How curved spacetime
changes Mercury’s orbit
Mercury’s orbit slowly turns. Relativity explains an extra 43 arcseconds per century beyond Newton’s prediction. First, build the picture: what curves, what moves, what takes longer, and what Earth’s viewpoint changes. The mathematics comes last.
First, what makes an orbit?
A planet has sideways velocity while gravity continually changes that velocity toward the Sun. It is always falling, but its sideways motion keeps it from simply falling straight in. In Newton’s model, gravity is an attractive force proportional to the inverse square of distance.
The Sun sits at a focus of the ellipse. The closest point is perihelion; the farthest is aphelion. A radial cycle means one complete near → far → near journey.
The velocity points along the orbit. Gravity points toward the Sun. Their directions are different.
- Distance from the Sun
- million km
- Orbital speed
- km/s
- Elapsed time
- days
Drag time toward the farthest point: Mercury moves more slowly there. The ellipse’s orientation stays fixed. Velocity and gravity arrows use separate length scales.
Gravity is central, so it exerts no torque about the Sun. Angular momentum stays constant: Mercury moves quickly near perihelion and slowly near aphelion. The line from the Sun to Mercury sweeps out equal areas in equal times. There is no outward force “balancing gravity” on the moving planet in this inertial description.
For two isolated point masses under Newton’s inverse-square law, one radial cycle takes exactly one full turn. The relative orbit closes. This precise match is what relativity slightly changes.
Newton already predicts precession in the real Solar System. The other planets disturb the two-body ellipse. Their effects account for about 532″ per century. Relativity supplies roughly 43″ more. All the orbit experiments here omit other planets to isolate that extra contribution.
How spacetime curvature changes the orbit
A freely falling planet keeps moving naturally in its local frame. The geometry changes how those local motions join into a whole orbit.
Imagine a tiny laboratory falling alongside Mercury. A loose object floats inside it, and an accelerometer on the freely falling laboratory reads zero. Zoom out, however, and different falling laboratories can drift toward or away from one another. Those tidal differences reveal curvature: one gravity-free coordinate frame cannot cover the whole region.
Spacetime geometry tells us how clocks and rulers at different places relate. Together with the law of free fall, it determines how a planet’s direction, distance, and timing change. Most of the result agrees with Newtonian gravity. The small correction changes the orbit’s pattern of repetition.
The key mismatch: one turn versus one near–far–near cycle
A Newtonian two-body ellipse reaches its next closest point after exactly one full turn. In GR, the planet reaches the original direction while it is still moving inward. It must travel a little farther around before it reaches minimum distance. That is the extra perihelion advance.
Try Newton, then exaggerated GR. The same model drives every panel below. Then compare B: one full turn with C: the next closest point.
Relativistic effects are exaggerated using a hypothetical compact source, about 706,000 times Mercury’s compactness. Closest and farthest radii stay fixed. All orbital angles are calculated.
A spatial slice only. Its artificial height lets a flat drawing represent curved ruler geometry.
The time part matters too: clocks at different radii do not agree on elapsed time.
3 · Together, these spacetime measurements determine free fall. The orbit below is calculated from the full spacetime geometry. No planet is rolling on the surface above.
At the next minimum distance, the radial cycle has completed. The direction has already gone beyond 360°.
- Extra angle per radial cycle
- Distance above the minimum
- % of orbit size
Blue dashed: the isolated Newtonian orbit. Rust: the relativistic orbit with the same closest and farthest distances. Those matched distances require slightly different initial velocities. Both paths stay in the orbital plane.
What to take from the pictures: curved ruler geometry and position-dependent clock rates change the free-fall rule. The radial cycle then takes more angular travel to finish. The next closest point lands farther around the Sun. The equation proving that connection is in the final section.
The surface is a picture of spatial distances, using an artificial height to make them visible. It is not a physical funnel that Mercury rolls down. A spatial surface by itself cannot tell us the motion of a planet; the time part of spacetime matters too.
This is not an inward spiral or a repeated energy boost. In the ideal model, the nearest and farthest distances repeat and energy is conserved. Only the direction of the closest point advances. Simply making Newton’s Sun heavier would still leave an isolated bound ellipse closed.
