Covariant Derivative and Parallel Transport

On flat spacetime the partial derivative ${\partial }_{\mu }{V}^{\nu }$ of a vector field is itself a tensor, and "constant vector field" has an unambiguous meaning. On a curved manifold neither is true: ${\partial }_{\mu }{V}^{\nu }$ fails to transform as a tensor, and there is no coordinate-free way to compare vectors at different points. Repairing this requires a connection — a rule for parallel transport — whose derivative operator is the covariant derivative $\nabla$. On a Lorentzian manifold the metric picks out a unique such connection, the Levi-Civita connection, whose geodesics are the worldlines of freely falling bodies. This realizes the equivalence principle dynamically.

We use $c=1$, mostly-plus signature.

Why ${\partial }_{\mu }$ is not enough

Under a coordinate change $x^{\mu }\to x^{\prime \mu }$, a vector transforms as $V^{\prime \mu }=\frac{\partial x^{\prime \mu }}{\partial x^{\nu }}{V}^{\nu }$. Differentiating,

$${\partial }_{\alpha }^{\prime }{V}^{\prime \mu }=\frac{\partial x^{\beta }}{\partial x^{\prime \alpha }}\frac{\partial x^{\prime \mu }}{\partial x^{\nu }}{\partial }_{\beta }{V}^{\nu }+\frac{\partial x^{\beta }}{\partial x^{\prime \alpha }}\frac{{\partial }^2{x}^{\prime \mu }}{\partial x^{\beta }\partial x^{\nu }}{V}^{\nu }.$$

The first term is the desired tensor law; the second term — nonzero whenever the coordinate change is nonlinear (as any change between curved-space charts must be) — spoils it. Geometrically, ${\partial }_{\mu }{V}^{\nu }$ tries to subtract vectors living in different tangent spaces $T_{p}M$ and $T_{q}M$, which is not defined until we say how to carry one to the other.

The covariant derivative

Define an operator ${\nabla }_{\mu }$ that repairs the transformation law by adding a correction built from the connection coefficients (Christoffel symbols) ${\Gamma }^{\lambda }{}_{\mu \nu }$:

$${\nabla }_{\mu }{V}^{\nu }={\partial }_{\mu }{V}^{\nu }+{\Gamma }^{\nu }{}_{\mu \lambda }{V}^{\lambda }.$$

For a covector the sign flips, and for a general tensor there is one $+\Gamma$ per upper index and one $-\Gamma$ per lower index:

$${\nabla }_{\mu }{T}^{\nu }{}_{\rho }={\partial }_{\mu }{T}^{\nu }{}_{\rho }+{\Gamma }^{\nu }{}_{\mu \lambda }{T}^{\lambda }{}_{\rho }-{\Gamma }^{\lambda }{}_{\mu \rho }{T}^{\nu }{}_{\lambda }.$$

The $\Gamma$'s are not tensors — their inhomogeneous transformation is exactly what cancels the offending second term above, so that ${\nabla }_{\mu }T$ is a tensor. A connection is any choice of $\Gamma$ making $\nabla$ well-defined; requiring it to be compatible with the metric fixes it uniquely.

The Levi-Civita connection

Two natural conditions single out one connection on $(M,g)$:

1. Metric compatibility: ${\nabla }_{\alpha }{g}_{\mu \nu }=0$. The metric is covariantly constant, so lengths and angles are preserved under parallel transport, and $\nabla$ commutes with raising/lowering indices.
2. Torsion-free (symmetric): ${\Gamma }^{\lambda }{}_{\mu \nu }={\Gamma }^{\lambda }{}_{\nu \mu }$. Equivalently ${\nabla }_{\mu }{\nabla }_{\nu }f={\nabla }_{\nu }{\nabla }_{\mu }f$ on scalars.

Fundamental theorem of (pseudo-)Riemannian geometry. There is a unique connection satisfying both, the Levi-Civita connection, with coefficients given by the metric and its first derivatives:

$$\boxed{{\Gamma }^{\lambda }{}_{\mu \nu }=\frac{1}{2}{g}^{\lambda \sigma }({\partial }_{\mu }{g}_{\sigma \nu }+{\partial }_{\nu }{g}_{\sigma \mu }-{\partial }_{\sigma }{g}_{\mu \nu }).}$$

(This is the Lorentzian-signature specialization of the Riemannian result in metric geometry; the derivation is identical.) Because $\Gamma \sim \partial g$, the Christoffel symbols play the role of the gravitational field strength: in the Newtonian limit, ${\Gamma }^i{}_{00}\approx {\partial }_i\Phi$, so the geodesic equation below reduces to ${\ddot{x}}^i=-{\partial }_i\Phi$ — Newton's law of gravity.

At a point one can always choose local inertial coordinates with ${\Gamma }^{\lambda }{}_{\mu \nu }(p)=0$: the connection, like the first derivatives of the metric, is locally removable. This is the EEP again — gravity (the $\Gamma$'s) can be transformed away at a point, but its derivatives (curvature) cannot.

