special conformal killing tensors in general relativity

7/29/2019 Special Conformal Killing Tensors in General Relativity

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Special conformal Killing tensors

Example


in general relativity

Liselotte De Groote

Department of Mathematical AnalysisGhent University

Belgium

Tuesday, April 06, 2009

L. De Groote Special conformal Killing tensors in GR
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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)

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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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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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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!


S i l f l Killi
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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.


S i l f l Killi t
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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.


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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.


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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.


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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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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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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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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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p g

Example

Petrov type D

R constant : Bertotti-Robinson;

Subcase 11 = R8 Plebanski-Hacyan, Garcia-Plebanski.


special conformal killing tensors in general relativity

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