Gravitational Time Dilation Calculator

Background.

A clock deeper in a gravitational field runs slow. Not apparently, not as a measurement artefact — it accumulates less proper time, permanently and cumulatively. This calculator gives the exact Schwarzschild answer: the ratio √(1 − r_s/r₁) ÷ √(1 − r_s/r₂) between two static clocks at different distances from a spherical mass, or between one static clock and a reference at infinity, expressed as a fractional rate, as microseconds per day, as seconds per year, and as the gravitational redshift of light travelling between them.

The page defaults to the example that made the effect an engineering problem rather than a curiosity. A GPS satellite at a semi-major axis of 26 561 750 m, compared with a clock on Earth's equator at 6 378 137 m, gains 45.65 microseconds per day. Left uncorrected, that error alone would move a GPS fix by about thirteen kilometres in a day. GPS satellites therefore ship with their clocks deliberately set slow — the interface specification IS-GPS-200 fixes the offset at Δf/f = −4.4647 × 10⁻¹⁰, so a 10.23 MHz oscillator is built to run at 10.22999999543 MHz.

That published number is not the one this page prints, and the difference is worth understanding before you use the result. **This calculator gives the gravitational term only.** GPS clocks are also moving, and kinematic special-relativistic dilation slows them by 7.21 microseconds a day, so the net is 38.58 µs/day — which is exactly what the −4.4647 × 10⁻¹⁰ factory offset absorbs. On top of that, the reference clock for GPS is not a point-mass surface but the rotating geoid, whose effective potential includes Earth's oblateness and the centrifugal term. Ashby's adopted value Φ₀/c² = 6.969290134 × 10⁻¹⁰ is 0.227% larger than the point-mass GM/(Rc²) = 6.953487 × 10⁻¹⁰ this page uses, which is why 45.65 here versus 45.79 in the GPS literature. Both numbers are correct for what they describe; the page states which one it is computing.

The formula used is the exact one, not the familiar first-order approximation. Textbooks usually write 1 − GM/(rc²), which is the leading term of √(1 − r_s/r). For Earth the two are indistinguishable — they differ by about 6 × 10⁻¹⁹ — but they diverge fast in a strong field. At ten Schwarzschild radii the approximation is wrong by 1.4 × 10⁻³, at three Schwarzschild radii by 2.1 × 10⁻², and at 1.0001 r_s it is wrong by a factor of 49, returning 0.50 where the exact answer is 0.0100. This calculator computes both, prints their relative difference, and warns you whenever you are inside ten horizon radii.

There are limits on what "static" means here. The Schwarzschild solution describes a non-rotating, uncharged mass, and this page describes observers who hover at a fixed radius rather than orbiting. A rotating body drags spacetime (the Kerr metric) and an orbiting clock carries a velocity term; neither is included. There is also a hard domain edge. No static observer can exist at or inside the event horizon, where the tick rate reaches zero and the redshift diverges, so a radius at or below r_s raises an error rather than returning a number — that is not a numerical guard, it is the physics.

Two independent measurements calibrate the page's arithmetic. Chou and colleagues at NIST measured a fractional clock-rate change of (4.1 ± 1.6) × 10⁻¹⁷ when one optical clock was raised 33 centimetres above another; this engine returns 3.5977 × 10⁻¹⁷ for exactly that geometry, inside their error bar and equal to the Newtonian gh/c². At the other extreme, the Sun's surface produces a gravitational redshift of z = 2.1225 × 10⁻⁶, the value spectroscopists have quoted for a century as an equivalent 636 m/s Doppler shift. One tabletop, one stellar, six orders of magnitude apart, both reproduced.

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 of a static, spherically symmetric mass, a clock held at radius r ticks at √(1 − r_s/r) times the rate of coordinate time, where r_s = 2GM/c² is the Schwarzschild radius. Comparing two such clocks gives the ratio this calculator returns.

It is not an illusion and it is not symmetric, unlike the reciprocal time dilation of special relativity between two inertial frames. Bring the two clocks back together and the one that spent time deeper in the well will genuinely show less elapsed time. This has been measured directly — with aircraft-borne caesium clocks (Hafele–Keating, 1971), with rocket-borne masers (Gravity Probe A, 1976), and in the laboratory over a height difference of 33 centimetres (Chou et al., 2010).