Build the intuition further with two traveling clocks
A clock measures the elapsed time experienced by its carrier: proper time. Start two clocks together at a station. Leave one there; send the other out and back. Compare them side by side when they reunite.
Near a mass, the traveler can coast outward and fall back like a ball thrown straight up. The station needs thrust to stay put. Without the mass, the station can remain at rest freely, while the round-trip traveler needs thrust to follow that route.
Reunion: both clocks are back at the station.
Station clock
s
Engines hold the station in place against gravity.
Traveler’s clock
s
After launch, it coasts freely outward and falls back. No thrust during the trip.
At reunion, the traveler’s clock has recorded 0.053038 seconds more. The clocks are side by side again, so they can be compared directly.
Try Without the mass: the same round trip now needs thrust, and the clock that stays put records more time. This is a hypothetical strong-field example, not a trip near Mercury. The seconds are calculated for the example’s chosen mass; positions are a coordinate map. No light-travel delay is shown.
How does this trip become a spacetime diagram?
Read from bottom to top. Every row is a later moment. Left and right show position.
The dashed station line is vertical: time passes while its position stays the same.
The traveler’s line bows right, then returns left: it goes out, then comes back.
That bow is not a bend in the spatial route. The actual trip follows one straight radial line, as the replay above shows.
The two lines join at the start and at the reunion. Those are the “same endpoints”: the same place and the same time.
The vertical coordinate is a shared reference time, normalized to a stationary clock far from the mass. It is different from the time each traveling clock records. The horizontal dotted line follows the replay.
Why this free-fall trip rather than a slightly different one?
Change how far the traveler goes while keeping the departure and reunion events fixed. The result below always compares the complete trips.
This graph is not a picture of motion. Each point stands for a different, complete round trip.
Left–right changes the trip’s size. Up–down shows how much time the traveler’s clock has recorded at reunion. The dashed horizontal line is the station clock’s final reading.
Near the mass, the freely coasting reference trip (1) gives the most clock time. Go a little less far or a little farther, and the clock records less. These altered trips require thrust.
The maximum is calculated from the clock-rate rule, rather than built into the graph. This tests one family of nearby paths; the geodesic principle applies to arbitrary small path changes.
Near the mass, spending time farther out lets the traveler’s clock accumulate more time; motion works in the opposite direction. For the freely coasting trip shown, the first effect wins. Without the mass, the stationary clock records more. Each clock still ticks normally for the person beside it.
GR’s general rule is stronger: for a sufficiently short trip between fixed events, free fall locally gives the most elapsed proper time. Small changes to that path reduce it. Changing the spacetime measurement rule changes which path has this property. The clock is revealing the free-fall path, not steering the traveler toward a future destination.
See the orbit as a history through time
A worldline records where the planet was and when. Here time is drawn upward. The planet’s spatial motion stays in one orbital plane.
Collapse the time axis: the worldline projects back onto the spatial path. Switch to Newton: it has a worldline too, even though its spatial ellipse closes.
GR’s strength is exaggerated as in the preceding orbit experiment. Time height is normalized for readability. Spatial positions are drawn as a coordinate map; the planet stays in its orbital plane. Dots mark perihelion events.
A Newtonian ellipse also winds upward when time is drawn this way. Adding a time axis does not cause precession. GR changes the spacetime measurement rule and the resulting motion; it does not add a new spatial direction for Mercury to move through.
Does Mercury speed up, or take longer?
There is no rule that it must speed up enough to keep the old orbital period.
Your intuition about the extra segment is right: after passing the original direction, Mercury needs additional time to reach its next closest point. But “finish an orbit” now has more than one meaning.
- One full turn: return to the starting direction around the Sun. In this experiment, start at perihelion.
- One radial cycle: go from closest, to farthest, to closest again. Astronomers call its duration the anomalistic period.
These finish together in Newton’s isolated ellipse. In GR they do not. Speed also changes, and is already higher near the Sun than far away in both models. The orbit’s shape alone does not tell us its elapsed time; we must calculate the motion using a specified clock.
Relativity is exaggerated so the finish times and positions separate visibly. The two models are advanced by the same reference clock.
The GR planet has reached its next closest point. The Newtonian planet has already completed its ellipse and started another lap.
