The Perihelion Precession of Mercury

A Newtonian orbit under an inverse-square force is a closed ellipse: after one revolution in azimuth the radius returns to its starting value and the orbit retraces itself exactly. This closure is a special property of the $1/ r$ potential, not a generic feature of central forces, and general relativity breaks it. The Schwarzschild geometry adds a single term to the orbit equation, cubic in the inverse radius, and the ellipse no longer closes. Its long axis rotates slowly in the orbital plane, tracing a rosette. For Mercury the rotation is $4 3^{''}$ per century, the residual that nineteenth-century celestial mechanics could not explain and that Einstein computed to be an automatic consequence of the field equations.

The orbit equation used here comes from the Schwarzschild geodesics; this lesson takes it as the starting point and extracts the precession. Geometrized constants are restored so every number carries units. The signature is $(-, +, +, +)$ .

The orbit equation and its relativistic term

A massive particle on a bound geodesic in the equatorial plane of the Schwarzschild geometry has two conserved quantities per unit mass, from the time and azimuthal Killing vectors:

\tilde{E} = (1 - \frac{2 GM}{c ^{2} r}) c^{2} \frac{d t}{d τ}, \tilde{L} = r^{2} \frac{d ϕ}{d τ} .

The normalization $g_{μν} \overset{x}{˙}^{μ} \overset{x}{˙}^{ν} = - c^{2}$ turns into a radial energy equation, and substituting the inverse radius $u = 1/ r$ with $d r / d τ = - \tilde{L} d u / d ϕ$ converts it into an equation for the shape $u (ϕ)$ of the orbit. Differentiating once in $ϕ$ removes the constant term and leaves

\frac{d ^{2} u}{d ϕ ^{2}} + u = \frac{GM}{L ~ ^{2}} + \frac{3 GM}{c ^{2}} u^{2} .

Every piece except the last reproduces the Newtonian orbit. Dropping the final term gives $u^{''} + u = GM / \tilde{L}^{2}$ , a harmonic oscillator in $ϕ$ driven by a constant, whose solution is the conic

u_{0} (ϕ) = \frac{GM}{L ~ ^{2}} (1 + e cos ϕ), r (ϕ) = \frac{L ~ ^{2} / GM}{1 + e cos ϕ} .

The perihelion (minimum $r$ , maximum $u$ ) sits at $ϕ = 0$ , and because the solution has period $2 π$ in $ϕ$ , the next perihelion is at $ϕ = 2 π$ : the ellipse closes.

The relativistic term $3 GM u^{2} / c^{2}$ is small for a planetary orbit. Its size relative to the Newtonian driving term is

\frac{3 GM u ^{2} / c ^{2}}{GM / L ~ ^{2}} = \frac{3 L ~ ^{2} u ^{2}}{c ^{2}} \approx \frac{3 v _{⊥}^{2}}{c ^{2}},

of order $(v / c)^{2}$ . For Mercury this ratio is about $3 \times 1 0^{- 8}$ , so the term cannot change the orbit's size or shape appreciably over one revolution. What it does instead is accumulate: a tiny frequency shift each orbit, secular in the azimuth.

A Newtonian orbit closes after one revolution; the extra cubic term shifts the angular frequency slightly below one, so the perihelion arrives a little past 2 pi and the ellipse advances into a slowly rotating rosette.

Perturbation and the advance per orbit

Treat the cubic term as a perturbation of the Newtonian conic. Insert $u_{0}$ into the right-hand side and keep only the part that drives the oscillator on resonance. With $u_{0} = A (1 + e cos ϕ)$ and $A = GM / \tilde{L}^{2}$ ,

\frac{3 GM}{c ^{2}} u_{0}^{2} = \frac{3 GM A ^{2}}{c ^{2}} (1 + 2 e cos ϕ + e^{2} cos^{2} ϕ) .

The constant and the $cos^{2} ϕ$ pieces only shift and slightly distort the orbit; the $2 e cos ϕ$ term is resonant with the homogeneous solution and produces a secular response. Keeping it,

\frac{d ^{2} u}{d ϕ ^{2}} + u = A + \frac{6 GM A ^{2} e}{c ^{2}} cos ϕ .

A driving term $\propto cos ϕ$ on an oscillator of natural frequency one grows without bound as $ϕ cos ϕ$ , the signature of resonance. That secular term is exactly the first-order expansion of a small frequency shift: to the same order,

u (ϕ) = A [1 + e cos ((1 - k) ϕ)], k = \frac{3 G ^{2} M ^{2}}{c ^{2} L ~ ^{2}},

because expanding $cos ((1 - k) ϕ) \approx cos ϕ + k ϕ sin ϕ$ reproduces the $ϕ cos ϕ$ -type secular growth with the correct coefficient. The perihelion now recurs when the argument advances by $2 π$ , i.e. at

ϕ_{peri} = \frac{2 π}{1 - k} \approx 2 π (1 + k) .

