Lie Derivatives |
In General > s.a. Derivatives.
* Idea: A notion of directional
derivative on an arbitrary differentiable manifold that depends on a vector
field va (even for the
value of the Lie derivative at a point x we need more than the vector
va at x), but not
on a choice of connection or metric (it is a concomitant).
* Useful formula: For any
p-form ω (p > 1),
£v ω = v ·dω + d(v · ω) .
* Lie derivatives of the coordinate basis elements:
£v(∂/∂xi) = −(∂vj/∂xi) (∂/∂xj) , £v (dxi) = (∂vi/∂xj) dxj .
* Other properties: Acting on forms, it commutes with taking the exterior derivative, d(£v ω) = £v(dω).
For Various Types of Fields
> s.a. spin coefficients; spin structures.
* Scalar functions:
It coincides with other notions of derivative,
£v f |x:= limt → 0 t−1 [f(\(\sigma_t(x)\)) − f(x)] = v(f)|x ≡ va ∂a f |x .
* Vector fields: Defined using the push-forward under the diffeomorphisms generated by va,
£v w|x:= limt → 0 t−1 [\(\sigma_t^{-1\prime}\)(w(σt(x))) −w(x)] ≡ [v, w]x .
* One-forms: Defined using the pull-back under the diffeomorphisms generated by va,
£v ω|x:= limt → 0 t−1 [σt*(ω(\(\sigma_t(x)\))) − ω(x)] .
* Arbitrary tensor fields: Defined implicitly by the product rule £v (M ⊗ N) = £v M ⊗ N + M ⊗ £v N; e.g.,
£v Mac:= vm ∇m Mac − Mmc ∇mva + Mam ∇cvm .
* Scalar / tensor densities of weight 1:
£v f = va ∂a f + f ∂a va = ∂a (va f) ; £v M = |g|1/2 £v M + (div v) M .
* With torsion: It includes additional terms; For example,
£v Xa = vm ∇m Xa − Xm ∇m va − vm Xn Tmna .
* Spinor fields: The definition (given by Lichnerowicz for the case when va is a Killing vector field, where the explicit antisymmetrization in a and b is not necessary, and extended by Kosmann to the general case) is
£v ψ:= va ∇a ψ − \(1\over4\)(∇[a vb]) γaγb ψ ,
where the γs are the gamma matrices.
Related Concepts
* Lie bracket: The
Lie derivative induces a Lie-bracket structure on vector fields,
[v, w] = £v w .
@ References: Crainic & Fernandes AM(03) [integrability].
References
@ General: Schouten 54 [red threads embedded in gelatin];
Yano 57;
in Kolář et al 93;
in Choquet-Bruhat & DeWitt-Morette 00.
@ For spinor fields: Lichnerowicz CRAS(63);
Kosmann AMPA(71);
Jhangiani FP(78),
FP(78) [geometrical significance];
in Penrose & Rindler 86 [for infinitesimal conformal isometries];
Bilyalov TMP(92) [and Noether theorem for spinor fields];
Bourguignon & Gauduchon CMP(92) [and "metric Lie derivative"];
Delaney MS(93);
Hurley & Vandyck JPA(94),
JPA(94),
JPA(95) [and covariant derivative];
Fatibene et al gq/96-proc;
Ortín CQG(02)ht,
comment Fatibene & Francaviglia a0904 [all spins];
Godina & Matteucci IJGMP(05)m.DG;
Palese & Winterroth mp/06-proc;
Sharipov a0801;
Leão et al a1411 [geometrically motivated approach];
Helfer PRS(16)-a1602 [and fermion stress-energies];
> s.a. spin structure.
@ Generalized framework: Trautman in(72) [for maps between manifolds];
Hurley & Vandyck JMP(01);
Godina & Matteucci JGP(03).
> Online resources:
see Wikipedia page;
MathWorld page [for spinors];
article available in A Trautman's webpage.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 17 aug 2019