The Geodesic Equation

A stone tossed off a cliff, a planet swinging around the Sun, a photon skimming the edge of a galaxy — general relativity insists that all of them are doing the same simple thing: travelling as straight as the geometry allows. The geodesic equation is the precise statement of "as straight as possible" on a curved manifold, and it is the sentence in which Einstein's theory cashes out the slogan gravity is not a force but the shape of spacetime. In flat space "straight" means the tangent vector never changes — the curve never turns. On a curved manifold there is no global notion of a constant vector to compare against, so the phrase has to be rebuilt from a more primitive ingredient: the affine connection, which supplies a rule — parallel transport — for carrying a vector from one point to an infinitesimally near one. A geodesic is then the curve that parallel-transports its own tangent, so that it is doing nothing but continuing in the direction it is already pointing. This article takes the parallel-transport rule and the covariant derivative built in the prerequisite articles and turns the single condition $\nabla_u u = 0$ into the second-order differential equation that actually governs orbits, light bending, and free fall. Two facts make it central. First, freely falling bodies — those feeling no non-gravitational force — move on geodesics, so the equation is the law of motion of gravity, with no force term at all. Second, for the metric-compatible torsion-free connection the same curves extremize proper time, so "straightest" and "longest-lived" turn out to be one class of paths. The conserved quantities and tidal deviations that follow are recalled here by name and developed in the articles downstream.

Key takeaways

A geodesic is a curve that parallel-transports its own tangent: $\nabla_u u = 0$ , the curved-spacetime analogue of a straight line, with no "force" on the right-hand side.
In coordinates this is $\ddot x^\mu + \Gamma^\mu_{\;\alpha\beta}\dot x^\alpha\dot x^\beta = 0$ , a second-order ODE for the worldline $x^\mu(\lambda)$ in which the Christoffel symbols carry all the gravity.
The clean form holds only for an affine parameter $\lambda$ (proper time for a massive particle); a bad parameter adds a term $f(\sigma)\,\dot x^\mu$ on the right.
Massive particles follow timelike geodesics, light follows null geodesics, and the tangent's causal type is conserved along the way — the content of the equivalence principle.
For the Levi-Civita connection the geodesics also extremize proper time, so the autoparallel ("straightest") and variational ("extremal") definitions agree.
Quantitatively it delivers Mercury's $43''$ per century, the Sun's $1.75''$ light bending, and Newton's $\ddot{\vec x}=-\nabla\Phi$ in the weak-field slow-motion limit.
Key idea: a geodesic is the integral curve of a vector field that parallel-transports itself, $\nabla_u u = 0$ ; the Christoffel symbols appear as the only "force," and because that force is a coordinate artifact removable at any single point, free fall along geodesics encodes gravity geometrically rather than dynamically.

The core insight

A geodesic is the integral curve of a vector field that parallel-transports itself, $\nabla_u u = 0$ ; in coordinates the connection coefficients (Christoffel symbols) are the only "force," and because that force is a coordinate artifact removable pointwise, free fall along geodesics encodes gravity as geometry rather than dynamics.

Where this lives today

Every image of a black hole is a portrait of null geodesics. The bright ring in the Event Horizon Telescope's 2019 picture of M87 and its 2022 picture of Sagittarius A is light wound around the hole on nearly closed photon orbits, and turning those pictures into mass and spin measurements means integrating the null geodesic equation, equation (2), through the Kerr metric millions of times per synthetic image. The same machinery drives gravitational-wave astronomy: the extreme-mass-ratio inspirals that the space detector LISA is being built to hear are, to leading order, a small body sweeping through tens of thousands of geodesic orbits of a supermassive black hole, with a tiny self-force nudging it off each geodesic onto the next — decoding the waveform is geodesic dynamics plus a correction. Closer to home, the geodesic equation's Newtonian-limit normalization $g_{00}=-(1+2\Phi/c^2)$ is what every GPS receiver silently corrects for, since satellite and ground clocks tick along different worldlines and drift by about $38\ \mu\mathrm{s}$ per day. And the equivalence-principle premise beneath it all — that all bodies share one geodesic regardless of composition — is now tested to a few parts in $10^{15}$ by the MICROSCOPE satellite, which flew two test masses of different materials in identical free fall and found them to trace, within that stunning precision, the very same path.

