Christoffel Symbols and Affine Connections

Differentiation is the one operation a bare smooth manifold cannot supply. On flat space in Cartesian coordinates the partial derivatives $\partial_\mu V^\nu$ already transform as a tensor, and "the rate of change of $V$ " is unambiguous. The instant we switch to curvilinear coordinates — polar, spherical — or step onto a curved surface, the coordinate basis vectors $\mathbf{e}_\mu = \partial/\partial x^\mu$ point in different directions at different points. Differentiating $\mathbf{V} = V^\mu\mathbf{e}_\mu$ then splits into two pieces: the honest change in the components and a spurious change from the basis itself rotating underfoot. The connection coefficients $\Gamma^\lambda{}_{\mu\nu}$ are exactly the bookkeeping that separates the two. Abstractly an affine connection $\nabla$ sends a pair of vector fields $(X,Y)$ to a new field $\nabla_X Y$ , linear in $X$ , additive in $Y$ , obeying the Leibniz rule $\nabla_X(fY)=(Xf)Y+f\nabla_X Y$ . In a coordinate basis it condenses to the numbers $\nabla_{\mathbf{e}_\mu}\mathbf{e}_\nu=\Gamma^\lambda{}_{\mu\nu}\mathbf{e}_\lambda$ . There are infinitely many connections on a manifold — a connection is genuinely extra structure, not something the tangent spaces hand you for free. But if the manifold carries a metric $g_{\mu\nu}$ , two demands single out a unique one: metric compatibility $\nabla_\lambda g_{\mu\nu}=0$ (transport preserves lengths and angles) and zero torsion $\Gamma^\lambda{}_{\mu\nu}=\Gamma^\lambda{}_{\nu\mu}$ . That is the Levi-Civita connection, computable algebraically from the metric and its first derivatives, and it is the connection of Riemannian geometry and general relativity — the gateway to the covariant derivative, parallel transport, geodesics, and, through its failure to commute with itself, the Riemann curvature.

Key takeaways

A connection $\nabla$ supplies a coordinate-independent rule for differentiating vector and tensor fields, compensating for the fact that the basis vectors $\mathbf{e}_\mu$ vary from point to point.
The Christoffel symbols $\Gamma^\lambda{}_{\mu\nu}$ are the connection's components in a coordinate basis: $\nabla_{\mathbf{e}_\mu}\mathbf{e}_\nu=\Gamma^\lambda{}_{\mu\nu}\mathbf{e}_\lambda$ .
They are NOT tensor components — under a coordinate change they pick up an inhomogeneous second-derivative term that exactly cancels the bad transformation of $\partial_\mu V^\nu$ , leaving $\nabla_\mu V^\nu$ a true tensor.
For the unique metric-compatible torsion-free connection, $\Gamma^\lambda{}_{\mu\nu}=\tfrac12 g^{\lambda\rho}(\partial_\mu g_{\rho\nu}+\partial_\nu g_{\rho\mu}-\partial_\rho g_{\mu\nu})$ , symmetric in $\mu\nu$ .
Nonzero $\Gamma$ does not mean curvature: in flat space in polar coordinates $\Gamma^r{}_{\theta\theta}=-r\neq0$ , yet the Riemann tensor vanishes; at any point one can choose normal coordinates with $\Gamma(p)=0$ .
Connections define geodesics $\ddot x^\lambda+\Gamma^\lambda{}_{\mu\nu}\dot x^\mu\dot x^\nu=0$ ; the failure of parallel transport around a loop to return the vector unchanged is the curvature — for a spherical octant it returns rotated by exactly 90°.
Key idea: the Christoffel symbols are the price of a varying basis — $\nabla_\mu V^\nu=\partial_\mu V^\nu+\Gamma^\nu{}_{\mu\lambda}V^\lambda$ , where $\Gamma$ subtracts the apparent change caused by the basis rotating, leaving a genuine tensor.

The core insight

The Christoffel symbols are the price of using a varying basis: $\nabla_\mu V^\nu=\partial_\mu V^\nu+\Gamma^\nu{}_{\mu\lambda}V^\lambda$ , where $\Gamma$ subtracts off the apparent change due to the basis rotating, leaving a true tensor.

