Parallel Transport and the Covariant Derivative

A tangent vector lives at a single point, and the tangent spaces at two different points of a manifold are distinct vector spaces with no canonical identification between them. Differentiating a vector field means comparing its value at nearby points, so it requires a rule for transporting a vector from one tangent space to another. That rule is the connection, and the derivative it defines is the covariant derivative. This lesson shows why the naive partial derivative fails to be tensorial, constructs the covariant derivative that fixes it, and derives the Christoffel symbols that the metric supplies.

Why the partial derivative is not a tensor

For a scalar field $f$ , the partial derivatives $\partial_{μ} f$ are the components of a genuine one-form: under a change of chart they transform with the inverse Jacobian, as a lower index should. For a vector field $V^{ν}$ , the object $\partial_{μ} V^{ν}$ fails this test. Transforming both indices and applying the chain rule,

\partial_{μ^{'}} V^{ν^{'}} = \frac{\partial x ^{μ}}{\partial x ^{' μ^{'}}} \frac{\partial x ^{' ν^{'}}}{\partial x ^{ν}} \partial_{μ} V^{ν} + \frac{\partial x ^{μ}}{\partial x ^{' μ^{'}}} \frac{\partial ^{2} x ^{' ν^{'}}}{\partial x ^{μ} \partial x ^{ν}} V^{ν} .

The first term is the correct tensor transformation. The second, carrying the second derivative of the coordinate transformation, is inhomogeneous and does not vanish in general. It is nonzero even in flat space: polar coordinates have $\partial^{2} x^{'} / \partial x^{2} \neq = 0$ , so the components of a constant vector field appear to change from point to point simply because the basis vectors rotate. The partial derivative confuses a real change in the field with a change in the coordinate basis.

In polar coordinates the basis vectors rotate from point to point, so a genuinely constant vector field has position-dependent components. The partial derivative registers this basis rotation as a spurious change in the field.

The connection and the covariant derivative

The remedy is to add a correction that cancels the inhomogeneous term. Define the covariant derivative of a vector field by

\nabla_{μ} V^{ν} = \partial_{μ} V^{ν} + Γ^{ν}_{μ λ} V^{λ},

where the $Γ^{ν}_{μ λ}$ are the connection coefficients. They are not the components of a tensor; their transformation law is designed to carry an inhomogeneous piece that exactly cancels the offending second-derivative term,

Γ^{ν^{'}}_{μ^{'} λ^{'}} = \frac{\partial x ^{' ν^{'}}}{\partial x ^{ν}} \frac{\partial x ^{μ}}{\partial x ^{' μ^{'}}} \frac{\partial x ^{λ}}{\partial x ^{' λ^{'}}} Γ^{ν}_{μ λ} - \frac{\partial x ^{μ}}{\partial x ^{' μ^{'}}} \frac{\partial x ^{λ}}{\partial x ^{' λ^{'}}} \frac{\partial ^{2} x ^{' ν^{'}}}{\partial x ^{μ} \partial x ^{λ}} .

With this law the combination $\nabla_{μ} V^{ν}$ transforms as a genuine $(1, 1)$ tensor. The connection acts on a one-form with the opposite sign,

\nabla_{μ} ω_{ν} = \partial_{μ} ω_{ν} - Γ^{λ}_{μν} ω_{λ},

so that the covariant derivative of the scalar $ω_{ν} V^{ν}$ reduces to its partial derivative. On a general tensor the rule extends index by index: one $+ Γ$ term for each upper index and one $- Γ$ term for each lower index, added to the partial derivative. The covariant derivative reduces to the partial derivative on scalars and satisfies the Leibniz rule, which are the two axioms a derivative operator must obey.

Parallel transport

The geometric content of the connection is a rule for moving a vector along a curve while keeping it as constant as the geometry allows. A vector $V^{μ}$ is parallel-transported along a curve $x^{μ} (λ)$ with tangent $t^{μ} = d x^{μ} / d λ$ if its covariant derivative along the curve vanishes,

t^{μ} \nabla_{μ} V^{ν} = 0 ⟺ \frac{d V ^{ν}}{d λ} + Γ^{ν}_{μ λ} t^{μ} V^{λ} = 0.

