Tetrad (index notation)

Tetrad (index notation)

In Riemannian geometry, we can introduce a coordinate system over the Riemannian 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. The dual basis for the tangent 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 the radius "R" and give it polar coordinates 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 and contravariant 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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