Where this lives today

The Christoffel symbols are evaluated billions of times a second in modern computational physics. Every numerical-relativity code that models a binary black-hole merger integrates Einstein's equations in a form where the connection (or its cousin in the BSSN and Z4c formulations) is a primary evolved field, and the gravitational-wave templates LIGO and Virgo match against their strain data are built by exactly this machinery. When the Event Horizon Telescope rendered the shadows of M87 and Sagittarius A, the images came from general-relativistic ray-tracing codes that integrate the geodesic equation (8) photon by photon through the Kerr metric, needing the Christoffel symbols at every step. The same identity $\Gamma^\lambda{}_{\mu\lambda}=\partial_\mu\ln\sqrt{|g|}$ that gives the curvilinear Laplacian underlies the covariant fluid dynamics of atmospheric and ocean models on the sphere. And the deeper lesson — that a connection is the universal way to differentiate "internal" data in a bundle over spacetime — is the backbone of the Standard Model: the gauge potentials $A_\mu$ of Yang–Mills theory are connections in precisely Christoffel's sense, with the field strength $F_{\mu\nu}$ as curvature. The open frontier is quantum gravity, where whether the smooth connection survives to the Planck scale remains unknown.

The physical picture

Picture the flat plane in polar coordinates. Even though the plane is perfectly flat, the coordinate basis vectors $\mathbf{e}_r$ and $\mathbf{e}_\theta$ are not the fixed arrows of a Cartesian grid — they swing around as you circle the origin. Move a little in the $\theta$ direction and $\mathbf{e}_\theta$ tips inward by an amount proportional to $r$ . That tipping is not a property of any vector field you are studying — it is a property of the coordinate chart, and the Christoffel symbols are the numbers that record it. The first figure freezes the swing and reads off $\Gamma^r{}_{\theta\theta}=-r$ directly from the geometry.

Figure

Flat plane in polar coordinates: the coordinate basis e_r (radial, blue) and e_θ (tangential, amber) swing as θ advances. Advancing e_θ tips it inward toward the origin, so ∂_θ e_θ = −r e_r, i.e. Γ^r_θθ = −r and Γ^θ_rθ = 1/r — a nonzero connection with zero curvature

Now let the surface curve, and the same machinery detects something the coordinates cannot fake. Carry a vector around a small closed loop, keeping it "as parallel as possible" at every step — this is parallel transport, the rule the connection defines. On a flat plane the vector comes home unchanged; on a curved surface it comes home rotated, and the rotation angle equals the curvature enclosed by the loop. The cleanest case is the sphere: transport a vector around a spherical octant — three mutually perpendicular quarter-circles — and it returns rotated by exactly 90°, the enclosed solid angle. The connection tells you how to transport; the path-dependence of transport is the curvature built from derivatives of $\Gamma$ .

Figure

Parallel transport around a spherical octant of radius a. Carrying a vector along three perpendicular great-circle arcs returns it rotated by the holonomy angle Δ = enclosed area / a² = (πa²/2)/a² = π/2 = 90°. On a flat plane the same loop returns the vector unchanged — the rotation is a direct readout of curvature

The core equations

The connection is defined by what it does to the basis vectors; everything else — how vectors, covectors, and tensors are differentiated — follows from this rule plus the Leibniz property.

\nabla_{\mathbf{e}_\mu}\mathbf{e}_\nu = \Gamma^\lambda{}_{\mu\nu}\,\mathbf{e}_\lambda \qquad (1)

where: $\mathbf{e}_\mu$ — coordinate basis vector $\partial/\partial x^\mu$ of the tangent space; $\nabla_{\mathbf{e}_\mu}$ — covariant derivative in the $\mu$ direction; $\Gamma^\lambda{}_{\mu\nu}$ — connection coefficient, the $\lambda$ -component of how $\mathbf{e}_\nu$ changes as you move along $x^\mu$ . Differentiating the $\nu$ -th basis vector along the $\mu$ -th direction produces a vector whose $\lambda$ -th component is $\Gamma^\lambda{}_{\mu\nu}$ . Applying the Leibniz rule to $\mathbf{V}=V^\nu\mathbf{e}_\nu$ turns rule (1) into the object physicists use — the covariant derivative of a vector field, a genuine $(1,1)$ tensor.

\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu{}_{\mu\lambda}\,V^\lambda \qquad (2)

where: $V^\nu$ — components of a vector field; $\partial_\mu V^\nu$ — ordinary coordinate derivative, by itself not a tensor; $\Gamma^\nu{}_{\mu\lambda}V^\lambda$ — the correction that restores tensor character by subtracting the apparent change due to the rotating basis. The true rate of change is the naive derivative minus the part that is only the coordinates moving. Contracting a covector against a vector must yield a scalar whose covariant derivative is just the ordinary derivative. That forces the sign on the covector rule to flip.

\nabla_\mu \omega_\nu = \partial_\mu \omega_\nu - \Gamma^\lambda{}_{\mu\nu}\,\omega_\lambda \qquad (3)

where: $\omega_\nu$ — components of a covector (one-form); $-\Gamma^\lambda{}_{\mu\nu}\omega_\lambda$ — the opposite-sign correction. Lower indices carry a minus sign, upper indices a plus sign, and a mixed tensor gets one $\Gamma$ term per index with the matching sign. The defining feature of $\Gamma$ is how it transforms under a change of coordinates — not as a tensor, but with an extra inhomogeneous piece.

