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Special conformal Killing tensors
Example
Special conformal Killing tensors
in general relativity
Liselotte De Groote
Department of Mathematical AnalysisGhent University
Belgium
Tuesday, April 06, 2009
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Special conformal Killing tensors
Example
Role
Study of
stationary flows of soliton equations
cofactor pair systems (= non-Lagrangian systems which
provide examples of completely integrable systems)
conditions under which one can find coordinates w.r.to
which Hamilton-Jacobi equation for geodesics is separable(= solvable by additive separation of variables)
L. De Groote Special conformal Killing tensors in GR
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Special conformal Killing tensors
Example
Role
Study of
projective equivalence (g, h, Riemannian manifolds areproj.equiv. if they have same geodesics up to reparametrization)
Lij =
det h
det g
1/(n+1)gikgjlh
kl
L is SCK of metric g, conversely given SCK L of g, followingdefines metric h projectively equivalent to g:
hij = (det L)1
gikgjlLkl
.
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Special conformal Killing tensors
Example
What?
Conformal Killing tensor (CK) L of metric g: symmetric type
(0, 2) tensor field if there is a 1-form such that
L(ij|k) = (igjk),
| denotes covariant derivative w.r.to Levi-Civita connection of g.
If exact: L is called CK of gradient type.
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Special conformal Killing tensors
Example
What?
Special conformal Killing tensor (SCK) L of metric g: symmetric
type (0, 2) tensor field if there is a 1-form such that
Lij|k =
1
2
igjk + jgik
.
trace on i,j = d (trL).
A SCK is a CK of gradient type.
Torsion of Nijenhuis tensor (of (1, 1)-type tensor) vanishes:important consequence!
L. De Groote Special conformal Killing tensors in GR
S i l f l Killi
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Special conformal Killing tensors
Example
What?
Torsion vanishes if eigenfunctions of L simple and functionally independent, orthonormal basis of TxM w.r.to which L(x) is diagonal:
at each point x of M, L(x) is an endomorphism of tangent spaceTxM;
n functions ui exist, such that ui(x) is eigenvalue of L(x) andJacobian ui/xj everywhere non-singular;
ui may be taken as local (orthogonal) coordinates:
L =
ui
ui dui
;
ui are orthogonal separation coordinates for Hamilton-Jacobiequation for the geodesics of the manifold.
L. De Groote Special conformal Killing tensors in GR
S i l f l Killi t
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Special conformal Killing tensors
Example
Propositions and theorems
a CK with functionally independent eigenfunctions, whose
torsion vanishes is a SCK;
w.r.to the coordinates adapted to a SCK with independenteigenfunctions Rijik = 0 for j= k (notice: Rijkl, components ofcurvature, with i,j, k, l all different vanish in anyorthogonal
coordinate system);
space of SCK is finite dimensional.
L. De Groote Special conformal Killing tensors in GR
Special conformal Killing tensors
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Special conformal Killing tensors
Example
Spaces of constant curvature
Space of SCK attains its maximal dimension (in (pseudo-)
Riemannian case) manifold is space of constantcurvature.
If a Riemannian manifold (M, g) admits two independentSCK L, K (no non-trivial common invariant subspaces), andif L has functionally independent eigenfunctions,
then (M, g) is a space of constant curvature.
L. De Groote Special conformal Killing tensors in GR
Special conformal Killing tensors
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Special conformal Killing tensors
Example
Killing tensors
If L is a (non-singular) SCK, and A is its cofactor tensor, then A
is a Killing tensor:
AijLj= (det L)gi;
Take covariant derivative and using definition
Aij|k = (det L)
LijLk
1
2LikLj
1
2LiLjk
.
If L is SCK, so is L + kg for any constant k
cofactor tensor A(k) of L + kI is a Killing tensor for every k.A(k) is a polynomial of degree (n 1) in k, whose coefficientsare tensors.
L. De Groote Special conformal Killing tensors in GR
Special conformal Killing tensors
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Special conformal Killing tensors
Example
Killing tensors
Let L, K be SCKs, then the cofactor of aL + bK is Killing, a, b(at least those for which aL + bK is non-singular).
If K = I and L has functionally independent eigenvectors, wegenerate from one SCK n independent Killing tensors, one of
which is g, another of which is cofactor L.
Eigenvectors of aL + bI are eigenvectors of L Killing tensorshave same eigenvectors, are simultaneously diagonal with L.
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Special conformal Killing tensors
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Special conformal Killing tensors
Example
Lacunae
There are lacunae in ranges of possible dimensions of spaces
of SCK on a Riemannian manifold.
theory of concircular vector fields provides model for
theory of SCK
X = I, = scalar
concircular vector fields can be used to construct SCK.
More details: Concircular vector fields and special conformal Killingtensors(Crampin)
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Special conformal Killing tensors
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Special conformal Killing tensors
Example
Petrov type D
Aligned SCK in Petrov type D space-time : (SCK = kg)
Newman-Penrose, Bianchi and SCK equations;
2 = 0 = i, i= 2;
SCK only m, m, k, l-contribution;
commutator equations.
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Special conformal Killing tensors
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p g
Example
Petrov type D
= = = = = = = = 0;
ij = 0 except 11;
2 = R12 , R= Ricci constant;
SCK is a constant Killing tensor.
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Special conformal Killing tensors
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p g
Example
Petrov type D
R constant : Bertotti-Robinson;
Subcase 11 = R8 Plebanski-Hacyan, Garcia-Plebanski.
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