- Time from GR’s full turn to its next closest point
- GR’s whole near–far–near cycle
- GR’s locally measured speed at launch
This comparison holds the Sun’s gravity parameter and the two turning radii fixed, which requires slightly different initial velocities. Durations use a shared Sun-centered reference time. Launch speeds use a local observer held at the closest radius. They are different, explicitly chosen measurements. These are model comparisons, not an extra observed timing anomaly.
A slightly higher speed can coexist with a longer cycle
In this controlled comparison, GR’s locally measured launch speed is slightly higher, yet its full near–far–near cycle takes longer. The orbit has extra angular travel, its speed varies along the route, and local clock readings differ from the common reference time. “Faster at one point” does not fix the duration of the whole journey.
At Mercury’s actual strength, with the closest and farthest radii matched, the local speed at perihelion is only about 1.25 mm/s higher out of roughly 59 km/s. In the chosen Sun-centered reference time, the next-perihelion interval is about 0.58 seconds longer than this matched Newtonian orbit. The final segment from the original direction to the next perihelion takes about 0.39 seconds.
Those tiny timing and speed differences depend on what we hold fixed between models and how we define time and speed. They are not a separate measured “Mercury is late” anomaly. The robust point is the mismatch between angular and radial repetition; the famous residual is an angle per century.
The long-term mean time per full turn, often called the azimuthal or sidereal period in this ideal model, also differs from the first 360° interval measured from a particular starting point, because the planet’s angular speed is nonuniform. The final section makes these definitions precise.
Could this be only an effect of watching from Earth?
Earth changes the apparent view. It does not create Mercury’s relativistic perihelion advance.
Because Earth moves too, the time until Mercury returns to the same alignment with Earth and the Sun is different from its time around the Sun. Mercury can even appear to reverse its direction against the stars during part of the viewing cycle while still moving forward around the Sun.
Compare 88 days with 116 days below. A lap around the Sun does not restore the viewing geometry from a moving Earth.
Mercury has completed one lap, but Earth has moved almost a quarter of its orbit. Their original alignment has not returned.
- Mercury laps around the Sun
- Earth’s angle around the Sun
- Apparent Mercury–Sun separation from Earth
Circular, coplanar orbits isolate the viewpoint effect; the real orbits are eccentric and inclined. Repeating the same Mercury–Sun alignment takes about 116 days on average. This is separate from Mercury’s roughly 88-day orbit and its 43″-per-century perihelion advance.
The roughly 116-day alignment cycle is called the synodic period. It describes a changing viewpoint, not an extra 28 days needed to finish a Mercury orbit.
The 43″ effect concerns the direction of Mercury’s closest approach to the Sun, measured in a specified celestial reference frame. It remains in the dynamics after Earth’s motion, light-travel effects, and ordinary planetary perturbations are modeled. A camera move cannot turn a genuinely precessing Sun-centered orbit into a closed Kepler ellipse.
How much is an extra 43 arcseconds?
The relativistic advance is about 0.1035 arcseconds per orbit. Mercury makes roughly 415 orbits in a century, accumulating about 43 arcseconds: only 0.012 degrees over a hundred Earth years.
The double-prime symbol ″ means an angle measured in arcseconds. It is not a number of seconds of time, or a fixed distance by which Mercury is ahead of schedule.
At the actual angular scale, the two directions look identical. The number resolves the tiny difference.
0.011939° after 100 years
43″ is the residual, not the whole observed precession
These are rounded contributions in a common reference convention. Precision ephemerides also include the Sun’s slight oblateness, its rotation, other bodies, and further corrections. Historical totals near 5,600″ include about 5,025″ from the precessing equinox reference direction; that is a reference-frame contribution, not extra physical turning caused by the Sun.
Spacetime geometry changes free fall. The inward–outward motion no longer repeats in exactly one turn. The next closest point advances. The motion and its duration follow together; Earth’s viewing cycle is a separate effect.
The mathematics behind the pictures
Now connect the intuitive picture to the calculation: metric → free-fall equation → orbit shape → perihelion advance and timing.
The Sun is modeled as spherical and nonrotating, and Mercury as a test particle. The clock-and-ruler rule below is the Schwarzschild metric outside that source.
