- Tetrad (index notation)
In
Riemannian geometry , we can introduce acoordinate system over theRiemannian manifold (at least, over a chart), giving "n" coordinates:xi, i=1,...,n
for an n-dimensional manifold. Locally, at least, this gives a basis for the 1-forms, dxi where d is the
exterior derivative . Thedual basis for thetangent space is ei.Now, let's choose an
orthonormal basis for the fibers of T. The rest is index manipulation.Example
Take a
3-sphere with theradius "R" and give itpolar coordinate s α, θ, φ.:e(eα)/R, :e(eθ)/R sin(α) and :e(eφ)/R sin(α) sin(θ)
form an orthonormal basis of T.
Call these e1, e2 and e3. Given the metric η, we can ignore the
covariant andcontravariant distinction for T.Then, the dreibein,
:e_1=R dalpha:e_2=R sin alpha d heta:e_3=R sin alpha sin heta dphi.
So,
:de_1=0:de_2=R cos alpha dalpha wedge d heta:de_3=R (cos alpha sin heta dalpha wedge dphi + sin alpha cos heta d heta wedge dphi).
from the relation
:d_mathbf{A} e=de+Awedge e=0,
we get
:A_{12}=-cos alpha d heta:A_{13}=-cos alpha sin heta dphi:A_{23}=-cos heta dphi.
(dAη=0 tells us A is antisymmetric)
So, mathbf{F}=dmathbf{A}+mathbf{A}wedge mathbf{A},
:F_{12}=sinalpha dalphawedge d heta:F_{13}=sin alpha sin heta dalphawedge dphi:F_{23}=sin^2 alpha sin heta d hetawedge dphi
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