Unit 4: Christoffel Symbols & Affine Connections

1. Introduction and Historical Context

In Euclidean space with Cartesian coordinates, the partial derivatives of basis vectors vanish identically ($\partial_j \mathbf{e}_i = 0$), so ordinary partial derivatives of vector fields transform as tensors. However, in curvilinear coordinates or on a curved Riemannian manifold, the coordinate basis vectors $\mathbf{e}_i = \frac{\partial \mathbf{r}}{\partial x^i}$ vary from point to point.

To quantify how the basis changes along coordinate directions, Elwin Bruno Christoffel (1869) introduced connection coefficients that express the derivatives of basis vectors in terms of the basis itself.

2. Christoffel Symbol of the First Kind $[ij, k]$

Definition 4.1 (Christoffel Symbol of the First Kind): Let $g_{ij}$ be the covariant metric tensor of an $n$-dimensional Riemannian manifold $V_n$. The Christoffel symbol of the first kind, denoted by $[ij, k]$ or $\Gamma_{k, ij}$, is defined by the three-index metric derivative combination:

Essential Properties:

1. Symmetry in Lower Indices: Because $g_{ik} = g_{ki}$ and $g_{jk} = g_{kj}$:

The symbol $[ij, k]$ is strictly symmetric in its first two indices $i$ and $j$.

2. Derivative of the Metric Tensor:

By adding $[ik, j]$ and $[jk, i]$:

Hence, the partial derivative of any metric component can be expressed as a sum of Christoffel symbols:

3. Christoffel Symbol of the Second Kind $\Gamma^k_{ij}$

Definition 4.2 (Christoffel Symbol of the Second Kind): The Christoffel symbol of the second kind (or Levi-Civita affine connection coefficients), denoted by $\Gamma^k_{ij}$ or $\begin{Bmatrix} k \\ ij \end{Bmatrix}$, is defined by raising the last index of $[ij, m]$ using the conjugate metric tensor $g^{km}$:

Conversely, lowering the upper index retrieves the symbol of the first kind:

Symmetry:

In a torsion-free (Levi-Civita) connection, $\Gamma^k_{ij}$ is symmetric in the lower indices $i$ and $j$. In an $n$-dimensional space, the number of independent components of $\Gamma^k_{ij}$ is:

For $n = 2$: $2 \times 3 = 6$ components. For $n = 3$: $3 \times 6 = 18$ components. For $n = 4$: $4 \times 10 = 40$ components.

1. Proof of the Inhomogeneous Transformation Law

A critical foundational result in differential geometry is that Christoffel symbols are NOT tensors. Under coordinate transformations, an inhomogeneous second-derivative term arises.

Theorem 4.1 (Non-Tensorial Transformation of $\Gamma^k_{ij}$): Under a smooth, invertible coordinate transformation $x^i \to \bar{x}^\alpha$, the Christoffel symbols of the second kind transform according to:

Alternatively, expressing in terms of derivatives of $\bar{x}$ on the right:

Rigorous Proof:

Start with the transformation law of the metric tensor:

Differentiating with respect to $\bar{x}^\mu$:

Forming the cyclic permutation combination for the first kind:

Substituting the derivatives of $\bar{g}$, the metric derivative terms combine to give $[ij, k] \frac{\partial x^i}{\partial \bar{x}^\alpha} \frac{\partial x^j}{\partial \bar{x}^\beta} \frac{\partial x^k}{\partial \bar{x}^\mu}$, while four of the second-derivative terms cancel symmetrically, leaving:

Now multiply by $\bar{g}^{\gamma\mu} = g^{lm} \frac{\partial \bar{x}^\gamma}{\partial x^l} \frac{\partial \bar{x}^\mu}{\partial x^m}$:

2. The Difference Between Two Connections is a Tensor

Although an individual connection $\Gamma^k_{ij}$ does not transform as a tensor, the difference between two arbitrary connections on the same manifold does:

Theorem 4.2: If $\Gamma^k_{ij}$ and $\tilde{\Gamma}^k_{ij}$ are two affine connections on $M$, their difference:

transforms as a true tensor of type $(1, 2)$.

Proof: Subtracting their transformation equations, the non-tensorial second-derivative terms $\frac{\partial \bar{x}^\gamma}{\partial x^l} \frac{\partial^2 x^l}{\partial \bar{x}^\alpha \partial \bar{x}^\beta}$ are identical and cancel out:

which is the exact tensor transformation law for a $(1, 2)$ tensor. $\blacksquare$

1. The Metric Compatibility Axiom

A Riemannian manifold $(M, g)$ possesses a unique affine connection $\nabla$ that is:

1. Torsion-Free: $\Gamma^k_{ij} = \Gamma^k_{ji}$ (or $\nabla_X Y - \nabla_Y X = [X, Y]$).

2. Metric Compatible: The inner product of any two parallel-transported vectors is preserved along any curve. In terms of the metric tensor:

This unique connection is known as the Levi-Civita connection.