The physical picture

Picture an ant walking on a curved surface under one rule: never turn the steering wheel — always step straight ahead. On a sphere the ant traces a great circle; on a saddle it traces something more exotic; but in every case its path feels perfectly straight from the ant's point of view even while it curves dramatically when seen from outside. That local sense of "straight ahead," carried faithfully forward step after step, is exactly parallel transport, and the curve it draws is a geodesic. The Christoffel symbols are the bookkeeping that records how "straight ahead" rotates relative to the coordinate grid as the grid itself bends beneath the ant's feet — which is why a term full of $\Gamma$ 's appears in the equation even though nothing is pushing the ant off its course.

Figure

A great circle is a geodesic on the sphere: its unit tangent is parallel-transported along the curve, ∇_u u = 0, so the path never turns — while a line of latitude must keep bending and is not a geodesic

The physics beneath the geometry is the equivalence principle. Step inside a freely falling laboratory — Einstein's famous elevator with its cable cut — and for a moment you feel no gravity at all; a released ball floats, and free particles drift in straight lines exactly as in special relativity. The geodesic equation is the global stitching-together of all these local straight lines. The connection term $\Gamma^\mu_{\;\alpha\beta}$ is precisely what survives when you patch the locally inertial frames together across a curved spacetime: at any one event you can rotate into a frame where it vanishes, but you cannot make it vanish everywhere at once, and that stubborn residue is gravity. The most spectacular demonstration is a photon: light has no rest frame and cannot be "pulled" by a Newtonian force in the ordinary sense, yet its null geodesic bends measurably near a mass — twice as much as a naive Newtonian corpuscle would — because it, too, is simply going straight through curved spacetime.

Figure

A star's light grazing the Sun follows a null geodesic bent by δ = 4GM/(c²b) = 1.75″ at the limb — twice the Newtonian value — so the star's apparent position is shifted outward, away from the Sun

The core equations

The whole subject is compressed into one covariant statement: a geodesic is a curve whose tangent is parallel-transported along itself. Parallel transport of any vector $V^\mu$ along the direction $u$ is the condition $u^\nu\nabla_\nu V^\mu = 0$ ; choosing $V^\mu = u^\mu$ , the tangent itself, says the direction of motion is carried forward unchanged — the curved-space meaning of "never turning."

\nabla_u u = 0 \qquad\Longleftrightarrow\qquad u^\nu \nabla_\nu u^\mu = 0 \qquad (1)

where: $u^\mu$ — tangent vector to the curve, $u^\mu = \mathrm{d}x^\mu/\mathrm{d}\lambda$ ; $\nabla_\nu$ — covariant derivative fixed by the affine connection; $\lambda$ — affine parameter. Physical reading: the covariant acceleration of a freely falling body is zero. There is no force; what looks like gravitational acceleration is entirely inside $\nabla$ . Expanding the covariant derivative in a coordinate basis converts the geometric statement into a concrete differential equation for the four coordinate functions of the worldline. This is the form you actually integrate on a computer to trace an orbit or a light ray.

\frac{\mathrm{d}^2 x^\mu}{\mathrm{d}\lambda^2} + \Gamma^\mu_{\;\alpha\beta}\frac{\mathrm{d}x^\alpha}{\mathrm{d}\lambda}\frac{\mathrm{d}x^\beta}{\mathrm{d}\lambda} = 0 \qquad (2)

where: $x^\mu(\lambda)$ — coordinate functions of the worldline; $\Gamma^\mu_{\;\alpha\beta}$ — Christoffel symbols (connection coefficients); $\lambda$ — affine parameter (proper time $\tau$ for a timelike curve). Physical reading: the first term is coordinate acceleration, the second is the "fictitious" gravitational term that would vanish in a truly inertial frame. For the Levi-Civita connection the connection coefficients are built entirely from the metric and its first derivatives. This is the formula that puts real gravity into equation (2): once you know $g_{\mu\nu}(x)$ , the geodesics are fixed.

\Gamma^\mu_{\;\alpha\beta} = \tfrac{1}{2}\, g^{\mu\nu}\!\left(\partial_\alpha g_{\nu\beta} + \partial_\beta g_{\nu\alpha} - \partial_\nu g_{\alpha\beta}\right) \qquad (3)

where: $g_{\mu\nu}$ — metric tensor; $g^{\mu\nu}$ — inverse metric, $g^{\mu\nu}g_{\nu\rho}=\delta^\mu_\rho$ ; $\partial_\alpha$ — coordinate derivative $\partial/\partial x^\alpha$ . Physical reading: gravity enters only through gradients of the metric, so a uniform metric (flat space in inertial coordinates) has no geodesic bending at all. The simple form of the equation is not reparametrization-invariant — it is a privilege of affine parameters. Choose a bad parameter $\sigma$ and a term proportional to the tangent reappears on the right, mimicking a velocity-dependent force that is pure parametrization artifact.

