---
title: Parallel Transport and the Covariant Derivative
module: Curved Spacetime
moduleNumber: 5
lessonNumber: 3
order: 503
summary: >
  The ordinary derivative of a vector field is not a tensor, because it subtracts
  vectors living in different tangent spaces. A connection supplies the missing
  comparison: the covariant derivative adds Christoffel-symbol correction terms
  that cancel the coordinate artefacts. Requiring the connection to be
  torsion-free and to preserve the metric fixes the Christoffel symbols uniquely
  in terms of derivatives of the metric, giving the Levi-Civita connection that
  general relativity uses.
topics: [Curved Spacetime]
draft: false
sources:
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §3 — Curvature"
  - book: Hartle
    ref: "Gravity, Ch. 20 — A Little More Math; Ch. 21 — Curvature and the Einstein Equation"
---

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 $\partial_\mu f$ 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 $V^\nu$, the object
$\partial_\mu V^\nu$ fails this test. Transforming both indices and applying the
chain rule,

$$
\partial_{\mu'} V^{\nu'}
= \frac{\partial x^\mu}{\partial x'^{\mu'}}\frac{\partial x'^{\nu'}}{\partial x^\nu}\,
\partial_\mu V^\nu
+ \frac{\partial x^\mu}{\partial x'^{\mu'}}\,
\frac{\partial^2 x'^{\nu'}}{\partial x^\mu \partial x^\nu}\,V^\nu.
$$

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
$\partial^2 x'/\partial x^2 \ne 0$, 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.

$$
% caption: In polar coordinates the basis vectors rotate from point to point, so a
% genuinely constant vector field has position-dependent components. The partial
% derivative registers this basis rotation as a spurious change in the field.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % origin
  \fill[black!70] (0,0) circle (1.6pt);
  % radial grid
  \foreach \a in {20,50,80} {
    \draw[black] (0,0) -- (\a:3.2);
  }
  \draw[black] (0,0) ++(20:1.2) arc (20:80:1.2);
  \draw[black] (0,0) ++(20:2.4) arc (20:80:2.4);
  % same constant vector at three points, but local basis differs
  \draw[->, acc, very thick] (20:1.5) -- ++(0.9,0);
  \draw[->, acc, very thick] (50:1.6) -- ++(0.9,0);
  \draw[->, acc, very thick] (80:1.5) -- ++(0.9,0);
  \node[anchor=west] at (20:1.5) {};
  % radial basis arrows (rotate)
  \draw[->, black] (20:1.5) -- ++(20:0.7);
  \draw[->, black] (50:1.6) -- ++(50:0.7);
  \draw[->, black] (80:1.5) -- ++(80:0.7);
  \node[black, anchor=west] at (2.6,2.3) {constant vectors};
  \node[black, anchor=west] at (2.6,1.9) {rotating basis};
\end{tikzpicture}
$$

## 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

$$
\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu{}_{\mu\lambda} V^\lambda,
$$

where the $\Gamma^\nu{}_{\mu\lambda}$ 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,

$$
\Gamma^{\nu'}{}_{\mu'\lambda'}
= \frac{\partial x'^{\nu'}}{\partial x^\nu}
\frac{\partial x^\mu}{\partial x'^{\mu'}}
\frac{\partial x^\lambda}{\partial x'^{\lambda'}}\,\Gamma^\nu{}_{\mu\lambda}
- \frac{\partial x^\mu}{\partial x'^{\mu'}}
\frac{\partial x^\lambda}{\partial x'^{\lambda'}}
\frac{\partial^2 x'^{\nu'}}{\partial x^\mu \partial x^\lambda}.
$$

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

$$
\nabla_\mu \omega_\nu = \partial_\mu \omega_\nu - \Gamma^\lambda{}_{\mu\nu}\omega_\lambda,
$$

so that the covariant derivative of the scalar $\omega_\nu V^\nu$ reduces to its
partial derivative. On a general tensor the rule extends index by index: one
$+\Gamma$ term for each upper index and one $-\Gamma$ 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 $V^\mu$ is
**parallel-transported** along a curve $x^\mu(\lambda)$ with tangent $t^\mu = \d
x^\mu/\d\lambda$ if its covariant derivative along the curve vanishes,

$$
t^\mu \nabla_\mu V^\nu = 0
\quad\Longleftrightarrow\quad
\frac{\d V^\nu}{\d\lambda} + \Gamma^\nu{}_{\mu\lambda}\,t^\mu V^\lambda = 0.
$$