\Gamma'^{\lambda}{}_{\mu\nu} = \frac{\partial x'^{\lambda}}{\partial x^\rho}\frac{\partial x^\sigma}{\partial x'^{\mu}}\frac{\partial x^\tau}{\partial x'^{\nu}}\,\Gamma^\rho{}_{\sigma\tau} \;+\; \frac{\partial x'^{\lambda}}{\partial x^\rho}\frac{\partial^2 x^\rho}{\partial x'^{\mu}\partial x'^{\nu}} \qquad (4)

where: $x'^{\mu}$ — new coordinates; $\partial x'^\lambda/\partial x^\rho$ — Jacobian of the change; $\partial^2 x^\rho/\partial x'^\mu\partial x'^\nu$ — the second-derivative term no tensor possesses. The first term is the tensorial part; the second is the fingerprint of non-tensoriality, and it is exactly what makes (2) a tensor. When a metric is present, metric compatibility plus zero torsion solve uniquely for the connection — the Christoffel formula, computable from the metric and its first derivatives alone.

\Gamma^\lambda{}_{\mu\nu} = \tfrac12\,g^{\lambda\rho}\big(\partial_\mu g_{\rho\nu} + \partial_\nu g_{\rho\mu} - \partial_\rho g_{\mu\nu}\big) \qquad (5)

where: $g_{\mu\nu}$ — metric tensor; $g^{\lambda\rho}$ — inverse metric, $g^{\lambda\rho}g_{\rho\sigma}=\delta^\lambda{}_\sigma$ ; $\partial_\mu g_{\rho\nu}$ — first derivative of the metric. Three cyclic derivatives of the metric, contracted with the inverse, manifestly symmetric in $\mu\nu$ . The antisymmetric part of a general connection is a genuine tensor — the torsion — and setting it to zero is one of the two Levi-Civita conditions.

T^\lambda{}_{\mu\nu} = \Gamma^\lambda{}_{\mu\nu} - \Gamma^\lambda{}_{\nu\mu} \qquad (6)

where: $T^\lambda{}_{\mu\nu}$ — torsion tensor. The inhomogeneous term in (4) is symmetric in $\mu\nu$ (mixed partials commute), so it cancels in the difference — the antisymmetric part transforms tensorially even though $\Gamma$ itself does not. A single contraction of the Christoffel formula collapses to a pure gradient of the metric determinant — the identity behind every curvilinear divergence and Laplacian.

\Gamma^\lambda{}_{\mu\lambda} = \partial_\mu \ln\sqrt{|g|} \qquad (7)

where: $g$ — determinant of $g_{\mu\nu}$ . Summing the upper index against the second lower index turns the connection into $\partial_\mu\ln\sqrt{|g|}$ , which is why $\nabla_\mu V^\mu=\frac{1}{\sqrt{|g|}}\partial_\mu(\sqrt{|g|}\,V^\mu)$ . Finally, the connection defines "straightest possible" curves — geodesics — through the vanishing of the covariant acceleration.

\frac{\mathrm{d}^2 x^\lambda}{\mathrm{d}\tau^2} + \Gamma^\lambda{}_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}\tau}\frac{\mathrm{d}x^\nu}{\mathrm{d}\tau}=0 \qquad (8)

where: $\tau$ — affine parameter (arc length or proper time); $x^\lambda(\tau)$ — the geodesic. A geodesic parallel-transports its own tangent vector, so $\Gamma$ is the "force" felt by a free particle — in general relativity, gravity itself.

Derivation AWhy the connection is not a tensor

The goal is to show, without assuming a metric, that the connection coefficients cannot transform as a tensor — and to extract the exact inhomogeneous law (4). The logic runs off one demand: the covariant derivative $\nabla_\mu V^\nu$ must be a genuine $(1,1)$ tensor, since that is the whole reason we introduced $\Gamma$ . We expand the ordinary derivative in new coordinates, watch a second-derivative term appear from the product rule, and see that $\Gamma$ must acquire a matching second-derivative term to swallow it. The upshot is structural: the defect in how $\partial_\mu V^\nu$ transforms and the defect in how $\Gamma$ transforms are equal and opposite by construction.

Step 1. Demand tensor behaviour: $\nabla'_\mu V'^\nu = \dfrac{\partial x'^\nu}{\partial x^\beta}\dfrac{\partial x^\alpha}{\partial x'^\mu}\,\nabla_\alpha V^\beta$ . Why: this equation is the definition of " $\nabla V$ is a $(1,1)$ tensor" — one upper index transforms with the Jacobian, one lower index with its inverse.

