Perihelion Precession from Schwarzschild Orbits

Derivation

Perihelion Precession from Schwarzschild Orbits

D-394 Home PU-403 Threads force · energy Depends on Schwarzschild Solution from Spherical Vacuum, Geodesic Equation from Extremal Proper Time

Statement

For a test particle on a timelike geodesic of the Schwarzschild exterior, the angle between successive perihelia exceeds \(2\pi\) by a small amount. Working to first order in the relativistic correction to the Newtonian orbit equation, the perihelion advances per orbit by \(\Delta\phi = \dfrac{6\pi G M}{c^2\,a(1-e^2)}\), where \(a\) is the semi-major axis and \(e\) the eccentricity of the underlying Keplerian ellipse.

Why it matters

This is the first quantitative triumph of general relativity: the residual \(43''\) per century in Mercury's perihelion motion — unexplained by Newtonian planetary perturbations — falls straight out of the Schwarzschild geometry with no free parameters. It converted a nagging anomaly of celestial mechanics into a confirmation of curved spacetime.

The same formula governs the periastron advance of binary pulsars and the \(\sim 12'\)-per-orbit precession of the star S2 around the Galactic-centre black hole, so the single result spans fourteen orders of magnitude in field strength. It also exposes the physical origin of the effect: the extra inverse-cube term in the effective potential, which breaks the special \(1/r\) degeneracy responsible for closed Keplerian ellipses.

Assumptions

Test particle on a geodesic.The orbiting body carries negligible mass and self-gravity; drop this and the source is no longer static Schwarzschild — you need the two-body problem with radiation reaction and post-Newtonian self-force corrections.

Exterior is Schwarzschild: static, vacuum, spherically symmetric.Drop spherical symmetry (a rotating source) and the metric is Kerr, which adds a distinct frame-dragging (Lense–Thirring) precession on top of the geodetic term derived here.

Equatorial motion, \(\theta=\pi/2\).By spherical symmetry any geodesic lies in a plane, which we align with the equator; keeping a general plane changes nothing but clutters the algebra with a constant \(\dot\theta=0\) reduction.

Weak-field, bound orbit: \(r \gg r_s = 2GM/c^2\).The precession is computed as a first-order perturbation in \(r_s/r\); near the source the series diverges and the orbit equation must be integrated exactly (elliptic integrals), with the closed-orbit picture failing entirely near the ISCO.

No other Newtonian perturbers.Solar oblateness and the other planets dominate the raw observed precession (\(\sim 5600''\)/century for Mercury); the Schwarzschild term is the \(43''\) residual after those are subtracted.

Derivation

\[ ds^2 = -\left(1-\frac{r_s}{r}\right)c^2\,dt^2 + \left(1-\frac{r_s}{r}\right)^{-1}dr^2 + r^2\,d\phi^2, \qquad r_s \equiv \frac{2GM}{c^2} \]

Schwarzschild line element restricted to the equatorial plane \(\theta=\pi/2\) (so \(d\theta=0\)); assumed as a prior result. A

\[ \tilde E \equiv c^2\left(1-\frac{r_s}{r}\right)\dot t, \qquad \tilde L \equiv r^2\dot\phi \]

Because \(t\) and \(\phi\) are cyclic (metric independent of them), the geodesic equation gives two constants of motion — the specific energy and specific angular momentum. Overdot is \(d/d\tau\). B

\[ g_{\mu\nu}\,u^\mu u^\nu = -c^2 \;\Longrightarrow\; -\left(1-\frac{r_s}{r}\right)c^2\dot t^2 + \left(1-\frac{r_s}{r}\right)^{-1}\dot r^2 + r^2\dot\phi^2 = -c^2 \]

Timelike normalization of the four-velocity: proper time parametrizes the worldline, so \(u^\mu u_\mu=-c^2\). B

\[ \dot r^2 = \frac{\tilde E^2}{c^2} - c^2 + \frac{2GM}{r} - \frac{\tilde L^2}{r^2} + \frac{2GM\,\tilde L^2}{c^2 r^3} \]

