General covariant transformations


In physics, general covariant transformations are symmetries of gravitation theory on a world manifold. They are gauge transformations whose parameter functions are vector fields on. From the physical viewpoint, general covariant transformations are treated as particular reference frame transformations in general relativity. In mathematics, general covariant transformations are defined as particular automorphisms of so-called natural fiber bundles.

Mathematical definition

Let be a fibered manifold with local fibered coordinates. Every automorphism of is projected onto a diffeomorphism of its base. However, the converse is not true. A diffeomorphism of need not give rise to an automorphism of.
In particular, an infinitesimal generator of a one-parameter Lie group of automorphisms of is a projectable vector field
on. This vector field is projected onto a vector field on, whose flow is a one-parameter group of diffeomorphisms of. Conversely, let be a vector field on. There is a problem of constructing its lift to a projectable vector field on projected onto. Such a lift always exists, but it need not be canonical. Given a connection on, every vector field on gives rise to the horizontal vector field
on. This horizontal lift yields a monomorphism of the -module of vector fields on to the -module of vector fields on, but this monomorphisms is not a Lie algebra morphism, unless is flat.
However, there is a category of above mentioned natural bundles which admit the functorial lift onto of any vector field on such that is a Lie algebra monomorphism
This functorial lift is an infinitesimal general covariant transformation of.
In a general setting, one considers a monomorphism of a group of diffeomorphisms of to a group of bundle automorphisms of a natural bundle. Automorphisms are called the general covariant transformations of. For instance, no vertical automorphism of is a general covariant transformation.
Natural bundles are exemplified by tensor bundles. For instance, the tangent bundle of is a natural bundle. Every diffeomorphism of gives rise to the tangent automorphism of which is a general covariant transformation of. With respect to the holonomic coordinates on, this transformation reads
A frame bundle of linear tangent frames in also is a natural bundle. General covariant transformations constitute a subgroup of holonomic automorphisms of. All bundles associated with a frame bundle are natural. However, there are natural bundles which are not associated with.