General covariant transformations
In physics, general covariant transformations are symmetries of gravitation theory on a world manifold [math]\displaystyle{ X }[/math]. They are gauge transformations whose parameter functions are vector fields on [math]\displaystyle{ X }[/math]. 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 [math]\displaystyle{ \pi:Y\to X }[/math] be a fibered manifold with local fibered coordinates [math]\displaystyle{ (x^\lambda, y^i)\, }[/math]. Every automorphism of [math]\displaystyle{ Y }[/math] is projected onto a diffeomorphism of its base [math]\displaystyle{ X }[/math]. However, the converse is not true. A diffeomorphism of [math]\displaystyle{ X }[/math] need not give rise to an automorphism of [math]\displaystyle{ Y }[/math].
In particular, an infinitesimal generator of a one-parameter Lie group of automorphisms of [math]\displaystyle{ Y }[/math] is a projectable vector field
- [math]\displaystyle{ u=u^\lambda(x^\mu)\partial_\lambda + u^i(x^\mu,y^j)\partial_i }[/math]
on [math]\displaystyle{ Y }[/math]. This vector field is projected onto a vector field [math]\displaystyle{ \tau=u^\lambda\partial_\lambda }[/math] on [math]\displaystyle{ X }[/math], whose flow is a one-parameter group of diffeomorphisms of [math]\displaystyle{ X }[/math]. Conversely, let [math]\displaystyle{ \tau=\tau^\lambda\partial_\lambda }[/math] be a vector field on [math]\displaystyle{ X }[/math]. There is a problem of constructing its lift to a projectable vector field on [math]\displaystyle{ Y }[/math] projected onto [math]\displaystyle{ \tau }[/math]. Such a lift always exists, but it need not be canonical. Given a connection [math]\displaystyle{ \Gamma }[/math] on [math]\displaystyle{ Y }[/math], every vector field [math]\displaystyle{ \tau }[/math] on [math]\displaystyle{ X }[/math] gives rise to the horizontal vector field
- [math]\displaystyle{ \Gamma\tau =\tau^\lambda(\partial_\lambda +\Gamma_\lambda^i\partial_i) }[/math]
on [math]\displaystyle{ Y }[/math]. This horizontal lift [math]\displaystyle{ \tau\to\Gamma\tau }[/math] yields a monomorphism of the [math]\displaystyle{ C^\infty(X) }[/math]-module of vector fields on [math]\displaystyle{ X }[/math] to the [math]\displaystyle{ C^\infty(Y) }[/math]-module of vector fields on [math]\displaystyle{ Y }[/math], but this monomorphisms is not a Lie algebra morphism, unless [math]\displaystyle{ \Gamma }[/math] is flat.
However, there is a category of above mentioned natural bundles [math]\displaystyle{ T\to X }[/math] which admit the functorial lift [math]\displaystyle{ \widetilde\tau }[/math] onto [math]\displaystyle{ T }[/math] of any vector field [math]\displaystyle{ \tau }[/math] on [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ \tau\to\widetilde\tau }[/math] is a Lie algebra monomorphism
- [math]\displaystyle{ [\widetilde \tau,\widetilde \tau']=\widetilde {[\tau,\tau']}. }[/math]
This functorial lift [math]\displaystyle{ \widetilde\tau }[/math] is an infinitesimal general covariant transformation of [math]\displaystyle{ T }[/math].
In a general setting, one considers a monomorphism [math]\displaystyle{ f\to\widetilde f }[/math] of a group of diffeomorphisms of [math]\displaystyle{ X }[/math] to a group of bundle automorphisms of a natural bundle [math]\displaystyle{ T\to X }[/math]. Automorphisms [math]\displaystyle{ \widetilde f }[/math] are called the general covariant transformations of [math]\displaystyle{ T }[/math]. For instance, no vertical automorphism of [math]\displaystyle{ T }[/math] is a general covariant transformation.
Natural bundles are exemplified by tensor bundles. For instance, the tangent bundle [math]\displaystyle{ TX }[/math] of [math]\displaystyle{ X }[/math] is a natural bundle. Every diffeomorphism [math]\displaystyle{ f }[/math] of [math]\displaystyle{ X }[/math] gives rise to the tangent automorphism [math]\displaystyle{ \widetilde f=Tf }[/math] of [math]\displaystyle{ TX }[/math] which is a general covariant transformation of [math]\displaystyle{ TX }[/math]. With respect to the holonomic coordinates [math]\displaystyle{ (x^\lambda, \dot x^\lambda) }[/math] on [math]\displaystyle{ TX }[/math], this transformation reads
- [math]\displaystyle{ \dot x'^\mu=\frac{\partial x'^\mu}{\partial x^\nu}\dot x^\nu. }[/math]
A frame bundle [math]\displaystyle{ FX }[/math] of linear tangent frames in [math]\displaystyle{ TX }[/math] also is a natural bundle. General covariant transformations constitute a subgroup of holonomic automorphisms of [math]\displaystyle{ FX }[/math]. All bundles associated with a frame bundle are natural. However, there are natural bundles which are not associated with [math]\displaystyle{ FX }[/math].
See also
References
- 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, https://archive.org/details/geometryofjetbun0000saun
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