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 R3: 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]; > 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) gabc 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].
> 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 = qijN–2 N iN 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 gq/00-proc [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].
@ 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.


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