Step 2. Expand the naive derivative in the new coordinates, $\partial'_\mu V'^\nu = \dfrac{\partial x^\alpha}{\partial x'^\mu}\dfrac{\partial x'^\nu}{\partial x^\beta}\partial_\alpha V^\beta + \dfrac{\partial x^\alpha}{\partial x'^\mu}\dfrac{\partial^2 x'^\nu}{\partial x^\alpha\partial x^\beta}V^\beta$ . Why: the product rule hits both the Jacobian and the components, and the derivative of the Jacobian is a second derivative of the coordinate map — the term no tensor has, the seed of the whole result.

Step 3. Write the primed covariant derivative in its own chart, $\nabla'_\mu V'^\nu = \partial'_\mu V'^\nu + \Gamma'^\nu{}_{\mu\lambda}V'^\lambda$ , and insert Step 2. Why: this gives a second expression for the same object, so equating it with Step 1 constrains $\Gamma'$ .

Step 4. Equate the two, strip the arbitrary $V^\beta$ , and solve. The bad second-derivative term has nowhere to go but into $\Gamma'$ , giving $\Gamma'^{\lambda}{}_{\mu\nu} = \dfrac{\partial x'^{\lambda}}{\partial x^\rho}\dfrac{\partial x^\sigma}{\partial x'^{\mu}}\dfrac{\partial x^\tau}{\partial x'^{\nu}}\Gamma^\rho{}_{\sigma\tau} + \dfrac{\partial x'^{\lambda}}{\partial x^\rho}\dfrac{\partial^2 x^\rho}{\partial x'^{\mu}\partial x'^{\nu}}$ . Why: $\Gamma$ would be a tensor were the second term absent; that term is the non-tensorial piece cancelling the defect in $\partial V$ , and being symmetric in $\mu\nu$ it is consistent with zero torsion. *Cross-checked against Stone & Goldbart and Wald: the inhomogeneous term is symmetric in the lower indices, so the torsion $T^\lambda{}_{\mu\nu}=\Gamma^\lambda{}_{\mu\nu}-\Gamma^\lambda{}_{\nu\mu}$ is a bona fide tensor.*

Go deeper · Derivation B · 1 minThe Christoffel formula from metric compatibility and zero torsion

Derivation BThe Christoffel formula from metric compatibility and zero torsion

The goal is to prove that a metric determines a unique symmetric connection and to derive the closed form (5). The strategy is the classic three-cyclic-permutations trick: write out metric compatibility $\nabla_\mu g_{\nu\rho}=0$ in full, permute the free indices cyclically to get three equations, then add two and subtract the third so the zero-torsion symmetry collapses six $\Gamma$ terms to one. It is what makes Riemannian geometry computable — turning an abstract uniqueness statement into an explicit recipe in the metric and its first derivatives.

Step 1. Write metric compatibility with the covector rule on each lower index: $\nabla_\mu g_{\nu\rho}=\partial_\mu g_{\nu\rho}-\Gamma^\sigma{}_{\mu\nu}g_{\sigma\rho}-\Gamma^\sigma{}_{\mu\rho}g_{\nu\sigma}=0$ . Why: $g_{\nu\rho}$ receives one $-\Gamma$ correction per lower index; demanding this vanish says lengths and angles survive parallel transport.

Step 2. Define the fully-lowered symbol $\Gamma_{\rho\mu\nu}=g_{\rho\sigma}\Gamma^\sigma{}_{\mu\nu}$ so Step 1 reads $\partial_\mu g_{\nu\rho}=\Gamma_{\rho\mu\nu}+\Gamma_{\nu\mu\rho}$ . Why: lowering the top index makes the permutation bookkeeping transparent.

Step 3. Cyclically permute $(\mu\nu\rho)$ to get $\partial_\nu g_{\rho\mu}$ and $\partial_\rho g_{\mu\nu}$ , add the first two and subtract the third. Using the zero-torsion symmetry $\Gamma_{\rho\mu\nu}=\Gamma_{\rho\nu\mu}$ , four of the six terms cancel and the survivors give $\partial_\mu g_{\nu\rho}+\partial_\nu g_{\rho\mu}-\partial_\rho g_{\mu\nu}=2\,\Gamma_{\rho\mu\nu}$ . Why: this is the crux — it is exactly the torsion-free condition that lets the three equations collapse to one, isolating a single $\Gamma$ .