The metric makes the rule quantitative
The metric is the rule that converts small coordinate changes into measured spacetime intervals. Outside an ideal spherical, nonrotating Sun, and within Mercury’s orbital plane, it is
t is the time coordinate normalized to stationary clocks far away. r is a circumference-based radius: a circle at that radius has circumference 2πr. φ is the angle around the Sun. rs = 2GM/c² is the Schwarzschild radius—about 2.95 km for the Sun, far inside its actual surface.
Hold the clock at rest: set dr = dφ = 0. Its elapsed proper time is dτ = √A dt.
Measure a radial distance at constant t: set dt = dφ = 0. The local ruler measures dℓ = dr/√A.
The two factors already differ from flat spacetime. Moving through this geometry adds the motion terms too. Along a massive traveler’s path, ds² = −c²dτ², so:
The clock experiment integrates the square root of this expression along each candidate path, with dφ = 0. The example at r = 5rₛ uses a hypothetical compact source to make the differences legible.
How the local rule advances the planet
GR’s law of motion says that a freely falling test body follows a timelike geodesic. The metric alone does not make every object follow that path: a rocket can accelerate away from it. For a freely falling planet, making proper time stationary gives the geodesic equation:
Read it as a recipe: current position + current velocity + the local metric → change in coordinate velocity. The x’s are the spacetime coordinates; the repeated α and β indices mean sum over them. The coefficients Γ are calculated from the metric and its first derivatives. Given an initial position and velocity, this equation advances the trajectory without consulting a future destination.
In a small freely falling frame at an event, the Γ terms can vanish and the planet moves inertially through that event. In the Sun-centered coordinates used for the whole orbit, they generally do not vanish, so the coordinate path bends. The planet’s accelerometer still reads zero. Γ themselves are coordinate-dependent; spacetime curvature is the deeper statement below.
Why this also gives Newton’s gravity
For a weak field and slow motion, expand the clock rate. With Φ = −GM/r and ordinary speed v,
Making ∫dτ stationary is therefore equivalent, at this order, to making the Newtonian action ∫(½v² − Φ)dt stationary. Its equation of motion is acceleration = −∇Φ, the usual gravitational acceleration.
Ordinary Newtonian falling is already contained in the geometry. Mercury’s anomalous precession comes from the next corrections beyond that approximation. “Clocks run slower near the Sun” is a useful beginning, but computing the 43″ requires the full spacetime rule at the required order.
From the metric to the orbit equation
We now apply the same geodesic principle to an orbit, so both radius and angle change. It is convenient to describe the shape using u = 1/r as a function of the angle φ. Primes below mean derivatives with respect to angle, not time.
h is conserved angular momentum per unit planet mass. In GR, h = r²dφ/dτ. In Newton’s equation the analogous derivative uses Newtonian time. The added term follows from the metric; it is not fitted to Mercury’s measured precession.
Derive the extra term in four steps
Use the symmetries
The metric does not depend explicitly on t or φ. The corresponding conserved quantities are
E is energy per unit rest energy; h is angular momentum per unit mass. These conservation laws and the proper-time constraint are enough to derive the shape equation.
Insert them into the metric
For a massive traveler, ds²/dτ² = −c². Substitute the expressions for dt/dτ and dφ/dτ:
Expand A = 1 − 2GM/(rc²), divide by 2, and rearrange:
The first three terms are the familiar radial kinetic energy, angular-momentum barrier, and Newtonian potential per unit mass. The rust term is the relativistic correction to the effective potential. The “barrier” is angular motion expressed in a radial equation, not an extra repulsive force. A dot here means d/dτ.
Find the radial acceleration
Differentiating the conserved-energy equation gives
This is an equation for the Schwarzschild radius coordinate with respect to proper time. It is not an accelerometer reading: the planet is still freely falling.
Trade time for angle
Since u = 1/r and dφ/dτ = hu², the chain rule gives ṙ = −hu′ and r̈ = −h²u²u″. Substituting those identities into the radial equation yields
That is the missing bridge: a rule for spacetime intervals produces a new relationship between radius and angle.
The Schwarzschild metric, conserved quantities, and geodesic calculation are developed in David Tong’s general relativity notes, §1.3. The algebra above uses E normalized by rest energy.
