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
> s.a. vector calculus; vector fields.
* On R\(^3\): 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).
* On other manifolds: If
π1(M) is non-trivial, there can be
an additional harmonic field – this is related to the Aharonov-Bohm effect.
* Hydrodynamic decomposition:
Given a vector field va,
its covariant derivative can be decomposed as
∇a vb = ωab + hab + va ab ,
where ωab
= ω[ab] is
the rotation, hab
= h(ab) the rate of
deformation, and aa
the acceleration.
@ References:
Presnov RPMP(08)
[on a Riemannian manifold of non-positive curvature];
Woodside AJP(09)may [3D Euclidean space and 4D Minkowski space];
Tudoran AAM(19)-a1711;
> s.a. MathWorld page on Helmholtz's Theorem.
Other Tensors
> s.a. tensors; tensor fields.
* 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
ω, we can write
ψab = ψ(ab) = ψtrab + ψttab + ψlongab ,
where ψtrab
= \(1\over3\)ψ gab,
ψlongab
= 2 ∇(a ωb)
− (2/3) gab ∇c
ω c;
The transverse traceless part ψttab
is defined as the rest.
@ 3-metric: Berger & Ebin JDG(69); York JMP(73), AIHP(74).
@ Metric perturbations: Buniy & Kephart PLB(09)-a0811 [scalar, vector, and tensor modes and applications].
@ Other tensors: Fecko JMP(97)gq [forms, with respect to an observer field];
Senovilla gq/00-proc [general tensor, electric/magnetic];
Matagne AdP(08)gq/05 [electromagnetic tensor];
Straumann AdP(97)-a0805 [on spaces with constant curvature];
Auchmann & Kurz JPA(14)-a1411 [relativistic electrodynamics, observer space];
De las Cuevas et al a1909 [invariant
decompositions of elements of tensor product spaces, with indices arranged on a simplicial complex].
> Online resources:
see Wikipedia page.
Spacetime Metric
> s.a. ADM formulation; canonical general relativity;
Gauss-Codazzi Equations; gravitational energy-momentum.
* 3+1:
In ADM (spatial metric + lapse + shift) variables,
ds2 = −N 2 dt 2 + (N i dt + dx i) (N j dt + dx j) qij ; g00 = −N−2, g0i = N−2 N i, gij = qij − N−2 N i N j .
* 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.
@ 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 proc(01)gq/00 [refs];
Delphenich gq/07 [in terms of tangent bundle structure].
@ 3+1, conformal-traceless form: Brown PRD(05)gq;
> s.a. canonical general relativity; initial-value
problem; numerical general relativity.
@ 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 [in terms of 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];
Gergely et al a2007 [2+1+1. Hamiltonian dynamics].
@ Higher-dimensional, brane-world:
Anderson PhD(04)gq [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];
Bini et al PRD(12)-a1203 [admissible coordinates and causality].
> Related topics:
see foliations and types of spacetimes
[decomposition into regions]; Space [as spacetime submanifold].
Related Topics > see matrices.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 4 jul 2020