General covariant transformations

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

In particular, an infinitesimal generator of a one-parameter Lie group of automorphisms of ${\displaystyle Y}$  is a projectable vector field

${\displaystyle u=u^{\lambda }(x^{\mu })\partial _{\lambda }+u^{i}(x^{\mu },y^{j})\partial _{i}}$ 

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

${\displaystyle \Gamma \tau =\tau ^{\lambda }(\partial _{\lambda }+\Gamma _{\lambda }^{i}\partial _{i})}$ 

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

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

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

This functorial lift ${\displaystyle {\widetilde {\tau }}}$  is an infinitesimal general covariant transformation of ${\displaystyle T}$ .

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

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

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

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

See also

• General covariance
• Gauge gravitation theory
• Fibered manifold

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