Relativity: Riemann tensor

1. Function-Operator Commutator

We will use the notation:
∂_ν = ∂/ ∂x_ν

The cmmutator [∂_ν, f(x)] is :
[∂_ν, f(x)] = ∂_ν. f(x) - ∂_νf(x)

so
[∂_ν, f(x)]V(x) = [∂_ν f(x) - f(x) ∂_ν]V(x) =
∂_ν (f(x) V(x)) - f(x) ∂_νV(x) =
f(x) ∂_νV(x) + V(x) ∂_νf(x) - f(x) ∂_νV(x) = V(x) ∂_νf(x)

[∂_ν, f(x)]V(x) = V(x) ∂_νf(x)

∂_ν = ∂/ ∂x_ν

[∂_ν, f(x)]V(x) = V(x) ∂_νf(x)

2. Parallel transport of a vector: Riemann tensor

If, along a curve, a vector transported parallely, comes back to its initial direction, there is no curvature. Otherwise, if the final direction is deviated with respect to its initial direction, there is a curvature which is expressed by the Riemann tensor.



The parallel transport of a vector V along the path ABCD is writtena as:

[(V_C - V_D) - (V_B V_A)] - [(V_C V_B) - ( V_D - V_A')] =
V_A - V_A' = ∂V

∂V = [∂μ ∂ν∇_μ ∇_ν - ∂μ ∂ν∇_ν ∇_μ]V
= ∂μ ∂ν[∇_ν, ∇_μ]V

∂V = ∂μ ∂ν[∇_ν, ∇_μ]V = ∂μ ∂ν R_νμV
With:
R_νμ = [∇_ν, ∇_μ]



R_νμ = [∇_ν,∇_μ]

More precisely:

∂V^α = ∂μ ∂ν R^α<sub>μνβV^β

∂V^α = ∂μ ∂ν R^αμνβ V^β

We have:
∇ν = ∂ν + Γν

so
∂V = ∂μ ∂ν∇[ (∂ν + Γν)(∂μ + Γμ) - (∂μ + Γμ)(∂ν + Γν) ]V

∂ν∂μ = ∂μ∂ν (the ordinary derivatives commute), so
∂V = ΓνΓμ - ΓμΓν - [∂μ, Γν] + [∂ν, Γμ]

We have:
[∂μ, Γν]V(x) = V(x) ∂μΓν
and
[∂ν, Γμ]V(x) = V(x) ∂νΓμ

Therefore:
∂V = ∂μ ∂ν∇[ΓνΓμ - ΓμΓν + ∂νΓμ - ∂μΓν]V(x)

With indexes:
∂V = ∂μ ∂ν∇[ Γ^ανδΓ^δμβ - Γ^αμδΓ^δνβ + ∂νΓ^δμβ - ∂μΓ^δνβ]V(x)

Let's write:
R^ανμβ = Γ^ανδΓ^δμβ - Γ^αμδΓ^δνβ + ∂νΓ^δμβ - ∂μΓ^δνβ


R^ανμβ = Γ^ανδΓ^δμβ - Γ^αμδΓ^δνβ + ∂νΓ^δμβ - ∂μΓ^δνβ


Which is the Riemann curvature tensor.

It can be written as:

∂Vα = ∂μ ∂ν Rαμνβ V^β</sub>

3. Properties of Riemann curvature tensor

1. In the Minkowski (4-dimentional flat space time) , covariant derivatives reduce to partial derivatives and give a zero Rieman tensor.

The curvature is zero in a flat space.

2. The Christoffel coefficients are:
Γ_βμν = (1/2)[∂g_μβ/∂x^ν + ∂g_jk/∂x^μ - ∂g_μν/∂x^β]

3. In the local frame, the basis vectors g_μ are constant, so the related Christoffel coefficients are zero. But their partial derivatives are not zero.

It remains then for the Riemann tensor:

R_αβμν = ∂_αΓ_μνβ - ∂_βΓ_μνα

With the expressions of the Christoffel symbols:
Γ_kij = (1/2)[∂g_ik/∂x^j + ∂g_jk/∂xⁱ - ∂g_ij/∂x^k]

we obtain:
R_αβμν = ∂_α[(1/2)[∂g_μβ/∂x^ν + ∂g_νβ/∂x^μ - ∂g_μν/∂x^β] -
∂_β[ (1/2)[∂g_μα/∂x^ν + ∂g_να/∂x^μ - ∂g_μν/∂x^α] =
[(1/2)[∂g_μβ/∂x^ν ∂_α + ∂g_νβ/∂x^μ∂_α - ∂g_μα/∂x^ν ∂_β - ∂g_να/∂x^μ ∂_β ]

4. R_αβμν is
antisymmetric with α and β exchange,
antisymmetric with μ and ν and exchange,
symetric with αβ and μν exchange, and
R_αβμν + R_ανβμ + R_ανμβ = 0

4. Ricci tensor, Ricci scalar

The Ricci tensor R_αμ is defined as the contraction of the Riemann tensor:

R_αμ = g^βνR_αβμν

which is symmetric like Rieman curvature tensor:

R_αμ = R_μα

The Ricci scalar R is defined as the contraction of the Ricci tensor R_αμ:

R = g^αμ R_αμ

Ricci tensor: R_αμ = g^βνR_αβμν

Ricci scalar: R = g^αμ R_αμ