\frac{\mathrm{d}^2 x^\mu}{\mathrm{d}\sigma^2} + \Gamma^\mu_{\;\alpha\beta}\frac{\mathrm{d}x^\alpha}{\mathrm{d}\sigma}\frac{\mathrm{d}x^\beta}{\mathrm{d}\sigma} = f(\sigma)\,\frac{\mathrm{d}x^\mu}{\mathrm{d}\sigma} \qquad (4)

where: $\sigma$ — arbitrary (non-affine) parameter; $f(\sigma)$ — function fixed by $\lambda(\sigma)$ , vanishing iff $\lambda = a\sigma + b$ is affine. Physical reading: the affine class is $\lambda \to a\lambda + b$ ; proper time and proper length are the canonical affine choices for timelike and spacelike geodesics. Because the connection is metric-compatible, the norm of the tangent is conserved along an affinely parametrized geodesic. A geodesic therefore cannot change its causal character: timelike stays timelike, null stays null.

g_{\mu\nu}\,u^\mu u^\nu = \varepsilon = \begin{cases} -1 & \text{timelike} \\ \phantom{-}0 & \text{null} \\ +1 & \text{spacelike}\end{cases} \qquad (5)

where: $\varepsilon$ — constant causal character (signature $-\,+\,+\,+$ ); $u^\mu$ — affinely parametrized tangent. Physical reading: a massive particle ( $\varepsilon=-1$ ) can never be accelerated onto the light cone by gravity alone, and a photon ( $\varepsilon=0$ ) stays exactly null. Finally, every continuous symmetry of the spacetime hands you a conserved quantity along every geodesic. If $\xi^\mu$ is a Killing vector — a direction along which the metric does not change — then its projection onto the tangent is constant, turning second-order equations into first-order conservation laws.

\frac{\mathrm{d}}{\mathrm{d}\lambda}\!\left(\xi_\mu u^\mu\right) = 0 \quad\text{whenever}\quad \nabla_{(\mu}\xi_{\nu)} = 0 \qquad (6)

where: $\xi^\mu$ — Killing vector generating an isometry; $\nabla_{(\mu}\xi_{\nu)}=0$ — Killing's equation. Physical reading: a timelike Killing vector gives conserved energy, a rotational one gives conserved angular momentum — the two constants that reduce the Schwarzschild orbit problem to a single quadrature.

Derivation AFrom parallel transport to the coordinate geodesic equation

The goal is to turn the one-line geometric definition $u^\nu\nabla_\nu u^\mu=0$ into the explicit second-order ordinary differential equation (2) that a numerical integrator can actually solve. The only inputs are the coordinate expression of the covariant derivative acting on a vector — inherited from the covariant-derivative and Christoffel-symbol prerequisites — and the observation that a derivative along the curve is a total derivative in the parameter. Nothing about the metric is used yet, so the result holds for any affine connection, metric or not.

Step 1. Write the covariant derivative of the tangent in components, $\nabla_\nu u^\mu = \partial_\nu u^\mu + \Gamma^\mu_{\;\nu\beta}\,u^\beta$ . Why: this is the defining action of the connection on a contravariant vector; the partial derivative alone is not a tensor, and the $\Gamma$ term is precisely the correction that makes it one.

Step 2. Contract with the tangent to build the directional (covariant) derivative along the curve, $u^\nu\nabla_\nu u^\mu = u^\nu\partial_\nu u^\mu + \Gamma^\mu_{\;\nu\beta}\,u^\nu u^\beta$ . Why: "the tangent does not change along its own direction" is a statement about the rate of change in the $u$ direction, which is exactly this contraction.

Step 3. Convert the directional derivative of $u^\mu$ into a plain $\mathrm{d}/\mathrm{d}\lambda$ : since $u^\mu = \mathrm{d}x^\mu/\mathrm{d}\lambda$ and $u^\nu\partial_\nu = (\mathrm{d}x^\nu/\mathrm{d}\lambda)\,\partial_\nu = \mathrm{d}/\mathrm{d}\lambda$ , the first term becomes $\mathrm{d}^2 x^\mu/\mathrm{d}\lambda^2$ . Why: the chain rule collapses the "gradient contracted with velocity" into an ordinary second derivative of position, which is where the familiar acceleration term comes from.

