Gravitational time dilation

Change the heights or choose an Earth example: the offset updates immediately. Δτ is B − A at the selected time on A; Δτ / d is the offset after 86400 seconds on A. Positive means B is ahead. The examples include gravity only.

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, reference radius and additional measurements are under Options. Reset restores the GPS-height example. Swapping clocks exchanges their heights; the duration and selected time still refer to A. The slider selects a time in 0 ≤ τA ≤ T without rounding a typed time. A zero duration disables it. A probe at the old endpoint follows a changed duration; an explicitly selected intermediate time stays fixed. Choosing an example or resetting sets the probe to the full duration.

The line shows the signed, stable offset over the chosen duration, with a marker for the selected time. Zero duration is a single point, and equal rates give a horizontal zero line. Invalid physical inputs clear the comparison; invalid probe input clears its readings and marker while retaining the curve and daily rate. Raw input text, including incomplete entries and the selected static time, is kept through navigation, reload and language changes. Results, plots and open menus are not saved.

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.