R μνρ
λ
∂ μ Γ
λ
νρ À ∂ ν Γ
λ
μρ þ Γ
λ
μσ Γ
σ
νρ À Γ
λ
νσ Γ
σ
μρ ,
T μν
λ
Γ
λ
μν À Γ
λ
νμ :
Consider a manifold with a metric structure g μν , then it can be proved that there is
only one connection, Γ
g
ð Þ λ
μν , compatible with the metric and torsionless, namely
∇ λ g μν ¼ 0, T μν
λ
¼ 0:
This connection is called the Levi-Civita connection of g μν , and it is completely
determined by the metric:
Γ
g
ð Þ λ
μν ¼
1
2
g
λσ
∂ μ g σν þ ∂ ν g μσ À ∂ σ g μν
À
Á :
In fact, given a metric, the Levi-Civita affine structure is the simplest connection
we can deal with. The metric compatibility and the nullity of the torsion simplify
many geometrical identities. Moreover, we are not introducing extra degrees of
freedom in the theory, just the ones that come from the metric.
Before starting with the physics, let us introduce a few useful definitions. A curve
γ(α) is a differentiable function γ : I ! M, where I is an interval of the real line. The
Fig. 5.2 An affine connection represents a notion of parallel in the manifold and allows to parallely
transport vectors along curves. The dashed line is, by definition, the parallel transport of V
μ
( p) from
p to q along the x
λ direction. This vector and V
μ (q) can be compared since both live in T q M
5 (Non-)Uniqueness of Einstein–Palatini Gravity
51
λ
∂ μ Γ
λ
νρ À ∂ ν Γ
λ
μρ þ Γ
λ
μσ Γ
σ
νρ À Γ
λ
νσ Γ
σ
μρ ,
T μν
λ
Γ
λ
μν À Γ
λ
νμ :
Consider a manifold with a metric structure g μν , then it can be proved that there is
only one connection, Γ
g
ð Þ λ
μν , compatible with the metric and torsionless, namely
∇ λ g μν ¼ 0, T μν
λ
¼ 0:
This connection is called the Levi-Civita connection of g μν , and it is completely
determined by the metric:
Γ
g
ð Þ λ
μν ¼
1
2
g
λσ
∂ μ g σν þ ∂ ν g μσ À ∂ σ g μν
À
Á :
In fact, given a metric, the Levi-Civita affine structure is the simplest connection
we can deal with. The metric compatibility and the nullity of the torsion simplify
many geometrical identities. Moreover, we are not introducing extra degrees of
freedom in the theory, just the ones that come from the metric.
Before starting with the physics, let us introduce a few useful definitions. A curve
γ(α) is a differentiable function γ : I ! M, where I is an interval of the real line. The
Fig. 5.2 An affine connection represents a notion of parallel in the manifold and allows to parallely
transport vectors along curves. The dashed line is, by definition, the parallel transport of V
μ
( p) from
p to q along the x
λ direction. This vector and V
μ (q) can be compared since both live in T q M
5 (Non-)Uniqueness of Einstein–Palatini Gravity
51