Step 4. Set the whole thing to zero and rename the dummy index $\nu\to\alpha$ : $\ \mathrm{d}^2 x^\mu/\mathrm{d}\lambda^2 + \Gamma^\mu_{\;\alpha\beta}\,(\mathrm{d}x^\alpha/\mathrm{d}\lambda)(\mathrm{d}x^\beta/\mathrm{d}\lambda) = 0$ . Why: this is equation (2); note the Christoffel symbol is automatically contracted with the symmetric product $u^\alpha u^\beta$ , so only its part symmetric in the lower indices ever matters — torsion drops out of the geodesic equation. *As a quick check: the equation collapses to $\ddot x^\mu = 0$ (straight lines) the instant $\Gamma=0$ , as it must in flat spacetime with inertial coordinates.*

Go deeper · Derivation B · 1 minGeodesics extremize proper time for the Levi-Civita connection

Derivation BGeodesics extremize proper time for the Levi-Civita connection

The goal is to show that the autoparallel curves of the metric connection are the same curves a variational principle would pick — the ones that make proper time stationary — thereby proving that "straightest" and "extremal" name a single family of paths. We extremize the proper-time functional of a timelike worldline, use the trick of the squared Lagrangian to avoid a square root, and read off that the Euler-Lagrange equations rebuild exactly the Christoffel symbol of equation (3). This is the bridge between the connection-based and metric-based pictures of gravity.

Step 1. Replace the proper-time integrand $L=\sqrt{-g_{\mu\nu}\dot x^\mu\dot x^\nu}$ by the squared Lagrangian $\tilde L = \tfrac12\, g_{\mu\nu}\dot x^\mu\dot x^\nu$ . Why: with an affine parameter $\tilde L$ and $L$ share the same extremal curves, and $\tilde L$ is smooth and free of the square root, so the algebra stays clean.

Step 2. Impose the Euler-Lagrange equation $\dfrac{\mathrm{d}}{\mathrm{d}\lambda}\!\left(\dfrac{\partial\tilde L}{\partial\dot x^\sigma}\right) - \dfrac{\partial\tilde L}{\partial x^\sigma} = 0$ . Why: this is Hamilton's principle for the worldline — stationary proper time is a genuine variational condition, the same machinery that gives Lagrange's equations in mechanics.

Step 3. Compute the pieces: $\partial\tilde L/\partial\dot x^\sigma = g_{\sigma\nu}\dot x^\nu$ and $\partial\tilde L/\partial x^\sigma = \tfrac12\,\partial_\sigma g_{\mu\nu}\dot x^\mu\dot x^\nu$ . Why: the momentum is the velocity with a lowered index, and the "force" comes only from the coordinate dependence of the metric — gravity lives in $\partial g$ .

Step 4. Expand the total derivative with the chain rule and collect: $g_{\sigma\nu}\ddot x^\nu + \partial_\mu g_{\sigma\nu}\dot x^\mu\dot x^\nu - \tfrac12\partial_\sigma g_{\mu\nu}\dot x^\mu\dot x^\nu = 0$ . Why: differentiating $g_{\sigma\nu}(x(\lambda))$ along the curve produces the middle term, and symmetrizing it over $\mu\nu$ readies it for the Christoffel combination.

Step 5. Symmetrize $\partial_\mu g_{\sigma\nu}\dot x^\mu\dot x^\nu = \tfrac12(\partial_\alpha g_{\sigma\beta}+\partial_\beta g_{\sigma\alpha})\dot x^\alpha\dot x^\beta$ and raise the index with $g^{\mu\sigma}$ to get $\ddot x^\mu + \tfrac12 g^{\mu\sigma}(\partial_\alpha g_{\sigma\beta}+\partial_\beta g_{\sigma\alpha}-\partial_\sigma g_{\alpha\beta})\dot x^\alpha\dot x^\beta = 0$ . Why: the bracket is exactly the Levi-Civita Christoffel symbol of equation (3), so the extremal equation IS the geodesic equation — autoparallel equals extremal for the metric connection. *The bracket in Step 5 matches equation (3) term for term — the autoparallel and extremal definitions coincide precisely because the connection is the Levi-Civita one; for torsionful or non-metric connections the two definitions can split, but not here.*

Go deeper · Derivation C · 1 minThe Newtonian limit and the birth of Newton's law

Derivation CThe Newtonian limit and the birth of Newton's law

The goal is to prove that the geodesic equation contains Newtonian gravity as a limiting case and, as a bonus, to fix the crucial normalization $g_{00}=-(1+2\Phi)$ that links the metric to the familiar potential. We assume a weak, static field and slow motion — the everyday regime of falling apples and planetary orbits — keep only leading terms, and watch $\ddot{\vec x}=-\nabla\Phi$ fall out. This is the correspondence check every theory of gravity must pass.