See the changed rhythm in the equation
To isolate the mechanism, start with a circular orbit of radius r₀ and give it a tiny radial disturbance, holding h fixed. Write u = u₀ + δu, where u₀ = 1/r₀. Expand the extra GR term:
The constant pieces balance for the circular orbit. Move the remaining term to the left. The perturbation obeys:
This is an oscillator measured against angle. Newton’s coefficient is 1. GR’s is slightly less than 1, so the frequency per unit angle is smaller:
The radial response falls behind the angular motion. The next closest approach therefore happens farther around the Sun. The dials above use the exact orbit’s radial phase, rather than this near-circular approximation, but expose the same mismatch.
Simply making the Sun heavier does not do this. A larger constant GM in Newton’s equation changes the constant on its right side; the coefficient of u is still 1. The new orbit is still a closed ellipse if it remains bound. GR adds a term that depends on u itself, changing the radial response as the distance changes.
The orbit does not spiral inward. In this stationary, spherical model, energy and angular momentum stay constant. Successive closest and farthest radii stay the same while the closest-approach direction changes. Trace eight cycles to see the resulting rosette.
Calculate the 43″ with Mercury’s parameters
For an eccentric orbit in a weak field, the leading extra angle per radial cycle is
a is the semimajor axis and e is eccentricity. This is a consequence of the orbit equation, not an extra assumption. The small parameter GM/[a(1 − e²)c²] is about 2.66 × 10−8 for Mercury.
Derive the eccentric-orbit formula
At Newtonian order, set p = h²/(GM) = a(1 − e²) and uN = (1 + e cos φ)/p. Insert uN into the small GR correction:
The cos φ term drives the orbit’s existing angular frequency. It generates the phase correction 3GMe φ sin φ/(p²c²). This is precisely the first-order term obtained by expanding
The other terms give small bounded shape changes. The accumulating phase determines precession. The next radial cycle finishes at φ = 2π/(1 − ε), so the excess over 2π is Δϖ ≈ 2πε = 6πGM/[a(1 − e²)c²].
| Sun’s gravitational parameter GM | 1.32712440018 × 1020 m³/s² |
|---|---|
| Semimajor axis a | 0.38709927 AU ≈ 5.7909 × 1010 m |
| Eccentricity e | 0.20563593 |
| Speed of light c | 299,792,458 m/s |
- Extra angle in one orbit5.019 × 10−7 radians = 0.103517″
- Mercury orbits in a Julian century36,525 days ÷ 87.96947 days ≈ 415.20
- Accumulated relativistic advance0.103517″ × 415.20 ≈ 42.98″ per century
- Mercury · extra angle per orbit
- Orbits per century
- Extra advance per century
Smaller a increases the advance each orbit and also makes the orbital period shorter. Mercury gets more extra angle per lap and more laps per century.
The rate scales as a−5/2/(1 − e²) for the same Sun. The reference bars stay fixed while you experiment. At e = 0 the perihelion direction is undefined; the formula gives the limit for a tiny radial disturbance.
Which orbital period, and which speed?
Let Tr be the coordinate time between successive perihelia, and let Δϖ be the advance during that cycle. Define the mean angular frequency Ωφ and radial frequency Ωr by
The mean azimuthal period is consequently
This inequality compares two periods within the same prograde GR orbit. It does not, by itself, compare either period to a Newtonian orbit.
Nor is Tφ necessarily the time of the first 360° turn starting at perihelion. For that event we solve φ(χB) = 2π, then integrate coordinate time only to χB. The timeline above uses this first-turn event. Angular velocity is nonuniform, so its duration can differ from the long-term mean.
For the matched-turning-radii Mercury example, the calculated differences from Newton are approximately +0.190 s for the first 360° turn, +0.581 s for Tr, and -0.026 s for the mean Tφ. These are different questions, not conflicting orbital periods. Their numerical values are properties of this specified comparison, not ephemeris-fit residuals.
A static local observer has proper time dτstat = √A dt and radial ruler element dℓr = dr/√A. Therefore the locally measured velocity components are
The launch-speed comparison uses the magnitude of this local velocity. At perihelion the radial part vanishes. A coordinate speed such as r dφ/dt uses a different clock convention, so “faster” must always specify the measurement.
