Killing Vector Fields |

**Killing Vector Fields on Manifolds** > s.a. models
in numerical relativity; noether symmetries; solutions
with symmetries.

$ __For metric isometries__:
A Killing vector is a vector
field *X*^{a} such that the metric
is invariant along its integral curves,

\(\cal L\)_{X}* g*_{ab}
= ∇_{(a}* X*_{b)}
= 0, where *X*_{b}:=
*g*_{bc}
*X*^{ c}.

* __Useful formulae__: First
derivative ∇_{a}
*X*_{b} = (1/2) *λ*^{–1}
*ε*_{abcd} *X*^{c}*ω*^{d}
+ *λ*^{–1}
*X*_{[b}* D*_{a]} *λ*;
Second derivative ∇_{a}∇_{b}
*X*_{c} = *R*_{dabc} *X*^{d} [@
Wald 84, 441ff].

* __Examples__: In *n*-dimensional
constant curvature spacetimes there are \(1\over2\)*n*(*n*+1) independent ones.

* __Conserved quantities__:
For every Killing vector field *X*^{a},
(i) *u*^{b} ∇_{b}
*X*^{a }*u*_{a} =
0 if *u*^{a}
is tangent to affinely parametrized geodesics (4-velocity), and
∇_{a}(*T*^{ a}_{b} *X*^{b})
= 0 if *T*_{ab}
is a valid stress-energy tensor (examples are energy, momentum, angular momentum).

$ __For general transformations__:
Given a one-parameter group of transformations *G*,
the Killing vector field with respect to *G* is the one which generates
these transformations.

@ __General references__: Ludwig CQG(02)gq [and
bivectors]; Fayos & Sopuerta
CQG(02)gq [and
local spacetime structure]; Hall CQG(03)gq [orbits];
Harvey et al AJP(06)nov-gq/05
[introduction, and application to redshifts]; Atkins a0808 [algebraic
procedure for finding the Lie algebra].

@ __Related topics__: Rácz CQG(99)gq/98, CQG(01)
[and evolution]; > s.a. horizons; killing tensors and forms.

> __Online resources__: see Wikipedia page.

**Specific Types of Metrics** > s.a. asymptotically flat spacetimes [asymptotic
Killing vector field]; axisymmetry; Newman-Tamburino Metrics.

@ __Asymptotically flat
spacetimes__: Beig & Chruściel JMP(96)gq/95;
Chruściel & Maerten JMP(06)gq/05.

@ __Other special cases__: Robertson & Noonan 68, p325ff [constant
curvature]; Castejón-Amenedo & MacCallum GRG(90)
[hypersurface-orthogonal]; Defever & Rosca JGP(99)
[skew-symmetric]; in Stephani
et al 03; Dobarro & Ünal a0801 [static
spacetimes]; O'Murchadha a0810 [two commuting Killing vectors]; Chruściel & Delay JGP(11) [stationary vacuum, extensions at boundaries]; Mihai RPMP(12) [two null Killing vector fields].

**Generalizations** > s.a. conformal
structures [conformal and homothetic Killing vector fields]; killing tensors, spinors and forms.

* __Approximate
Killing vectors__: For a given metric, they can be found minimizing an "action" functional that depends on a vector field, which leads to an equation involving the Killing Laplacian", 2 ∇_{a}∇^{ (a} *X*^{ b)}; & Beetle & Wilder.

$ __Approximate symmetry__:
A vector field satisfying the almost-Killing equation
∇_{b}∇^{(a} *X*^{ b)} = 0; For positive-definite
metrics, this is equivalent to the Killing equation.

$ __Semi-Killing vector field__:
A vector field satisfying *X*^{a} ∇_{(a}* X*_{b)} =
0 and ∇_{a} *X*^{a} =
0 [@ Komar PR(62), PR(63)].

$ __Almost symmetry__: A vector field
minimizing *λ*[*X*]:=
(∫ *X*^{ (a;b)} *X*_{(a;b)}d*v*)
/ (∫ *X*^{a} *X*_{a}d*v*),
i.e., satisfying
∇_{b}∇^{(b}
*X*^{a)} + *λ*_{(i)}*X*^{a} =
0.

@ __General references__: Tintareanu-Mircea MPLA(11)-a1012 [and conserved currents].

@ __Approximate
Killing vectors__:
Beetle a0808; Beetle & Wilder CQG(14)-a1401 [Riemannian metric, small perturbations].

@ __Approximate symmetry__: Yano & Bochner 53;
York AIHP(74).

@ __Almost symmetry__: Matzner JMP(68),
JMP(68);
Isaacson PR(68),
PR(68) [and gravitational waves];
Zalaletdinov in(00)gq/99.

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