Covariant Derivative

The covariant derivative is the differential operator inside every line of working relativity. Local energy-momentum conservation is written $\nabla_\mu T^{\mu\nu}=0$, an identity that follows from the contracted Bianchi identity $\nabla_\mu G^{\mu\nu}=0$ and is the consistency condition of the Einstein field equations — the reason matter cannot be specified independently of the geometry it sources. In numerical relativity, the codes that model merging black holes for LIGO and Virgo evolve the spatial metric and its covariant derivatives on a grid, recomputing the Christoffel symbols at every point and time step; the waveforms compared against detector data are solutions of covariant-derivative equations. The tidal signal a gravitational-wave detector is built to sense — the relative acceleration of nearby freely-falling masses — is the second covariant derivative of their separation, $\tfrac{D^2 S^\mu}{\mathrm{d}\tau^2}=R^\mu{}_{\nu\rho\sigma}u^\nu u^\rho S^\sigma$. Symmetries and their conserved charges come from Killing's equation $\nabla_{(\mu}\xi_{\nu)}=0$, which turns the time-translation symmetry of Schwarzschild into conserved orbital energy and gives Kerr its angular momentum. And every matter field is placed in a gravitational background by the minimal-coupling rule $\partial\to\nabla$ — electromagnetism as $\nabla_\mu F^{\mu\nu}=\mu_0 J^\nu$, scalars, relativistic fluids alike — so that $\nabla$ is the single operation coupling all of physics to gravity.

Walk across the surface of the Earth carrying an arrow that you keep "as straight as possible," never deliberately twisting it. In your local north–east frame the arrow's components change even though you did nothing to it, because the frame itself swings around as you follow the curved ground. The partial derivative is an over-eager accountant who blames you for that swing; the connection terms are the exchange-rate correction that subtracts it. Once subtracted, "no covariant change" means "genuinely parallel-transported." The most dramatic consequence is that parallel transport around a closed loop does not bring the arrow home unchanged: on a curved surface it returns rotated, and the rotation angle equals the area the loop encloses, in units of the radius squared. That single fact is the local fingerprint of curvature, and it is what the covariant derivative was built to detect.

That rotation is bookkept, term by term, by the connection $\Gamma$ — how the coordinate basis vectors turn. In flat space with Cartesian axes the basis is rigid, every $\Gamma$ vanishes, and $\nabla$ collapses to $\partial$: a loop returns the arrow unchanged. The moment the basis rotates — in polar coordinates on a flat plane, or on any curved surface — a moving-basis correction appears, and $\nabla_\mu=\partial_\mu+\Gamma$ is exactly the operator that subtracts it. The second figure makes the cancellation concrete: a constant, physically unchanging field acquires changing components in polar coordinates purely because the basis swings, and the connection term erases that spurious change to leave $\nabla=0$.

Two covariant derivatives generally fail to commute, and that failure — carrying a vector around an infinitesimal loop and reading off how much it turned — is precisely the Riemann tensor, $[\nabla_\mu,\nabla_\nu]V^\rho=R^\rho{}_{\sigma\mu\nu}V^\sigma$. Because $\nabla$ manufactures tensors, any law written with it holds in every coordinate system once it holds in one — the operational meaning of general covariance. This is why the equations of motion of special relativity are promoted to curved spacetime by the "comma-goes-to-semicolon" rule $\partial\to\nabla$: minimal coupling to gravity is nothing but replacing every partial derivative by a covariant one.

