General covariant transformations

Template:Short description Template:No footnotes 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 $\mathrm{\Gamma }$ on $Y$, every vector field $\tau$ on $X$ gives rise to the horizontal vector field

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

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

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

$[\tilde{\tau },{\tilde{\tau }}^{\prime }]=\widetilde{[\tau ,{\tau }^{\prime }]}.$

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

In a general setting, one considers a monomorphism $f\to \tilde{f}$ of a group of diffeomorphisms of $X$ to a group of bundle automorphisms of a natural bundle $T\to X$. Automorphisms $\tilde{f}$ 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 $\tilde{f}=Tf$ 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

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

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

• General covariance
• Gauge gravitation theory
• Fibered manifold

References

• Kolář, I., Michor, P., Slovák, J., Natural operations in differential geometry. Springer-Verlag: Berlin Heidelberg, 1993. Template:ISBN, Template:ISBN.
• Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing: Saarbrücken, 2013. Template:ISBN; Template:ArXiv
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