- 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,
:::.
So,
:::.
from the relation
:,
we get
:::.
(dAη=0 tells us A is antisymmetric)
So, ,
:::
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