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 $FX$ .

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

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General covariant transformations

Mathematical definition

See also

References