Gravitational Time Dilation Calculator — Schwarzschild clock rate

The gravitational time dilation calculator above uses the Schwarzschild metric to work out how fast a clock ticks at a given distance from a spherical mass, compared with one far away or at a second radius you specify. It reports the rate factor, the fractional slowdown and the accumulated difference over whatever interval you enter.

Arb Digital builds free physics calculators that state their regime of validity. The time dilation calculator covers the special-relativistic case, where the effect comes from relative motion and nothing else. This page covers the gravitational case, which is a general-relativistic effect and exists even when nothing is moving. The optional speed input lets you see both at once, which matters because for anything in orbit they pull in opposite directions.

What This Gravitational Time Dilation Calculator Does

In the Schwarzschild solution, the proper time measured by a stationary clock at radius r runs slower than coordinate time by a factor of the square root of (1 − r_s/r), where r_s is the Schwarzschild radius of the body. Coordinate time is the time kept by a clock infinitely far away, so that factor is directly the ratio of tick rates between the two.

The tool computes that factor, converts it into a fractional slowdown and into an accumulated lag over your chosen interval, and reports the Schwarzschild radius alongside the ratio r/r_s so you can see immediately whether you are in the weak-field regime or not. It also handles a second radius, which is the practically useful case: nobody has a clock at infinity, but comparing a satellite against the ground is a real measurement.

Because the fractional effect on Earth is around one part in a billion, the calculation is done with a series expansion when the field is weak. Working out 1 − √(1 − 1.4 × 10⁻⁹) by direct subtraction throws away most of the available precision; the series gives full accuracy, and the tool switches to the exact expression only where the field is strong enough to need it.

How to Use It

1. Pick a body or enter a mass. The presets load a mass and a surface radius; everything stays editable.
2. Set the radial coordinate. Measure from the centre of the body, not from its surface. For a clock on a mountain, add the mountain height to the planetary radius.
3. Choose the reference. Infinity gives the textbook factor; a second radius gives the difference between two real locations, which is what any experiment actually measures.
4. Enter an interval. The accumulated difference over a year or a day is far easier to reason about than a factor differing from one in the tenth decimal place.
5. Add a speed if the clock is moving. The kinematic term always slows the clock, so for a satellite it subtracts from the gravitational gain rather than adding to it.

The Formula: How the Rate Factor Is Calculated

For a stationary observer at radius r outside a non-rotating spherical mass, the Schwarzschild metric gives dτ/dt = √(1 − 2GM/rc²) = √(1 − r_s/r). Georgia State University's HyperPhysics page on gravitational time dilation gives this relation, notes that the effect at the Earth's surface is about one part in 10⁹, and describes the 1976 Scout rocket experiment that confirmed it to within 0.01 per cent by comparing a maser clock at 10,000 km altitude against one on the ground.

The constants are the 2022 CODATA values: NIST gives the Newtonian constant of gravitation as 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻², with a relative uncertainty of 2.2 × 10⁻⁵ that is by far the largest source of error in any result here. The speed of light is exactly 299,792,458 m/s by definition, so it contributes no uncertainty at all.

Between two radii the ratio of rates is √((1 − r_s/r₁) ÷ (1 − r_s/r₂)). In the weak field this reduces to a difference of gravitational potentials divided by c², which is the form used in practice: Δf/f ≈ ΔΦ/c², giving about 1.09 × 10⁻¹⁶ per metre of height near the Earth's surface.

Work the defaults by hand. Earth has a mass of 5.9722 × 10²⁴ kg, so r_s = 2 × 6.6743 × 10⁻¹¹ × 5.9722 × 10²⁴ ÷ (2.99792458 × 10⁸)² = 8.870 mm. At the surface radius of 6,371 km, r_s/r = 1.3923 × 10⁻⁹, so the fractional slowdown is half of that, 6.961 × 10⁻¹⁰. Over a Julian year of 31,557,600 seconds that accumulates to 0.02197 seconds — about 22 milliseconds a year that a ground clock loses relative to one far from the Earth.

Where the Weak-Field Approximation Applies and Where It Fails

When r_s/r is very small — which for the Earth means about 10⁻⁹, and for the Sun about 10⁻⁶ — the square root can be expanded and the whole effect collapses to Δt/t = GM/rc², a Newtonian potential divided by c squared. That approximation is excellent everywhere in the solar system, and it is what satellite navigation, geodesy and precision timekeeping actually use.