Substitute the constants of Step 2 for \(\dot t,\dot\phi\), then multiply through by \(\left(1-r_s/r\right)\) and use \(c^2 r_s = 2GM\). Pure algebra. B

\[ \tfrac{1}{2}\dot r^2 + V_{\rm eff}(r) = \frac{\tilde E^2-c^4}{2c^2}, \qquad V_{\rm eff}(r) = -\frac{GM}{r} + \frac{\tilde L^2}{2r^2} - \frac{GM\,\tilde L^2}{c^2 r^3} \]

Recast Step 4 as an energy equation. The first two terms are Newtonian gravity plus the centrifugal barrier; the \(1/r^3\) term is the pure-GR correction that will drive the precession. B

\[ u \equiv \frac{1}{r}, \quad \dot r = -\tilde L\,\frac{du}{d\phi} \;\Longrightarrow\; \left(\frac{du}{d\phi}\right)^2 = \frac{\tilde E^2/c^2 - c^2}{\tilde L^2} + \frac{2GM}{\tilde L^2}u - u^2 + \frac{2GM}{c^2}u^3 \]

Standard substitution: \(\dot r = \dfrac{dr}{d\phi}\dot\phi = \dfrac{dr}{d\phi}\dfrac{\tilde L}{r^2} = -\tilde L\,u'\). Divide Step 4 by \(\tilde L^2\). This is a first integral of the orbit. B

\[ \frac{d^2u}{d\phi^2} + u = \frac{GM}{\tilde L^2} + \frac{3GM}{c^2}\,u^2 \]

Differentiate Step 6 in \(\phi\) and divide by the common factor \(2u'\) (valid wherever \(u'\neq0\)). The Newtonian orbit equation is recovered without the final term. B

\[ u_0(\phi) = A\left(1 + e\cos\phi\right), \qquad A \equiv \frac{GM}{\tilde L^2} = \frac{1}{a(1-e^2)} \]

Zeroth-order solution: dropping the small \(3GM u^2/c^2\) term leaves the linear Kepler equation, whose bound solution is the ellipse of semi-latus rectum \(p=1/A=a(1-e^2)\). A

\[ u_1'' + u_1 = \frac{3GM}{c^2}u_0^2 = \frac{3GM A^2}{c^2}\Big[\underbrace{1+\tfrac{e^2}{2}}_{\text{constant}} + \underbrace{2e\cos\phi}_{\text{resonant}} + \underbrace{\tfrac{e^2}{2}\cos2\phi}_{\text{bounded}}\Big] \]

Write \(u=u_0+u_1\) and linearize, keeping the source at zeroth order. Expand \(\cos^2\phi=\tfrac12(1+\cos2\phi)\). Only the \(\cos\phi\) term is resonant with the operator \(d^2/d\phi^2+1\). C

\[ u_{1,\rm sec} = \frac{3GM A^2}{c^2}\,e\,\phi\sin\phi \;\Longrightarrow\; u \approx A\Big[1 + e\cos\big((1-\varepsilon)\phi\big)\Big], \quad \varepsilon = \frac{3GM A}{c^2} \]

The particular solution of \(u_1''+u_1 = C\cos\phi\) is \(\tfrac{C}{2}\phi\sin\phi\) (secular). Absorbing it, \(\cos\phi+\varepsilon\phi\sin\phi\approx\cos[(1-\varepsilon)\phi]\) to first order — a slow shift of the angular frequency. C

\[ \Delta\phi = \frac{2\pi}{1-\varepsilon} - 2\pi \approx 2\pi\varepsilon = \frac{6\pi G M A}{c^2} = \frac{6\pi GM}{c^2\,a(1-e^2)} \]

Perihelion (\(u\) maximal) recurs when the argument advances by \(2\pi\), i.e. after \(\phi=2\pi/(1-\varepsilon)\). Insert \(A=1/[a(1-e^2)]\). B

Result

\[ \boxed{\;\Delta\phi = \frac{6\pi GM}{c^2\,a(1-e^2)} = \frac{3\pi\,r_s}{a(1-e^2)}\;}\qquad\text{radians per orbit} \]

Reading. Each orbit the ellipse rotates prograde (in the direction of motion) by an angle set entirely by the ratio of the Schwarzschild radius \(r_s=2GM/c^2\) to the semi-latus rectum \(p=a(1-e^2)\). Tighter orbits (small \(p\)), more massive centres, and higher eccentricity all amplify the advance; the effect vanishes as \(c\to\infty\), recovering the closed Keplerian ellipse.

