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];
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, 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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send feedback and suggestions to bombelli at olemiss.edu – modified 22 dec 2017