyl tensor. The formula
obtained is similar to the functional obtained from the Bel--Robinson tensor
and is examined in Kerr spacetime. A new interpretation of the conserved
quantity obtained is proposed.Comment: 29 page
No more CKY two-forms in the NHEK
We show that in the near-horizon limit of a Kerr-NUT-AdS black hole, the
space of conformal Killing-Yano two-forms does not enhance and remains of
dimension two. The same holds for an analogous polar limit in the case of
extremal NUT charge. We also derive the conformal Killing-Yano -form
equation for any background in arbitrary dimension in the form of parallel
transport.Comment: 36 pages, 12 pdf figures, v2: minor change
Hidden symmetries and Killing tensors on curved spaces
Higher order symmetries corresponding to Killing tensors are investigated.
The intimate relation between Killing-Yano tensors and non-standard
supersymmetries is pointed out. In the Dirac theory on curved spaces,
Killing-Yano tensors generate Dirac type operators involved in interesting
algebraic structures as dynamical algebras or even infinite dimensional
algebras or superalgebras. The general results are applied to space-times which
appear in modern studies. One presents the infinite dimensional superalgebra of
Dirac type operators on the 4-dimensional Euclidean Taub-NUT space that can be
seen as a twisted loop algebra. The existence of the conformal Killing-Yano
tensors is investigated for some spaces with mixed Sasakian structures.Comment: 12 pages; talk presented at Group 27 Colloquium, Yerevan, Armenia,
August 200
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