Units check. \(GM\) has units \(\mathrm{m^3\,s^{-2}}\), \(c^2\) has \(\mathrm{m^2\,s^{-2}}\), and \(a(1-e^2)\) has \(\mathrm{m}\). Hence \(\dfrac{GM}{c^2\,a(1-e^2)} = \dfrac{\mathrm{m^3\,s^{-2}}}{(\mathrm{m^2\,s^{-2}})(\mathrm{m})}\) is dimensionless — an angle in radians, as required.

Limiting cases

Newtonian limit \(c\to\infty\): \(\Delta\phi\to0\), the ellipse closes — the Bertrand-theorem statement that a pure \(1/r\) potential gives closed orbits.
Circular orbit \(e=0\): \(\Delta\phi = 6\pi GM/(c^2 a) = 6\pi\,(v_{\rm circ}/c)^2\), since \(v_{\rm circ}^2\simeq GM/a\) — a clean measure of orbital velocity in units of \(c\).
Weak field \(r_s/p\ll1\): the exact result reduces to the linear form \(\Delta\phi\approx 3\pi\,r_s/p\) used here.
High eccentricity \(e\to1\): \(p=a(1-e^2)\to0\) sharply enhances the advance, but the perturbation breaks if perihelion approaches \(r_s\).

Breaks when

Strong field, \(r\sim\) a few \(r_s\). The first-order series in \(r_s/r\) diverges; approaching the innermost stable circular orbit \(r_{\rm ISCO}=6GM/c^2\) the advance per orbit grows without bound and bound non-circular orbits cease to be well approximated by a precessing ellipse.
Rotating (Kerr) source. Frame-dragging adds a separate Lense–Thirring precession, with sign depending on prograde/retrograde motion, that Schwarzschild cannot reproduce.
Non-vacuum or oblate interior. A solar quadrupole moment \(J_2\) or any mass inside the orbit produces an additional Newtonian precession; the Schwarzschild formula assumes a pure vacuum exterior.
Comparable masses. For a binary the test-particle geodesic fails; one must use the total mass and higher post-Newtonian terms (and eventually radiation reaction).

Failure modes

Dropping the factor of 3 in the \(3GM u^2/c^2\) term — the precession then comes out three times too small.
Treating the Schwarzschild \(r\) as a proper radial distance; it is the areal radius, defined so that spheres have area \(4\pi r^2\).
Comparing the theoretical \(43''\) to the total observed precession (\(\sim 5600''\)/century) instead of the residual after planetary and oblateness perturbations are removed.
Confusing specific angular momentum \(\tilde L\) (per unit mass) with total angular momentum — a hidden mass factor that corrupts the units.
Getting the sign wrong: expanding \(2\pi/(1-\varepsilon)\) as \(2\pi(1-\varepsilon)\) gives a spurious regression instead of an advance.
Evaluating \(\cos[(1-\varepsilon)\phi]\) with \(\phi\) in degrees, or forgetting that \(\Delta\phi\) emerges in radians before conversion to arcseconds.
Using single-body \(M\) in a binary where the total mass \(M=m_1+m_2\) is required.

Discussion

The physical crux is the extra \(-GM\tilde L^2/(c^2 r^3)\) term in \(V_{\rm eff}\). Newtonian gravity's \(1/r\) potential is special: its radial oscillation frequency exactly equals its orbital frequency, so the radius returns to perihelion after precisely \(2\pi\) of azimuth and the ellipse closes. The relativistic inverse-cube term detunes these two frequencies slightly, and the accumulated mismatch is the perihelion advance. Precession is therefore not an exotic add-on but the generic behaviour of any potential that is not exactly \(1/r\) (or \(r^2\)).

Notice the effect is purely geometric — it survives for a chargeless, non-radiating point mass — and scales as \((v/c)^2\), the leading post-Newtonian order. This is why Mercury, the fastest and most eccentric inner planet with the smallest \(p\), shows the largest planetary signal, and why the same \(6\pi GM/c^2 p\) reappears in the Hulse–Taylor pulsar at \(4.2^\circ\) per year.