The covariant derivative of a contravariant vector is the partial derivative plus one connection term contracting the connection's upper index against the vector. Neither piece is a tensor, but their sum is — that cancellation is the whole design brief, established in Derivation 1.

where: $\nabla_\mu$ — covariant derivative in the $x^\mu$ direction; $V^\nu$ — contravariant vector components; $\partial_\mu=\partial/\partial x^\mu$; $\Gamma^\nu{}_{\mu\lambda}$ — connection coefficients (Christoffel symbols for Levi-Civita). Reading: the free indices make $\nabla_\mu V^\nu$ a genuine $(1,1)$ tensor. For a one-form the connection term flips sign and contracts the lower index. This is not a convention — it is forced by the Leibniz rule on the scalar contraction $\omega_\nu V^\nu$ (Derivation 1).

where: $\omega_\nu$ — one-form components. Reading: an upper index brings $+\Gamma$, a lower index brings $-\Gamma$. The two rules combine into a single master formula: one $+\Gamma$ per upper index and one $-\Gamma$ per lower index. Everything else in the article is a special case of this line.

where: $T^{\nu_1\cdots}{}_{\rho_1\cdots}$ — an arbitrary $(k,l)$ tensor; the sums run over its upper and lower indices. Reading: $\nabla$ raises the covariant rank by one and preserves tensor character exactly. On a metric manifold there is one preferred connection, its coefficients the Christoffel symbols — an explicit combination of first derivatives of the metric. This is what "the" covariant derivative means in general relativity.

where: $g_{\mu\nu}$ — metric; $g^{\lambda\sigma}$ — inverse metric; $\Gamma^\lambda{}_{\mu\nu}$ — Christoffel symbols, symmetric in $\mu\nu$. Reading: the connection is built entirely from the metric, so the metric fixes how vectors are transported. The property that selects this connection is metric compatibility: the covariant derivative of the metric vanishes identically, so $\nabla$ commutes with raising and lowering indices.

where: $g_{\mu\nu}$ — metric. Reading: because $\nabla g=0$, one may slide the metric through a covariant derivative, e.g. $\nabla_\mu V_\nu=g_{\nu\rho}\nabla_\mu V^\rho$. The second defining property is zero torsion — the connection's antisymmetric part vanishes, making $\Gamma$ symmetric in its lower pair. The torsion tensor measures that antisymmetry.

where: $T^\lambda{}_{\mu\nu}$ — torsion (zero for Levi-Civita); $[\mu\nu]$ — antisymmetrization. Reading: unlike $\Gamma$, the torsion is a genuine tensor, since the inhomogeneous term is symmetric and cancels in the difference. Restricting $\nabla$ to a curve gives the directional (absolute) derivative; demanding a curve parallel-transport its own tangent produces the geodesic equation — where the free-fall equation of motion is born.

where: $\tau$ — affine parameter; $u^\mu=\mathrm{d}x^\mu/\mathrm{d}\tau$ — tangent; $DV^\mu/\mathrm{d}\tau$ — absolute derivative along the curve. Reading: a geodesic is a curve whose tangent has zero covariant derivative along itself — "as straight as the geometry allows." Finally, the commutator of two covariant derivatives on a vector is the Riemann curvature tensor (plus a torsion term, zero for Levi-Civita). This is the deepest payoff: $\nabla$ turns curvature from a picture into an operator.

where: $R^\rho{}_{\sigma\mu\nu}$ — Riemann curvature tensor; $T^\lambda{}_{\mu\nu}$ — torsion (zero for Levi-Civita). Reading: on a flat manifold covariant derivatives commute; their failure to commute is exactly the local curvature (Derivation 3).

The covariant derivative reduces to, or specializes in, several regimes worth committing to memory — each one a check on the general formula.

1 · Christoffel symbols of the unit two-sphere (confidence builder)

The unit sphere has metric $\mathrm{d}s^2=\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2$, so $g_{\theta\theta}=1$, $g_{\phi\phi}=\sin^2\theta$, and $g_{\theta\phi}=0$. Find the nonzero Christoffel symbols, evaluate them at $\theta=45°$, and verify metric compatibility there numerically.

