Time Dilation Calculator

dilated time = Lorentz factor × proper time, where the Lorentz factor is 1 ÷ √(1 − v²/c²)

v

The speed of the moving clock as a fraction of the speed of light, so 0.9 means nine tenths of c. Entering it this way is deliberate: light speed is the unit here, not a constant you have to type. The field runs from 0 to 1, and exactly 1 is not a value the page can use, because the whole formula divides by something that reaches zero there

Δτ

The proper time — what the travelling clock itself reads, in hours, minutes, seconds or days. It is the smaller of the two times and the one the traveller experiences. This page has two clocks and this is the one you have to keep separate from the other: fill in the trip as the crew would experience it, not as mission control would

γ

The Lorentz factor, 1 ÷ √(1 − v²/c²), printed to four decimals with no unit because it is a ratio. It is the whole subject of this page: at any speed a person or a vehicle reaches it agrees with 1 to eleven decimal places — 1000 m/s still gives only 1.0000000000056 — so the page prints 1.0000; at 0.1 c it is 1.005; and only past about 0.9 c does it start to look like a number worth a table. Everything else on the page is this factor multiplied by a time or subtracted from one

Δt

The dilated time — what a clock that stayed behind reads between the same two events. It is always the larger of the two, and the page shows it as a duration rather than a count of seconds: five hours aboard at 0.9 c comes back as 11 hours 28 minutes 15 seconds. There is no case where it is smaller, because the factor never drops below one

Δt − Δτ

The gap between the two clocks, also shown as a duration. It is not the same as the dilated time and not a multiple of it — it is what is left over after the traveller's own hours are subtracted, and it is the number the twin paradox is actually asking about. At 0.9 c over five hours the gap is 6 hours 28 minutes 15 seconds

Use this page when you want to know what a fast trip costs in time rather than in distance: how much a crew would age on a journey at some respectable fraction of light speed, what a muon's lifetime looks like from the ground, or why a clock on a satellite has to be corrected. The factor is the part worth looking at, because the answer surprises people in both directions — at 0.1 c, which is faster than anything ever built, clocks disagree by half a percent, while the last stretch from 0.99 c to 0.999999 c multiplies the effect a hundredfold. The outbound half of the twin paradox lives here: send one twin away at 0.9 c and the other twin's calendar runs more than twice as fast.

1. The defaults: 0.9 c for five hours aboard

   1. Speed 0.9 c, proper time 5 hours
   2. Lorentz factor: 1 ÷ √(1 − 0.9²) = 1 ÷ √0.19 = 2.294157, which prints as 2.2942
   3. Five hours is 5 × 3600 = 18000 seconds aboard
   4. Dilated time: 2.294157 × 18000 = 41294.83 seconds, rounded to 41295 — that is 11 hours 28 minutes 15 seconds
   5. Gap: 41295 − 18000 = 23295 seconds, or 6 hours 28 minutes 15 seconds

   The two duration rows are the reason this page is not just a multiplication. The travelling clock reads five hours; the clock at home reads eleven hours twenty-eight minutes; and the difference between them, six hours twenty-eight minutes, is the third number and the one the twin paradox is about. Note that the gap is not a multiple of the proper time either — it is 23295 seconds, which is 1.294 times the 18000 the traveller measured, not 2.29 times. Subtracting γ times the time from the time does not give the difference; subtracting the two times does.

2. Half light speed for one hour: the effect is real but modest

   1. Speed 0.5 c, proper time 1 hour
   2. Lorentz factor: 1 ÷ √(1 − 0.25) = 1 ÷ √0.75 = 1.154701, which prints as 1.1547
   3. One hour aboard is 3600 seconds
   4. Dilated time: 1.154701 × 3600 = 4156.92 seconds, rounded to 4157 — 1 hour 9 minutes 17 seconds
   5. Gap: 4157 − 3600 = 557 seconds, or 9 minutes 17 seconds

   Half the speed of light sounds extreme and, for the clock, is not: a year aboard would be a year and two months at home. This case is here because of the rounding rather than the physics — a truncating implementation instead of a rounding one reports 4156 and 556, one second short on both rows, and that second is invisible unless you check the arithmetic the way the steps above do. It is also the case that shows the factor is symmetric in a way people do not expect: 0.5 c gives 1.1547, and so does 0.5 c in any direction, because the formula contains v² and not v.