The perturbative treatment used here is a two-timing / Poincaré–Lindstedt calculation in disguise: the resonant \(\cos\phi\) forcing produces a secular term \(\phi\sin\phi\) that a naive expansion would report as a growing, unbounded amplitude. Recognizing it instead as the first-order Taylor expansion of a shifted frequency \((1-\varepsilon)\phi\) resums the secularity into a bounded, slowly precessing orbit. Exactly, the orbit equation integrates to Jacobi elliptic functions with a period \(4K(k)\) that reduces to \(2\pi(1+\varepsilon+\dots)\) in the weak-field limit, confirming the linear result and supplying its higher-order corrections.

Common misconceptions. The precession is not caused by time dilation slowing Mercury's clock, nor by the mass of Mercury, nor by an unseen planet (Vulcan). It is the curvature of the vacuum exterior alone; a massless would-be "planet" on the same geodesic would precess identically. Nor is it a coordinate artefact — the angle between perihelia is a gauge-invariant, directly observable quantity.

Worked examples

Example 1 — Mercury around the Sun.

\[ p = a(1-e^2), \qquad \Delta\phi_{\rm orbit} = \frac{6\pi GM_\odot}{c^2 p} \]

Assemble symbols before numbers: the per-orbit advance depends only on \(GM_\odot\) and the semi-latus rectum. A

\[ a = 5.791\times10^{10}\,\mathrm{m}, \; e=0.2056 \;\Rightarrow\; p = 5.791\times10^{10}(1-0.0423)=5.546\times10^{10}\,\mathrm{m} \]

Insert Mercury's orbital elements; \(1-e^2 = 0.9577\). A

\[ \Delta\phi_{\rm orbit} = \frac{6\pi(1.327\times10^{20})}{(8.988\times10^{16})(5.546\times10^{10})} = 5.02\times10^{-7}\,\mathrm{rad} = 0.1035'' \]

With \(GM_\odot=1.327\times10^{20}\,\mathrm{m^3 s^{-2}}\) and \(c^2=8.988\times10^{16}\,\mathrm{m^2 s^{-2}}\); convert via \(1\,\mathrm{rad}=206265''\). B

\[ N = \frac{100\,\mathrm{yr}}{0.2408\,\mathrm{yr}} = 415.2\ \text{orbits/century}, \quad \Delta\phi_{\rm cy} = 0.1035''\times415.2 \]

Multiply the per-orbit advance by the number of Mercury orbits per century (period \(0.2408\) yr). A

\[ \Delta\phi_{\rm cy} \approx 42.98''\ \text{per century} \]

Reading. Matches the observed GR residual of \(\approx 43''\)/century to within measurement error. Units check. Radians (dimensionless) × orbits → radians, converted to arcseconds.

Example 2 — The star S2 around Sgr A*.

\[ \Delta\phi_{\rm orbit} = \frac{6\pi GM_{\bullet}}{c^2\,a(1-e^2)}, \qquad M_\bullet = 4.30\times10^{6}\,M_\odot \]

Same formula with the Galactic-centre black-hole mass. A

\[ GM_\bullet = 4.30\times10^{6}\times1.327\times10^{20} = 5.71\times10^{26}\,\mathrm{m^3 s^{-2}} \]

Scale \(GM_\odot\) by the mass ratio. A

\[ a = 970\,\mathrm{AU}=1.451\times10^{14}\,\mathrm{m},\; e=0.884 \Rightarrow p = 1.451\times10^{14}(0.2185)=3.17\times10^{13}\,\mathrm{m} \]

S2 orbital elements; \(1-e^2=0.2185\). A

\[ \Delta\phi_{\rm orbit} = \frac{6\pi(5.71\times10^{26})}{(8.988\times10^{16})(3.17\times10^{13})} = 3.77\times10^{-3}\,\mathrm{rad} \]

Evaluate. Convert: \(3.77\times10^{-3}\,\mathrm{rad}\times(180/\pi)\times60 = 12.97'\). B

\[ \Delta\phi_{\rm orbit} \approx 13'\ \text{(arcminutes) per orbit} \]

Reading. Consistent with the GRAVITY-collaboration detection of Schwarzschild precession of S2 (\(\sim 12'\)/orbit) — the same physics as Mercury, but \(\sim10^4\) times stronger because \(r_s/p\) is far larger. Units check. Dimensionless radians, then converted to arcminutes.