It begins to fail as r approaches r_s. At r = 10r_s the exact factor is 0.9487 while the first-order approximation gives 0.95, a difference already visible in the third decimal place. At r = 2r_s the exact factor is 0.7071 against an approximate 0.75, and the approximation is simply wrong. Compact objects are where the full expression is mandatory: the surface of a typical neutron star sits at roughly three Schwarzschild radii, where clocks run at about 80 per cent of the distant rate.

At r = r_s the factor reaches zero, and the calculation stops meaning anything. This is the event horizon. A distant observer sees infalling signals redshift without limit and never quite sees anything cross; the Schwarzschild coordinates themselves break down there, though nothing physically singular happens to an infalling observer. Inside r_s no stationary observer can exist at all — remaining at fixed r would require moving faster than light — so the formula has nothing to describe, and the tool refuses rather than returning a number. Our Schwarzschild radius calculator covers the horizon itself in more depth.

GPS: The Standard Real-World Example

Satellite navigation is the case where both relativistic effects show up together, in opposite directions, at a magnitude that matters commercially. A GPS satellite orbits at a radius of about 26,560 km, well above the Earth's surface, so it sits higher in the potential well and its clock runs faster than a ground clock — by roughly 45.7 microseconds per day on the figures this calculator produces.

It is also moving, at about 3,874 metres per second. Special-relativistic dilation slows a moving clock, and that term costs about 7.2 microseconds per day. The two work against each other, and the net effect is a satellite clock gaining roughly 38.5 microseconds every day relative to the ground.

Thirty-eight microseconds sounds negligible until you multiply by the speed of light. Light travels about 11.5 kilometres in that time, so an uncorrected system would drift by that much in a single day. The satellite oscillators are deliberately offset in frequency before launch so that they tick at the correct rate once in orbit, and further corrections handle the small eccentricity of each orbit. Relativity is not a refinement in this system; without it, satellite navigation would not work at all. Our orbital velocity calculator gives the speed term for any orbit radius.

What This Calculation Leaves Out

The Schwarzschild solution describes a non-rotating, spherically symmetric mass in vacuum. Real bodies rotate, and rotation drags spacetime around with it, which the Kerr solution describes and this page does not. For the Earth the frame-dragging contribution to clock rates is far below the terms computed here, but for a rapidly spinning compact object it is not.

Real bodies are also not spherical. The Earth is an oblate spheroid whose gravitational potential varies with latitude, and for precision timekeeping the reference is the geoid rather than a mean radius. That is why the figure this page gives for a GPS satellite differs slightly from the value quoted in the navigation literature: the published number is referenced to the rotating geoid and includes the Earth's rotation, while this calculation uses a static, spherical, non-rotating model.

Finally, the formula applies outside the mass, in vacuum. Inside a body the interior solution is different, because only the mass enclosed within your radius contributes in the same way. At the centre of the Earth the dilation is larger than at the surface, not zero, and this page does not compute that case. For the related redshift of light climbing out of a potential well, see the redshift calculator, and for the Newtonian gravitational quantities the gravitational force calculator and the escape velocity calculator.

Common Mistakes to Avoid

• Measuring r from the surface — the radial coordinate runs from the centre of mass. Using an altitude instead of a full radius changes the answer enormously.
• Adding the two relativistic effects with the same sign — for an orbiting clock, height speeds it up and motion slows it down. They subtract.
• Using the weak-field approximation near a compact object — at a few Schwarzschild radii the first-order form is visibly wrong and the exact square root is required.
• Expecting the formula to work at or inside the horizon — at r equal to the Schwarzschild radius the rate factor is zero, and inside it no stationary observer exists for the formula to describe.
• Applying it inside a massive body — the exterior Schwarzschild solution is valid only in vacuum. The interior needs a different solution entirely.

Related Free Tools From Arb Digital

For dilation caused by motion alone, use the time dilation calculator, and for the horizon of a compact object the Schwarzschild radius calculator. Orbital speeds come from the orbital velocity calculator, escape conditions from the escape velocity calculator, and Newtonian attraction from the gravitational force calculator. The frequency shift of light in a potential well is covered by the redshift calculator, and relativistic energies by the relativistic kinetic energy calculator. Everything Arb Digital publishes is listed on the free online tools hub.

Frequently Asked Questions

This tool is provided for educational and study use. It applies the exterior Schwarzschild solution for a non-rotating spherical mass in vacuum, ignoring rotation, oblateness, nearby masses and interior structure, so treat its output as a physics teaching result rather than a timekeeping or navigation figure.