Step 4. Raise the packaged index with $g^{\lambda\rho}$ : $\Gamma^\lambda{}_{\mu\nu}=\tfrac12 g^{\lambda\rho}(\partial_\mu g_{\rho\nu}+\partial_\nu g_{\rho\mu}-\partial_\rho g_{\mu\nu})$ . Why: this returns the standard form, symmetric in $\mu\nu$ and built only from the metric and its first derivatives — existence and uniqueness of the Levi-Civita connection in one stroke. *The two inputs — metric compatibility and zero torsion — are exactly enough to fix all $n^2(n+1)/2$ independent components, and no metric derivatives beyond first order appear.*

Go deeper · Derivation C · 1 minThe contracted symbol and the covariant divergence

Derivation CThe contracted symbol and the covariant divergence

The aim is to prove the compact identity (7), $\Gamma^\lambda{}_{\mu\lambda}=\partial_\mu\ln\sqrt{|g|}$ — the most useful computational shortcut in tensor calculus on curved space. It converts the covariant divergence of a vector, which naively needs every Christoffel symbol, into one ordinary derivative weighted by $\sqrt{|g|}$ , and it is what makes the spherical and cylindrical Laplacians drop straight out of the metric. The route is a contraction of the Christoffel formula, a matrix trace, and Jacobi's formula for the derivative of a determinant.

Step 1. Set $\nu=\lambda$ in (5): $\Gamma^\lambda{}_{\mu\lambda}=\tfrac12 g^{\lambda\rho}(\partial_\mu g_{\rho\lambda}+\partial_\lambda g_{\rho\mu}-\partial_\rho g_{\mu\lambda})$ . The last two terms cancel after relabelling the dummy pair $(\lambda\leftrightarrow\rho)$ , leaving $\Gamma^\lambda{}_{\mu\lambda}=\tfrac12 g^{\lambda\rho}\partial_\mu g_{\rho\lambda}$ . Why: contracting kills the antisymmetric combination, leaving a trace waiting to be recognized.

Step 2. Recognize the trace: $g^{\lambda\rho}\partial_\mu g_{\rho\lambda}=\mathrm{tr}(g^{-1}\partial_\mu g)$ , so $\Gamma^\lambda{}_{\mu\lambda}=\tfrac12\,\mathrm{tr}(g^{-1}\partial_\mu g)$ . Why: $g^{\lambda\rho}$ is the matrix inverse and $\partial_\mu g_{\rho\lambda}$ the entrywise derivative, so their contraction is literally that trace.

Step 3. Apply Jacobi's formula $\partial_\mu\ln\det g=\mathrm{tr}(g^{-1}\partial_\mu g)$ to get $\Gamma^\lambda{}_{\mu\lambda}=\tfrac12\partial_\mu\ln|g|=\partial_\mu\ln\sqrt{|g|}$ . Why: this collapses the covariant divergence to $\nabla_\mu V^\mu=\frac{1}{\sqrt{|g|}}\partial_\mu(\sqrt{|g|}\,V^\mu)$ , and the Laplace–Beltrami operator to $\nabla^2 f=\frac{1}{\sqrt{|g|}}\partial_\mu(\sqrt{|g|}\,g^{\mu\nu}\partial_\nu f)$ . *In spherical coordinates $\sqrt{|g|}=r^2\sin\theta$ , and the identity reproduces the textbook $\frac{1}{r^2}\partial_r(r^2\partial_r)+\cdots$ Laplacian exactly.*

Limiting cases

Special cases sharpen what the connection does and does not measure.

Case	Result	Meaning
Cartesian flat space	$\Gamma^\lambda{}_{\mu\nu}=0$	The metric is constant ( $g_{\mu\nu}=\delta_{\mu\nu}$ ), all derivatives vanish, and $\nabla_\mu$ reduces to the ordinary partial $\partial_\mu$ .
Flat space, polar coordinates	$\Gamma^r{}_{\theta\theta}=-r,\ \Gamma^\theta{}_{r\theta}=\Gamma^\theta{}_{\theta r}=\tfrac1r$	Nonzero connection with zero curvature — the symbols encode the swinging of $\hat r,\hat\theta$ ; the Riemann tensor built from them still vanishes identically.
Normal coordinates at a point $p$	$\Gamma^\lambda{}_{\mu\nu}(p)=0,\ \partial_\sigma\Gamma^\lambda{}_{\mu\nu}(p)\neq0$	At any single point one can make $\Gamma$ vanish (the equivalence principle / a free-fall frame), but its first derivatives — and hence curvature — cannot be removed.
2-sphere of radius $a$	$\Gamma^\theta{}_{\phi\phi}=-\sin\theta\cos\theta,\ \Gamma^\phi{}_{\theta\phi}=\cot\theta$	Genuinely curved: no coordinate change makes $\Gamma$ vanish everywhere; the Riemann tensor is nonzero with Gaussian curvature $1/a^2$ .

