The Levi-Civita Connection

A bare smooth manifold is a set of points with no way to compare a vector here with a vector there — the derivative of a vector field is not even defined until you supply extra structure. That structure is an affine connection, and it is wildly non-unique: on any manifold there are infinitely many, each giving a different rule for parallel transport and a different covariant derivative. The pivotal simplification of (pseudo-)Riemannian geometry — the one that makes General Relativity a theory of geometry rather than a theory of a geometry plus a connection — is that once a metric $g_{ab}$ is fixed, there is exactly one connection that respects it in the two ways physics demands. This is the Levi-Civita connection, named for Tullio Levi-Civita, who with Gregorio Ricci-Curbastro built the absolute differential calculus that Einstein needed. The two demands are torsion-freedom and metric compatibility. Torsion-freedom, $T^a{}_{bc}=\Gamma^a{}_{bc}-\Gamma^a{}_{cb}=0$ , says infinitesimal parallelograms close and the coefficients are symmetric in their lower pair. Metric compatibility, $\nabla_a g_{bc}=0$ , says the inner product of two vectors is unchanged when both are parallel transported — lengths, proper times, and angles all survive the trip. Either demand alone leaves an infinity of connections; together they leave precisely one. The fundamental theorem of Riemannian geometry proves this and hands you the coefficients explicitly — the Christoffel symbols, built algebraically from the metric and its first derivatives. Every geometric object of gravitation, from geodesics to the Einstein tensor, descends from this one connection, and therefore from the metric alone.

Key takeaways

The Levi-Civita connection is the unique affine connection on a (pseudo-)Riemannian manifold that is simultaneously torsion-free ( $\Gamma^a{}_{bc}=\Gamma^a{}_{cb}$ ) and metric-compatible ( $\nabla_a g_{bc}=0$ ).
Its uniqueness is the fundamental theorem of Riemannian geometry, which also yields the explicit Christoffel formula $\Gamma^a{}_{bc}=\tfrac12 g^{ad}(\partial_b g_{dc}+\partial_c g_{db}-\partial_d g_{bc})$ — the connection from the metric and nothing else.
Metric compatibility means parallel transport preserves lengths and angles; torsion-freedom means the connection carries no independent twist and closes small parallelograms.
It turns the metric — a device for measuring intervals — into a device for differentiating, defining geodesics, curvature, and ultimately the gravitational field equations.
The connection coefficients are NOT tensor components: they can be set to zero at any point (local inertial frame), which is the equivalence principle; what survives is the curvature built from their derivatives.
Key idea: On any (pseudo-)Riemannian manifold there is exactly one connection that is both torsion-free and metric-compatible; its coefficients are the Christoffel symbols determined algebraically by the metric, so geometry alone fixes how to differentiate, define free fall, and measure curvature.

Where this lives today

The Levi-Civita connection is not a historical artefact but the object every practicing relativist computes first. In numerical relativity — the simulations that produced the gravitational-waveform templates LIGO and Virgo matched against the 2015 detection of merging black holes — the Christoffel symbols of an evolving spacetime metric are recomputed at every grid point and timestep, the coefficients in the equations that push the geometry forward. In precision solar-system tests the same $\Gamma^r{}_{tt}$ of Worked Example 2, carried to next order in $r_s/r$ , delivers Mercury's $43''$ -per-century perihelion precession and the Shapiro delay of radar grazing the Sun, and the same geometry sets the clock corrections without which GPS positions would drift kilometers per day. And the frontier question of whether spacetime carries torsion — the true connection being Levi-Civita or Levi-Civita plus a spin-sourced contorsion $K^a{}_{bc}$ (Einstein-Cartan theory) — is tested wherever extreme spin densities occur, so far with no deviation found.

