Geodesic Equation

Definition

The geodesic equation is the equation of motion in general relativity. It says that an object in free fall — feeling no forces other than gravity — traces the straightest possible worldline through curved spacetime:

d²xᵘ/dτ² + Γᵘ_αβ (dxᵃ/dτ)(dxᵇ/dτ) = 0

Here xᵘ(τ) is the particle's worldline (its path through the four dimensions of spacetime), τ is proper time, and Γᵘ_αβ are the Christoffel symbols. The repeated indices α and β are summed over (Einstein summation), so the middle term is a sum of products of velocity components.

Read the equation as a sentence: the coordinate acceleration of a free-falling body is whatever the geometry forces it to be. The first term, d²xᵘ/dτ², is the apparent acceleration. The second term is the geometric correction. There is no force on the right-hand side — it is zero. Gravity is geometry, not a force.

How it works

Imagine a heavy ball resting on a stretched rubber sheet. It dimples the sheet downward. Now roll a marble across the sheet: even though you launched it in a straight line, it curves toward the dimple. The marble is not being pulled by a string — it is simply following the contours of the surface, the straightest path it can take given the curvature beneath it. Spacetime is that sheet, mass is the ball, and the marble's curved track is the geodesic.

The mathematics that turns "straightest possible path" into a precise rule is parallel transport. On a flat plane, "keep going straight" means "keep your velocity vector pointing the same way." On a curved surface there is no global notion of "the same way" — the coordinate axes themselves twist from point to point. The Christoffel symbols are the bookkeeping that tells you, when you move an infinitesimal step, how much the axes rotated underneath your velocity vector. A geodesic is a worldline that parallel-transports its own tangent vector: it never turns relative to the local geometry, even though it looks bent from far away.

The Christoffel symbols are computed entirely from the metric tensor g_μν — the object that stores all the distances and angles of the spacetime:

Γᵘ_αβ = ½ gᵘᵛ (∂_α g_βᵛ + ∂_β g_αᵛ − ∂_ᵛ g_αβ)

Two facts about Γ are worth burning in. First, it is built only from first derivatives of the metric, so it captures how the geometry tilts but not yet how it bends back on itself (that is curvature, the Riemann tensor, which uses derivatives of Γ). Second, the Christoffel symbols are not a tensor. You can always choose a local frame — the free-falling frame — in which they vanish at a point. That is the whole content of the equivalence principle in equation form: a falling observer can make gravity disappear locally, because Γ = 0 in their own reference frame.

The flat-spacetime check

The fastest sanity check on any equation of motion is: what happens with no gravity? Switch off curvature — choose flat Minkowski spacetime in Cartesian coordinates — and every metric derivative is zero, so every Christoffel symbol is zero. The geodesic equation collapses to:

d²xᵘ/dτ² = 0

That is Newton's first law: constant velocity in a straight line. The geodesic equation is therefore a strict generalization — it is "an object in motion stays in motion," upgraded so that "straight line" means "straightest line the geometry permits." Newton's first law was never wrong; it was just written in coordinates that happened to be flat.

Worked example — bending starlight at the Sun

This is the calculation that made Einstein famous. A ray of light from a distant star skims the edge of the Sun on its way to Earth. In the weak field outside the Sun the relevant deflection comes from integrating the geodesic equation for a null (lightlike) path through the Schwarzschild geometry. The result is beautifully compact:

δ = 4GM / (c²·b)

where G = 6.674×10⁻¹¹ m³/(kg·s²), M is the Sun's mass 1.989×10³⁰ kg, c = 2.998×10⁸ m/s, and b is the impact parameter — for a grazing ray, the Sun's radius, 6.96×10⁸ m. Plugging in:

δ = 4 × (6.674e-11) × (1.989e30) / [ (2.998e8)² × (6.96e8) ]
  ≈ 8.49 × 10⁻⁶ radians
  ≈ 1.75 arcseconds

The punchline lives in the factor of 4. A naive Newtonian "photon as a tiny mass" calculation gives half as much — 0.875 arcseconds — because it only accounts for the curvature of time. General relativity's geodesic, threading curved space as well, doubles the deflection to 1.75 arcseconds. When Arthur Eddington's 1919 eclipse expeditions measured a deflection near the full GR value and not the Newtonian half, it settled the matter. The bending of light is direct evidence that geodesics, not forces, govern motion.

Variants and regimes

Regime	Geodesic type	Parameter	What you see
Weak field, slow motion	Timelike	Proper time τ ≈ t	Newtonian gravity, g = −∇Φ
Massive body, strong field	Timelike	Proper time τ	Orbital precession, e.g. Mercury 43″/century
Light / photons	Null (ds² = 0)	Affine λ (τ = 0)	Light bending, 1.75″ at the Sun; lensing
Free fall toward a black hole	Timelike, plunging	Proper time τ	Finite τ to cross horizon; infinite coordinate t
Spinning mass (Kerr)	Timelike / null	τ or λ	Frame dragging — geodesics get swept sideways
Cosmological expansion (FLRW)	Timelike comoving	Proper time τ	Galaxies coast apart; cosmological redshift

The single geodesic equation covers every row. What changes is the metric you feed it — Schwarzschild for a static star, Kerr for a rotating one, Friedmann–Lemaître–Robertson–Walker (FLRW) for the expanding universe — and whether the worldline is timelike (massive) or null (light).

