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 $$FX$$.