The correction term encodes how the coordinate basis turns along the path, so the condition says the vector's true direction is held fixed while its components adjust. On a flat plane in Cartesian coordinates all $Γ = 0$ and parallel transport just carries the arrow rigidly. On a curved manifold the result of parallel transport depends on the path: carrying a vector around a closed loop generally returns it rotated relative to its start. That path-dependence is the fingerprint of curvature and is made quantitative by the Riemann tensor two lessons on.

Parallel transport of a vector around a closed loop on the sphere. A vector carried along a triangle of great-circle arcs returns rotated relative to its starting direction; the rotation angle equals the enclosed area over the squared radius, a direct measure of curvature.

Metric compatibility and no torsion

The covariant-derivative construction works for any connection satisfying the transformation law above, and there are infinitely many. General relativity selects a unique one by imposing two conditions.

Torsion-free (symmetric). The connection is symmetric in its lower indices, $Γ^{λ}_{μν} = Γ^{λ}_{ν μ}$ . Equivalently, the antisymmetric part, the torsion tensor $T^{λ}_{μν} = Γ^{λ}_{μν} - Γ^{λ}_{ν μ}$ , is set to zero. This guarantees that covariant second derivatives of a scalar commute, $\nabla_{μ} \nabla_{ν} f = \nabla_{ν} \nabla_{μ} f$ , and that infinitesimal parallelograms close.
Metric-compatible. The covariant derivative of the metric vanishes, $\nabla_{λ} g_{μν} = 0$ . Parallel transport then preserves lengths and inner products: two vectors carried along the same curve keep their dot product, and the operations of raising and lowering indices commute with $\nabla$ .

The Christoffel symbols are built entirely from first derivatives of the metric, so they vanish wherever the metric is constant. At any single event a local inertial frame can be chosen in which $g_{μν} = η_{μν}$ and $\partial g = 0$ , whence all $Γ = 0$ at that point — the covariant derivative reduces to the partial derivative locally, the mathematical face of the equivalence principle. The symbols cannot in general be made to vanish over an extended region, precisely when curvature is present.

The Christoffel symbols are assembled from first derivatives of the metric by the Levi-Civita formula. Two requirements pin them down: no torsion (symmetric lower indices) and metric compatibility (parallel transport preserves lengths).

Covariant differentiation in practice

Two computational facts make the machinery usable. First, the contracted Christoffel symbol simplifies to a logarithmic derivative of the metric determinant,

Γ^{μ}_{μν} = \partial_{ν} ln ∣ g ∣, g \equiv det g_{μν},

which turns the covariant divergence of a vector into an ordinary divergence,

\nabla_{μ} V^{μ} = \frac{1}{∣ g ∣} \partial_{μ} (∣ g ∣ V^{μ}) .

This is the curved-space form of the divergence theorem and is how conservation laws $\nabla_{μ} T^{μν} = 0$ are integrated. Second, evaluating the covariant derivative of a general tensor is mechanical: one $+ Γ$ per contravariant index, one $- Γ$ per covariant index. For a $(1, 1)$ tensor,

\nabla_{λ} T^{μ}_{ν} = \partial_{λ} T^{μ}_{ν} + Γ^{μ}_{λσ} T^{σ}_{ν} - Γ^{σ}_{λ ν} T^{μ}_{σ} .

The covariant derivative of a tensor adds one connection term per index: a plus-Gamma correction for each upper index and a minus-Gamma for each lower index, appended to the ordinary partial derivative. The corrections cancel the basis-rotation artefacts.

The covariant derivative supplies the notion of constant along a curve that the partial derivative lacked. The straightest possible worldlines — those whose own tangent vector is parallel-transported along themselves — are geodesics, and they turn out to be the paths of freely falling particles. That construction, and its reduction to Newtonian gravity in the weak field, is the next lesson.