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 (holonomic) reference frame transformations in General Relativity.
In mathematics, general covariant transformations are defined as particular automorphisms of so-called natural fiber bundles. Let be a fiber bundle coordinated by . 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 .
- Kolář, I., Michor, P., Slovák, J., Natural operations in differential geometry. Springer-Verlag: Berlin Heidelberg, 1993. ISBN 3-540-56235-4, ISBN 0-387-56235-4.
- Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing: Saarbrücken, 2013. ISBN 978-3-659-37815-7; arXiv: 0908.1886