Problems

(A) Earth's precession. Using \(a=1.496\times10^{11}\,\mathrm{m}\), \(e=0.0167\), find the Schwarzschild perihelion advance per orbit and per century.
Solution
\(p=a(1-e^2)=1.496\times10^{11}(0.99972)=1.4956\times10^{11}\,\mathrm{m}\). Then \(\Delta\phi_{\rm orbit}=6\pi GM_\odot/(c^2 p)=18.85\times1.327\times10^{20}/(8.988\times10^{16}\times1.4956\times10^{11})=1.86\times10^{-7}\,\mathrm{rad}=0.0384''\). Earth completes \(100\) orbits per century, so \(\Delta\phi_{\rm cy}\approx 3.84''\)/century — small because Earth is farther out and nearly circular.
(B) Two useful re-expressions. Show that \(\Delta\phi = 3\pi r_s/p\) and, for a circular orbit, \(\Delta\phi = 6\pi(v/c)^2\). Then verify Mercury numerically using \(r_{s,\odot}=2953\,\mathrm{m}\).
Solution
Since \(r_s=2GM/c^2\), \(\dfrac{6\pi GM}{c^2 p}=\dfrac{3\pi(2GM/c^2)}{p}=\dfrac{3\pi r_s}{p}\). For a circular orbit \(p=a\) and \(v^2=GM/a\), so \(\dfrac{6\pi GM}{c^2 a}=6\pi\dfrac{v^2}{c^2}\). Mercury: \(\Delta\phi=3\pi(2953)/(5.546\times10^{10})=5.02\times10^{-7}\,\mathrm{rad}\), matching Example 1.
(C) Hulse–Taylor binary pulsar. Total mass \(M=2.828\,M_\odot\), \(e=0.617\), orbital period \(P_b=27907\,\mathrm{s}\). Using \(M\to M_{\rm total}\), find the periastron advance per year.
Solution
Kepler: \(a^3=GM P_b^2/(4\pi^2)\), with \(GM=2.828\times1.327\times10^{20}=3.75\times10^{20}\). Then \(a^3=3.75\times10^{20}\times(27907)^2/39.48=7.40\times10^{27}\Rightarrow a=1.95\times10^{9}\,\mathrm{m}\). \(p=a(1-e^2)=1.95\times10^{9}(0.619)=1.21\times10^{9}\,\mathrm{m}\). \(\Delta\phi_{\rm orbit}=6\pi GM/(c^2 p)=18.85\times3.75\times10^{20}/(8.988\times10^{16}\times1.21\times10^{9})=6.5\times10^{-5}\,\mathrm{rad}=3.7\times10^{-3}\,\text{deg}\). Orbits per year \(=3.156\times10^{7}/27907=1131\), giving \(\approx 4.2^\circ\)/year — matching the observed \(4.226^\circ\)/yr.
(B) Velocity form. A star moves on a near-circular orbit at \(v=0.01c\). Find the precession per orbit in arcminutes.
Solution
\(\Delta\phi=6\pi(v/c)^2=6\pi(10^{-2})^2=1.885\times10^{-3}\,\mathrm{rad}\). Converting, \(1.885\times10^{-3}\times(180/\pi)=0.108^\circ=6.48'\) per orbit.
(C, t3) Breakdown radius. In the orbit equation, at what radius does the relativistic term reach \(10\%\) of the linear \(u\) term for a near-circular orbit? Express in units of \(r_s\), and relate to the ISCO.
Solution
The ratio of the GR term to the linear term is \(\dfrac{3GM u^2/c^2}{u}=\dfrac{3GM}{c^2 r}=\dfrac{3}{2}\dfrac{r_s}{r}\). Setting this to \(0.1\) gives \(r_s/r=0.0667\), i.e. \(r=15\,r_s\). Well inside this, at \(r_{\rm ISCO}=6GM/c^2=3\,r_s\), the ratio is \(0.5\) and the perturbative "precessing ellipse" picture collapses: stable bound non-circular orbits no longer exist and the per-orbit advance is order unity or larger.