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

*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 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].

@ __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 31 mar 2019