Step 1. In slow motion the spatial velocities are tiny compared with $\mathrm{d}t/\mathrm{d}\tau$ , so in the sum $\Gamma^\mu_{\;\alpha\beta}\dot x^\alpha\dot x^\beta$ only the $\alpha=\beta=0$ term survives: $\ddot x^\mu + \Gamma^\mu_{\;00}(\mathrm{d}t/\mathrm{d}\tau)^2 \approx 0$ . Why: dropping the $v/c$ -suppressed terms is the whole content of "slow motion," and it isolates a single Christoffel component.

Step 2. For a weak static field $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $\partial_t h=0$ , equation (3) gives $\Gamma^i_{\;00} = -\tfrac12\,\partial_i h_{00}$ to first order in $h$ . Why: with time derivatives of the metric switched off, only the spatial gradient of $h_{00}$ enters, and it enters through a single clean derivative.

Step 3. The time component is trivial: $\Gamma^0_{\;00}\approx 0$ gives $\mathrm{d}^2 t/\mathrm{d}\tau^2\approx 0$ , so $\mathrm{d}t/\mathrm{d}\tau\approx\text{const}$ and coordinate time may replace proper time, $\mathrm{d}/\mathrm{d}\tau\to\mathrm{d}/\mathrm{d}t$ . Why: this lets us write the spatial equation directly in terms of the ordinary time $t$ that a Newtonian would use.

Step 4. The spatial equation becomes $\mathrm{d}^2 x^i/\mathrm{d}t^2 = \tfrac12\,\partial_i h_{00}$ ; identifying it with Newton's $\ddot x^i=-\partial_i\Phi$ forces $h_{00}=-2\Phi$ , i.e. $g_{00}=-(1+2\Phi)$ . Why: matching to the known Newtonian answer is what pins the otherwise-free normalization of the metric, and it is the same $g_{00}$ that later governs gravitational redshift and GPS clock rates. *At Earth's surface $\Phi/c^2 = GM_\oplus/(R_\oplus c^2)\approx 7\times10^{-10}\ll1$ , comfortably inside the weak-field regime, and $\partial_i\Phi$ reproduces $g\approx9.8\ \mathrm{m/s^2}$ .*

Limiting cases

The geodesic equation specializes cleanly, and each specialization recovers a piece of physics you already trust.

Case	Result	Meaning
Flat spacetime, Cartesian coordinates	$\mathrm{d}^2 x^\mu/\mathrm{d}\lambda^2 = 0$	Every $\Gamma^\mu_{\;\alpha\beta}=0$ , so geodesics are straight lines $x^\mu = a^\mu\lambda + b^\mu$ — Newton's first law, inertial motion.
Weak field, slow motion	$\mathrm{d}^2 x^i/\mathrm{d}t^2 \approx -\partial_i\Phi$	Only $\Gamma^i_{\;00}\approx\partial_i\Phi$ survives; Newtonian gravity re-emerges and fixes $g_{00}=-(1+2\Phi)$ .
Null geodesics (light)	$g_{\mu\nu}\dot x^\mu\dot x^\nu = 0$	Photons follow geodesics with a null tangent; proper time is identically zero, so an affine $\lambda$ is mandatory. Light bending and lensing follow.
Static spacetime, timelike Killing vector	$E = -\,\xi_\mu u^\mu = \text{const}$	Symmetry gives a conserved energy per unit mass; with a rotational Killing vector too, the orbit reduces to first-order quadratures.
Local inertial frame (single event)	$\Gamma^\mu_{\;\alpha\beta}(p)=0$	Riemann normal coordinates zero the connection at $p$ — free fall looks force-free — but $\partial\Gamma\neq0$ (curvature) remains as tidal effects.

Common trapHeavier objects follow "stronger" geodesics and fall faster

They do not, and the reason is written into equation (2): the mass of the test particle appears nowhere in it. The geodesic equation is a statement about the geometry of spacetime and the worldline's own tangent, with no coupling to how much stuff is being carried along. A bowling ball and a feather released together in vacuum trace the same geodesic and land together — Galileo's Leaning-Tower result, promoted to a theorem. This is the universality of free fall (the weak equivalence principle), and it is exactly what lets gravity be geometry rather than a force: a genuine force like electromagnetism produces an acceleration $F/m$ that depends on the body's mass and charge, so different bodies follow different paths, and you could never absorb it into the shape of spacetime. Gravity's mass-independence is the loophole. (Two riders: the body must be a test particle — light enough not to distort the metric itself — and it must feel no non-gravitational force. A charged particle radiating, or a rocket firing, leaves its geodesic.)

