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Time Dilation Calculator

Physics

Calculation mode

Special relativity (speed → dilated time)

A clock moving past you ticks slow by the Lorentz factor. Feed in a speed and a proper time — the time the moving clock itself reads — and the stationary observer's reading comes back.

γ = 1/√(1 − v²/c²), Δt = γ·Δt₀

The specification's headline example. γ = 2.0281848, so the stay-at-home observer measures 2.0281848 years and the clocks end 375.544 days apart.

Inputs

Must be strictly below the speed of light
Drag to sweep the speed. The scale is cubed so the interesting region near c is reachable.
The interval measured by the moving or deep clock itself

Optional add-ons. Both reuse the γ already computed above; leave a box blank and its rows stay hidden.

Reports the contracted length L = L₀/γ
Reports E = γm₀c² and the kinetic energy (γ − 1)m₀c²
0 to 10. Near-unity factors are always given extra digits so they are not flattened to 1.
Lorentz factor γ
2.028185

Dilated time 2.028185 years

Lorentz factor γ

1/√(1 − β²) — the multiplier between proper time and coordinate time

2.028185

β = v/c

260819.438460 km/s

0.870000

Dilated time Δt = γ·Δt₀

2.028185 years

Time difference Δt − Δt₀

375.544493 days — computed as (γ − 1)·Δt₀, never as γ·Δt₀ − Δt₀

1.028185 years

Moving clock runs slow by

(1 − 1/γ) × 100

50.6948 %

Drift per day

Negative: the moving clock falls behind the stationary one

−12.166759 h

Contracted length L = L₀/γ

A proper length of 100.000000 m measured in the moving frame

49.305172 m

The same drift on every basis

Per second-0.5069483 s/s
Per day−12.166759 h
Per year (Julian)−185.162859 days

γ against β

why speed feels free until it isn't
1246800.250.50.751v = cγ = 2.02820

Working

β = v / c

2.608194385e+8 / 299792458 = 0.8700000000

γ = 1 / √(1 − β²)

1 / √(1 − 0.7569) = 2.02818479

γ − 1 = β² / (1 + √(1 − β²)) ÷ √(1 − β²)

1.02818478579 — the cancellation-free route

Δt = γ · Δt₀

2.02818479 × 1.000000 years = 2.028185 years

Constants used

Speed of light c299792458 m/s (exact by definition)
Gravitational constant G6.6743e-11 m³ kg⁻¹ s⁻²
Julian year31557600 s (365.25 days)
Day86400 s

Current selection: v = 260819.438460 km/s (0.870000 c (β)), Δt₀ = 31557600.000000 s (years selected), L₀ in m.

About This Tool

Time Dilation Calculator — Moving Clocks and Deep Clocks

Two clocks that start together and meet again do not have to agree. This time dilation calculator works out by how much, for both reasons it happens: special relativistic dilation, where a clock moving past you ticks slow by the Lorentz factor γ = 1/√(1 − v²/c²), and gravitational time dilation, where a clock deeper in a potential well ticks slow by √(1 − 2GM/rc²). Six modes cover the problems these produce: a moving clock, a deep clock, a satellite that is both at once, the inverse question, a twin-paradox trip plan, and the cosmic-ray muon.

The Lorentz factor, and one number worth checking

At 0.87c the arithmetic runs: 0.87² = 0.7569, 1 − 0.7569 = 0.2431, √0.2431 = 0.49305, and so γ = 2.0281848. Over one Julian year of the traveller's time the stay-at-home observer measures 2.0281848 years, a difference of 375.544 days. The moving clock runs slow by (1 − 1/γ) × 100 = 50.6948 %, and a 100 m rod carried aboard measures 49.305 m to the observer it flies past. Several printed worksheets give 2.0278 here, which propagates into a 375.34-day answer; the tool implements the formula rather than carrying a rounded number forward.

Gravitational dilation and the Schwarzschild radius

Earth's Schwarzschild radius is r_s = 2GM/c² = 8.870 mm. Divided by the 6371 km surface radius that is 1.392 × 10⁻⁹, so the gravitational factor is 0.999999999304 and a sea-level clock loses 22.0 ms per yearagainst one infinitely far away. On the Sun's photosphere the same calculation gives about 67 seconds a year — the gravitational redshift Einstein predicted in 1911, since a radiating atom is itself a clock. Leave the second-radius box blank to compare against infinity, or fill it in to compare two real altitudes directly.

Why GPS needs both effects at once

A GPS satellite at r = 26 560 km is moving fast, which slows its clock, and sitting high in a weaker field, which speeds it up. The second effect wins. From a two-radius Schwarzschild model with dτ/dt = √(1 − r_s/r − v²/c²) and a 6371 km ground station, speed contributes −7.213 µs/day, altitude contributes +45.717 µs/day, and the net is +38.504 µs/day. Left uncorrected that is about 11 km of positioning error accumulated every day, which is why the satellite oscillators are deliberately detuned before launch.

Computed here, cited elsewhere
Textbooks quote the GPS split as −7.2, +45.9 and +38.6 µs/day. Those come from the full relativistic model, which also carries Earth's rotation, the J₂ oblateness term of the geopotential and the true mean orbital radius. The calculator prints both columns side by side and labels which is which, rather than quietly presenting a literature figure as though the model had produced it.

