Differential geometry MOC

Vielbein

A vielbein1 is a (local) orthonormal basis for a Semi-Riemannian manifold . #m/def/geo/diff This amounts to being a local frame satisfying

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

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 , this is the appropriate way to raise and lower concrete indices.

Curvature

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

We let and . The compatibility with the metric becomes

The torsion-free condition becomes

Bianchi identity I becomes

and Bianchi identity II becomes

Letting be the 2-form part of the Riemannian curvature, we have

Proof

#missing/proof

Generality

This can be generalized to other connexions.

#state/develop | #lang/en | #SemBr

Footnotes

  1. Physicists often call this a non-coördinate basis, even though not being a coördinate basis is insufficient to guarantee this property. ↩

  2. If is a Riemannian manifold we take the Kronecker delta. Otherwise is chosen so that its signature matches . ↩