Relativity: Christoffel symbols

1. Gradient operator

The gradient of a scalar φ is defined as vector:
∇ φ = (∂φ/∂x, ∂φ/∂y, ∂φ/∂z)
or:
∇ φ = (∂φ/∂x¹, ∂φ/∂x², ∂φ/∂x³)

or more generally:
∇ φ = ∂φ/∂xⁱ
In the contravariant basis vector gⁱ, it has the following expression:
∇ φ = ∂φ/∂xⁱ gⁱ

Using the definition of the del operator =
∇ = gⁱ ∂/∂xⁱ , we can write : ∇ φ = del (φ)

∂φ/∂xⁱ is the i^th covariant component of the gradient vector.

2. Divergence operator for a vector field

A vector field is the vector with components in the basis vector gⁱ or g_i which vary with position.

The divergence of a normal vector A = (A1, A2, A3) is
∇ A = d A1/dx1 + d A2/dx2 + d A3/dx3

Let's consider the vector V in the covariant basis vector g_i, the
V = Vⁱ g_i,
The divergence of the vector field V is :
∇_i V = gⁱ ∂/∂xⁱ (V^j g_j)

The derivative operates both on the vector components V^j and the basis vectors g_j .

We have then:
∇_i V = gⁱ . [(∂V^j /∂xⁱ) g_j + V^j (∂g_j/∂xⁱ)]

The term ∂g_j/∂xⁱ is a vector of components Γ_ji^k written in the basis vectors covariants g_k as:

∂g_j/∂xⁱ = Γ^k_ji g_k

The coefficients Γ^k _ji are called Christoffel symbols.

Christoffel symbols: Γ^k_ij :
∂g_i/∂x^j = Γ^k_ij g_k

The form Γ^k_ij of the Christoffel symbols are called of the second kind.

∇_iV is called the covariant derivative of the vector field V.

We can express the covariant derivative of V in terms of Chrisytoffel symbols as:

∇_i V = gⁱ . [(∂V^j /∂xⁱ) g_j + V^j Γ^k_ji g_k]
= gⁱ . [(∂V^k /∂xⁱ) g_k + V^j Γ^k_ji g_k]
= gⁱ . [(∂V^k /∂xⁱ) + V^j Γ^k_ji] g_k

∇_i V = gⁱ . [(∂V^k /∂xⁱ) + V^j Γ^k_ji] g_k

∇_i V = gⁱ . [(∂V^k/∂xⁱ) + V^j Γ^k_ji] g_k

Setting
∂ /∂xⁱ = ∂xⁱ,
we can write:
∇_i = gⁱ [∂xⁱ + Γ^k_ji]g_k =
[∂xⁱ + Γ^k_ji] δⁱ_k = ∂xⁱ + Γⁱ_ji = ∂xⁱ + Γ_j
∇_i = ∂xⁱ + Γ_j

For short: ∇_μ = ∂x^μ + Γ_μ
∇_μ = ∂x^μ + Γ_μ

3. Christoffel symbols of first kind

Dot multiplying the above equation by g^l, we obtain:
∂g_i/∂x^j g^l = Γ^k_ij(g_k g^l) =
Γ^k_ij^k g_k^l = Γ^k_ij δg^l_k = Γ^l_ij

Γ^l_ij = ∂g_i/∂x^j g^l

Γ^l_ij = g^l ∂g_i/∂x^j    (3.1)

In terms of derivatives of the position vector r:
g_i = ∂r/∂xⁱ .

The Christoffel symbol; becomes:
Γ^l_ij = g^l ∂²r ∂xⁱ∂x^j

That shows the symmetry with respect to the two indexes i an j:
Γ^l_ij = Γ^l_ji

Γ^l_ij = Γ^l_ji

Using the metric g_ij to lower an index, we can write:
g_kl Γ^l_ij = Γ_kij

Γ_kij = g_kl Γ^l_ij    (3.2)

The term Γ_kij is called the Christoffel symbol of first kind . We have the same symmetry for this kind:

Γ_kij = Γ_jik

Γ_kij = Γ_jik

4. Christoffel symbols in terms of the metric g_μν

Using the expression (3) and (4):

Γ^l_ij = g^l ∂g_i/∂x^j
g_kl Γ^l_ij = Γ_kij

we obtain:
Γ_kij = g_kl g^l . ∂g_i/∂x^j = g_k . ∂g_i/∂x^j

Γ_kij = g_k . ∂g_i/∂x^j    (4.1)

The spatial derivative of the metric g_ij is:

d(g_ij)/∂x^k = d(g_i. g_j)/∂x^k
g_i . d(g_j)/∂x^k + g_j . d(g_i)/∂x^k

In terms of the Christoffel symbols:
d(g_ij)/∂x^k = Γ_ijk + Γ_jik    (4.2)
similarly
d(g_ik)/∂x^j = Γ_ikj + Γ_kij    (4.3)
and
d(g_jk)/∂xⁱ = Γ_jki + Γ_kji    (4.4)

Adding (4.4) to (4.3) gives:
d(g_ik)/∂x^j + d(g_jk)/∂xⁱ = Γ_ikj + Γ_kij + Γ_jki + Γ_kji

Sutracting (4.2) gives:
d(g_ik)/∂x^j + d(g_jk)/∂xⁱ - d(g_ij)/∂x^k = Γ_ikj + Γ_kij + Γ_jki + Γ_kji - Γ_ijk - Γ_jik

d(g_ik)/∂x^j + d(g_jk)/∂xⁱ - d(g_ij)/∂x^k = Γ_kij + Γ_kji

Since, we have by symmetry:
Γ_ikj - Γ_ijk = 0
Γ_jki - Γ_jik = 0
Γ_kij + Γ_kji = 2 Γ_kij

We obtain:

2 Γ_kij = d(g_ik)/∂x^j + d(g_jk)/∂xⁱ - d(g_ij)/∂x^k

That is:

Γ_kij = (1/2)[dg_ik/∂x^j + dg_jk/∂xⁱ - dg_ij/∂x^k]    (4.5)

We have
g^kl Γ_kij = Γ^l_ij
or
Γ^k_ij = g^lk Γ_lij = g^kl Γ_lij

Therefore:
Γ^k_ij = g^kl (1/2)[dg_il/∂x^j + dg_jl/∂xⁱ - dg_ij/∂x^l]
Γ^k_ij = g^kl (1/2)[dg_il/∂x^j + dg_jl/∂xⁱ - dg_ij/∂x^l]



Γ^k_ij = (1/2) g^kl [dg_il/∂x^j + dg_jl/∂xⁱ - dg_ij/∂x^l]