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 $\Gamma = 0$ 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.

$$
% caption: Parallel transport of a vector around a closed loop on the sphere. A
% vector carried along a triangle of great-circle arcs returns rotated relative to
% its starting direction; the rotation angle equals the enclosed area over the
% squared radius, a direct measure of curvature.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, very thick] (0,0) circle (2);
  \draw[black, dashed] (0,0) ellipse (2 and 0.6);
  % spherical triangle: pole to two equator points
  \fill[black!70] (0,2) circle (1.6pt);
  \node[black!70, anchor=south] at (0,2.05) {start};
  \draw[black, thick] (0,2) .. controls (-0.4,1.0) and (-1.2,0.35) .. (-2,0);
  \draw[black, thick] (0,2) .. controls (0.9,1.0) and (1.6,0.35) .. (2,0);
  \draw[black, thick] (-2,0) .. controls (-1.0,-0.35) and (1.0,-0.35) .. (2,0);
  % transported vectors at three corners, differing directions
  \draw[->, acc, very thick] (0,2) -- (0.7,1.75);
  \draw[->, acc, very thick] (-2,0) -- (-1.35,0.35);
  \draw[->, acc, very thick] (2,0) -- (1.45,0.55);
  \node[acc, anchor=west] at (2.15,1.3) {returns rotated};
\end{tikzpicture}
$$

## 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,
  $\Gamma^\lambda{}_{\mu\nu} = \Gamma^\lambda{}_{\nu\mu}$. Equivalently, the
  antisymmetric part, the torsion tensor $T^\lambda{}_{\mu\nu} =
  \Gamma^\lambda{}_{\mu\nu} - \Gamma^\lambda{}_{\nu\mu}$, is set to zero. This
  guarantees that covariant second derivatives of a scalar commute,
  $\nabla_\mu\nabla_\nu f = \nabla_\nu\nabla_\mu f$, and that infinitesimal
  parallelograms close.
- **Metric-compatible.** The covariant derivative of the metric vanishes,
  $\nabla_\lambda g_{\mu\nu} = 0$. 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 $\nabla$.

> **Theorem (Fundamental theorem of Riemannian geometry).** On a manifold with
> metric $g_{\mu\nu}$ there is a unique connection that is both torsion-free and
> metric-compatible. Its coefficients, the **Christoffel symbols**, are
> $$
> \Gamma^\lambda{}_{\mu\nu}
> = \tfrac{1}{2}\,g^{\lambda\sigma}\big(
> \partial_\mu g_{\sigma\nu} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu}
> \big).
> $$

> **Proof.** Write metric compatibility $\nabla_\lambda g_{\mu\nu} = 0$ three
> times with the indices cyclically permuted:
> $$
> \partial_\lambda g_{\mu\nu} = \Gamma^\sigma{}_{\lambda\mu} g_{\sigma\nu}
> + \Gamma^\sigma{}_{\lambda\nu} g_{\sigma\mu},
> $$
> and similarly for $\partial_\mu g_{\nu\lambda}$ and $\partial_\nu g_{\lambda\mu}$.
> Add the first two and subtract the third. Using the symmetry
> $\Gamma^\sigma{}_{\mu\nu} = \Gamma^\sigma{}_{\nu\mu}$, six of the resulting terms
> cancel in pairs and the survivors combine into
> $2\,\Gamma^\sigma{}_{\mu\nu} g_{\sigma\lambda}
> = \partial_\mu g_{\nu\lambda} + \partial_\nu g_{\lambda\mu} - \partial_\lambda g_{\mu\nu}$.
> Contracting with the inverse metric $g^{\lambda\rho}$ isolates
> $\Gamma^\rho{}_{\mu\nu}$ and gives the stated formula. Existence is the formula
> itself; uniqueness follows because torsion-freedom and metric compatibility fix
> every component.