The precision problem, and how it is solved

Every result on this page is a deviation from one. For an airliner at 900 km/h, γ − 1 is 3.5 × 10⁻¹³. Computing γ and then subtracting 1 destroys that number: a double has about sixteen significant digits, and the leading 1 consumes most of them. The usual workaround is a series expansion, but there is something better — the exact identity

1 − √(1 − x) ≡ x / (1 + √(1 − x))

which holds for every xfrom 0 to 1, has no cancellation anywhere, and needs no threshold to switch on. The numerator is the input untouched; the denominator sits near 2 and dilutes the square root's rounding rather than amplifying it. Pass β² for the kinematic case, r_s/r for the gravitational one, or r_s/r + β² for both together. Everything else — γ − 1, 1 − 1/γ, the satellite drift — is derived from that single function by exact algebra, so no two panels can disagree.

The twin paradox and the muon

A round trip to Proxima Centauri, 4.37 light-years away at 0.5c, takes 17.480 years of Earth time and 15.138 years aboard, an age gap of 2.342 years. The situation is not symmetric because the traveller turns round and changes inertial frames; the stay-at-home twin never does. The same γ explains why cosmic-ray muons reach sea level: with a rest-frame lifetime of 2.197 µs at 0.9994c, γ = 28.872 stretches the lab-frame lifetime to 63.43 µs and the mean travel distance to 19.005 km. Classically they would manage 658 m and never arrive.

Reading the results honestly

The inverse mode answers the science-fiction question directly: γ = 10, one shipboard year per decade at home, needs β = 0.994987. Trip figures assume a constant cruise speed with no acceleration phase, so they are a lower bound on the traveller's ageing. Particle lifetimes are means, not limits — decay is exponential, and a good fraction of any beam survives several times longer. The relativistic energy rows appear only once you supply a rest mass, because E = γm₀c² needs one and the tool will not invent it.

Frequently Asked Questions

Is the Time Dilation Calculator free?

Yes, Time Dilation Calculator is totally free :)

Can I use the Time Dilation Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Time Dilation Calculator?

Yes, any data related to Time Dilation Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

How does this time dilation calculator work?

Every input is normalised to SI first — metres per second, seconds, kilograms, metres — so a speed typed in mph and a radius typed in Earth radii meet on the same footing. The active mode then applies one closed-form relation: γ = 1/√(1 − v²/c²) for the kinematic effect, √(1 − 2GM/rc²) for the gravitational one, and √(1 − r_s/r − v²/c²) when a clock is both moving and deep in a well. Conversion back into your chosen display unit happens only at the last step, so no two panels on the page can ever disagree.

Why does the calculator give γ = 2.0281848 at 0.87c when my worksheet says 2.0278?

Because 2.0278 is wrong. Work it through: 0.87² = 0.7569, 1 − 0.7569 = 0.2431, √0.2431 = 0.49305, and 1/0.49305 = 2.0281848. The rounded figure propagates into everything downstream — the correct time difference over one Julian year is 375.544 days rather than 375.34, the clock slows by 50.6948 % rather than 50.68 %, and a 100 m rod contracts to 49.305 m rather than 49.32. This tool implements the formula and does not carry a printed number forward.

Why is the GPS drift here 38.504 µs/day and not the 38.6 everyone quotes?

The literature figure comes from the full relativistic model of the GPS constellation, which folds in Earth's rotation, the J₂ oblateness term of the geopotential and the true mean orbital radius. This calculator uses a two-radius Schwarzschild model, so from the standard inputs — M = 5.972 × 10²⁴ kg, r_sat = 26 560 km, r_ground = 6371 km, v = √(GM/r) — it gives −7.213 µs/day from speed, +45.717 µs/day from altitude and +38.504 µs/day net. The tool prints the canonical values beside its own and labels them as cited rather than computed, so you can see exactly how much the simplified model costs.

How does the tool keep nanosecond-per-day effects from vanishing into rounding?

Every number this tool reports is a deviation from 1, and for an aircraft that deviation is 3.5 × 10⁻¹³. Computing γ and then subtracting 1 destroys it — a double has only about sixteen digits to spend, and the leading 1 eats most of them. So the arithmetic never subtracts: it uses the exact identity 1 − √(1 − x) ≡ x / (1 + √(1 − x)), where the numerator is the input untouched and the denominator sits harmlessly near 2. That is not a series approximation, it holds for every x from an airliner's 10⁻¹² to a black hole's 0.33, and every other near-unity quantity on the page is derived from it by exact algebra.

Why does the relativistic energy row only appear when I enter a rest mass?

Because E = γm₀c² needs a mass and the tool will not invent one for you. Type a rest mass — in kilograms, atomic mass units, MeV/c², electron masses or proton masses — and the rest energy, total energy and kinetic energy appear, all built from the same γ − 1 as the rest of the page rather than from γE − E. Leave the box blank and the rows stay hidden, because a number with no stated input behind it is not a result.

Is gravitational time dilation the same thing as gravitational redshift?

They are two readings of one fact. A clock deeper in a potential well ticks slow by √(1 − r_s/r) as seen from higher up, so any light it emits arrives with its frequency lowered by exactly that factor — the emitting atom is itself a clock. The Sun's photosphere loses about 67 seconds a year against a distant observer, which is a redshift of 2.1 × 10⁻⁶ and was the effect Einstein predicted in 1911; the far larger shift from the white dwarf Sirius B was measured spectroscopically in 1925 and is one of the earliest confirmations of general relativity.