Английская Википедия:General covariant transformations
Шаблон:Short description Шаблон:No footnotes In physics, general covariant transformations are symmetries of gravitation theory on a world manifold <math>X</math>. They are gauge transformations whose parameter functions are vector fields on <math>X</math>. 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 <math>\pi:Y\to X</math> be a fibered manifold with local fibered coordinates <math> (x^\lambda, y^i)\,</math>. Every automorphism of <math>Y</math> is projected onto a diffeomorphism of its base <math>X</math>. However, the converse is not true. A diffeomorphism of <math>X</math> need not give rise to an automorphism of <math>Y</math>.
In particular, an infinitesimal generator of a one-parameter Lie group of automorphisms of <math>Y</math> is a projectable vector field
- <math> u=u^\lambda(x^\mu)\partial_\lambda + u^i(x^\mu,y^j)\partial_i </math>
on <math>Y</math>. This vector field is projected onto a vector field <math>\tau=u^\lambda\partial_\lambda</math> on <math>X</math>, whose flow is a one-parameter group of diffeomorphisms of <math>X</math>. Conversely, let <math>\tau=\tau^\lambda\partial_\lambda</math> be a vector field on <math>X</math>. There is a problem of constructing its lift to a projectable vector field on <math>Y</math> projected onto <math>\tau</math>. Such a lift always exists, but it need not be canonical. Given a connection <math>\Gamma</math> on <math>Y</math>, every vector field <math>\tau</math> on <math>X</math> gives rise to the horizontal vector field
- <math>\Gamma\tau =\tau^\lambda(\partial_\lambda +\Gamma_\lambda^i\partial_i) </math>
on <math>Y</math>. This horizontal lift <math>\tau\to\Gamma\tau</math> yields a monomorphism of the <math>C^\infty(X) </math>-module of vector fields on <math>X</math> to the <math>C^\infty(Y) </math>-module of vector fields on <math>Y</math>, but this monomorphisms is not a Lie algebra morphism, unless <math>\Gamma</math> is flat.
However, there is a category of above mentioned natural bundles <math>T\to X</math> which admit the functorial lift <math>\widetilde\tau</math> onto <math>T</math> of any vector field <math>\tau</math> on <math>X</math> such that <math>\tau\to\widetilde\tau</math> is a Lie algebra monomorphism
- <math> [\widetilde \tau,\widetilde \tau']=\widetilde {[\tau,\tau']}.</math>
This functorial lift <math>\widetilde\tau</math> is an infinitesimal general covariant transformation of <math>T</math>.
In a general setting, one considers a monomorphism <math>f\to\widetilde f</math> of a group of diffeomorphisms of <math>X</math> to a group of bundle automorphisms of a natural bundle <math>T\to X</math>. Automorphisms <math>\widetilde f</math> are called the general covariant transformations of <math>T</math>. For instance, no vertical automorphism of <math>T</math> is a general covariant transformation.
Natural bundles are exemplified by tensor bundles. For instance, the tangent bundle <math>TX</math> of <math>X</math> is a natural bundle. Every diffeomorphism <math>f</math> of <math>X</math> gives rise to the tangent automorphism <math>\widetilde f=Tf</math> of <math>TX</math> which is a general covariant transformation of <math>TX</math>. With respect to the holonomic coordinates <math>(x^\lambda, \dot x^\lambda) </math> on <math>TX</math>, this transformation reads
- <math>\dot x'^\mu=\frac{\partial x'^\mu}{\partial x^\nu}\dot x^\nu. </math>
A frame bundle <math>FX</math> of linear tangent frames in <math>TX</math> also is a natural bundle. General covariant transformations constitute a subgroup of holonomic automorphisms of <math>FX</math>. All bundles associated with a frame bundle are natural. However, there are natural bundles which are not associated with <math>FX</math>.
See also
References
- Kolář, I., Michor, P., Slovák, J., Natural operations in differential geometry. Springer-Verlag: Berlin Heidelberg, 1993. Шаблон:ISBN, Шаблон:ISBN.
- Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing: Saarbrücken, 2013. Шаблон:ISBN; Шаблон:ArXiv
- Шаблон:Citation
