8 research outputs found
Complete Integrability of Geodesic Motion in General Kerr-NUT-AdS Spacetimes
We explicitly exhibit n-1 constants of motion for geodesics in the general
D-dimensional Kerr-NUT-AdS rotating black hole spacetime, arising from
contractions of even powers of the 2-form obtained by contracting the geodesic
velocity with the dual of the contraction of the velocity with the
(D-2)-dimensional Killing-Yano tensor. These constants of motion are
functionally independent of each other and of the D-n+1 constants of motion
that arise from the metric and the D-n = [(D+1)/2] Killing vectors, making a
total of D independent constants of motion in all dimensions D. The Poisson
brackets of all pairs of these D constants are zero, so geodesic motion in
these spacetimes is completely integrable.Comment: 4 pages. We have now found that the geodesic motion is not just
integrable, but completely integrabl
Killing-Yano Tensors, Rank-2 Killing Tensors, and Conserved Quantities in Higher Dimensions
From the metric and one Killing-Yano tensor of rank D-2 in any D-dimensional
spacetime with such a principal Killing-Yano tensor, we show how to generate
k=[(D+1)/2] Killing-Yano tensors, of rank D-2j for all j=0,...,k-1, and k
rank-2 Killing tensors, giving k constants of geodesic motion that are in
involution. For the example of the Kerr-NUT-AdS spacetime (hep-th/0604125) with
its principal Killing-Yano tensor (gr-qc/0610144), these constants and the
constants from the k Killing vectors give D independent constants in
involution, making the geodesic motion completely integrable (hep-th/0611083).
The constants of motion are also related to the constants recently obtained in
the separation of the Hamilton-Jacobi and Klein-Gordon equations
(hep-th/0611245).Comment: 7 page
Higher-Dimensional Black Holes: Hidden Symmetries and Separation of Variables
In this paper, we discuss hidden symmetries in rotating black hole
spacetimes. We start with an extended introduction which mainly summarizes
results on hidden symmetries in four dimensions and introduces Killing and
Killing-Yano tensors, objects responsible for hidden symmetries. We also
demonstrate how starting with a principal CKY tensor (that is a closed
non-degenerate conformal Killing-Yano 2-form) in 4D flat spacetime one can
"generate" 4D Kerr-NUT-(A)dS solution and its hidden symmetries. After this we
consider higher-dimensional Kerr-NUT-(A)dS metrics and demonstrate that they
possess a principal CKY tensor which allows one to generate the whole tower of
Killing-Yano and Killing tensors. These symmetries imply complete integrability
of geodesic equations and complete separation of variables for the
Hamilton-Jacobi, Klein-Gordon, and Dirac equations in the general
Kerr-NUT-(A)dS metrics.Comment: 33 pages, no figures, updated references and corrected typo