The physical picture

Imagine walking across a curved surface carrying a rigid measuring rod and a protractor, refusing to twist them gratuitously. Metric compatibility is the promise that the rod never changes length and the protractor never loses calibration: the geometry itself tells you how to slide the instruments so lengths and angles are conserved. Torsion-freedom is the second promise — that you never spin your frame about your direction of travel, so an infinitesimal parallelogram of two transported steps closes on itself. Two prohibitions, no remaining freedom: that unique rule is the Levi-Civita connection. The first figure separates the two conditions to show what each forbids.

Figure

The two conditions that pin down the connection — metric compatibility keeps the transported vector's length and the angle between vectors fixed along a curve, while the torsion-free condition makes the connection symmetric so that stepping along e₂ then e₁ lands at the same point as stepping along e₁ then e₂, closing the small parallelogram

A general connection in $n$ dimensions has $n^3$ independent coefficients at each point. Metric compatibility contributes $\tfrac12 n^2(n+1)$ equations and torsion-freedom the constraints $\Gamma^a{}_{[bc]}=0$ ; count them and they exactly match the unknowns — neither over- nor under-determined — which is the arithmetic behind the fundamental theorem. The famous cyclic-permutation trick is not a conjuring act but the explicit solution of this square linear system: write $\nabla g=0$ three times with the indices rotated, add two and subtract one, and the antisymmetric pieces cancel by torsion-freedom, leaving the connection as an algebraic expression in $\partial g$ . The second figure shows the connection's deepest payoff — carry a vector around a closed loop and it comes back rotated, by an angle computable from the geometry alone.

Figure

Parallel transport around a circle of colatitude θ₀ = 60° on a unit sphere returns the vector rotated by the holonomy angle Δ = 2π(1 − cos θ₀) = π = 180°, which equals the solid angle of the polar cap the loop encloses — the connection turns global geometry into a measurable rotation

The core equations

The Levi-Civita connection is defined not by a formula but by two properties; the formula is what those properties force. Start with the two defining conditions, then read off the coefficients they determine, and finally the machinery — covariant derivative, geodesics, divergence — that the connection powers. Indices run over the $n$ coordinates and the signature is left general, so the results hold for Riemannian and Lorentzian manifolds alike. The first condition, metric compatibility, says the metric is covariantly constant: differentiating lengths and angles with this connection gives zero, so parallel transport is an isometry.

\nabla_a g_{bc}=0 \qquad (1)

where: $\nabla_a$ — the covariant derivative built from the connection; $g_{bc}$ — the metric tensor components. Reading: transporting two vectors keeps their inner product fixed. The second condition, torsion-freedom, kills the antisymmetric part of the connection in its lower indices. The torsion is a genuine tensor (the inhomogeneous transformation terms cancel in the difference), and setting it to zero is a real physical assumption, not a convention.

T^a{}_{bc}=\Gamma^a{}_{bc}-\Gamma^a{}_{cb}=0 \qquad (2)

where: $T^a{}_{bc}$ — the torsion tensor; $\Gamma^a{}_{bc}$ — the connection coefficients in a coordinate basis. Reading: the lower pair is symmetric, $\Gamma^a{}_{bc}=\Gamma^a{}_{cb}$ , so infinitesimal parallelograms close. Together, (1) and (2) determine the connection completely; solving the linear system (Derivation 1) gives the Christoffel symbols, built entirely from the metric and its first derivatives.

\Gamma^a{}_{bc}=\tfrac12\, g^{ad}\left(\partial_b g_{dc}+\partial_c g_{db}-\partial_d g_{bc}\right) \qquad (3)

where: $g^{ad}$ — the inverse metric, $g^{ad}g_{db}=\delta^a{}_b$ ; $\partial_b$ — the coordinate partial derivative. Reading: only $\partial g$ appears, no $\partial\Gamma$ — the connection is algebraic in the metric's first derivatives, which is why it can be set to zero at a point. With the coefficients in hand, the covariant derivative of a vector adds a connection term that repairs the non-tensorial partial derivative.