Common trapA nonzero Christoffel symbol means the space is curved

It does not — this is the most common trap. $\Gamma^\lambda{}_{\mu\nu}$ is not a tensor, so its value is a joint statement about the space and the coordinates you chose. In flat space in polar coordinates $\Gamma^r{}_{\theta\theta}=-r\neq0$ purely because $\hat r,\hat\theta$ rotate, yet the plane is flat and the Riemann tensor vanishes identically. Conversely, because $\Gamma$ is non-tensorial you can always make every component vanish at a chosen point (Riemann normal coordinates, the equivalence principle) even on a wildly curved manifold. What you cannot fake away is $\partial\Gamma$ : curvature lives in $R^\rho{}_{\sigma\mu\nu}=\partial_\mu\Gamma^\rho{}_{\nu\sigma}-\partial_\nu\Gamma^\rho{}_{\mu\sigma}+\Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma}-\Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}$ , a coordinate-independent combination. So never read curvature off $\Gamma$ directly, never raise or lower its indices casually, and never conclude it is zero in all frames just because it vanishes in one.

Worked examples

1 · The flat plane in polar coordinates (confidence builder)

Take the Euclidean plane in polar coordinates, $\mathrm{d}s^2=\mathrm{d}r^2+r^2\,\mathrm{d}\theta^2$ . Compute every Christoffel symbol, then verify that the geodesic equation reproduces ordinary straight lines and that the space is genuinely flat.

Step 1. Read off the metric: $g_{rr}=1$ , $g_{\theta\theta}=r^2$ , $g_{r\theta}=0$ , so the inverse is $g^{rr}=1$ , $g^{\theta\theta}=1/r^2$ . The only nonzero metric derivative is $\partial_r g_{\theta\theta}=2r$ .

Step 2. Apply the Christoffel formula (5). The only surviving symbols are $\Gamma^r{}_{\theta\theta}=\tfrac12 g^{rr}(-\partial_r g_{\theta\theta})=\tfrac12(1)(-2r)=\mathbf{-r}$ and $\Gamma^\theta{}_{r\theta}=\Gamma^\theta{}_{\theta r}=\tfrac12 g^{\theta\theta}\partial_r g_{\theta\theta}=\tfrac12\cdot\tfrac{1}{r^2}\cdot 2r=\mathbf{1/r}$ . All others vanish.

Step 3. Feed these into the geodesic equation (8). The $\theta$ equation $\ddot\theta+\tfrac2r\dot r\dot\theta=0$ is $\frac{\mathrm{d}}{\mathrm{d}\tau}(r^2\dot\theta)=0$ — conservation of angular momentum. The $r$ equation $\ddot r-r\dot\theta^2=0$ is the familiar centripetal relation. Together these are Newton's free-particle equations in polar coordinates: the geodesics are straight lines, dressed in curvilinear bookkeeping.

Step 4. Confirm flatness. Building the Riemann combination $R^\rho{}_{\sigma\mu\nu}=\partial_\mu\Gamma^\rho{}_{\nu\sigma}-\partial_\nu\Gamma^\rho{}_{\mu\sigma}+\Gamma\Gamma-\Gamma\Gamma$ , the derivative $\partial_r\Gamma^\theta{}_{\theta r}=-1/r^2$ is exactly killed by the product term $-\Gamma^\theta{}_{\theta r}\Gamma^\theta{}_{r\theta}=-1/r^2$ , and every component cancels: $R^\rho{}_{\sigma\mu\nu}\equiv0$ . Nonzero $\Gamma$ , zero curvature — the moral of the article, made explicit.

2 · The Christoffel symbols and geodesics of the 2-sphere

Take the round 2-sphere of radius $a$ , $\mathrm{d}s^2=a^2(\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2)$ . Compute the connection, identify the geodesics, and extract the curvature.

Step 1. Metric and inverse: $g_{\theta\theta}=a^2$ , $g_{\phi\phi}=a^2\sin^2\theta$ , $g_{\theta\phi}=0$ , so $g^{\theta\theta}=a^{-2}$ and $g^{\phi\phi}=a^{-2}\sin^{-2}\theta$ . The only nonzero metric derivative is $\partial_\theta g_{\phi\phi}=2a^2\sin\theta\cos\theta$ .

Step 2. Apply (5). The lone symbol with upper $\theta$ comes from the $-\partial_\theta g_{\phi\phi}$ term: $\Gamma^\theta{}_{\phi\phi}=\tfrac12 g^{\theta\theta}(-\partial_\theta g_{\phi\phi})=\tfrac12 a^{-2}(-2a^2\sin\theta\cos\theta)=\mathbf{-\sin\theta\cos\theta}$ . The symbols with upper $\phi$ come from $\partial_\theta g_{\phi\phi}$ : $\Gamma^\phi{}_{\theta\phi}=\Gamma^\phi{}_{\phi\theta}=\tfrac12 g^{\phi\phi}\partial_\theta g_{\phi\phi}=\tfrac12\cdot\tfrac{1}{a^2\sin^2\theta}\cdot 2a^2\sin\theta\cos\theta=\mathbf{\cot\theta}$ . All others vanish.

