General covariant transformations – Wikipedia

Symmetries in a gravitational theory

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[edit]

Let

${displaystyle pi :Yto X}$

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

${displaystyle Tto X}$

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

${displaystyle fto {widetilde {f}}}$

of a group of diffeomorphisms of

${displaystyle X}$

to a group of bundle automorphisms of a natural bundle

${displaystyle Tto X}$

. 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[edit]

References[edit]

• 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