\nabla_a V^b=\partial_a V^b+\Gamma^b{}_{ac}V^c \qquad (4)

where: $V^b$ — a contravariant vector field; $\partial_a V^b$ — the ordinary partial derivative, not a tensor by itself. The $\Gamma V$ term cancels the coordinate junk in $\partial_a V^b$ , so $\nabla_a V^b$ is a genuine $(1,1)$ tensor. The autoparallels of this connection — curves that transport their own tangent — are the geodesics, which because the connection is metric are also the extremal-proper-time worldlines of free fall.

\frac{\mathrm{d}^2 x^a}{\mathrm{d}\lambda^2}+\Gamma^a{}_{bc}\frac{\mathrm{d}x^b}{\mathrm{d}\lambda}\frac{\mathrm{d}x^c}{\mathrm{d}\lambda}=0 \qquad (5)

where: $x^a(\lambda)$ — the curve's coordinates; $\lambda$ — an affine parameter (proper time for a massive particle). Reading: the $\Gamma$ term is the gravitational "force" per unit mass in disguise, vanishing in a local inertial frame. Finally, one contraction of the connection is worth memorizing: its trace is a pure gradient of $\ln\sqrt{|g|}$ , which collapses covariant divergences into ordinary ones.

\Gamma^a{}_{ba}=\partial_b\ln\sqrt{|g|}=\frac{1}{\sqrt{|g|}}\,\partial_b\sqrt{|g|} \qquad (6)

where: $g$ — the determinant of $g_{ab}$ ; $|g|$ — its absolute value (needed for Lorentzian signature). Reading: contracting formula (3) on $a$ leaves $\tfrac12\partial_b\ln|g|$ by Jacobi's formula, so this identity turns $\nabla_a V^a$ into $\tfrac{1}{\sqrt{|g|}}\partial_a(\sqrt{|g|}\,V^a)$ , recovering the divergence formulas of vector calculus.

Derivation AThe fundamental theorem of Riemannian geometry

The goal is to prove that the two conditions do their job: that a torsion-free, metric-compatible connection exists on every (pseudo-)Riemannian manifold and is unique, and to extract its coefficients as an explicit function of the metric. Uniqueness is the important half — it is what lets General Relativity speak of "the" connection and hence "the" curvature of spacetime with no further choice — so the strategy is to treat metric compatibility as a linear system for the unknown coefficients and show torsion-freedom makes that system square and invertible.

Step 1. Write metric compatibility out with the connection acting on the lower-index metric: $\partial_a g_{bc}=\Gamma^d{}_{ab}g_{dc}+\Gamma^d{}_{ac}g_{db}$ . This is $n$ copies (one per free index $a$ ) of $\tfrac12 n(n+1)$ equations, linear in the unknown $\Gamma$ 's. Why: metric compatibility is not a vague physical wish but a concrete set of linear constraints, and counting them against the $n^3$ unknown coefficients is the whole game.

Step 2. Rotate the free indices cyclically to write the same relation for $\partial_b g_{ca}$ and $\partial_c g_{ab}$ , then form the combination $\partial_a g_{bc}+\partial_b g_{ca}-\partial_c g_{ab}$ . Substituting the three expanded forms, the connection terms pair up and cancel wherever they carry the antisymmetric index combination. Why: this specific add-two-subtract-one pattern is designed to isolate one symmetric connection term; the cancellation only works because of the next step.

Step 3. Impose torsion-freedom, $\Gamma^d{}_{ab}=\Gamma^d{}_{ba}$ . With the lower pair symmetric, all the cross terms in Step 2 cancel and the survivor is $\partial_a g_{bc}+\partial_b g_{ca}-\partial_c g_{ab}=2\,\Gamma^d{}_{ab}\,g_{dc}$ . Why: torsion-freedom is exactly the extra input that turns an under-determined system into a solvable one — without it the antisymmetric part $\Gamma^a{}_{[bc]}$ would be free.

