Killing Tensor Fields and Killing Forms| Killing Tensor Fields and Killing Forms |
In General
> s.a. killing vector fields / conservation laws.
$ Killing tensor: An n-th rank
Killing tensor is a symmetric covariant tensor, Kab...
c = K(ab... c), such that
∇(m Kab...
c) = 0.
$ Killing form: A differential
form ωab... c
= ω[ab...
c] whose covariant derivative is totally skew-symmetric.
* Conserved quantities:
While Killing vectors give the linear first integrals of the geodesic equations,
Killing tensors give the quadratic, cubic, and higher-order first integrals;
For every Killing tensor Kab... c,
um ∇m
(Kab... c
ua ub
··· uc) = 0,
if ua is tangent to
affinely parametrized geodesics.
* Example: The metric itself
is always a rank-2 Killing tensor; The associated conserved quantity is
the norm squared gab
uaub
= uaua.
* Applications: Integrability
of geodesics in Kerr-Newman spacetime.
@ General references:
Sommers JMP(73) [and particle constants of motion];
Dolan et al GRG(89) [significance];
Benn JMP(06) [and mechanics];
Coll et al JMP(06)gq [spectral decomposition];
Garfinkle & Glass CQG(10)-a1003 [method, in spacetimes with symmetries];
Cariglia et al CQG(14);
Houri et al CQG(18)-a1704 [integrability].
@ Special types of manifolds:
Rosquist & Uggla JMP(91) [2D spacetimes];
Smirnov & Yue JMP(04)mp [constant curvature pseudo-Riemannian];
Belgun et al DG&A(06) [symmetric spaces];
Cariglia & Galajinsky PLB(15)-a1503 [Ricci-flat spacetimes];
Vollmer JGP(17)-a1602 [in stationary and axisymmetric spacetimes].
@ From conformal Killing vectors:
Koutras CQG(92);
Barnes et al gq/02-proc;
Rani et al CQG(03).
@ Second-rank:
Walker & Penrose CMP(70) [Kerr spacetime];
{Bombelli & Rosquist};
Baleanu gq/98,
NC-gq/98;
Chanu et al JMP(06) [2D flat manifold];
Brink PRD(10);
Oota & Yasui IJMPA(10)
[generalized Kerr-NUT-de Sitter spacetime];
Keane & Tupper CQG(10) [pp-wave spacetimes].
@ Third-rank: Rosquist & Goliath GRG(98);
Karlovini & Rosquist GRG(99)gq/98 [1+1 dimensions];
Baleanu G&C(99).
Killing-Yano Tensors / Forms > s.a. Taub-NUT Metric.
$ Def: An n-th rank
Killing-Yano tensor is an n-form ηab...
c, such that ∇(m
ηa)b... c = 0.
* Relationships: A Killing 2-tensor can
be defined from a Killing-Yano tensor by Kmn:=
ηmb... c
ηnb... c.
* Example: The alternating
tensor εab... c
is a Killing-Yano tensor; The corresponding rank-2 Killing tensor is (proportional to) the metric.
@ General references: Yano AM(52);
Kastor et al CQG(07)-a0705
[conditions for graded Lie algebra with respect to the Schouten-Nijenhuis bracket];
Batista CQG(14)-a1405 [Killing-Yano tensors of order n – 1].
@ In gravitation:
Dietz & Rüdiger PRS(81),
PRS(82);
Hall IJTP(87);
Baleanu NCB(99)gq/98 [and Nambu tensors];
Ferrando & Sáez GRG(03) [Rainich problem];
Açık et al GRG(10) [and conserved gravitational currents].
@ Special types of spacetimes:
Howarth & Collinson GRG(00) [spherical static];
Jezierski & Lukasik CQG(06) [Kerr];
Kubizňák a0909-proc [black holes];
Acik et al JMP(10) [spherically symmetric];
Houri et al CQG(12) [with torsion, classification];
Garfinkle & Glass JMP(13)-a1302 [spacetimes admitting a hypersurface-orthogonal Killing vector];
Frolov et al PRD(18)-a1712;
> s.a. kerr spacetime.
@ Conformal Killing-Yano tensors: Jezierski APPB(08)-a0705 [asymptotically AdS];
Kubizňák & Krtouš PRD(07)-a0707 [for Plebański-Demiański type-D solutions].
Killing Spinors
* Examples: Manifolds with
Killing spinors include nearly Kähler 6-manifolds, nearly parallel
G2-manifolds in dimension 7,
Sasaki-Einstein manifolds, and 3-Sasakian manifolds.
@ General references:
Baum m.DG/02 [conformal];
Bohle JGP(03) [on Lorentzian manifolds];
in Cariglia CQG(04)ht/03 [and Yano tensors];
Harland & Nölle JHEP(12) [instantons on manifolds with Killing spinors];
Cole & Valiente Kroon CQG(16)-a1601 [implications of their existence];
Gutowski & Sabra a1905 [in 4D supergravity];
Cortés et al a1911 [new framework].
@ Specific types of spacetimes: Van den Bergh CQG(10)-a0908 [spacetimes admitting non-null valence-two Killing spinors];
Van den Bergh CQG(11) [homogeneous Petrov-type D Killing spinor spacetimes];
Batista PRD(16)-a1512 [in 6D spacetime].
> Online resources:
see Wikipedia page.
Other Variations and Generalizations
$ Killing form on a Lie algebra:
Given a Lie algebra \(\cal G\), its Killing form is the bilinear form
B(X, Y) = tr[Ad(X), Ad(Y)].
@ Generalized Killing tensors:
Collinson & Howarth GRG(00);
Howe & Lindström JHEP(16)-a1511 [in superspace].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
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