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 Xa such that the metric is invariant along its integral curves,

\(\cal L\)X gab = ∇(a Xb) = 0,   where   Xb:= gbc X c.

* Useful formulae: First derivative ∇a Xb = (1/2) λ–1 εabcd Xcωd + λ–1 X[b Da] λ; Second derivative ∇ab Xc = Rdabc Xd [@ 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 Xa, (i) ubb Xa ua = 0 if ua is tangent to affinely parametrized geodesics (4-velocity), and ∇a(T ab Xb) = 0 if Tab 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 Xa(a Xb) = 0 and ∇a Xa = 0 [@ Komar PR(62), PR(63)].
$ Almost symmetry: A vector field minimizing λ[X]:= (∫ X (a;b) X(a;b)dv) / (∫ Xa Xadv), i.e., satisfying ∇b(b Xa) + λ(i)Xa = 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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