Vielbein

A vielbein¹ is a (local) orthonormal basis for a Semi-Riemannian manifold $𝑀$ . diff This amounts to being a local frame $(𝑒 𝜇) 𝑎$ satisfying

(𝑒 𝜇) 𝑎 (𝑒 𝜈) 𝑎 = 𝜂 𝜇 𝜈

for an appropriate diagonal “Minkowski” metric $𝜂$ .² It immediately follows

(𝑒 𝜇) 𝑎 (𝑒 𝜈) 𝑏 = 𝜂 𝜇 𝜈 (𝑒 𝜇) 𝑎 (𝑒 𝜈) 𝑏 = 𝛿 𝑎 𝑏 .

Note that such a local frame is only unique up to a local Lorentz transformation or analogous group.

Index notation

This Zettel mixes Latin indices for abstract index notation and Greek indices for concrete indices with Einstein summation convention, sometimes within the same expression. Note that since the metric in this frame is $𝜂 𝜇 𝜈$ , this is the appropriate way to raise and lower concrete indices.

Curvature

Suppose $\nabla$ is the Levi-Civita connexion. The connexion 1-forms are defined by

𝜔 𝜇 𝜈 𝑎 = (𝑒 𝜇) 𝑏 \nabla 𝑎 (𝑒 𝜈) 𝑏 \in Ω 1 (𝑀) .

We let $𝝎 𝜇 𝜈 = 𝜔 𝜇 𝜈 𝑎$ and $𝒆 𝜇 = (𝑒 𝜇) 𝑎$ . The compatibility with the metric becomes

𝝎 𝜇 𝜈 = - 𝝎 𝜈 𝜇 .

The torsion-free condition becomes

d 𝒆 𝜇 + 𝝎 𝜇 𝜈 \land 𝒆 𝜈 = 0

^B1 becomes

𝑹 𝜇 𝜈 \land 𝒆 𝜈 = 0

and ^B2 becomes

d 𝑹 𝜇 𝜈 + 𝝎 𝜇 𝜎 \land 𝑹 𝜎 𝜈 - 𝑹 𝜇 𝜎 \land 𝝎 𝜎 𝜈 .

Letting $𝑹 𝜇 𝜈 = 𝑅 𝜇 𝜈 𝑎 𝑏 \in Ω 2 (𝑀)$ be the 2-form part of the Riemannian curvature, we have

𝑹 𝜇 𝜈 = d 𝝎 𝜇 𝜈 + 𝝎 𝜇 𝜎 \land 𝝎 𝜎 𝜈 .

Proof

proof

Generality

This can be generalized to other connexions.

develop | en | SemBr

Physicists often call this a non-coördinate basis, even though not being a coördinate basis is insufficient to guarantee this property. ↩
If $𝑀$ is a Riemannian manifold we take the Kronecker delta. Otherwise $𝜂$ is chosen so that its signature matches $𝑔$ . ↩

Table of Contents

Vielbein

Curvature

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Table of Contents

Vielbein

Curvature

Footnotes

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