Example: General Relativity
We can describe geometries in general relativity in terms of a tetrad field instead of the usual metric tensor field. The metric tensor gives the inner product in the tangent space directly:
The tetrad may be seen as a (linear) map from the tangent space to Minkowski space that preserves the inner product. This lets us find the inner product in the tangent space by mapping our two vectors into Minkowski space and taking the usual inner product there:
Here and range over tangent-space coordinates, while and range over Minkowski coordinates. The tetrad field defines a metric tensor field via the pullback .
Read more about this topic: Cartan Formalism (physics)
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