Tensor Fields |

**In General**
> s.a. tensors; types of fiber bundles [tensor bundles].

$ __Def__: An element of the space
⊗^{p} T*_{x}* X*
⊗^{q}
T_{x} *X* of multilinear
forms on T_{x} *X* ⊗ ...
⊗ T_{x} *X* ⊗
T*_{x} *X* ⊗ ...
⊗ T*_{x} *X* (*p*
copies of T_{x} *X*, and
*q* copies of T*_{x} *X*),
for all *x* ∈ *X*.

* __And other structure__:
The set of all tensor fields on *X* forms an algebra,
\(\cal D\)(*X*).

* __Tensors and physical theories__:
A commonplace attributed to Kretschmann states that any local physical theory
can be represented in arbitrary coordinates using tensor calculus, but this may
not be true for theories with spinors.

@ __General references__:
Wasserman 09 [and physics applications].

@ __Related topics__: Brännlund et al IJMPD(10)-a1003
[covariant averaging procedure, and Weitzenböck connection].

> __Online resources__:
see Wikipedia page.

**Tensor Density** > s.a. projective structures.

$ __Def__: A tensor density (sometimes
called a relative tensor) of weight *w* on a manifold is an object that
transforms as

**T**'^{ a...
b}_{c... d} =
|∂*x*/∂*x*'|^{w}
{∂*x*'^{a}/∂*x*^{m}}
··· {∂*x*'^{b}/∂*x*^{n}}
{∂*x*^{p}/∂*x*'^{c}} ··· {∂*x*^{q}/∂*x*'^{d}} **T**'^{ m...
n}_{p... q} .

* __With a metric__: A tensor density
of weight *w* on a manifold can be expressed as

**T**^{ a... b}_{c... d}
= |*g*|^{w/2}* T*^{ a...
b}_{c... d} ,

where *T*^{a... b}_{c... d}
is a tensor and does not depend on the choice of a volume element *ε*_{abcd} .

* __Special cases__: The ones with *w* = 1 are
the tensor densities proper; The ones with *w* = –1 are sometimes called tensor capacities.

@ __References__: in Dalarsson & Dalarsson 05 ["relative tensors"].

> __In physics__: see ADM, connection, other
formulations, and actions for general relativity; canonical
quantum theory.

**Types of Tensor Fields** > s.a. 3D geometries [transverse
traceless]; decomposition; forms; vector
field [vertical].

$ __Horizontal__: Given a fibration
of a manifold, a covariant tensor field is horizontal if any contraction of it
with a vector tangent to a fiber vanishes; With a metric, the definition can
be extended to contravariant tensor fields.

@ __Generalizations__: Akhmedov TMP(05) [non-abelian],
TMP(06) [non-abelian, gauge transformations and curvature];
Gallego a1207 [higher-order, and applications to electrodynamics];
Navarro JMP(14)-a1306 [second-order, divergence-free tensors].

**Derivatives, Tensor Calculus** > s.a. analysis; Calculus;
connections; lie derivative.

* __Covariant derivative__: For a covariant/contravariant vector field,
it is given respectively by

∇_{a} *k*_{b}
= ∂_{a}* k*_{b} − Γ^{c}_{ab} *k*_{c}
, ∇_{a} *k*^{b} = ∂_{a}* k*^{b}
+ Γ^{b}_{ac} *k*^{c}
.

* __Weak derivative__: A locally integrable tensor field *T* has
a weak derivative if there exists a tensor field *X* such that their associated distributions are related
by *X*_{ma... c}^{b... d}
= ∇_{m} *T*_{a...
c}^{b... d}.

@ __General references__: Frederiks & Friedmann 24; Spivak 65; Synge & Schild 69; Dodson & Poston 91 [geometry]; Akivis & Goldberg 03; Hackbusch 12 [numerical].

@ __Related topics__: Ashtekar et al GRG(82) [generalization];
Geroch & Traschen PRD(87) [weak];
Hall JMP(91) [covariantly constant, and holonomy groups];
Thiffeault JPA(01)n.CD [time derivatives];
Tapia gq/04 [differential invariants];
Boulanger JMP(05)ht/04 [Weyl-covariant].

**Spacetime Tensors** > s.a. Potential for a Field.

@ __Averaging__: Mars & Zalaletdinov JMP(97)dg;
Boero & Moreschi a1610.

> __As dynamical fields__:
see gravity theories; types of field theories.

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