Step 4. Contract with the inverse metric $g^{ec}$ to lift the lowered index, giving $\Gamma^e{}_{ab}=\tfrac12 g^{ec}(\partial_a g_{bc}+\partial_b g_{ca}-\partial_c g_{ab})$ . Because $g^{ec}$ is invertible, the solution is unique — no freedom remains, proving uniqueness. Why: an invertible linear system has exactly one solution, so uniqueness is automatic once the system is square, which the index count guaranteed.

Step 5. Close the existence half by running the logic backwards: define $\Gamma$ by the boxed formula and substitute it into $\nabla_a g_{bc}$ ; direct expansion returns identically zero, and the expression is manifestly symmetric in $(a,b)$ , hence torsion-free. So a connection with both properties not only is unique but does exist. Why: uniqueness alone would be vacuous if no such connection existed; the explicit formula is the constructive existence proof. *As a check: substituting $\Gamma^a{}_{bc}=\tfrac12 g^{ad}(\partial_b g_{dc}+\partial_c g_{db}-\partial_d g_{bc})$ back into $\nabla_a g_{bc}$ returns identically 0, and $\Gamma^a{}_{[bc]}=0$ by construction — confirming both defining conditions are satisfied.*

Go deeper · Derivation B · 1 minMetric compatibility preserves lengths and angles

Derivation BMetric compatibility preserves lengths and angles

The goal is to show that the abstract equation $\nabla_a g_{bc}=0$ carries the concrete physical meaning advertised in the intuition — that the inner product of two vectors is conserved when both are parallel transported along any curve, so a transported rod keeps its length and a transported pair of arrows keeps its angle. This is the property that makes the Levi-Civita connection the one physics wants: it guarantees that proper time and measured angles are transport-invariant, without which "length" and "clock rate" would drift as you moved.

Step 1. Take a curve $x^a(\lambda)$ with tangent $u^a=\mathrm{d}x^a/\mathrm{d}\lambda$ , and two vector fields $V$ and $W$ that are parallel transported along it, meaning $u^c\nabla_c V^a=0$ and $u^c\nabla_c W^b=0$ . Their inner product $g_{ab}V^aW^b$ is a scalar, so its rate of change along the curve is $\tfrac{\mathrm{d}}{\mathrm{d}\lambda}(g_{ab}V^aW^b)=u^c\nabla_c(g_{ab}V^aW^b)$ . Why: for a scalar the directional covariant derivative equals the ordinary derivative, letting us swap $\mathrm{d}/\mathrm{d}\lambda$ for $u^c\nabla_c$ and bring the connection into play.

Step 2. Apply the Leibniz rule, since $\nabla$ obeys the product rule on tensor products: $u^c\nabla_c(g_{ab}V^aW^b)=u^c[(\nabla_c g_{ab})V^aW^b+g_{ab}(\nabla_c V^a)W^b+g_{ab}V^a(\nabla_c W^b)]$ . Why: splitting the derivative across the three factors is what exposes the metric-compatibility term as a separate, killable piece.

Step 3. Kill each term in turn. The first vanishes by metric compatibility, $\nabla_c g_{ab}=0$ . The second and third contain $u^c\nabla_c V^a$ and $u^c\nabla_c W^b$ , which vanish because $V$ and $W$ are parallel transported. Every term is gone. Why: the two hypotheses — compatibility and transport — attack different terms, and together they leave nothing.

Step 4. Conclude $\tfrac{\mathrm{d}}{\mathrm{d}\lambda}(g_{ab}V^aW^b)=0$ : the inner product is constant along the curve. Setting $W=V$ shows $|V|^2=g_{ab}V^aV^b$ is conserved (lengths), and with $V\neq W$ the angle $\cos\theta=g_{ab}V^aW^b/(|V||W|)$ is conserved (angles). Why: length and angle are just special and general cases of the same conserved inner product, so one calculation delivers both. *As a check: the argument follows directly from the Leibniz rule and the two axioms — a null geodesic transporting its own tangent keeps $g_{ab}u^au^b=0$ , while a timelike one keeps it at $-1$ .*

Limiting cases

The single formula (3) specializes cleanly, and the specializations are the fastest way to build intuition — and to see what is coordinate artefact versus real geometry.

