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. Let ${\displaystyle \pi :Y\to X}$ be a fiber bundle coordinated by ${\displaystyle (x^{\lambda },y^{i})}$. 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 ${\displaystyle FX}$.

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