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 e^a_I , with I a label and a a spacetime index, satisfying

e^a_I e^b_J g_ab = η_IJ ,     e^a_I e^b_J η^IJ = g^ab .

$ Dual frame: The n-tuple of covectors, denoted by θ^I_a (or sometimes simply e^I_a), satisfying θ_a ^Ie^a_J = δ^I_J .
* Holonomic tetrad: One for which there is a coordinate system x^I such that e^a_I = (∂/∂x^I)^a.
* Structure coefficients: Defined for each tetrad, by [e_I , e_J] =: C_IJ^K e_K; 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),

dθ^I = ω^I_J ∧ θ^J = Ω^I_JK θ^J ⊗ θ^K ,    subject to   ω_IJ:= η_IK ω^K_J = ω_{[IJ ]} ;

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

F^I_J = dω^I_J + ω^I_K ∧ ω^K_J ;

Finally, calculate the Riemann curvature tensor by R_abcd = F_IJcd θ^I_a θ^J_b .
* Abstract index notation: If the Ricci rotation coefficients are denoted by Ω_IJK:= η_IL Ω^L_JK = Ω_[IJ]K,

Ω_IJ^K:= e_I^a e_J^b ∂_[a θ_b]^K ,    T_ab^I:= 2 (∂_[a θ_b]^I − θ_[a^J ω_b]J^I) ,    F_abI^J = 2 (∂_[a ω_b]I^J − ω_[aI^K ω_b]K^J) ,

where, if the torsion T_ab^I = 0, the connection

ω_aIJ = \(1\over2\)θ_a^K (Ω_KIJ − Ω_IJK + Ω_JKI) .

Covariant Differentiation
* Ricci rotation coefficients: They are defined by γ_IJK:= −\(1\over2\)(Ω_IJK − Ω_KIJ − Ω_JIK).
* Covariant derivatives: In terms of Ricci rotation coefficients, they are a^a_;b = ω_b(a^a) + γ^a_cb a^c, or

∇_a ψ = ω_a(ψ) − Γ_a ψ ,     where    Γ_a := −\(1\over4\)γ_mna γ^m 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]; Cordeiro dos Santos a1711 [intro, and curvature calculations].
@ 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 IJGMP(16)-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 l^a, and two spacelike vector fields x^a and y^a, 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 n^a such that n · l = 1 and n · m = n · m = 0.
* Interpretation: The real vectors l^a and n^a span a timelike 2-plane; The complex null vectors m^a 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.

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