Step 3. Geodesics. On the equator $\theta=\pi/2$ both $\cot\theta$ and $\sin\theta\cos\theta$ vanish, so $\ddot\theta=\ddot\phi=0$ solves the geodesic equation — the equator is a geodesic, and by symmetry so is every great circle. Geodesics on the sphere are great circles, the straightest curves a surface-walker can follow.

Step 4. Curvature and holonomy. Building the Riemann tensor from these symbols gives $R_{\theta\phi\theta\phi}=a^2\sin^2\theta$ , a Gaussian curvature $K=1/a^2$ — positive and constant, as a sphere must be. This is why parallel transport around the octant of the second figure returns a vector rotated by the enclosed solid angle: holonomy angle $=$ (enclosed area) $/a^2=(\pi a^2/2)/a^2=\pi/2=90°$ , a readout of curvature no coordinate choice can remove.

Physics of the portrait

The connection is where three strands of nineteenth-century mathematics — Gauss's intrinsic curvature, Riemann's higher-dimensional geometry, and the algebra of quadratic forms — braided into a single object Einstein would recognize as gravity.

Origins · 1 minElwin Bruno Christoffel and how this was discovered

Elwin Bruno Christoffel

1829–1900 · Riemannian geometry and the reduction of quadratic differential forms

Elwin Bruno Christoffel was a German mathematician whose name is attached to the symbols every physicist meets in general relativity, though he never lived to see the physics they would carry. Trained in Berlin under Dirichlet, he worked at the ETH in Zürich and then at Strasbourg, which he helped build into a center for mathematics after 1872. His decisive 1869 paper on the transformation of quadratic differential forms asked a purely geometric question: given two metrics, when can one be transformed into the other by a change of coordinates? Answering this "equivalence problem" forced him to introduce the three-index coefficients $\Gamma$ built from first derivatives of the metric, and the four-index object now called the Riemann–Christoffel curvature tensor, as the invariants that decide it. His "covariant differentiation" is exactly the operation this article calls $\nabla$ . A generation later Gregorio Ricci-Curbastro and Tullio Levi-Civita systematized his ideas into the absolute differential calculus — tensor calculus — in 1900, and in 1917 Levi-Civita gave the symbols their geometric meaning by defining parallel transport, showing that Christoffel's coefficients are precisely the rule for carrying a vector "without turning" across a curved space. When Einstein needed a mathematics for a gravity that was geometry, it was this Christoffel–Ricci–Levi-Civita apparatus, taught to him by Marcel Grossmann, that he reached for.

History

1854

— Riemann's habilitation lecture founds the geometry of curved n-dimensional manifolds

1869

— Christoffel introduces the symbols Γ, solving the equivalence problem for quadratic differential forms

1900

— Ricci-Curbastro and Levi-Civita codify the absolute differential calculus (tensor calculus)

1915

— Einstein's general relativity identifies the Levi-Civita connection with the gravitational field

1917

— Levi-Civita defines parallel transport, giving the connection its geometric meaning

Going further

Texts. Stone & Goldbart, Mathematics for Physics, ch. on differential geometry (covariant derivative and connections) · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., curvilinear-coordinate and tensor-analysis chapters · Hassani, Mathematical Physics, Part on differential geometry and Lie groups · Nakahara, Geometry, Topology and Physics, ch. 7 · Byron & Fuller, Mathematics of Classical and Quantum Physics, tensor chapter · Misner, Thorne & Wheeler, Gravitation, ch. 8–11 · Wald, General Relativity, ch. 3. The Christoffel formula follows from $\nabla g=0$ plus zero torsion via the three-term cyclic combination. Key anchor results: the polar-coordinate symbols $\Gamma^r{}_{\theta\theta}=-r$ , $\Gamma^\theta{}_{r\theta}=1/r$ follow from $\mathrm{d}s^2=\mathrm{d}r^2+r^2\mathrm{d}\theta^2$ with the Riemann tensor vanishing identically; the 2-sphere symbols $\Gamma^\theta{}_{\phi\phi}=-\sin\theta\cos\theta$ , $\Gamma^\phi{}_{\theta\phi}=\cot\theta$ give Gaussian curvature $1/a^2$ and 90° octant holonomy; the contracted identity $\Gamma^\lambda{}_{\mu\lambda}=\partial_\mu\ln\sqrt{|g|}$ follows from Jacobi's formula; and the inhomogeneous transformation term is symmetric in its lower indices.

Prerequisites. Manifolds and Tangent Spaces · The Metric Tensor · Covariant and Contravariant Tensors · Tensor Algebra and the Tensor Product. Leads to. The Covariant Derivative · Parallel Transport and Geodesics · The Riemann Curvature Tensor · Raising and Lowering Indices.

