Killing 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,

_X g_ab = _(a X_b) = 0,   where   X_b:= g_bc X ^c.

* Useful formulae: First derivative _a X_b = (1/2) ^–1 _abcd X^cd + ^–1 X_[b D_a] ; Second derivative _ab X_c = R_dabc X^d [@ Wald 84, 441ff].
* Examples: In n-dimensional constant curvature spacetimes there are n(n+1)/2 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 wrt 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)gq/05 [intro, and application to redshifts].
@ Asymptotically flat spacetimes: Beig & Chrusciel JMP(96)gq/95; Chrusciel & Maerten JMP(06)gq/05.
@ Other special cases: Robertson & Noonan 68, p325ff [constant curvature]; in Kramer et al 80; Castejón-Amenedo & MacCallum GRG(90) [hypersurface-orthogonal]; Defever & Rosca JGP(99) [skew-symmetric]; Dobarro & Unal a0801 [static spacetimes].
@ And evolution: Rácz CQG(99)gq/98, CQG(01).
> For specific metrics: see asymptotically flat [asymptotic Killing vector field]; Newman-Tamburino.

Killing Tensors and Forms
$ Killing tensor: An n-th rank Kt is a symmetric covariant tensor, K_ab...c = K_{(ab... c)}, such that _(m K_{ab... c)} = 0.
$ Killing form: A differential form _{ab... c} = _{[ab... c]} whose covariant derivative is totally skew-symmetric.
* Conserved quantities: While Killing vectors give the linear first integrals of the geodesic equations, Killing tensors give the quadratic, cubic, and higher-order first integrals; For every Kt K_{ab... c}, u^m _m (K_{ab... c} u^au^b ··· u^c) = 0, if u^a is tangent to affinely parametrized geodesics.
* Example: The metric itself is always a rank-2 Killing tensor; The associated conserved quantity is the norm squared g_ab u^au^b = u^au_a.
* Applications: Integrability of geodesics in Kerr-Newman spacetime.
@ General references: Sommers JMP(73) [and particle constants of motion]; Dolan et al GRG(89) [significance]; Collinson & Howarth GRG(00) [generalized]; Benn JMP(06) [and mechanics]; Coll et al JMP(06)gq [spectral decomposition].
@ Special types of manifolds: Rosquist & Uggla JMP(91) [2D spacetimes]; Smirnov & Yue JMP(04)mp [constant curvature pseudo-Riemannian]; Belgun et al DG&A(06) [symmetric spaces].
@ From conformal Killing vectors: Koutras CQG(92); Barnes et al gq/02-in; Rani et al CQG(03).
@ Second-rank: Walker & Penrose CMP(70) [Kerr spacetime]; Baleanu gq/98, gq/98/NC; Chanu et al JMP(06) [2D flat manifold]; {Bombelli & Rosquist}.
@ Third-rank: Rosquist & Goliath GRG(98); Karlovini & Rosquist GRG(99)gq/98 [1+1 dimensions]; Baleanu G&C(99).

Killing Horizon > s.a. quantum field theory in curved spacetime [vacua].
* Idea: The locus of points in spacetime where a Killing vector field X^a is null.
* Remark: It is often a source of coordinate singularities (similarly to the case when the Killing vector field vanishes), if one uses coordinates adapted to the action of the isometry group generated by the Killing vector field.
* Special cases: It is called degenerate when the surface gravity vanishes, _a(X_m X^m) = 0; If the Killing horizon is non-degenerate, the Killing vector field has to change character from timelike to spacelike across the Killing horizon; In general, non-degenerate Kh's cross each other–they are bifurcate; The Killing horizon is called bifurcate if it is the union of two null surfaces which intersect in a codimension-2 spacelike surface (e.g., Rindler space, Schwarzschild, de Sitter).
* Examples: Spacetimes that have Killing horizons are some black holes, Rindler, de Sitter, Taub-NUT and Taub-Bolt spaces.
@ References: Griffiths GRG(05)gq [Killing-Cauchy horizons for colliding plane waves, instability]; Jacobson & Parentani a0806 [surface gravity, as expansion rate].

Killing Spinors
@ References: Baum m.DG/02 [conformal]; Bohle JGP(03) [on Lorentzian manifolds]; in Cariglia CQG(04)ht/03 [and Yano tensors].

Generalizations > s.a. conformal structures [conformal Killing vector field, homothetic].
$ Semi-Killing vector field: A vector field satisfying X^a _(a X_b) = 0 and _a X^a = 0 [@ Komar PR(62), PR(63)].
$ 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.
$ Almost symmetry: A vector field minimizing [X]:= (X ^(a;b) X_(a;b)dv) / (X^a X_adv), i.e., satisfying _b^(b X^a) + _(i)X^a = 0.
@ Approximate symmetry: Yano & Bochner 53; York AIHP(74).
@ Almost symmetry: Matzner JMP(68), JMP(68); Isaacson PR(68), PR(68) [and gravitational waves]; Zalaletdinov gq/99-in.

Killing-Yano Tensors > s.a. Taub-NUT Metric.
$ Def: An n-th rank Killing-Yano tensor is an n-form _{ab... c}, such that _{(m a)b... c} = 0.
* Relationships: A Killing 2-tensor can be defined from a KYt by K_mn:= _{mb... c n}^{b... c}.
* Example: The alternating tensor _{ab... c} is a KY tensor; The corresponding rank-2 Killing tensor is (proportional to) the metric.
@ General references: Yano AM(52); Kastor et al CQG(07)-a0705 [conditions for graded Lie algebra wrt Schouten-Nijenhuis bracket].
@ In general relativity: Dietz & Rüdiger PRS(81), PRS(82); Hall IJTP(87); Baleanu NCB(99)gq/98 [and Nambu tensors]; Ferrando & Sáez GRG(03) [Rainich problem].
@ Special types of spacetimes: Howarth & Collinson GRG(00) [spherical static]; Jezierski & Lukasik CQG(06) [Kerr]; > s.a. kerr.
@ Conformal Killing-Yano tensors: Jezierski a0705 [asymptotically AdS]; Kubiznak & Krtous a0707 [for Plebanski-Demianski type-D solutions].

Killing Form on a Lie Algebra
$ Def: Given a Lie algebra , its Killing form is the bilinear form B(X, Y) = tr[Ad(X), Ad(Y)].

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