2. Fundamental Derivative Formulae

From metric compatibility and the definition of Christoffel symbols, we have several fundamental identities frequently used in tensor computations:

Identity 1: Derivative of Covariant Metric

Identity 2: Derivative of Contravariant Metric

Differentiating the relation $g_{im} g^{mj} = \delta_i^j$ with respect to $x^k$:

Contracting with $g^{il}$:

Substituting $\frac{\partial g_{im}}{\partial x^k} = g_{sm} \Gamma^s_{ik} + g_{is} \Gamma^s_{mk}$:

Renaming indices:

3. Local Geodesic (Riemann Normal) Coordinates

By virtue of the inhomogeneous term in the transformation law:

At any given point $P$, one can choose a coordinate transformation such that:

In this special coordinate frame (called Riemann normal coordinates centered at $P$):

This is the mathematical embodiment of Einstein's Equivalence Principle: at any single spacetime point, the effects of gravitation (curvature) can be locally transformed away into a locally flat inertial frame.

1. Jacobi's Determinant Formula

Let $g = \det(g_{ij})$. From Jacobi's formula for the derivative of a matrix determinant:

Proof:

By Laplace expansion along row $i$:

where $G^{ij}$ is the cofactor. Since $g^{ij} = \frac{G^{ji}}{g} = \frac{G^{ij}}{g}$, we have $G^{ij} = g g^{ij}$. Differentiating with respect to $g_{ij}$:

Applying the multivariable chain rule:

2. The Contracted Christoffel Symbol $\Gamma^i_{ik}$

Theorem 4.3 (Contracted Connection Identity): The contracted Christoffel symbol with upper index contracted against a lower index satisfies:

Complete Proof:

From definition 4.2:

Distribute the contraction:

In the second term, rename the dummy indices in $g^{im} \frac{\partial g_{ik}}{\partial x^m}$: swap $i \leftrightarrow m$. Since $g^{im} = g^{mi}$, the term becomes $g^{mi} \frac{\partial g_{mk}}{\partial x^i} = g^{im} \frac{\partial g_{km}}{\partial x^i}$. Thus, the second and third terms cancel identically:

We are left with:

Using Jacobi's formula $g^{im} \frac{\partial g_{im}}{\partial x^k} = \frac{1}{g} \frac{\partial g}{\partial x^k}$:

3. Divergence of a Contravariant Vector

This identity allows us to compute the invariant divergence of a vector field $A^i$ without needing individual Christoffel symbols:

This proves that the divergence $\text{div}(\mathbf{A}) = \frac{1}{\sqrt{|g|}} \partial_i (\sqrt{|g|} A^i)$ is an invariant scalar!

1. Geodesics as Extremals of Arc Length

In Riemannian geometry, a geodesic is the curve that extremizes the distance between two fixed points $P$ and $Q$. Let the curve be parameterized by an arbitrary parameter $\lambda \in [\lambda_0, \lambda_1]$:

Instead of the square root, it is standard and dynamically equivalent to extremize the action / energy functional:

2. Derivation of the Geodesic Equation

Theorem 4.4 (The Geodesic Differential Equations): If an affine parameter $s$ (the arc length parameter) is used, the Euler-Lagrange equations for the Lagrangian $\mathcal{L} = g_{ij} \frac{dx^i}{ds} \frac{dx^j}{ds}$ yield:

Step-by-Step Variational Derivation:

1. Compute the functional derivatives:

1. Apply the Euler-Lagrange equation $\frac{d}{ds}\left( \frac{\partial \mathcal{L}}{\partial \dot{x}^k} \right) - \frac{\partial \mathcal{L}}{\partial x^k} = 0$:

Expanding the total derivative via the chain rule $\frac{d}{ds} g_{ki} = \frac{\partial g_{ki}}{\partial x^j} \dot{x}^j$:

1. Symmetrize the second term: $2 \frac{\partial g_{ki}}{\partial x^j} \dot{x}^i \dot{x}^j = \left( \frac{\partial g_{ki}}{\partial x^j} + \frac{\partial g_{kj}}{\partial x^i} \right) \dot{x}^i \dot{x}^j$.

Grouping terms:

Recognizing the term in parentheses as $2 [ij, k]$:

1. Contract with the conjugate metric $g^{km}$:

Renaming index $m \to k$:

3. Geodesic Deviation & Tidal Forces

When two neighboring geodesics $x^i(s)$ and $x^i(s) + \xi^i(s)$ propagate with separation vector $\xi^i$, the relative acceleration of the geodesics is governed by the Jacobi equation (Equation of Geodesic Deviation):

where $R^k_{imj}$ is the Riemann curvature tensor.

• In flat space ($R = 0$), $\frac{D^2 \xi^k}{ds^2} = 0$: initially parallel geodesics remain parallel forever (Euclid's 5th postulate).
• In curved space ($R \ne 0$), geodesics converge or diverge. In General Relativity, this relative acceleration is observed physically as gravitational tidal force!