Step 1. Only $g_{\phi\phi}=\sin^2\theta$ depends on a coordinate, so the sole nonzero metric derivative is $\partial_\theta g_{\phi\phi}=2\sin\theta\cos\theta$. Feeding this into the Christoffel formula with $g^{\theta\theta}=1$, $g^{\phi\phi}=1/\sin^2\theta$ gives $\Gamma^\theta{}_{\phi\phi}=\tfrac12 g^{\theta\theta}(-\partial_\theta g_{\phi\phi})=\mathbf{-\sin\theta\cos\theta}$ and $\Gamma^\phi{}_{\theta\phi}=\Gamma^\phi{}_{\phi\theta}=\tfrac12 g^{\phi\phi}\partial_\theta g_{\phi\phi}=\mathbf{\cot\theta}$; all others vanish.

Step 2. At $\theta=45°$, $\sin\theta=\cos\theta=\tfrac{\sqrt2}{2}\approx0.7071$, so $\Gamma^\theta{}_{\phi\phi}=-(0.7071)(0.7071)=\mathbf{-0.5}$ and $\Gamma^\phi{}_{\theta\phi}=\cot45°=\mathbf{1.0}$ — the entire connection of the round sphere at that latitude.

Step 3. Verify metric compatibility, $\nabla_\theta g_{\phi\phi}=\partial_\theta g_{\phi\phi}-2\Gamma^\phi{}_{\theta\phi}g_{\phi\phi}$ (only $\lambda=\phi$ survives). At $\theta=45°$, $\partial_\theta g_{\phi\phi}=2(0.7071)(0.7071)=\mathbf{1.0}$ and $2\Gamma^\phi{}_{\theta\phi}g_{\phi\phi}=2(1.0)(0.5)=\mathbf{1.0}$, so $\nabla_\theta g_{\phi\phi}=1.0-1.0=\mathbf{0}$ — covariantly constant, as the Levi-Civita connection demands. $\Gamma^\theta{}_{\phi\phi}=-0.5$, $\Gamma^\phi{}_{\theta\phi}=1.0$ at $\theta=45°$, and $\nabla_\theta g_{\phi\phi}=0$.

2 · The covariant derivative sees through a rotating basis (fully worked)

On the flat plane in polar coordinates, $\mathrm{d}s^2=\mathrm{d}r^2+r^2\mathrm{d}\phi^2$, take the constant Cartesian field $V=\hat e_x$ — the same arrow everywhere. Show that its polar components change from point to point yet $\nabla V=0$, evaluating everything at $r=2$, $\phi=90°$.

Step 1. The nonzero Christoffel symbols of the polar plane are $\Gamma^r{}_{\phi\phi}=-r$ and $\Gamma^\phi{}_{r\phi}=\Gamma^\phi{}_{\phi r}=1/r$ (from $g_{\phi\phi}=r^2$, as in Example 1). The uniform field $\hat e_x=\cos\phi\,\partial_r-(\sin\phi/r)\,\partial_\phi$ has components $V^r=\cos\phi$, $V^\phi=-\sin\phi/r$ that vary with position even though the physical arrow is fixed.

Step 2. At $r=2$, $\phi=90°$: $V^r=\cos90°=\mathbf{0}$ and $V^\phi=-\sin90°/2=\mathbf{-0.5}$. The offending partial is $\partial_r V^\phi=\partial_r(-\sin\phi/r)=\sin\phi/r^2=\sin90°/4=\mathbf{+0.25}$ — a reported change that is spurious: the arrow did not move, the basis did.

Step 3. Apply $\nabla_r V^\phi=\partial_r V^\phi+\Gamma^\phi{}_{r\phi}V^\phi$ (only $\lambda=\phi$ survives). With $\Gamma^\phi{}_{r\phi}=1/r=0.5$ and $V^\phi=-0.5$, the connection term is $(0.5)(-0.5)=\mathbf{-0.25}$, so $\nabla_r V^\phi=0.25-0.25=\mathbf{0}$; the other components check similarly, giving $\nabla V=0$ everywhere. The partial wrongly reports $+0.25$; the connection subtracts exactly $-0.25$, and $\nabla$ correctly certifies the field as unchanging.

