Tetrad (index notation)

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{{#invoke:Hatnote|hatnote}} In Riemannian geometry, we can introduce a coordinate system over the Riemannian manifold (at least, over a chart), giving n coordinates

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 T 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 (triad),

.

So,

.

from the relation

,

we get

.

(dAη=0 tells us A is antisymmetric)

So, ,

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