Worked examples

1 · Light bending at the Sun's limb (fully numeric)

A ray from a distant star just grazes the edge of the Sun. Using the null-geodesic deflection $\delta = 4GM/(c^2 b)$ with impact parameter $b = R_\odot$ , compute the deflection in arcseconds and compare with the Newtonian corpuscle prediction.

Step 1. Assemble the constants: $GM_\odot = 1.327\times10^{20}\ \mathrm{m^3/s^2}$ , $c^2 = 8.988\times10^{16}\ \mathrm{m^2/s^2}$ , and the grazing impact parameter $b = R_\odot = 6.96\times10^{8}\ \mathrm{m}$ . These are the only inputs the geodesic result needs — the deflection depends on the mass and the closest approach, nothing else.

Step 2. Form the characteristic length $4GM/c^2 = 4(1.327\times10^{20})/(8.988\times10^{16}) = \mathbf{5.91\times10^{3}\ m}$ — about $5.9$ km, twice the Sun's Schwarzschild radius. Divide by the impact parameter: $\delta = 5.91\times10^{3}/6.96\times10^{8} = 8.49\times10^{-6}\ \mathrm{rad}$ .

Step 3. Convert to arcseconds with $1\ \mathrm{rad} = 206265''$ : $\delta = 8.49\times10^{-6}\times206265 = \mathbf{1.75''}$ . The naive Newtonian corpuscle calculation gives exactly half, $0.87''$ , because it misses the equal contribution from the curvature of space (not just time). The general-relativistic null geodesic predicts 1.75″, and Eddington's 1919 eclipse measurement of ≈ 1.6–1.9″ confirmed it, not the Newtonian 0.87″.

2 · Mercury's perihelion precession (fully numeric)

The timelike geodesic in the Schwarzschild field precesses by $\Delta\varphi = 6\pi GM/[c^2 a(1-e^2)]$ per orbit. Evaluate it for Mercury ( $a = 5.79\times10^{10}\ \mathrm{m}$ , $e = 0.206$ ) and convert to arcseconds per century.

Step 1. Compute the relativistic correction factor per orbit. With $1-e^2 = 1-0.206^2 = 0.958$ , the denominator is $c^2 a(1-e^2) = (8.988\times10^{16})(5.79\times10^{10})(0.958) = 4.98\times10^{27}$ . The numerator is $6\pi GM = 6\pi(1.327\times10^{20}) = 2.50\times10^{21}$ . This ratio is the extra angle the perihelion advances each time Mercury rounds the Sun.

Step 2. Divide: $\Delta\varphi = 2.50\times10^{21}/4.98\times10^{27} = \mathbf{5.02\times10^{-7}\ rad/orbit}$ . It is tiny because $GM/(c^2 a)\sim10^{-8}$ — the field at Mercury's orbit is weak — but it accumulates.

Step 3. Mercury completes one orbit in $0.241$ yr, so it makes $100/0.241 = 415$ orbits per century. The total is $\Delta\varphi_{\text{cy}} = 5.02\times10^{-7}\times415 = 2.08\times10^{-4}\ \mathrm{rad}$ , and $2.08\times10^{-4}\times206265 = \mathbf{43''\ per\ century}$ . This 43″/century — the residual left after Newtonian planetary perturbations are subtracted — was the first quantitative triumph of general relativity, matching the long-standing observational anomaly to within the errors.

3 · The Newtonian-limit sanity check (numeric)

Verify that the normalization $g_{00}=-(1+2\Phi/c^2)$ from Derivation 3 is consistent with everyday gravity at Earth's surface, and that the weak-field assumption holds.

Step 1. The surface potential is $\Phi = GM_\oplus/R_\oplus = (6.67\times10^{-11})(5.97\times10^{24})/(6.37\times10^{6}) = 6.26\times10^{7}\ \mathrm{m^2/s^2}$ . This is the depth of Earth's gravity well per unit mass.

Step 2. The dimensionless field strength is $\Phi/c^2 = 6.26\times10^{7}/8.988\times10^{16} = \mathbf{6.96\times10^{-10}}$ , so $g_{00} = -(1 + 1.39\times10^{-9})$ — a departure from flat spacetime of about one part in a billion, safely $\ll1$ , confirming the weak-field expansion.

