In comparison with General Relativity, dynamic variables of metric-affine gravitation theory are both a pseudo-Riemannian metric and a general linear connection on a world manifold
X
. Metric-affine gravitation theory has been suggested as a natural generalization of Einstein–Cartan theory of gravity with torsion where a linear connection obeys the condition that a covariant derivative of a metric equals zero.
Metric-affine gravitation theory straightforwardly comes from gauge gravitation theory where a general linear connection plays the role of a gauge field. Let
T
X
be the tangent bundle over a manifold
X
provided with bundle coordinates
(
x
μ
,
x
˙
μ
)
. A general linear connection on
T
X
is represented by a connection tangent-valued form
Γ
=
d
x
λ
⊗
(
∂
λ
+
Γ
λ
μ
ν
x
˙
ν
∂
˙
μ
)
.
It is associated to a principal connection on the principal frame bundle
F
X
of frames in the tangent spaces to
X
whose structure group is a general linear group
G
L
(
4
,
R
)
. Consequently, it can be treated as a gauge field. A pseudo-Riemannian metric
g
=
g
μ
ν
d
x
μ
⊗
d
x
ν
on
T
X
is defined as a global section of the quotient bundle
F
X
/
S
O
(
1
,
3
)
→
X
, where
S
O
(
1
,
3
)
is the Lorentz group. Therefore, on can regard it as a classical Higgs field in gauge gravitation theory. Gauge symmetries of metric-affine gravitation theory are general covariant transformations.
It is essential that, given a pseudo-Riemannian metric
g
, any linear connection
Γ
on
T
X
admits a splitting
Γ
μ
ν
α
=
{
μ
ν
α
}
+
S
μ
ν
α
+
1
2
C
μ
ν
α
in the Christoffel symbols
{
μ
ν
α
}
=
−
1
2
(
∂
μ
g
ν
α
+
∂
α
g
ν
μ
−
∂
ν
g
μ
α
)
,
a nonmetricity tensor
C
μ
ν
α
=
C
μ
α
ν
=
∇
μ
Γ
g
ν
α
=
∂
μ
g
ν
α
+
Γ
μ
ν
α
+
Γ
μ
α
ν
and a contorsion tensor
S
μ
ν
α
=
−
S
μ
α
ν
=
1
2
(
T
ν
μ
α
+
T
ν
α
μ
+
T
μ
ν
α
+
C
α
ν
μ
−
C
ν
α
μ
)
,
where
T
μ
ν
α
=
1
2
(
Γ
μ
ν
α
−
Γ
α
ν
μ
)
is the torsion tensor of
Γ
.
Due to this splitting, metric-affine gravitation theory possesses a different collection of dynamic variables which are a pseudo-Riemannian metric, a non-metricity tensor and a torsion tensor. As a consequence, a Lagrangian of metric-affine gravitation theory can contain different terms expressed both in a curvature of a connection
Γ
and its torsion and non-metricity tensors. In particular, a metric-affine f(R) gravity, whose Lagrangian is an arbitrary function of a scalar curvature
R
of
Γ
, is considered.
A linear connection
Γ
is called the metric connection for a pseudo-Riemannian metric
g
if
g
is its integral section, i.e., the metricity condition
∇
μ
Γ
g
ν
α
=
0
holds. A metric connection reads
Γ
μ
ν
α
=
{
μ
ν
α
}
+
1
2
(
T
ν
μ
α
+
T
ν
α
μ
+
T
μ
ν
α
)
.
For instance, the Levi-Civita connection in General Relativity is a torsion-free metric connection.
A metric connection is associated to a principal connection on a Lorentz reduced subbundle
F
g
X
of the frame bundle
F
X
corresponding to a section
g
of the quotient bundle
F
X
/
S
O
(
1
,
3
)
→
X
. Restricted to metric connections, metric-affine gravitation theory comes to the above-mentioned Einstein – Cartan gravitation theory.
At the same time, any linear connection
Γ
defines a principal adapted connection
Γ
g
on a Lorentz reduced subbundle
F
g
X
by its restriction to a Lorentz subalgebra of a Lie algebra of a general linear group
G
L
(
4
,
R
)
. For instance, the Dirac operator in metric-affine gravitation theory in the presence of a general linear connection
Γ
is well defined, and it depends just of the adapted connection
Γ
g
. Therefore, Einstein – Cartan gravitation theory can be formulated as the metric-affine one, without appealing to the metricity constraint.
In metric-affine gravitation theory, in comparison with the Einstein - Cartan one, a question on a matter source of a non-metricity tensor arises. It is so called hypermomentum, e.g., a Noether current of a scaling symmetry.
