Gravitational Time Dilation Calculator

Background.

In 2010 a group at NIST raised one aluminium-ion clock 33 centimetres above an identical one and measured it running faster by a fractional (4.1 ± 1.6) × 10⁻¹⁷. That is gravitational time dilation on a laboratory bench. A clock sitting deeper in a gravitational field accumulates less time than one higher up, and the deficit is not a quirk of how the clocks are compared: bring them back together and the lower one shows less elapsed time.

Near Earth the effect is tiny but not negligible. A clock on the equator runs 60.08 microseconds a day slower than one infinitely far away. A GPS satellite, 26,561,750 m from Earth's centre, gains 45.65 µs a day on a ground clock from gravity alone, and since positioning works by timing radio signals, an error that size multiplied by the speed of light is 13.69 km of range in a single day.

The calculation uses general relativity's exact solution for a spherical, non-rotating mass, comparing two clocks held still at different distances from the centre, or one clock against a reference infinitely far away. It counts gravity only: a GPS satellite's orbital motion and Earth's flattened, spinning shape both change the published GPS figures, as the questions below explain.

What is gravitational time dilation calculator?

Gravitational time dilation is the difference in the rate at which two clocks accumulate proper time because they sit at different gravitational potentials. In the Schwarzschild geometry around a static, spherical mass, a clock held at a fixed radius r ticks at √(1 − r_s/r) times the rate of a clock infinitely far away, where r_s = 2GM/c² is the Schwarzschild radius. Unlike the time dilation between two moving observers in special relativity, it is not reciprocal: both clocks agree which one is slow. Light climbing from the lower clock to the higher one is redshifted by the same factor.

How to use this calculator.

1. 01Choose whether to compare your clock with a second clock at another radius or with a clock infinitely far away.
2. 02Enter the mass of the body in kilograms: 5.97217 × 10²⁴ for Earth, 1.98841 × 10³⁰ for the Sun, 1.89812 × 10²⁷ for Jupiter.
3. 03Enter each radius from the centre of the body, not as an altitude. For a point 400 km above Earth's equator, add 400,000 m to the equatorial radius of 6,378,137 m.
4. 04In two-clock mode, enter the reference clock's radius. If it is lower than your clock, your clock runs fast and the results come out positive.
5. 05Set the elapsed period in days if you want a running total; the per-day and per-year figures do not depend on it.

The formula.

A clock held at rest at radius r keeps proper time τ at the rate dτ/dt = √(1 − r_s/r), where t is the time of a clock at infinity and r_s = 2GM/c². For two clocks the coordinate time cancels: ratio = √(1 − r_s/r₁) ÷ √(1 − r_s/r₂). G is the CODATA 2022 value 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² and c is exactly 299,792,458 m/s. The fractional difference is ratio − 1; multiplied by 86,400 × 10⁶ it becomes microseconds per day, and by 31,557,600, the seconds in a Julian year, it becomes seconds per year. The redshift is z = 1/ratio − 1, positive for light climbing out of the field and negative for light falling in.

Expanding the square root to first order gives the textbook form 1 − GM/(rc²), or 1 − (GM/c²)(1/r₁ − 1/r₂) for two clocks, and both values are reported with their relative difference. Around Earth they agree to better than 10⁻¹⁸. Near a black hole they part company: at ten Schwarzschild radii the approximation is out by 1.4 × 10⁻³, at three Schwarzschild radii, the innermost stable circular orbit, the exact rate is √(2/3) = 0.8165 against an approximate 0.8333, and at 1.0001 r_s the approximation is wrong by a factor of 49. Clocks inside ten Schwarzschild radii get a strong-field warning, an edge chosen for this calculator rather than a published threshold.

At the horizon itself the tick rate falls to zero and the redshift becomes infinite, and inside it no object can hold a fixed radius at all, so any radius at or below r_s is refused. The model also assumes the body does not spin; a rotating mass drags spacetime around with it and needs the Kerr solution instead.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Ashby, N. (2003), "Relativity in the Global Positioning System", Living Reviews in Relativity 6, 1; open-access full text retrieved 2026-07-29 via PubMed Central. Source of the adopted geoid potential Φ₀/c² = 6.969290134 × 10⁻¹⁰ used for Terrestrial Time, of the GPS semi-major axis a = 2.656175 × 10⁷ m used as this page's default orbital radius, and of the statement that "the standard clock in orbit is beating too fast, primarily because its frequency is gravitationally blueshifted", with the adjusted proper frequency of 10.229 999 995 43 MHz. Peer-reviewed; open access.
2. [2]Chou, C. W., Hume, D. B., Rosenband, T. and Wineland, D. J. (2010), "Optical Clocks and Relativity", Science 329, 1630–1633. Measured a fractional clock-rate change of (4.1 ± 1.6) × 10⁻¹⁷ for a 33 cm change in height between two aluminium-ion optical clocks. This is the second, independent authority used to verify this calculator at the weak-field extreme: the page returns 3.5977 × 10⁻¹⁷ for exactly that geometry. Peer-reviewed; paywalled at Science, abstract freely visible; the NIST press release describing the same result is open.
3. [3]US Space Force / GPS Directorate, Interface Specification IS-GPS-200, "Navstar GPS Space Segment / Navigation User Interfaces", §3.3.1.1 SV Carrier and Clock Rates. Specifies that the space-vehicle clock rates are offset by Δf/f = −4.4647 × 10⁻¹⁰, equivalent to Δf = −4.5674 × 10⁻³ Hz on the 10.23 MHz reference, giving 10.229 999 995 43 MHz, "to compensate for relativistic effects". A government signal specification and therefore independent of the physics literature; consulted 2026-07-29 via the official GPS.gov / NAVCEN document set. Freely published.
4. [4]NIST/CODATA 2022 fundamental constants, retrieved 2026-07-29: G = 6.674 30(15) × 10⁻¹¹ m³ kg⁻¹ s⁻² (relative standard uncertainty 2.2 × 10⁻⁵) and c = 299 792 458 m s⁻¹ (exact). Standards body; open access.
5. [5]Particle Data Group (Workman et al.), Review of Particle Physics, "Astrophysical Constants and Parameters", Table 2.1, revised August 2023; 2024 PDF retrieved 2026-07-29. Source of the default Earth mass 5.97217(13) × 10²⁴ kg and the solar mass 1.98841(4) × 10³⁰ kg, and independently prints the Schwarzschild radii of both bodies (8.870056 mm and 2.9532501 km), which this page reproduces. Open access.
6. [6]Pound, R. V. and Rebka, G. A. (1960), "Apparent Weight of Photons", Physical Review Letters 4, 337. The first terrestrial measurement of the gravitational frequency shift, over 22.5 m. Peer-reviewed; paywalled at APS. Historical reference — no number is taken from it.
7. [7]Vessot, R. F. C., et al. (1980), "Test of Relativistic Gravitation with a Space-Borne Hydrogen Maser", Physical Review Letters 45, 2081. Gravity Probe A, confirming the gravitational redshift to about 1.4 × 10⁻⁴. Peer-reviewed; paywalled at APS. Cited for the measurement history only.