Parallel transport

A vector $V^{\mu }$ is parallel-transported along a curve $x^{\mu }(\lambda )$ with tangent $u^{\mu }=d{x}^{\mu }/d\lambda$ if its covariant derivative along the curve vanishes:

$$\frac{D{V}^{\mu }}{d\lambda }\equiv u^{\alpha }{\nabla }_{\alpha }{V}^{\mu }=\frac{d{V}^{\mu }}{d\lambda }+{\Gamma }^{\mu }{}_{\alpha \beta }{u}^{\alpha }{V}^{\beta }=0.$$

This is the curved-space notion of "carrying a vector without turning it". Metric compatibility guarantees that parallel transport preserves inner products: transported vectors keep their lengths and mutual angles.

Path dependence = curvature. On a curved manifold the result of parallel transport depends on the path. Carry a vector around a closed loop and it generally returns rotated; the mismatch, per unit area, is the Riemann curvature. (Carry a vector around a spherical triangle on the globe and it comes back turned by the enclosed solid angle — the paradigm example.) This holonomy is the coordinate-free meaning of curvature and the reason no global notion of "same direction" exists.

Geodesics: the straightest worldlines

A geodesic is a curve that parallel-transports its own tangent vector — the curved-space "straight line", the path that turns as little as possible:

$$u^{\alpha }{\nabla }_{\alpha }{u}^{\mu }=0⟺\boxed{\frac{d^2{x}^{\mu }}{d{\lambda }^2}+{\Gamma }^{\mu }{}_{\alpha \beta }\frac{d{x}^{\alpha }}{d\lambda }\frac{d{x}^{\beta }}{d\lambda }=0.}$$

Here $\lambda$ is an affine parameter (for a massive particle, proper time $\tau$). Two complementary characterizations:

• Straightest. The tangent is parallel-transported — no "sideways" acceleration; the only bending is that of the geometry itself.
• Extremal. Timelike geodesics extremize (in fact locally maximize) the proper time $\tau =\int d\tau$ between two events — the twin paradox generalized: the freely falling worldline is the longest proper time, not the shortest. Varying $\int \sqrt{-g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }}d\lambda$ reproduces the geodesic equation, giving a practical route to the $\Gamma$'s.

Geodesics are free fall

This is the dynamical core of general relativity, replacing $\mathbf{F}=m\mathbf{a}$ for gravity:

A freely falling body (one acted on by no non-gravitational force) moves on a timelike geodesic; light moves on a null geodesic.

There is no gravitational force: unmodeled "gravity" is the $\Gamma$ term, which is not a tensor and vanishes in the local inertial frame. What we feel as weight is the non-gravitational (normal) force preventing us from following the geodesic into the floor; the freely falling observer feels nothing (the EEP). Massive particles follow timelike geodesics ($d{\tau }^2>0$), photons null geodesics ($d{s}^2=0$), and the universality is automatic because the geodesic equation contains no reference to the body's mass.

For massless particles proper time is not available; one uses an affine parameter and imposes the null condition $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=0$. The bending of starlight and the Shapiro delay are null geodesics in the Sun's metric.

Divergence, the covariant d'Alembertian, and conservation

Two identities recur throughout the theory. For the Levi-Civita connection,

$${\Gamma }^{\mu }{}_{\mu \nu }={\partial }_{\nu }\ln \sqrt{-g},$$

which streamlines covariant divergences of vectors and antisymmetric tensors:

$${\nabla }_{\mu }{V}^{\mu }=\frac{1}{\sqrt{-g}}{\partial }_{\mu }\left(\sqrt{-g}{V}^{\mu }\right),\square f=g^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu }f=\frac{1}{\sqrt{-g}}{\partial }_{\mu }\left(\sqrt{-g}{g}^{\mu \nu }{\partial }_{\nu }f\right).$$

The conservation law of the stress–energy tensor, ${\nabla }_{\mu }{T}^{\mu \nu }=0$, is written with $\nabla$, not $\partial$ — and the extra $\Gamma$ terms encode the exchange of energy–momentum between matter and the gravitational field. Unlike the flat-space case, ${\nabla }_{\mu }{T}^{\mu \nu }=0$ does not give a globally conserved integral charge in general, precisely because gravity itself carries energy that the local matter tensor omits.

Summary

• ${\partial }_{\mu }$ is not tensorial on a curved manifold; the covariant derivative ${\nabla }_{\mu }={\partial }_{\mu }+\Gamma$ repairs it.
• Metric compatibility + torsion-free ⇒ the unique Levi-Civita connection, with $\Gamma \sim \partial g$ acting as the gravitational field strength; it vanishes at a point in a local inertial frame (EEP).
• Parallel transport carries vectors without turning; its path-dependence is curvature.
• Geodesics (${\ddot{x}}^{\mu }+{\Gamma }^{\mu }{}_{\alpha \beta }{\dot{x}}^{\alpha }{\dot{x}}^{\beta }=0$) are the straightest / longest-proper-time worldlines, and are the trajectories of freely falling bodies and light — general relativity's replacement for the gravitational force law.

Next

Parallel transport around a loop returns a vector rotated by an amount measuring curvature. The next page builds the Riemann tensor from the commutator of covariant derivatives, connects it to tidal forces via geodesic deviation, and contracts it into the Ricci and Einstein tensors that source the field equations.