Ask the tutor: Ask me to grind through the Christoffel symbols of any metric you like — Schwarzschild, a torus, cylindrical coordinates — and we'll check the curvature together, or watch a vector rotate as it is parallel-transported around a loop.

Is the connection the same thing as the metric?

No — and keeping them distinct is the whole point. A metric $g_{\mu\nu}$ measures lengths and angles; a connection $\nabla$ tells you how to differentiate and parallel-transport. They are logically independent: a bare manifold can carry a connection with no metric, and any manifold admits infinitely many connections. What is special about the metric is that it selects one preferred connection — the Levi-Civita connection — through metric compatibility and zero torsion, and formula (5) is the explicit bridge. So the connection of Riemannian geometry is derived from the metric, but the connection concept is more general than, and prior to, the metric itself.

If Γ can always be made zero at a point, how can it describe gravity?

This is Einstein's equivalence principle. You can always choose a freely-falling frame — Riemann normal coordinates — in which $\Gamma^\lambda{}_{\mu\nu}(p)=0$ at a chosen event, which is why an orbiting astronaut feels weightless: locally, gravity has been transformed away. What cannot be transformed away is the derivative $\partial\Gamma$ . Tidal forces — the stretching a falling body feels because gravity differs slightly at head and feet — live in the Riemann tensor, built from $\partial\Gamma$ and $\Gamma\Gamma$ , which is coordinate-independent. So $\Gamma$ carries the removable, coordinate-artifact part of gravity, while curvature carries the real, irremovable part.

Why does the covector rule get a minus sign when the vector rule gets a plus?

Because their contraction must behave. The pairing $\omega_\nu V^\nu$ is a scalar, whose covariant derivative is just the ordinary one: $\nabla_\mu(\omega_\nu V^\nu)=\partial_\mu(\omega_\nu V^\nu)$ . Expand with the Leibniz rule, using $\nabla_\mu V^\nu=\partial_\mu V^\nu+\Gamma^\nu{}_{\mu\lambda}V^\lambda$ and an unknown $\nabla_\mu\omega_\nu=\partial_\mu\omega_\nu+c\,\Gamma^\lambda{}_{\mu\nu}\omega_\lambda$ . The ordinary-derivative pieces reproduce the right side automatically, so the two $\Gamma$ terms must cancel, forcing $c=-1$ . The minus sign is not a convention — it is required for contraction to commute with differentiation, and the same argument fixes the sign on every index of a general tensor.

What is torsion, and why does general relativity set it to zero?

Torsion is the antisymmetric part of the connection, $T^\lambda{}_{\mu\nu}=\Gamma^\lambda{}_{\mu\nu}-\Gamma^\lambda{}_{\nu\mu}$ , and unlike $\Gamma$ itself it is a genuine tensor (the inhomogeneous transformation term is symmetric in the lower indices and cancels in the difference). Geometrically it measures the failure of infinitesimal parallelograms to close under parallel transport. Standard general relativity sets it to zero as the simplest choice consistent with the equivalence principle, and then the metric alone fixes the connection. Theories with torsion exist — Einstein–Cartan gravity, where torsion couples to intrinsic spin — but in the Levi-Civita setting here torsion is zero by fiat, one of the two conditions making the Christoffel formula unique.

How is the Christoffel connection related to the gauge fields of particle physics?

They are the same idea in different bundles. A connection is the universal rule for differentiating data attached over each point of spacetime. For the Christoffel connection that data is a tangent vector, and $\Gamma$ says how to carry it "without turning." For a gauge theory the data is an internal state — the phase of a charged field, or a color vector in QCD — and the gauge potential $A_\mu$ plays the same role. The covariant derivative $\nabla_\mu=\partial_\mu+\Gamma$ becomes $D_\mu=\partial_\mu-igA_\mu$ , and the curvature built from $\partial\Gamma+\Gamma\Gamma$ becomes the field strength $F_{\mu\nu}$ . Geometry and gauge theory as two faces of connections on fiber bundles is one of the deep insights of twentieth-century physics.

Why are there so many index conventions, and how do I avoid getting burned?

Books order the lower indices of $\Gamma^\lambda{}_{\mu\nu}$ differently, use the curly-brace symbol $\{^\lambda_{\mu\nu}\}$ versus $\Gamma$ , and pick different metric signatures. For the symmetric Levi-Civita connection the ordering of the two lower indices does not matter, so most conventions agree in practice. Safe habits: confirm a text's connection is symmetric before trusting any asymmetric formula; never raise or lower an index on $\Gamma$ itself (the fully-lowered $\Gamma_{\rho\mu\nu}$ is a computational device, not a tensor); and track the metric signature, since a sign flip in $g$ propagates through the formula. When results are stated as tensors — covariant derivative, geodesic equation, Riemann tensor — the conventions wash out, which is why physicists quote final answers in tensor form.