From geodesics back to Newton

To see how the geodesic equation contains Newtonian gravity, take the weak-field limit. Write the metric as flat space plus a tiny perturbation, g_μν = η_μν + h_μν, with the time–time piece set by the Newtonian potential, g_tt ≈ −(1 + 2Φ/c²). For a slow particle the four-velocity is dominated by its time component, and only one Christoffel symbol survives:

Γⁱ_tt ≈ (1/c²) ∂ᵢΦ

The spatial part of the geodesic equation then reads d²xⁱ/dt² + c² Γⁱ_tt = 0, which simplifies to:

d²xⁱ/dt² = −∂ᵢΦ   →   g = −∇Φ

Exactly Newton's law of gravity. The "force" of gravity was always just the Γⁱ_tt term in disguise — the curvature of time, which a slow-moving object spends almost all its worldline traversing. Speed the object up toward c, let it sweep through curved space too, and the extra terms switch on; that is precisely why light gets twice the deflection.

Common pitfalls and misconceptions

• Thinking the rubber-sheet analogy is the real physics. The trampoline is a 2D spatial cartoon that smuggles in ordinary downward gravity to do its work. The actual curvature is in four-dimensional spacetime, and — crucially — it is the curvature of the time direction that produces everyday gravity. A ball "falls" mostly because its worldline is being bent in the time direction, not the space directions.
• Believing the Christoffel symbols are physical. They are coordinate artifacts. They can be nonzero in flat space (just use polar coordinates) and zero in curved space at any chosen point (use the local free-fall frame). Real, coordinate-independent curvature lives in the Riemann tensor, built from derivatives of Γ.
• Using coordinate time for everything. The geodesic equation for a massive particle is parametrized by proper time τ, the clock the particle itself carries. Mixing up τ and the distant observer's t is the most common source of paradoxes near black holes, where they diverge wildly.
• Asking what proper time a photon experiences. Zero. A null geodesic has ds² = 0, so you cannot use τ; you must parametrize by an affine parameter λ. Plugging dτ into the photon's geodesic equation gives a 0/0 mess.
• Confusing "straightest" with "shortest." Timelike geodesics actually maximize proper time (the longest-aged path between two events), not minimize length — that is the resolution of the twin paradox. "Straightest," meaning the path that parallel-transports its own tangent, is the safe phrasing.
• Assuming geodesics are closed orbits. In curved spacetime a bound orbit generally does not close — it precesses. Mercury's perihelion advances 43 arcseconds per century purely from the non-closing geodesic, a discrepancy Newtonian gravity could never account for.

Applications

• GPS and satellite navigation. Orbiting clocks tick faster than ground clocks by ~38 microseconds per day — a geodesic effect (gravitational time dilation plus velocity). Ignore it and GPS positions drift by ~10 km/day.
• Gravitational lensing. Light from distant galaxies follows null geodesics around foreground mass, producing Einstein rings, multiple images, and the magnification astronomers use to weigh dark matter.
• Black-hole orbits and accretion. The innermost stable circular orbit, photon sphere, and plunge trajectories are all just specific geodesics of the Schwarzschild or Kerr metric.
• Spacecraft trajectory design. Interplanetary "coasting" between burns is free-fall — a geodesic of the Sun's field. Relativistic corrections matter for missions like the BepiColombo flybys and for pulsar-timing navigation.
• Cosmology. Galaxies recede along comoving geodesics of the expanding FLRW universe; the redshift of their light is a geodesic effect, not a Doppler shift through space.
• Precision tests of relativity. Binary-pulsar timing (e.g. Hulse–Taylor) tracks geodesics whose decay matches gravitational-wave emission to better than 0.1%.

Derivation and validation analysis

There are two routes to the geodesic equation, and it is reassuring that they agree.

Route 1 — extremal aging. Demand that the worldline make the proper time τ = ∫√(−g_μν dxᵘ dxᵛ)/c stationary. Applying the Euler–Lagrange equations to that action drops out the geodesic equation directly, with the Christoffel symbols appearing automatically from the metric derivatives. This is the "longest-aged path" picture.

Route 2 — parallel transport. Demand that the four-velocity be parallel-transported along itself: uᵛ ∇_ᵛ uᵘ = 0. Expanding the covariant derivative ∇ in terms of partial derivatives plus Christoffel symbols reproduces the same equation. This is the "never turns relative to the local geometry" picture.

How well does it hold up? The empirical scorecard is one of the strongest in physics:

Test	Geodesic prediction	Measured	Agreement
Light deflection at Sun's limb	1.75″	1.75″ (Eddington 1919; VLBI to 0.02%)	< 0.1%
Mercury perihelion advance	43.0″/century	43.1″/century	< 0.5%
Shapiro time delay (radar)	~200 µs past Sun	matches	~0.001% (Cassini)
GPS clock rate	+38 µs/day	+38 µs/day	routine engineering
Binary-pulsar orbital decay	geodesic + GW emission	Hulse–Taylor PSR B1913+16	< 0.2%
Gravitational redshift	Δf/f = Φ/c²	Pound–Rebka, GP-A	~0.007% (GP-A)

Every entry is the geodesic equation fed a different metric. The factor-of-2 in light bending, the leftover 43 arcseconds of Mercury, and the microsecond drift of orbiting clocks are not separate theories patched together — they are one equation, d²xᵘ/dτ² + Γᵘ_αβ (dxᵃ/dτ)(dxᵇ/dτ) = 0, evaluated in different places.