3. 0.999999 c for one hour: twenty-nine days at home

   1. Speed 0.999999 c, proper time 1 hour
   2. Lorentz factor: 1 ÷ √(1 − 0.999998000001) = 1 ÷ √(1.999999 × 10⁻⁶) = 707.106958, which prints as 707.107
   3. One hour aboard is 3600 seconds
   4. Dilated time: 707.106958 × 3600 = 2545585.05 seconds, rounded to 2545585 — 29 days 11 hours 6 minutes 25 seconds
   5. Gap: 2545585 − 3600 = 2541985 seconds, or 29 days 10 hours 6 minutes 25 seconds

   One hour of travel and twenty-nine days of home time, and that is what the last digit of the input is buying: at 0.999 c the factor is 22.37, and at 0.999999 c it is 707.11, from a change of one part in a thousand in the speed. That steepness is the single most useful thing to take away from this page, and it is why the table's six speeds are bunched at the top rather than spread evenly. The arithmetic in the second step is the one place where the square root is easy to get wrong by hand: √2 ≈ 1.4142 would suggest 707.1068, and the printed value has a zero in the fourth decimal, not an eight.

4. At rest: the two clocks agree exactly

   1. Speed 0 c, proper time 1 hour
   2. Lorentz factor: 1 ÷ √(1 − 0) = 1, which prints as 1.0000
   3. Dilated time: 1 × 3600 = 3600 seconds, which is 1 hour
   4. Gap: 3600 − 3600 = 0 seconds
   5. Nothing moves, nothing runs slow

   This row is the anchor for everything above it: with no relative motion there is no effect at all, and the factor is exactly 1. It is also the row that shows why four decimals are printed rather than two — at 0.05 c the factor is 1.00125, which two decimals would print as 1.00, indistinguishable from standing still, while at four decimals the 0.1 c row reads 1.005 and the difference is there to see. The printed 1.0000 is not padding either; it says the disagreement is below the fourth decimal, which is a fact about zero speed rather than a formatting choice.

Only the outbound leg is covered: a clock that turns round and comes back has to be treated as two separate constant-velocity legs, and it is the traveller's clock that ends up behind. Gravity is not modelled, though a clock lower in a gravitational field also runs slow, by a different formula. The speed is measured relative to whoever is watching, and the trip is assumed to be at that speed throughout, so nothing here accounts for the acceleration at either end.

What is the time dilation formula?

Dilated time equals the Lorentz factor times the proper time, Δt = γΔτ, where γ is 1 divided by the square root of 1 − v²/c². Five hours aboard a ship at 0.9 c becomes 11 hours 28 minutes 15 seconds on the ground, because γ is 2.2942 there. The factor depends only on the speed, so changing the trip's duration scales both times together and leaves γ untouched.

Why does the Lorentz factor stay near 1 for so long?

Because the formula contains v²/c², so at low speed the thing being subtracted from 1 is tiny and the square root of a number near 1 is near 1. At 0.1 c — thirty million metres per second, faster than any spacecraft — the factor is 1.005, a half a percent. Even at half the speed of light it is only 1.1547. The steep part arrives late: 7.09 at 0.99 c, 22.37 at 0.999 c, and 707 at 0.999999 c.

How is this different from the twin paradox?

This page gives the outbound half of it. If one twin leaves at 0.9 c and the other stays, the stay-at-home twin's clock runs 2.2942 times as fast, so five years aboard is eleven and a half years at home. The full paradox needs the traveller to turn round and come back, which is two legs rather than one and involves acceleration — that is what breaks the symmetry between the two twins, and it is outside what this page models.

Does gravity cause time dilation too?

Yes, and this page does not cover it. A clock deeper in a gravitational field runs slow relative to one higher up, which is a separate effect with its own formula and is why satellite clocks need correcting for both. What this page covers is the special-relativity effect, which depends only on relative speed and applies even with no gravity at all. The two are added together in real navigation systems.

What happens at exactly the speed of light?

The page refuses the input. At v = c the quantity 1 − v²/c² is zero and the Lorentz factor divides by zero, so there is no answer rather than an infinite one. Above the speed of light the square root becomes the root of a negative number. The field accepts 0 to 1 and the calculator rejects exactly 1, which is the honest split: reaching light speed is not a very large number on this page, it is the point where the formula stops being defined.

• Time Dilation (College Physics 2e, §28.2) — the Lorentz factor, the two clocks, and the muon and satellite evidence for the effect — OpenStax
• Time dilation — the derivation of γ = 1 ÷ √(1 − v²/c²), the twin paradox, and the difference between this effect and gravitational time dilation — Wikipedia
• NIST Guide for the Use of the International System of Units (SI), Appendix B.8 — Factors for Units Listed Alphabetically: the hour, minute and day rows (1 h = 3600 s) the two duration rows are ultimately counted in — NIST