Case	Result	Meaning
Flat space, Cartesian coordinates	$g_{ab}=\eta_{ab}=\text{const}\Rightarrow\Gamma^a{}_{bc}=0$	A constant metric has no derivatives, so every Christoffel symbol vanishes and $\nabla$ reduces to $\partial$ .
Flat space, polar coordinates	$\Gamma^r{}_{\theta\theta}=-r,\ \ \Gamma^\theta{}_{r\theta}=\Gamma^\theta{}_{\theta r}=\tfrac1r$	The connection is nonzero even though the plane is flat — Christoffel symbols are coordinate artefacts; the Riemann tensor is still zero.
The 2-sphere of radius $a$	$\Gamma^\theta{}_{\phi\phi}=-\sin\theta\cos\theta,\ \ \Gamma^\phi{}_{\theta\phi}=\cot\theta$	From $g_{\theta\theta}=a^2,\ g_{\phi\phi}=a^2\sin^2\theta$ ; these drive great-circle geodesics and the $180°$ holonomy of the portrait.
Weak-field spacetime	$\Gamma^i{}_{00}\approx\partial_i\Phi$ (Newtonian potential $\Phi$ )	The connection reduces to the gradient of the gravitational potential — the geodesic equation becomes $\ddot x^i=-\partial_i\Phi$ , Newton's law.
Metric-compatible with torsion	$\Gamma^a{}_{bc}=\{{}^a_{bc}\}+K^a{}_{bc}$	A general metric connection is Levi-Civita plus the contorsion $K$ built from torsion; Levi-Civita is the unique $K=0$ representative.

Common trapThe Christoffel symbols measure gravity, so where they vanish gravity is gone.

This is the single most common trap, and it is false. The Christoffel symbols are not tensor components — under a coordinate change they pick up an inhomogeneous term with second derivatives of the transformation, $\Gamma'^a{}_{bc}=\tfrac{\partial x'^a}{\partial x^d}\tfrac{\partial x^e}{\partial x'^b}\tfrac{\partial x^f}{\partial x'^c}\Gamma^d{}_{ef}+\tfrac{\partial x'^a}{\partial x^d}\tfrac{\partial^2 x^d}{\partial x'^b\partial x'^c}$ . That extra term lets you set every $\Gamma^a{}_{bc}=0$ at any chosen point by moving to local inertial (freely falling) coordinates — precisely the equivalence principle, gravity transformed away locally. Yet a freely falling frame in a black hole's tidal field has zero $\Gamma$ at a point with very real gravity nearby. Only the curvature tensor $R^a{}_{bcd}$ , built from derivatives of $\Gamma$ , is a coordinate-invariant measure of gravity; the connection alone is not.

Worked examples

1 · The 2-sphere, its connection, and a 180° holonomy (confidence builder)

The round 2-sphere of radius $a$ has metric $g_{\theta\theta}=a^2$ , $g_{\phi\phi}=a^2\sin^2\theta$ (diagonal, $g_{\theta\phi}=0$ ). Compute its Christoffel symbols, and find the angle by which a vector is rotated after parallel transport once around the circle of colatitude $\theta_0=60°$ .

Step 1. Invert the metric: since it is diagonal, $g^{\theta\theta}=1/a^2$ and $g^{\phi\phi}=1/(a^2\sin^2\theta)$ . The only nonconstant metric component is $g_{\phi\phi}=a^2\sin^2\theta$ , so the only nonzero derivative is $\partial_\theta g_{\phi\phi}=2a^2\sin\theta\cos\theta$ . This single derivative feeds every nonzero Christoffel symbol.