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 $g_{\mu\nu} = \eta_{\mu\nu}$ and $\partial
g = 0$, whence all $\Gamma = 0$ 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.

$$
% caption: The Christoffel symbols are assembled from first derivatives of the
% metric by the Levi-Civita formula. Two requirements pin them down: no torsion
% (symmetric lower indices) and metric compatibility (parallel transport preserves
% lengths).
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % metric box
  \draw[black, thick] (0,1.2) rectangle (2.2,2.4);
  \node[black, anchor=center] at (1.1,1.8) {metric g};
  % derivatives box
  \draw[black, thick] (0,-0.6) rectangle (2.2,0.6);
  \node[black, anchor=center] at (1.1,0) {metric gradients};
  \draw[->, black] (1.1,1.15) -- (1.1,0.65);
  % conditions
  \draw[black] (3.4,1.4) rectangle (6.4,2.4);
  \node[black, anchor=center] at (4.9,2.05) {no torsion};
  \node[black, anchor=center] at (4.9,1.65) {metric compatible};
  % Christoffel box
  \draw[acc, thick, fill=acc!10] (3.4,-0.6) rectangle (6.4,0.6);
  \node[acc, anchor=center] at (4.9,0) {connection terms};
  \draw[->, black] (2.25,0) -- (3.35,0);
  \draw[->, black] (4.9,1.35) -- (4.9,0.65);
  \node[black, anchor=south] at (2.8,0.05) {combine};
\end{tikzpicture}
$$

## 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,

$$
\Gamma^\mu{}_{\mu\nu} = \partial_\nu \ln\sqrt{\lvert g\rvert},
\qquad g \equiv \det g_{\mu\nu},
$$

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

$$
\nabla_\mu V^\mu = \frac{1}{\sqrt{\lvert g\rvert}}\,
\partial_\mu\!\big(\sqrt{\lvert g\rvert}\,V^\mu\big).
$$

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

$$
\nabla_\lambda T^\mu{}_\nu = \partial_\lambda T^\mu{}_\nu
+ \Gamma^\mu{}_{\lambda\sigma} T^\sigma{}_\nu
- \Gamma^\sigma{}_{\lambda\nu} T^\mu{}_\sigma.
$$

> **Worked example.** On the 2-sphere of radius $a$ with metric $\d s^2 = a^2\,
> \d\theta^2 + a^2\sin^2\theta\,\d\phi^2$, the nonzero metric derivative is
> $\partial_\theta g_{\phi\phi} = 2a^2\sin\theta\cos\theta$. The Levi-Civita
> formula gives the nonvanishing Christoffel symbols
> $$
> \Gamma^\theta{}_{\phi\phi} = -\sin\theta\cos\theta,
> \qquad
> \Gamma^\phi{}_{\theta\phi} = \Gamma^\phi{}_{\phi\theta} = \cot\theta.
> $$
> These terms are what keep a parallel-transported vector pointing in
> a fixed direction as it is carried across the sphere; their nonvanishing is why
> transport around a loop rotates a vector, and feeding them into the curvature
> formula returns the sphere's nonzero Riemann tensor.

$$
% caption: The covariant derivative of a tensor adds one connection term per index:
% a plus-Gamma correction for each upper index and a minus-Gamma for each lower
% index, appended to the ordinary partial derivative. The corrections cancel the
% basis-rotation artefacts.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black] (-0.2,-0.55) rectangle (1.4,0.55);
  \node[black!70, anchor=center] at (0.6,0) {partial};
  % upper-index correction box, descriptor placed above it
  \draw[black, thick] (2.3,0.35) rectangle (4.3,1.15);
  \node[black, anchor=center] at (3.3,0.75) {plus Gamma};
  \node[black, anchor=south] at (3.3,1.2) {per upper index};
  % lower-index correction box, descriptor placed below it
  \draw[black, thick] (2.3,-1.15) rectangle (4.3,-0.35);
  \node[black, anchor=center] at (3.3,-0.75) {minus Gamma};
  \node[black, anchor=north] at (3.3,-1.2) {per lower index};
  \draw[->, black] (1.45,0.2) -- (2.25,0.7);
  \draw[->, black] (1.45,-0.2) -- (2.25,-0.7);
  \node[black, anchor=west] at (4.5,0) {sum = covariant derivative};
\end{tikzpicture}
$$

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.

[^carroll-conn]: **Carroll**, _Lecture Notes on General Relativity_, §3 — the failure of the partial derivative to be tensorial, the covariant derivative and connection coefficients, parallel transport, torsion-freedom and metric compatibility, and the Christoffel symbols of the Levi-Civita connection.

[^hartle-curv]: **Hartle**, _Gravity_, Ch. 20–21 — covariant differentiation in curved spacetime, the Christoffel symbols computed from the metric, and their role in geodesic motion and curvature.