Killing Vector Fields| 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
Xc
ωd
+ λ−1
X[b
Da] λ;
Second derivative ∇a∇b
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) ub ∇b
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];
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 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];
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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 4 may 2019