General covariant transformations explained
. 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.
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
[\widetilde\tau,\widetilde\tau']=\widetilde{[\tau,\tau']}.
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.
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
| | \mu= | \partialx'\mu | | \partialx\nu |
|
| |
\nu.
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
.
See also
References
- Kolář, I., Michor, P., Slovák, J., Natural operations in differential geometry. Springer-Verlag: Berlin Heidelberg, 1993., .
- Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing: Saarbrücken, 2013. ;
