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
.
