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,
X gab
=
(a Xb)
= 0, where Xb:=
gbc
X c.
* Useful formulae: First
derivative
a
Xb = (1/2)
–1
abcd Xc
d
+
–1
X[b Da]
;
Second derivative
a
b
Xc = Rdabc Xd [@
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 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 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)nov-gq/05
[introduction, and application to redshifts]; Atkins a0808 [algebraic
procedure for finding Lie algebra].
@ 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]; O'Murchadha a0810 [two commuting Killing vectors].
@ Related topics: Rácz CQG(99)gq/98,
CQG(01)
[and evolution]; Beetle a0808 [approximate
Killing vectors].
> For specific metrics:
see asymptotically flat spacetimes [asymptotic
Killing
vector field]; Newman-Tamburino Metrics.
Killing Tensors and Forms
$ Killing tensor: An n-th
rank Killing tensor is a symmetric covariant tensor, Kab...c
= K(ab... c),
such that
(m Kab...
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
Killing tensor Kab... c, um
m
(Kab... c uaub ···
uc) = 0, if ua 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 gab uaub = uaua.
* 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 Xa 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(Xm Xm)
= 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 CQG(08)-a0806 [surface
gravity, as expansion rate]; > s.a. Kundt
Spacetimes.
Killing Spinors
@ References: Baum m.DG/02 [conformal];
Bohle JGP(03)
[on Lorentzian manifolds]; in Cariglia CQG(04)ht/03 [and
Yano tensors]; Van den Bergh a0908 [spacetimes admitting non-null valence-two
Killing spinors].
Generalizations > s.a. conformal
structures [conformal Killing vector field, homothetic].
$ Semi-Killing vector field:
A vector field satisfying Xa
(a Xb) =
0 and
a Xa =
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)
/ (
Xa Xadv),
i.e., satisfying
b
(b
Xa) +
(i)Xa =
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 Killing-Yano tensor by Kmn:=
mb...
c
nb...
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]; Kubiznák a0909-in
[black
holes]; > s.a. kerr
spacetime.
@ Conformal Killing-Yano tensors: Jezierski APPB(08)-a0705 [asymptotically
AdS]; Kubiznak & Krtous PRD(07)-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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oct 2009