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];
Nozawa & Tomoda a1902
[counting the number of Killing vectors in a 3D spacetime];
> 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];
Peterson & Bonder a1904 [with torsion, T-Killing vectors].

@ __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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