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.


Take a 3-sphere with the radius R and give it polar coordinates α, θ, φ.

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),




from the relation


we get


(dAη=0 tells us A is antisymmetric)

So, ,