Frame Fields in General Relativity - Generalizations

Generalizations

This article has focused on the application of frames to general relativity, and particularly on their physical interpretation. Here we very briefly outline the general concept. In an n-dimensional Riemannian manifold or pseudo-Riemannian manifold, a frame field is a set of orthonormal vector fields which forms a basis for the tangent space at each point in the manifold. As before, frames can be specified in terms of a given coordinate basis, and in a non-flat region, some of their pairwise Lie brackets will fail to vanish.

In fact, given any inner-product space, we can define a new space consisting of all tuples of orthonormal bases for . Applying this construction to each tangent space yields the orthonormal frame bundle of a (pseudo-)Riemannian manifold and a frame field is a section of this bundle. More generally still, we can consider frame bundles associated to any vector bundle, or even arbitrary principal fiber bundles. The notation becomes a bit more involved because it is harder to avoid distinguishing between indices referring to the base, and indices referring to the fiber. Many authors speak of internal components when referring to components indexed by the fiber.