Step 2. Apply formula (3). For $\Gamma^\theta{}_{\phi\phi}$ the only surviving term is $-\tfrac12 g^{\theta\theta}\partial_\theta g_{\phi\phi}=-\tfrac12(1/a^2)(2a^2\sin\theta\cos\theta)=\mathbf{-\sin\theta\cos\theta}$ . For $\Gamma^\phi{}_{\theta\phi}=\Gamma^\phi{}_{\phi\theta}$ it is $\tfrac12 g^{\phi\phi}\partial_\theta g_{\phi\phi}=\tfrac12(1/(a^2\sin^2\theta))(2a^2\sin\theta\cos\theta)=\mathbf{\cot\theta}$ ; all others vanish. Note the radius $a$ cancels — a sphere's connection coefficients are scale-free.

Step 3. Parallel transport a tangent vector once around the latitude circle $\theta=\theta_0$ . Integrating $u^a\nabla_a V^b=0$ with $u^a$ along $\partial_\phi$ , using the symbols above, rotates the vector relative to the local geodesic frame by the holonomy angle $\Delta=2\pi(1-\cos\theta_0)$ — which by Gauss-Bonnet equals the enclosed cap's solid angle $\iint K\,\mathrm{d}A$ with Gaussian curvature $K=1/a^2$ .

Step 4. Put in $\theta_0=60°$ : $\cos 60°=0.5$ , so $\Delta=2\pi(1-0.5)=\pi\ \text{rad}=\mathbf{180°}$ . Carry an arrow around the 60° latitude and it comes back pointing exactly the opposite way — a purely geometric rotation, computed from the connection alone, with no reference to any embedding. Γᵃ from one metric derivative; holonomy at 60° colatitude = 180°.

2 · The Schwarzschild connection is the gravitational field (fully worked)

The Schwarzschild metric (geometrized units, $c=1$ ) is $g_{tt}=-(1-r_s/r)$ , $g_{rr}=(1-r_s/r)^{-1}$ with $r_s=2GM$ . Compute $\Gamma^r{}_{tt}$ , show the radial geodesic reduces to Newton's law in the weak-field slow-motion limit, and evaluate it at the surface of the Sun.

Step 1. Only $g_{tt}$ and $g_{rr}$ vary with $r$ , and $g_{tt}$ depends on no coordinate but $r$ . Formula (3) then gives $\Gamma^r{}_{tt}=\tfrac12 g^{rr}(2\partial_t g_{tr}-\partial_r g_{tt})=-\tfrac12 g^{rr}\partial_r g_{tt}$ . With $g^{rr}=(1-r_s/r)$ and $\partial_r g_{tt}=-r_s/r^2$ , this is $\Gamma^r{}_{tt}=-\tfrac12(1-r_s/r)(-r_s/r^2)=\dfrac{r_s}{2r^2}\left(1-\dfrac{r_s}{r}\right)$ .

Step 2. Restore constants and identify the Newtonian piece: $r_s=2GM/c^2$ , so $\Gamma^r{}_{tt}=\dfrac{GM}{r^2c^2}\left(1-\dfrac{r_s}{r}\right)$ in the $t$ -in-seconds convention, and the leading factor $GM/r^2$ is exactly Newton's gravitational acceleration. The parenthetical $(1-r_s/r)$ is the general-relativistic correction that switches on near the horizon.

Step 3. Write the radial geodesic (5): $\dfrac{\mathrm{d}^2 r}{\mathrm{d}\tau^2}+\Gamma^r{}_{tt}\left(\dfrac{\mathrm{d}t}{\mathrm{d}\tau}\right)^2+\cdots=0$ . For a slowly moving particle far from the horizon, $\mathrm{d}t/\mathrm{d}\tau\approx1$ and $r_s/r\ll1$ , so $\dfrac{\mathrm{d}^2r}{\mathrm{d}\tau^2}\approx-\Gamma^r{}_{tt}\approx-\dfrac{GM}{r^2}$ . The connection term IS the gravitational force per unit mass; free fall along a geodesic reproduces Newton exactly at leading order.

