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 θa
IeaJ
= 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 ∂[a θb]K , TabI:= 2 (∂[a θb]I − θ[aJ ωb]JI) , FabIJ = 2 (∂[a ωb]IJ − ω[aIK ωb]KJ) ,
where, if the torsion TabI = 0, the connection
ωaIJ = \(1\over2\)θaK (ΩKIJ − 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];
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 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.
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send feedback and suggestions to bombelli at olemiss.edu – modified 22 dec 2017