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.
Let π : Y → X be a fibered manifold with local fibered coordinates ( x λ , 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 λ ( x μ ) ∂ λ + u i ( x μ , y j ) ∂ i on Y . This vector field is projected onto a vector field τ = u λ ∂ λ on X , whose flow is a one-parameter group of diffeomorphisms of X . Conversely, let τ = τ λ ∂ λ be a vector field on X . There is a problem of constructing its lift to a projectable vector field on Y projected onto τ . Such a lift always exists, but it need not be canonical. Given a connection Γ on Y , every vector field τ on X gives rise to the horizontal vector field
Γ τ = τ λ ( ∂ λ + Γ λ i ∂ i ) on Y . This horizontal lift τ → Γ τ yields a monomorphism of the C ∞ ( X ) -module of vector fields on X to the C ∞ ( Y ) -module of vector fields on Y , but this monomorphisms is not a Lie algebra morphism, unless Γ is flat.
However, there is a category of above mentioned natural bundles T → X which admit the functorial lift τ ~ onto T of any vector field τ on X such that τ → τ ~ is a Lie algebra monomorphism
[ τ ~ , τ ~ ′ ] = [ τ , τ ′ ] ~ . This functorial lift τ ~ is an infinitesimal general covariant transformation of T .
In a general setting, one considers a monomorphism f → f ~ of a group of diffeomorphisms of X to a group of bundle automorphisms of a natural bundle T → X . Automorphisms 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 T X of X is a natural bundle. Every diffeomorphism f of X gives rise to the tangent automorphism f ~ = T f of T X which is a general covariant transformation of T X . With respect to the holonomic coordinates ( x λ , x ˙ λ ) on T X , this transformation reads
x ˙ ′ μ = ∂ x ′ μ ∂ x ν x ˙ ν . A frame bundle F X of linear tangent frames in T X also is a natural bundle. General covariant transformations constitute a subgroup of holonomic automorphisms of F X . All bundles associated with a frame bundle are natural. However, there are natural bundles which are not associated with F X .