The perihelion has moved forward by $Δ ϕ = 2 π k$ per revolution:

Δ ϕ = \frac{6 π G ^{2} M ^{2}}{c ^{2} L ~ ^{2}} = \frac{6 π GM}{c ^{2} a ( 1 - e ^{2} )},

using $\tilde{L}^{2} = GM a (1 - e^{2})$ . The advance is per orbit, in the direction of motion, and depends only on the mass of the central body and the orbit's size and shape.

The Newtonian ellipse (dashed) returns its perihelion to the same direction each orbit; the Schwarzschild orbit (solid) carries the perihelion forward by a fixed angle every revolution.

Mercury's residual

Mercury has semi-major axis $a = 5.79 \times 1 0^{10} m$ and eccentricity $e = 0.206$ , so $a (1 - e^{2}) = 5.55 \times 1 0^{10} m$ . The Sun contributes $G M_{⊙} / c^{2} = 1.477 \times 1 0^{3} m$ , the gravitational radius of the Sun. Then the advance per orbit is

Δ ϕ = \frac{6 π ( 1.477 \times 1 0 ^{3} )}{5.55 \times 1 0 ^{10}} = 5.02 \times 1 0^{- 7} rad .

Mercury completes $415.2$ orbits per Julian century (orbital period $87.97$ days). The precession per century is

415.2 \times 5.02 \times 1 0^{- 7} rad = 2.08 \times 1 0^{- 4} rad = 43. 0^{''},

converting with $1 rad = 20626 5^{''}$ . The measured value, once the two larger classical effects are removed, is $(43.11 \pm 0.45)^{''}$ per century.

Worked example. The observed precession of Mercury's perihelion relative to the fixed stars is about $560 0^{''}$ per century. Two Newtonian effects dominate it:

Precession of the equinoxes — the slow rotation of the Earth-based coordinate frame in which the observation is recorded, about $502 5^{''}$ per century. This is a property of the measuring frame, not of Mercury.
Perturbations by the other planets — chiefly Venus, Jupiter, and Earth, totaling about $53 2^{''}$ per century by Newtonian celestial mechanics.

Subtracting both leaves a residual near $4 3^{''}$ per century that no Newtonian source accounts for. Le Verrier isolated this discrepancy in 1859 and attributed it to an unseen inner planet, Vulcan, never found. General relativity supplies the missing $4 3^{''}$ with no free parameters: it is $6 π G M_{⊙} / (c^{2} a (1 - e^{2}))$ summed over a century.

The measured perihelion motion decomposes into a large frame-precession term, a planetary-perturbation term, and a small residual; the residual matches the general-relativistic prediction. Bar lengths use a compressed scale so the 43-arcsecond residual stays visible beside the 5000-arcsecond terms.

The residual perihelion advance and the general-relativistic prediction, both near 43 arcseconds per century, agree within the measurement uncertainty shown as an error bar.

Frame dragging

The perihelion advance is a static effect: it uses only the mass $M$ in the Schwarzschild metric and needs no rotation of the source. A rotating mass produces a second, smaller relativistic effect, the dragging of inertial frames (Lense–Thirring), which the Kerr geometry describes exactly. A gyroscope orbiting a spinning body precesses even with no torque on it, because the local inertial frames themselves are dragged around in the direction of the spin.

Two experiments have measured this in the weak field of the Earth.

Gravity Probe B carried four gyroscopes in a $642 km$ polar orbit and measured two precessions: the geodetic (de Sitter) precession from the Earth's mass, $6. 6^{''}$ per year, and the much smaller frame-dragging precession from the Earth's rotation, $0.03 9^{''}$ per year. Both matched general relativity to their measured precision.
LAGEOS and LAGEOS 2, dense laser-ranged satellites, register the Lense–Thirring dragging of their orbital planes at the level of tens of milliarcseconds per year, again consistent with the prediction.

The perihelion precession, the geodetic precession, and frame dragging probe successively finer features of the metric: the static mass term, the same term's effect on a transported spin vector, and the off-diagonal $g_{tϕ}$ term that only a rotating source produces.¹²

Hartle, Gravity: An Introduction to Einstein's General Relativity, §10.2 (perihelion precession) and §14.5 (gyroscopes in curved spacetime, frame dragging). ↩
Carroll, Lecture Notes on General Relativity, §7 (orbits in the Schwarzschild geometry and the precession of the perihelion), arXiv:gr-qc/9712019. ↩