Tetrad or Vielbein Formalism  

In General
* Idea: A basis for the tangent space at each point in a manifold (Cartan's "repère mobile" or "moving frame"), not necessarily arising from a coordinate system.
$ Vielbein: (Vierbein or tetrad in 4D) An orthonormal frame, or n-tuple of vector fields eaI , with I a label and a a spacetime index, satisfying

eaI ebJ gab = ηIJ ,    eaI ebJ ηIJ = gab .

$ Dual frame: The n-tuple of covectors, denoted by θIa (or sometimes simply eIa), satisfying θaI eaJ = δIJ .
* Holonomic tetrad: One for which there is a coordinate system xI such that eaI = (∂/∂xI)a.
* Structure coefficients: Defined for each tetrad, by [eI , eJ] =: CIJK eK; They vanish if the tetrad is holonomic.

Use for Calculating Curvature > s.a. riemann curvature tensor; Structure Equations.
* Form notation: Pick a vielbein; The connection 1-forms are obtained by solving the first structure equation (this is easy),

dθI = ωIJ ∧ θJ = ΩIJK θJ ⊗ θK ,    subject to  ωIJ:= ηIK ωKJ = ω[IJ ] ;

Then calculate the curvature 2-forms from the second structure equation,

FIJ = dωIJ + ωIK ∧ ωKJ ;

Finally, calculate the Riemann curvature tensor by Rabcd = FIJcd θIa θJb .
* Abstract index notation: If the Ricci rotation coefficients are denoted by ΩIJK:= ηIL ΩLJK = Ω[IJ]K,

ΩIJK:= eIa eJb[aθb]K ,    TabI:= 2 (∂[a θb]Iθ[aJωb]JI) ,    FabIJ = 2 (∂[aωb]IJω[aIKωb]KJ) ,

where, if the torsion TabI = 0, the connection

ωaIJ = \(1\over2\)θaKKIJ – ΩIJK + ΩJKI) .

Covariant Differentiation
* Ricci rotation coefficients: They are defined by γIJK:= –\(1\over2\)(ΩIJK – ΩKIJ – ΩJIK).
* Covariant derivatives: In terms of Ricci rotation coefficients, they are aa;b = ωb(aa) + γacb ac, or

a ψ = ωa(ψ) – Γa ψ ,    where    Γa := –\(1\over4\)γmna γm γn .

References
@ General: in Weinberg 72; in Carter in(73); in Chandrasekhar 83; in Wald 84 [clear and simple, directly on how to use for curvature]; {in KK notes (KK4, KK10)}; Rodrigues & Gomes de Souza IJMPD(05)mp/04 [the "tetrad postulate" is ambiguous]; Fukuyama MPLA(09) [obvious comment].
@ And gravity: Garat JMP(05)gq/04 [for Einstein-Maxwell equations], gq/06, a1207 [in Yang-Mills geometrodynamics]; Vacaru IJTP(10)-a0909 [integration of Einstein's equation]; > s.a. formulations of general relativity [vierbein variables]; approaches to canonical quantum gravity; torsion in physical theories [for f(T) gravity].
@ Related topics: Collinson GRG(90) [isometry groups leaving tetrads invariant]; Borowiec et al a1602 [non-commutative tetrads, and quantum spacetimes].
> Related topics and examples: see equivalence principle [adapted to free fall]; sphere [complex dyad on the 2-sphere].

Null Tetrads > s.a. Alternating Tensor; spin coefficients [Newman-Penrose formalism].
* Construction: A null tetrad field (l, m, m*, n) in a region U of spacetime can be obtained by picking a null geodesic congruence in U with tangent la, and two spacelike vector fields xa and ya, orthogonal to each other and normal to l, and defining the complex combinations

m:= \(1\over\sqrt2\)(x + i y)   and   m*:= \(1\over\sqrt2\)(x – i y) ;

finally determine the unique null vector field na such that n · l = 1 and n · m = n · m = 0.
* Interpretation: The real vectors la and na span a timelike 2-plane; The complex null vectors ma and m*a span the perpendicular spacelike subspace.
* Transverse tetrad: One for which the NP scalars representing pure gauge vanish, Ψ1 = Ψ3 = 0; They do not always exist, but they do for the interesting types of algebraically special spacetimes.


main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 2 feb 2016