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.

Mathematical definition

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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  .

See also

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References

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  • 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
  • Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN 0-521-36948-7