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.