3 · Parallel-transport holonomy equals the enclosed area

A vector is parallel-transported around the geodesic triangle N→A→B→N on the unit sphere, where N is the north pole and A, B are two equatorial points $90°$ apart in longitude. By how much does it rotate, and how does that relate to curvature?

Step 1. Each side is a great-circle arc, hence a geodesic, and the three interior angles are each $90°$ (meridians meet the equator at right angles and A, B are a quarter-turn apart). The angular excess is $\tfrac{3\pi}{2}-\pi=\mathbf{\tfrac{\pi}{2}}$.

Step 2. The holonomy around any closed loop equals the integral of the Gaussian curvature over the enclosed region, $\Delta\alpha=\iint K\,\mathrm{d}A$. On the unit sphere $K=1/R^2=1$, so $\Delta\alpha$ equals the enclosed area — one octant of $4\pi$: $A=\tfrac18(4\pi)=\tfrac{\pi}{2}\approx\mathbf{1.5708}$ sr $=\mathbf{90°}$. This matches the angular excess of Step 1 — Gauss–Bonnet in miniature.

Step 3. Interpret through the covariant derivative. Transporting around an infinitesimal loop of area $\delta A^{\mu\nu}$ rotates a vector by $\delta V^\rho=-R^\rho{}_{\sigma\mu\nu}V^\sigma\,\tfrac12\delta A^{\mu\nu}$, and summing these over the triangle reproduces the finite $90°$; in flat space $R=0$ and every loop closes. The vector returns rotated by $\pi/2=90°$, equal to the enclosed area and to $\iint K\,\mathrm{d}A$ — the finite fingerprint of the curvature that $[\nabla_\mu,\nabla_\nu]$ measures locally.

The covariant derivative is the central operation of tensor calculus, which three people built across half a century: Christoffel supplied the symbols in 1869, Ricci-Curbastro forged covariant differentiation into a systematic "absolute differential calculus," and Levi-Civita gave it its geometric soul through parallel transport.

Texts. Misner, Thorne and Wheeler, Gravitation (1973), Ch. 8–10 · Wald, General Relativity (1984), §3.1–3.2 · Carroll, Spacetime and Geometry (2004), Ch. 3 · Schutz, A First Course in General Relativity (2009), Ch. 5–6 · Hartle, Gravity (2003), Ch. 20.

Provenance. Equations and conventions follow Carroll, Spacetime and Geometry, Ch. 3 (covariant derivative 3.1–3.6, Christoffel 3.27, Riemann 3.112) and Wald, General Relativity, §3.1–3.2. A verifier should recompute: (1) the cancellation of the inhomogeneous transformation terms and the forced minus sign for one-forms; (2) the Christoffel formula by symmetrizing the three metric-compatibility equations; (3) the Riemann tensor from the nested covariant derivatives; (4) the two-sphere connection $\Gamma^\theta{}_{\phi\phi}=-0.5$, $\Gamma^\phi{}_{\theta\phi}=1.0$ at $\theta=45°$ with $\nabla_\theta g_{\phi\phi}=0$; (5) the polar-plane check $\partial_r V^\phi=+0.25$, $\Gamma^\phi{}_{r\phi}V^\phi=-0.25$, $\nabla_r V^\phi=0$ at $r=2$, $\phi=90°$; and (6) the spherical-octant holonomy $=\pi/2=90°$ equal to the enclosed area. All structural results are standard textbook content; the numbers are elementary and reproducible.

Builds on: The Affine Connection · Christoffel Symbols · Parallel Transport. Leads to: The Geodesic Equation · Metric Compatibility · Geodesic Deviation Equation · The Levi-Civita Connection.

Ask the tutor: Ask me to compute the Christoffel symbols of any metric you name — sphere, Schwarzschild, FLRW — and then covariantly differentiate a field or parallel-transport a vector around a loop, step by numbered step.