Step 4. Evaluate at the Sun's surface, $r=R_\odot=6.96\times10^{8}\ \text{m}$ , with $GM_\odot=1.327\times10^{20}\ \text{m}^3\text{s}^{-2}$ : the leading term is $GM_\odot/R_\odot^2=\mathbf{274\ m\,s^{-2}}$ , the Sun's surface gravity. The relativistic correction is $r_s/R_\odot=(2953\ \text{m})/(6.96\times10^{8}\ \text{m})=\mathbf{4.2\times10^{-6}}$ — utterly negligible, which is why Newtonian gravity works there to six figures. Γʳ_tt → GM/r² = 274 m/s² at the Sun, with a 4×10⁻⁶ GR correction.

Physics of the portrait

The Levi-Civita connection crystallized a fifty-year effort to make "differentiation on a curved space" coordinate-independent — running from Christoffel's index gymnastics through the Italian school's absolute differential calculus to Einstein's geometric theory of gravity.

Origins · 1 minTullio Levi-Civita and how this was discovered

Tullio Levi-Civita

1873–1941 · the absolute differential calculus behind general relativity

Tullio Levi-Civita, born in Padua in 1873, was the student and then collaborator of Gregorio Ricci-Curbastro; together in 1900 they published Méthodes de calcul différentiel absolu, the paper that gave the world the tensor calculus. When Einstein needed exactly this language for general covariance it was waiting for him, and Levi-Civita corresponded with him over the field equations. His own decisive contribution came in 1917, when he introduced parallel transport (parallelismo) on a general Riemannian manifold, giving Christoffel's symbols their geometric meaning at last: the rule for sliding a vector along a curve without turning it, defining the unique connection compatible with the metric and free of torsion. A committed anti-fascist, he was stripped of his professorship under Mussolini's 1938 racial laws and died in Rome in 1941 — but the connection bearing his name had already become the backbone of gravitational physics.

History

1869

— Christoffel publishes the three-index symbols in his study of how quadratic differential forms transform

1887

— Ricci-Curbastro begins developing the absolute differential calculus in a series of papers

1900

— Ricci and Levi-Civita codify the tensor calculus in Méthodes de calcul différentiel absolu

1915

— Einstein builds general relativity on the unique metric connection, making gravity geometry

1917

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

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §8.5, §10 · Wald, General Relativity (1984), §3.1 · Carroll, Spacetime and Geometry (2004), §3.2 · Schutz, A First Course in General Relativity (2009), §5 · Hartle, Gravity (2003), §8.4 · do Carmo, Riemannian Geometry (1992), Ch. 2. The Christoffel formula (3) is re-derivable from $\nabla g=0$ plus torsion symmetry, and $\nabla_a g_{bc}=0$ is confirmable by direct substitution. The 2-sphere symbols ( $\Gamma^\theta{}_{\phi\phi}=-\sin\theta\cos\theta$ , $\Gamma^\phi{}_{\theta\phi}=\cot\theta$ ) and polar-coordinate values ( $\Gamma^r{}_{\theta\theta}=-r$ , $\Gamma^\theta{}_{r\theta}=1/r$ ) follow from the stated metrics; the holonomy $\Delta=2\pi(1-\cos\theta_0)$ at $\theta_0=60°$ gives $180°$ (Gauss-Bonnet with $K=1/a^2$ ). The Schwarzschild $\Gamma^r{}_{tt}=\tfrac{r_s}{2r^2}(1-r_s/r)$ reduces to $GM/r^2$ ; at the Sun this is $274\ \text{m s}^{-2}$ with correction $r_s/R_\odot=4.2\times10^{-6}$ ( $GM_\odot=1.327\times10^{20}\ \text{m}^3\text{s}^{-2}$ , $R_\odot=6.96\times10^{8}\ \text{m}$ , $r_s=2953\ \text{m}$ ).

Prerequisites. The Affine Connection · The Covariant Derivative · Metric Compatibility · The Torsion Tensor · Parallel Transport. Leads to. Christoffel Symbols · The Geodesic Equation · Geodesics as Extremal Paths · The Geodesic Deviation Equation · Killing Vectors.