Step 3. The geodesic term reproduces surface gravity: $\Gamma^i_{\;00} = \partial_i\Phi/c^2$ , and multiplying back by $c^2$ gives $\partial_i\Phi \approx GM_\oplus/R_\oplus^2 = 9.8\ \mathrm{m/s^2}$ . The tiny metric perturbation $10^{-9}$ nonetheless integrates, through the geodesic equation, into the everyday $g = 9.8\ \mathrm{m/s^2}$ — and the same $g_{00}$ produces the gravitational time dilation that GPS must correct.

Physics of the portrait

The geodesic equation is the meeting point of two lineages: Riemann's geometry of curved manifolds and Einstein's physics of gravitation. But the specific piece that makes the equation mean something — the rule for carrying a direction forward so that "straightest" has content — was supplied by one mathematician a year after the field equations.

Origins · 2 minTullio Levi-Civita and how this was discovered

Tullio Levi-Civita

1873–1941 · Italian mathematician who founded tensor calculus and defined parallel transport

Tullio Levi-Civita, born in Padua in 1873, gave the geodesic equation the very structure it stands on. Working with his teacher Gregorio Ricci-Curbastro, he built the "absolute differential calculus" — what we now simply call tensor calculus — and laid it out in their 1900 memoir Méthodes de calcul différentiel absolu et leurs applications. When Einstein needed a mathematical language for a metric theory of gravity, this was the language he and Marcel Grossmann adopted; Einstein later wrote to Levi-Civita in open admiration and, more than once, to argue over signs. Levi-Civita's own decisive contribution came in 1917, in the paper on the nozione di parallelismo: he defined parallel transport on a general Riemannian manifold, giving a direct geometric meaning to the Christoffel symbols and the covariant derivative that had until then been mere formal machinery. Parallel transport is exactly the operation the geodesic equation asks a tangent vector to obey, and the unique torsion-free, metric-compatible connection whose autoparallels are the metric's geodesics now bears his name — the Levi-Civita connection. He held chairs at Padua and then Rome, and worked broadly, from celestial mechanics to hydrodynamics. In 1938 Mussolini's racial laws expelled him, as a Jew, from his professorship and from the academies he had adorned; increasingly isolated and in failing health, he died in Rome in 1941. The connection, the symbol, and the idea of "straightest possible" all still carry his name.

History

1868

— Christoffel introduces the three-index symbols Γ while studying transformations of quadratic differential forms

1900

— Ricci-Curbastro and Levi-Civita publish the absolute differential calculus, the tensor calculus of curved manifolds

1913

— Einstein and Grossmann adopt tensor calculus for the Entwurf theory, making a metric the potential for gravity

1915

— Einstein's field equations make freely falling bodies follow spacetime geodesics, with no gravitational force

1917

— Levi-Civita defines parallel transport, giving the connection and the geodesic equation their geometric meaning

1918

— Weyl generalizes the idea to a general affine connection, separating the connection from the metric

1919

— Eddington's eclipse expeditions confirm that starlight bends on null geodesics near the Sun

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §13.4 and §16.4 · Wald, General Relativity (1984), §3.3 · Carroll, Spacetime and Geometry (2004), §3.3 · Hartle, Gravity (2003), Ch. 8 · Schutz, A First Course in General Relativity, 2nd ed. (2009), §6.4. Key results used throughout: equation (2) follows from $u^\nu\nabla_\nu u^\mu=0$ ; the variational derivation rebuilds the Levi-Civita symbol of equation (3); the Newtonian limit fixes $g_{00}=-(1+2\Phi)$ . The anchor numbers are: Sun deflection $4GM/c^2b=1.75''$ , Mercury precession $6\pi GM/[c^2a(1-e^2)]\approx43''$ /century, GPS clock drift $\approx38\ \mu$ s/day, and the weak equivalence principle tested to $\sim10^{-15}$ by MICROSCOPE (2022). Signature $(-,+,+,+)$ throughout.

Builds on: Parallel Transport · The Affine Connection · Covariant Derivative · Christoffel Symbols. Leads to: Geodesics as Extremal Paths · Affine Parameters · Geodesic Deviation Equation · Killing Vectors · Tidal Forces in General Relativity.

Ask the tutor: ask me to trace a specific geodesic — a Schwarzschild orbit, a light ray past the Sun, or free fall at Earth's surface — and I will set up the metric, the Christoffel symbols, and integrate the equation with you step by step.

