Tetrad or Vielbein Formalism

In General
* Idea: A basis for the tangent space at each point in a manifold, that does not necessarily arise from a coordinate system; Cartan's "repère mobile" or "moving frame".
$ 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 = _IJ ,   e^a_I e^b_J ^IJ = g^ab .

$ Dual frame: The n-tuple of covectors, denoted by ^I_a (or sometimes simply e^I_a), satisfying

_a^I e^a_J = ^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; Structure Equations.
* Form notation: Pick a vielbein; The connection 1-forms are obtained by solving the first structure equation (this is easy),

d^I = ^I_J  ^J = ^I_JK ^{J  K} ,   subject to _IJ:= _IK ^K_J = _{[IJ ]} ;

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

F^I_J = d^I_J + ^I_K  ^K_J ;

Finally, calculate the Riemann curvature tensor by R_abcd = F_IJcd ^I_a ^J_b .
* Abstract index notation: If the Ricci rotation coefficients are denoted by _IJK:= _IL ^L_JK = _[IJ]K,

_IJ^K:= e_I^a e_J^b _[ab]^K ,   T_ab^I:= 2 (_{[a b]}^I – _[a^J_b]J^I) ,   F_abI^J = 2 (_[ab]I^J – _[aI^K_b]K^J) ,

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

_aIJ =  _a^K (_KIJ – _IJK + _JKI) .

Covariant Differentiation
* Calculation: One has to calculate the Ricci rotation coefficients

_IJK:= – (_IJK – _KIJ – _JIK) ;

then the covariant derivatives are a^a_;b = _b(a^a) + ^a_cb a^c, or

_a = _a() – _a ,   where   _a := – _mna ^{m n} .

References > s.a. formulations of general relativity.
@ 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].
@ Related topics: Collinson GRG(90) [isometry groups leaving tetrads invariant]; Garat JMP(05) [for Einstein-Maxwell equations].
> Related topics and examples: see equivalence principle [adapted to free fall]; sphere [complex dyad on 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 st 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:= 2^–1/2(x + i y)   and   m*:= 2^–1/2(x – i y) ;

finally determine the unique null vector field n^a such that n · l = 1, 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, ₁ = ₃ = 0; They do not always exist, but they do for the interesting types of algebraically special spacetimes.

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 16 jun 2008