General covariant transformations

In physics, general covariant transformations are symmetries of gravitation theory on a world manifold X. They are gauge transformations whose parameter functions are vector fields on X. 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 \pi:Y\to X be a fibered manifold with local fibered coordinates (x^\lambda, y^i)\,. Every automorphism of Y is projected onto a diffeomorphism of its base X. However, the converse is not true. A diffeomorphism of X need not give rise to an automorphism of Y.

In particular, an infinitesimal generator of a one-parameter Lie group of automorphisms of Y is a projectable vector field

u=u^\lambda(x^\mu)\partial_\lambda + u^i(x^\mu,y^j)\partial_i

on Y. This vector field is projected onto a vector field \tau=u^\lambda\partial_\lambda on X, whose flow is a one-parameter group of diffeomorphisms of X. Conversely, let \tau=\tau^\lambda\partial_\lambda be a vector field on X. There is a problem of constructing its lift to a projectable vector field on Y projected onto \tau. Such a lift always exists, but it need not be canonical. Given a connection \Gamma on Y, every vector field \tau on X gives rise to the horizontal vector field

\Gamma\tau =\tau^\lambda(\partial_\lambda +\Gamma_\lambda^i\partial_i)

on Y. This horizontal lift \tau\to\Gamma\tau yields a monomorphism of the C^\infty(X) -module of vector fields on X to the C^\infty(Y) -module of vector fields on Y, but this monomorphisms is not a Lie algebra morphism, unless \Gamma is flat.

However, there is a category of above mentioned natural bundles T\to X which admit the functorial lift \widetilde\tau onto T of any vector field \tau on X such that \tau\to\widetilde\tau is a Lie algebra monomorphism

[\widetilde \tau,\widetilde \tau']=\widetilde {[\tau,\tau']}.

This functorial lift \widetilde\tau is an infinitesimal general covariant transformation of T.

In a general setting, one considers a monomorphism f\to\widetilde f of a group of diffeomorphisms of X to a group of bundle automorphisms of a natural bundle T\to X. Automorphisms \widetilde f are called the general covariant transformations of T. For instance, no vertical automorphism of T is a general covariant transformation.

Natural bundles are exemplified by tensor bundles. For instance, the tangent bundle TX of X is a natural bundle. Every diffeomorphism f of X gives rise to the tangent automorphism \widetilde f=Tf of TX which is a general covariant transformation of TX. With respect to the holonomic coordinates (x^\lambda, \dot x^\lambda) on TX, this transformation reads

\dot x'^\mu=\frac{\partial x'^\mu}{\partial x^\nu}\dot x^\nu.

A frame bundle FX of linear tangent frames in TX also is a natural bundle. General covariant transformations constitute a subgroup of holonomic automorphisms of FX. All bundles associated with a frame bundle are natural. However, there are natural bundles which are not associated with FX.

See also

References

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