Is a geodesic the shortest path, like a straight line?

Not quite, and the distinction matters. A geodesic is an extremal path — one that makes the length or proper-time functional stationary — not necessarily the shortest. For a spacelike geodesic on a sphere the short great-circle arc is a minimum, but the long way around the same great circle is also a geodesic and is a maximum. For a timelike geodesic in spacetime the extremum is a maximum of proper time: the freely falling worldline between two events is the one that ages the most, which is the deep resolution of the twin paradox. So "straightest" (parallel-transporting its tangent) is the primary definition; "extremal" is an equivalent one for the Levi-Civita connection, and "shortest" is only sometimes true.

Why does the parametrization matter so much?

Because the tidy form $\ddot x^\mu + \Gamma^\mu_{\;\alpha\beta}\dot x^\alpha\dot x^\beta = 0$ is not invariant under relabelling the curve. If you switch from an affine parameter $\lambda$ to some arbitrary $\sigma$ , a term $f(\sigma)\,\mathrm{d}x^\mu/\mathrm{d}\sigma$ sprouts on the right-hand side — a phantom "force" pointing along the motion that is pure bookkeeping, not physics. It vanishes exactly when $\lambda$ and $\sigma$ are linearly related, which is what defines the affine class. The privileged affine parameters are proper time (for massive particles) and proper length (for spacelike curves), with an arbitrary rescaling and shift allowed. Choosing one keeps the tangent's norm constant and the equation in its simplest, force-free shape.

If Γ can be zeroed at a point, is gravity just a coordinate illusion?

Locally, yes; globally, no — and the difference is the whole theory. At any single event you can choose Riemann normal coordinates in which every $\Gamma^\mu_{\;\alpha\beta}=0$ , so free fall looks exactly like inertial motion in special relativity: this is the equivalence principle, and it means the value of the gravitational "force" is a coordinate artifact. What you cannot remove is the variation of $\Gamma$ from point to point. Those derivatives assemble into the Riemann curvature tensor, which is a genuine tensor: if it is nonzero in one frame it is nonzero in all of them. Curvature is what drives the relative acceleration of nearby geodesics — tidal forces — and no choice of coordinates can transform a real tide away. So a uniform field is fake, but a real, tidal, curved field is not.

Why can't light's worldline be parametrized by proper time?

Because along a light ray the proper time is identically zero. Proper time is defined by $\mathrm{d}\tau^2 = -g_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu$ , and a null curve is precisely one where $g_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu = 0$ everywhere, so $\mathrm{d}\tau = 0$ at every step — there is no clock that ticks along a photon's path. Trying to divide the geodesic equation by $\mathrm{d}\tau$ would be dividing by zero. Instead one uses an affine parameter $\lambda$ , chosen so that $\nabla_u u = 0$ keeps its simple form and the tangent stays null, $g_{\mu\nu}u^\mu u^\nu=0$ , all along the ray. Physically $\lambda$ can be taken proportional to the photon's energy-weighted path length; it plays for light the organizing role that proper time plays for matter.

How does one equation give both orbits and light bending?

It is the same equation (2) with two different values of the tangent's norm. Feed it a timelike initial tangent ( $g_{\mu\nu}u^\mu u^\nu=-1$ ) and integrate in the Schwarzschild metric, and you get planetary orbits — including the extra $1/r^3$ term that makes Mercury's ellipse precess. Feed it a null tangent ( $g_{\mu\nu}u^\mu u^\nu=0$ ) and you get light rays — including the $1.75''$ bend at the Sun's limb and, in strong fields, the photon rings the Event Horizon Telescope images. Nothing about the differential equation changes; only the causal character of the starting velocity does. That single unification — matter and light, orbits and lensing, all as geodesics of one metric — is a large part of why the geodesic equation is considered the heart of the theory.

Do the Christoffel symbols form a tensor?

No, and this is a common and important trap. Under a coordinate change $\Gamma^\mu_{\;\alpha\beta}$ picks up an inhomogeneous term — a second-derivative piece — on top of the usual tensor transformation, which is exactly why it can be made to vanish at a point (a tensor that is zero in one frame is zero in all). That non-tensorial character is a feature, not a bug: it is what lets the connection represent a "force" that different observers disagree about. What is tensorial is the antisymmetrized, differentiated combination that builds the Riemann curvature tensor, and the difference of two connections is also a genuine tensor. So $\Gamma$ is the right object for writing the geodesic equation, but the wrong object for making frame-independent statements about curvature.