Ask the tutor: Want to see the full set of Christoffel symbols for the Schwarzschild or FLRW metric worked out term by term, or watch how metric compatibility forces the Einstein tensor to be covariantly conserved?

Why is torsion-freedom a physical assumption and not just a convenience?

Torsion is a genuine tensor — the antisymmetric part $\Gamma^a{}_{[bc]}$ survives coordinate changes intact — so setting it to zero is a real statement about nature, not a coordinate choice. Physically, torsion would couple to the intrinsic spin of matter: in Einstein-Cartan theory the connection is metric-compatible but carries a spin-sourced torsion. General Relativity assumes torsion-freedom — the reason its connection is Levi-Civita. The assumption is well tested (no torsion detected), but it remains an assumption that theories beyond GR relax. So torsion-freedom is where a physical hypothesis enters what otherwise looks like pure geometry.

How can the Christoffel symbols vanish at a point yet gravity be present?

Because the symbols are not tensor components. Their transformation law carries an inhomogeneous term built from second derivatives of the coordinate change, and you can always choose that term to cancel the symbols at any single chosen point — these are local inertial (freely falling) coordinates, the mathematical form of the equivalence principle. What you cannot cancel are the derivatives of the symbols, which assemble into the Riemann curvature tensor. Gravity that is real — tidal force, geodesic deviation — lives in $R^a{}_{bcd}$ , and no coordinate choice sets a nonzero tensor to zero. So " $\Gamma=0$ here" means "free fall here," not "no gravity anywhere."

Why does the connection depend only on first derivatives of the metric?

Look at formula (3): it contains $\partial g$ and the algebraic inverse $g^{-1}$ , but no $\partial^2 g$ . This is forced by the two defining conditions, which are algebraic and first-order in the connection, so solving them can only bring in first derivatives of the metric. The consequence matters: because only $\partial g$ appears, and you can always arrange $\partial g=0$ at a point by a smart coordinate choice, the connection is locally removable. Curvature, by contrast, involves $\partial\Gamma$ and hence $\partial^2 g$ , which cannot generally be made to vanish at a point — that second derivative is the irreducible core of gravity.

What exactly is the difference between a connection and a metric?

A metric $g_{ab}$ measures — it gives lengths and angles at each point. A connection $\Gamma^a{}_{bc}$ transports — it tells you how to carry a vector to a neighboring point so you can compare and differentiate. These are logically independent: a bare manifold can carry a connection with no metric, or a metric paired with a connection that mangles it. The Levi-Civita construction is the statement that a metric secretly contains a preferred connection — the unique one leaving its measurements invariant under transport and adding no twist. That is why in General Relativity you specify only the metric: the connection, and everything built from it, comes for free.

Are geodesics the straightest paths or the extremal-length paths, and why do they agree?

They are defined as the straightest paths — autoparallels that transport their own tangent, $u^a\nabla_a u^b=0$ . Separately, one can seek the extremal-proper-time paths by varying $\int\sqrt{|g_{ab}\dot x^a\dot x^b|}\,\mathrm{d}\lambda$ . For a general connection these two families differ. They coincide precisely because the Levi-Civita connection is metric-compatible and torsion-free: compatibility keeps a transported tangent at constant norm, and the transport and variational conditions then produce the same equation (5). With torsion or non-metricity the "straightest" and "longest" curves split apart, so their agreement is a special feature of the metric connection.

Does the sign convention or index placement in the Christoffel formula matter?

The symbol $\Gamma^a{}_{bc}$ is symmetric in its lower pair $(b,c)$ by torsion-freedom, so their order never matters for Levi-Civita. Formula (3) is convention-independent once you fix the metric signature and the definition $\nabla_a V^b=\partial_a V^b+\Gamma^b{}_{ac}V^c$ ; textbooks using $\{{}^a_{bc}\}$ notation land on the same object. What matters is consistency — mixing signature conventions or swapping the lower slots of the covariant-derivative term flips signs downstream in the Riemann and Einstein tensors. Pick one convention and hold it throughout.