Dilatation gravitationnelle

Two stationary clocks

This model compares two stationary clocks outside a spherical, nonrotating mass. The chosen duration T is explicitly proper time on reference clock A. The two clock readings and their offset refer to the same Schwarzschild coordinate time. Positive B − A means B is ahead; negative means B is behind. Both clocks are held stationary, which generally requires acceleration.

R₀ is a freely chosen reference radius. Heights hA and hB are nonnegative radial coordinate differences, not proper distances. The clock locations rA = R₀ + hA and rB = R₀ + hB are Schwarzschild areal radii. Both must lie strictly outside rs = 2GM/c². Stationary clocks at or inside the horizon are excluded. R₀ alone does not specify a material surface or an interior mass distribution.

Rates and stable offsets

Relative to coordinate time at infinity, fA = √((rA − rs)/rA) and fB = √((rB − rs)/rB). The clock-rate ratio is q = dτB/dτA = fB/fA. At time τA, τB = qτA and the common coordinate time is t = τA/fA.

The small rate difference is evaluated stably as deltaRate = [rs(hB − hA)/(rA rB)] / [fA(fA + fB)]. The offset is Δτ = deltaRate · τA. This avoids subtracting nearly equal rates or forming q − 1 when both values round close to one. A day in the rate result is exactly 86400 seconds on A. The nanosecond offset is authoritative when the large displayed absolute clock readings round to the same value. Extra displayed digits are not a promise of measurement accuracy.

The weak-field reference is GM(hB − hA)/(c²rA rB), approximately gΔh/c² near Earth's surface. Equal heights and zero mass give exactly zero offset. Swapping A and B reciprocates the rate ratio; the specified duration then refers to the newly chosen A.

Earth examples

The presets use R₀ = 6371000 m and M = 5.9722 × 10²⁴ kg, with A at hA = 0 and duration 86400 s on A. G = 6.67430 × 10⁻¹¹ in SI units and c = 299792458 m/s. Mass is entered in multiples of this model Earth mass.

GPS height 20200000 m gives about +45724.426697 ns per day on A; ISS height 420000 m gives +3719.806585 ns. A mountain height of 4000 m gives +37.738497 ns. At a laboratory height difference of 0.33 m the model gives about +0.003115380601 ns per day. The height fields retain these exact input values.

GPS and ISS presets describe only the gravitational contribution for stationary clocks at those heights. For GPS, the commonly quoted gravitational effect is about +45 μs/day, while orbital motion contributes about −7 μs/day, giving about +38 μs/day together. Motion, Earth's rotation, oblateness and the geoid are not included here. The 33 cm experiment demonstrates the physical effect; this simple spherical Earth model does not reproduce that local experiment to measurement accuracy.

Controls and plot

Mass ranges from 0 to 10¹² model Earth masses, R₀ from 0.001 to 10¹² m, each height from 0 to 10¹² m, and duration from 0 to 3155760000 s. Inputs must also satisfy the exterior-radius condition. Presets change inputs without calculating. Each calculation starts the time probe at the full duration. Type an exact time or use the proportional slider; the slider is a navigation aid and does not round typed values. It is disabled at zero duration.

The line plots the stable offset only for 0 ≤ τA ≤ T. Zero duration appears as a single computed point; at zero rate the offset remains zero. Axes can be expanded for visibility without claiming calculated times beyond that interval. Invalid time input hides probe readings and its marker while leaving the full curve. Main input changes clear the comparison. Navigation preserves the prepared result and probe; reloading or changing language restores valid settings and waits for Calculate.

Sources

NIST: Putting Einstein to the Test; NIST: the 2010 clock experiment; OIST: general relativity, lecture 8, section VII; Princeton: stationary Schwarzschild clock rates and motion corrections.