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 (holonomic) reference frame transformations in general relativity.
In mathematics, general covariant transformations are defined as particular automorphisms of so-called natural fiber bundles.
is projected onto a diffeomorphism of its base
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
, whose flow is a one-parameter group of diffeomorphisms of
There is a problem of constructing its lift to a projectable vector field on
Such a lift always exists, but it need not be canonical.
gives rise to the horizontal vector field on
, but this monomorphisms is not a Lie algebra morphism, unless
However, there is a category of above mentioned natural bundles
which admit the functorial lift
is a Lie algebra monomorphism This functorial lift
is an infinitesimal general covariant transformation of
In a general setting, one considers a monomorphism
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
gives rise to the tangent automorphism
which is a general covariant transformation of
With respect to the holonomic coordinates
, this transformation reads A frame bundle
of linear tangent frames in
General covariant transformations constitute a subgroup of holonomic automorphisms of