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 / ∂xⁿ} {∂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 Torromé a1207 [higher-order, and applications to electrodynamics]; Navarro JMP(14)-a1306 [second-order, divergence-free tensors]; Nigsch & Vickers a1910 [distributional].

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.

main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 26 dec 2020