Ordinary partial derivatives compare tensor components at neighboring coordinate points. In curved spaces, or even in curvilinear coordinates, those components can change partly because the coordinate basis itself changes.

A covariant derivative corrects for that basis variation.

Connection coefficients

For a vector field $V^\mu$, the covariant derivative is written

$$\nabla_\nu V^\mu=\partial_\nu V^\mu+\Gamma^\mu_{\nu\rho}V^\rho.$$

The quantities $\Gamma^\mu_{\nu\rho}$ are connection coefficients. For the standard torsion-free, metric-compatible connection used in general relativity, they are the Christoffel symbols determined by the metric:

$$\Gamma^\rho_{\mu\nu}=\frac12 g^{\rho\sigma} \left(\partial_\mu g_{\nu\sigma}+\partial_\nu g_{\mu\sigma}-\partial_\sigma g_{\mu\nu}\right).$$

Christoffel symbols are not tensor components themselves; in suitable coordinates they can vanish at one event even when spacetime is curved.

Parallel transport

A vector is parallel transported along a curve when its covariant derivative along that curve vanishes. This gives a geometric rule for carrying directions from one tangent space to the next.

In curved geometry, transporting a vector around a closed loop can return a different direction. That path dependence is related to curvature.

Geodesics

A freely falling test particle follows a timelike geodesic. In coordinates its path $x^\mu(\lambda)$ satisfies

$$\frac{d^2x^\rho}{d\lambda^2}+\Gamma^\rho_{\mu\nu} \frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}=0.$$

For light, the corresponding free trajectories are null geodesics.

Geometry rather than a force law

The coordinate acceleration in the geodesic equation can resemble a force term, but its origin is geometric: it is produced by the connection associated with the spacetime metric.

Locally freely falling coordinates can remove the Christoffel symbols at one event. What cannot generally be removed over a finite region is spacetime curvature, which controls the relative acceleration of neighboring geodesics.