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}^{I }*e*^{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),

d*θ*^{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} =
d*ω*^{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} *γ*^{n} .

**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 *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, Ψ_{1} = Ψ_{3} = 0; They do not
always exist, but they do for the interesting types of algebraically special spacetimes.

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feb 2016