2022-02-20   00418

Diffeomorphisms, Tensor Fields, and General Covariance

Tensor fields live on smooth manifolds. The concept of a tensor field is independent of any coordinate system: changing the coordinate system has no effect on the tensor field itself, even though it changes the way the tensor field is represented in terms of coordinates. In contrast, given any smooth rearrangement of the manifold's points (that is, any diffeomorphism from the manifold to itself), a corresponding transformation of tensor fields may be defined in a natural way. This article defines that transformation and calls it a fieldomorphism. Unlike a coordinate transformation, a fieldomorphism does change the tensor fields. In general relativity, any fieldomorphism of a solution of the equations of motion gives another solution of the equations of motion, a property often called general covariance. This is a gauge symmetry in general relativity, which means that two solutions related by a fieldomorphism are physically equivalent to each other.

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www.cphysics.org updated 2024-06-01