The same factor governs gravitational redshift. Light emitted at radius r₁ with proper frequency ν arrives at radius r₂ with its frequency scaled by exactly the clock-rate ratio, so z = 1/ratio − 1. Light climbing out of a potential well is redshifted; light falling in is blueshifted. Pound and Rebka measured this over 22.5 metres of a Harvard tower in 1959.

What it is not: it is not kinematic (velocity) time dilation, which is a separate special-relativistic effect that this page deliberately excludes; it is not frame dragging, which requires the Kerr metric; and the radius it takes is a distance from the centre of the mass, not an altitude above a surface.

How to use this calculator.

1. 01Choose whether to compare two clocks at different radii, or one clock against a reference at infinity.
2. 02Enter the mass of the gravitating body in kilograms.
3. 03Enter the radius of your clock, measured from the centre of the mass — not its altitude above the surface.
4. 04In the two-clock route, enter the reference radius as well. A reference below your clock gives a positive result: your clock runs fast.
5. 05Set the elapsed period if you want a total in seconds; the microseconds-per-day and fractional outputs do not depend on it.
6. 06Read the fractional rate difference rather than the ratio — the ratio is so close to 1 in any weak field that a floating-point number cannot show its digits.
7. 07Check the conventions note before quoting anything. It states that only the gravitational term is included, and flags strong-field results where the familiar approximation fails.

The formula.

The Schwarzschild line element gives the proper time of a clock at rest at radius r as dτ = √(1 − r_s/r) dt, where t is the coordinate time kept by a clock infinitely far away and r_s = 2GM/c². Comparing two static clocks divides one such factor by the other, and the coordinate time cancels: ratio = √(1 − r_s/r₁) ÷ √(1 − r_s/r₂). Setting r₂ to infinity reduces the denominator to 1. Everything else on the page is a restatement: the fractional difference is ratio − 1, the microseconds per day is that times 86 400 × 10⁶, the seconds per year is that times a Julian year of exactly 31 557 600 s, and the redshift is z = 1/ratio − 1 because a light wave's period is stretched by exactly the inverse of the emitting clock's rate.

Expanding the square root to first order in r_s/r gives the familiar textbook form 1 − GM/(rc²). The calculator computes this alongside the exact value and reports the relative difference between them, so the approximation's failure is visible rather than assumed. At Earth's surface the difference is about 2 × 10⁻¹⁹, which is far below anything measurable. At ten Schwarzschild radii it is 1.4 × 10⁻³. At three Schwarzschild radii — the innermost stable circular orbit — the exact factor is √(2/3) = 0.8165 while the approximation says 0.8333, a 2.1% error. At 1.0001 r_s the exact factor is 0.009999 and the approximation says 0.5000, wrong by a factor of 49.

Units and dimensions. Mass in kilograms, both radii in metres from the centre of the mass, elapsed period in days. The ratio, the fractional difference, r/r_s and z are all dimensionless; r_s comes out in metres because GM/c² has dimensions (m³ kg⁻¹ s⁻² × kg) ÷ (m² s⁻²) = m. The dimensional guard the test suite runs is that at exactly four Schwarzschild radii the ratio must equal √3/2 and at three it must equal √(2/3), both pure numbers with no constants in them.

Rounding stage. No intermediate rounding: forty significant digits throughout, rounded once at the return boundary. The two near-unity ratios round to eighteen decimal places, which is already beyond what an IEEE-754 double can represent — a double near 1.0 resolves about 2 × 10⁻¹⁶, so for Earth-scale problems the ratio's meaningful digits do not survive at all. That is precisely why the fractional difference is computed as a separate quantity in high precision and returned at twelve significant digits, and why the page tells you to read it instead of the ratio.

Significant figures. Five digits at most. Everything inherits G's 2.2 × 10⁻⁵ relative uncertainty, and any real-world application inherits much larger modelling error from the effects this page excludes.

Invalid-domain behaviour. A radius at or inside the Schwarzschild radius raises a field error naming the horizon radius, because the tick rate is exactly zero at r = r_s, the redshift is infinite there, and inside it no observer can hover at any cost — the radial direction is timelike. Zero or negative mass, zero or negative radius, and a zero or negative elapsed period each raise their own field error. One threshold attaches a note rather than an error: a clock inside ten Schwarzschild radii is flagged as strong-field, with the exact and approximate values shown side by side. That ten-radius edge is a presentation convention chosen for this page, not a published threshold.

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 the engine 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.