Why Earth’s alignment cycle is about 116 days
For coplanar circular motion, Mercury gains on Earth at the difference of their angular rates. The mean synodic period S obeys
Using approximately 87.97 days and 365.26 days gives 115.88 days. The real interval varies because the actual orbits are not circular. This viewing-cycle calculation uses no relativistic perihelion advance.
What the curved surface represents
The ruler panel is a Flamm embedding of the equatorial, constant-t spatial metric. Introduce an artificial Euclidean height zemb:
Its radial slope makes the embedded surface’s Euclidean line element equal that spatial metric. The height is neither a real direction of motion nor the full spacetime geometry. The orbital curve is separately computed as a timelike Schwarzschild geodesic; no rolling-ball dynamics are used.
Evidence, assumptions, and further detail
How this is measured in the real Solar System
The unexplained advance was known before Einstein’s 1915 calculation. Astronomers did not see a literal ellipse painted in space; they compared dynamical calculations with observations, including transit timings. Modern orbit fitting uses radar ranging and spacecraft radio tracking as well.
A fitted ephemeris includes initial conditions, planetary masses, relativistic dynamics, and smaller perturbations. Mercury’s agreement with GR is one test among many, not a unique proof by a single number. The INPOP15a analysis using MESSENGER data is an example of testing for additional perihelion advances beyond a relativistic model. For the history, see Janssen and Renn’s account of Einstein’s Mercury calculation.
What the interactive models calculate
Orbit geometry. The code uses the exact bound Schwarzschild test-particle relations. With p = a(1 − e²), η = GM/(pc²), and radial anomaly χ:
Each cycle uses 4,096 composite-Simpson intervals. The excess angle ∫(dφ/dχ − 1)dχ is integrated in a cancellation-safe form. Playback follows the coordinate-time integral, not a dot moving at uniform angular speed. The χ dial is a radial phase coordinate, not a separate physical clock.
Fixed turning radii. Newton and GR share the nearest and farthest radii; their initial velocities are not identical. For strong examples, a and e label those turning radii, not an exact GR ellipse. The orbit experiment changes compactness to 0.018 while retaining Mercury’s eccentricity; the tiny-angle experiment changes only its drawing scale.
Clock experiment. In units rₛ = c = 1, a shooting method integrates the radial coordinate-time geodesic equation r″ = −A/(2r²) + 3(r′)²/(2r²A) using fourth-order Runge–Kutta steps. It finds the launch velocity that returns to r = 5 at t = 16. An altered trip multiplies the entire radial excursion r(t) − 5, and its velocity, by a factor from 0 to 2. The same coordinate path is used when switching to flat spacetime; it then generally requires thrust. The example chooses a mass for which rₛ/c = 1 second, and retains those coordinate units in the flat comparison. Thus the reunion is at reference time t = 16 seconds, while the station clock near the mass records 16√0.8 seconds. Playback interpolates radius and velocity together and integrates the clock rate up to the selected moment. Clock time is then independently integrated as ∫√[A − (dr/dt)²/A]dt. The selected segment is locally maximizing; timelike geodesics need not globally maximize proper time for arbitrarily long trips.
Scope. The Sun is spherical and nonrotating; the orbiting planet is a test particle. Other planets, solar oblateness and spin, Mercury’s back-reaction, radiation, and optical/light-travel effects are omitted. These demonstrations explain the relativistic contribution; they are not a precision ephemeris.
Reference data and sources
- David Tong, Cambridge · Geodesics and planetary orbits — the metric, orbital equations, and planetary perturbations.
- Einstein Online · Gravity: from weightlessness to curvature — local free fall and tidal effects.
- Andrew Hamilton, JILA · Schwarzschild geometry — clock rates, rulers, and spatial embeddings.
- NASA JPL · Reference planetary elements and astrodynamic parameters — a, e, units, and constants.
- Berche and Medina · The advance of Mercury’s perihelion — historical reference-frame accounting.
The reference elements are approximate J2000 values. The model’s solar GM is a fixed reference value; modern fitted values differ slightly without affecting the rounded 42.98″ result. A century here is 100 Julian years, or 36,525 days. The exact reference solver yields 0.103517315″ per radial cycle and 42.980477″ per century.