Decomposition
of Functions and Tensor Fields| Decomposition
of Functions and Tensor Fields |
Functions
> Complete
sets: see bessel
functions; legendre
polynomials;
Special Functions [including Minkowski];
Visscher Basis; wave
equations.
Vector Fields
* Idea: Any vector field va can
be decomposed into a gradient and a divergenceless part Ba (
Hodge
theorem),
va = Da f + Ba ;
The decomposition is unique up to f ' = f + c
(we assume that the metric is positive-definite); If 
1(M)
is non-trivial, there can be an additional harmonic
field – related
to the Aharonov-Bohm effect.
* Hydrodynamic decomposition: Given a vector field va, its covariant
derivative can be decomposed as
a vb = 
ab + hab + va ab ,
where ab = [ab] is
the rotation, hab =
h(ab) the rate
of deformation, and aa the
acceleration.
Other Tensors > s.a. Wikipedia page.
* 3D rank-2 symmetric, conformally
invariant decomposition: Assume we
have a (+,+,+) metric on a closed M; Then, under a certain condition
for the
existence
of an appropriate vector field ,
we can write
ab = (ab) =
trab + ttab +
longab ,
where trab =
gab,
longab =
2 (a b)
– (2/3) gab c c;
The transverse traceless part ttab is
defined as the rest.
@ 3-metric: Berger & Ebin JDG(69); York JMP(73), AIHP(74).
@ Other tensors:
Fecko JMP(97)gq [forms,
wrt observer field]; Senovilla gq/00-in
[general tensor, electric/magnetic]; Matagne gq/05 [electromagnetic
tensor]; Straumann AdP-a0805 [on
spaces with constant curvature].
Spacetime Metric > s.a. canonical
general relativity; Gauss-Codazzi
Equations; Space [as spacetime submanifold].
* 3+1: In ADM variables,
ds2 = –N 2 dt 2 + (N i dt + dx i) (N j dt + dx j) qij .
* 2+2: General relativity is describable as a Yang-Mills theory defined
on the (1+1)-dimensional base manifold, whose local gauge symmetry is
the
group of the diffeomorphisms of the two-dimensional fibre manifold.
* Threading/Fermat geometry:
The spatial part is fij:=
N –2 qij;
It
can be defined without hypersurfaces, on the instantaneous 3-space of each
observer. & Abramowicz; > s.a. canonical general
relativity and modified
forms, Optical
Geometry; {
SS
29.11.1995}.
@ General references: York JMP(73), AIHP(74);
D'Eath AP(76);
Fischer & Marsden in(79); Choquet-Bruhat
et al in(79); in Stewart CQG(90);
Bini & Jantzen gq/00-in
[refs]; Brown PRD(05)gq [3+1,
conformal-traceless]; Delphenich gq/07 [ito
tangent bundle structure].
@ 2+2 form: d'Inverno & Stachel JMP(78);
Brady et al CQG(96)gq/95;
Yoon PLB(99)gq/00 [Kaluza-Klein-type];
d'Inverno et al CQG(06)gq,
CQG(06)gq [ito
complex self-dual 2-forms]; > s.a.
quasilocal general relativity.
@ More general forms: Mc Manus GRG(92) [m+n, generalised Gauss-Codazzi
equations]; Lau CQG(96)gq/95 [1+2+1
slicings].
@ Higher-dimensional, brane-world: Anderson
gq/04-PhD
[including brane world]; Gergely & Kovács PRD(05)gq.
@ Threading: van Elst & Ellis CQG(96)gq/95 [applications];
van Elst & Uggla
CQG(97)gq/96 [and
slicing]; Fecko JMP(97)gq;
Harris & Low CQG(01)gq [shape
of space];
Larsson ht/01 [quantum
gravity, p-jets
on world-line]; Ahmadi
et al JCAP(08)
[application, quantum gravity phenomenology].
> Related topics:
see foliations and types of spacetimes [decomposition
into regions].
Related Topics > see matrices.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
24 jun 2008