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__

*v*^{a} = *D*^{a}
*f* + *B*^{a} ;

The decomposition is unique up to *f '* = *f* + *c*
(we assume that the metric is positive-definite).

* __On other manifolds__: If π_{1}(*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 *v*^{a}, its covariant
derivative can be decomposed as

∇_{a}* v*_{b} = *ω*_{ab} + *h*_{ab} + *v*_{a }*a*_{b} ,

where *ω*_{ab} = *ω*_{[ab]} is
the rotation, *h*_{ab} =
*h*_{(ab)} the rate
of deformation, and *a*_{a} 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 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

*ψ*^{ab} = *ψ*^{(ab)} =
*ψ*_{tr}^{ab} + *ψ*_{tt}^{ab} +
*ψ*_{long}^{ab} ,

where *ψ*_{tr}^{ab} =
\(1\over3\)*ψ* g^{ab},
*ψ*_{long}^{ab} =
2 ∇^{(a }*ω*^{b)}
– (2/3) *g*^{ab} ∇_{c} *ω*^{ c};
The transverse traceless part *ψ*_{tt}^{ab} 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,

d*s*^{2} = –*N*^{ 2} d*t*^{ 2} +
(*N*^{ i} d*t* +
d*x*^{ i})
(*N *^{j} d*t* + d*x*^{ j}) *q*_{ij} ; *g*^{00} = –*N*^{ –2}, *g*^{0i} = *N*^{–2} *N ^{ i}*,

* __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 *f*_{ij}:=
*N*^{ –2}* q*_{ij};
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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nov 2017