What is the time dilation formula?

The Lorentz factor, gamma = 1/√(1 - β²), where β is the speed as a fraction of the speed of light. Coordinate time is the proper time multiplied by gamma, so at beta 0.600 gamma is 1.250000 and one year on the moving clock is 1.250 yr in the frame watching it. At beta 0.99 gamma is 7.088812.

What is the difference between proper time and coordinate time?

Proper time is what a clock reads along its own path, between two events that happen in the same place for that clock. Coordinate time is what the frame watching it measures between those same two events, using clocks at rest in that frame. The conversion runs one way, coordinate = gamma times proper, which is why this tool asks which of the two you entered and labels both in the answer.

Why does gamma come out as exactly 1 at ordinary speeds?

Because 1/sqrt(1 - v^2/c^2) evaluated left to right in double precision throws the answer away. At 30 m/s, beta squared is about 1.0e-14, and subtracting that from 1 leaves only a couple of its significant figures; at 1 m/s the subtraction returns exactly 1, and gamma with it. This page computes gamma minus 1 through expm1 and log1p instead, so 30 m/s comes back as 1 + 5.007e-15, a gap of 158.007 ns over a Julian year of 31,557,600 seconds.

Does this include the gravitational time dilation that GPS corrects for?

No. This is special relativity between inertial frames, so it accounts for relative speed and nothing else. A satellite moving at 3,874 m/s runs 7.214 µs a day slow from its speed alone, which this tool works out, but gravitational time dilation runs the other way, is larger, and is not computed here.

How fast is 0.6 c in metres per second?

179,875,474.8 m/s. c is exactly 299,792,458 m/s by definition, because the metre has been defined from it since 1983, so turning a fraction of c into m/s adds no measurement error of its own. The tool prints that figure beside the answer whichever unit you type the speed in.

What happens at or above the speed of light?

The speed is refused with the reason, rather than printed as an infinity. At exactly c the formula divides by zero, and above c there is no inertial frame to convert between. Every speed a double can tell apart from c is accepted, including one a hair under it: at beta 0.9999999999 gamma is 70710.675195.

Time Dilation Calculator

Enter a speed, as a fraction of c or in m/s, km/h, km/s, mph or ft/s, with one time interval and which clock measured it. The page returns gamma, beta, the proper time and the coordinate time together: at beta 0.600 gamma is 1.250000, so 1.000 yr on the moving clock is 1.250 yr in the frame watching it, and 0.600 c is 179,875,474.8 m/s because c is exactly 299,792,458 m/s by definition. The arithmetic is arranged so that ordinary speeds survive it: evaluating 1/sqrt(1 - v^2/c^2) directly leaves a car's gamma with about two significant figures and 1 m/s with none at all, while this page works out gamma minus 1 first and reports 1 + 5.007e-15 at 30 m/s, a gap of 158.007 ns over a year. Special relativity between inertial frames only, with no gravitational time dilation anywhere in it.

Gamma1.250000beta 0.600 · 1.000 yr proper becomes 1.250 yr coordinate

Speed

Speed in

Interval

Interval in

That interval is the

Enter a speed, or a fraction of c, and one interval. Say which clock that interval belongs to: the proper time is what the moving clock reads, and the coordinate time is what the frame watching it reads.

Gamma

1.250000

Beta

0.600

Proper time

1.000 yr

Coordinate time

1.250 yr

gamma = 1/√(1 - (0.600)²) = 1.250000. 1.000 yr of proper time is 1.250 yr of coordinate time, a gap of 0.250 yr. The moving clock advances 0.800000 seconds for every second in the frame watching it.

179,875,474.8 m/s is beta 0.600: c is exactly 299,792,458 m/s by definition, so that conversion adds no error of its own.

A year here is the Julian year: 365.25 days, or 31,557,600 seconds exactly. The ratio itself does not depend on the unit.

Three cases, worked

Two speeds a physics problem picks, and rest. Every line below is computed by the code that answers the calculator above, so the examples and the tool can never disagree.

beta = 0.6gamma = 1/√(1 - β²)
1.250000
beta = 0.99the same formula
7.088812
beta = 0no relative motion
gamma 1, the two clocks agree

Accuracy. Exact special-relativistic arithmetic: gamma is 1/√(1 - β²) for one inertial frame moving at constant velocity relative to another, and the page labels which interval is the proper time and which is the coordinate time, because the conversion runs one way only. A speed at or above c is rejected rather than returned as infinity. Gravity and accelerating frames are outside this model and are not approximated here.

Common questions

What is the time dilation formula?: The Lorentz factor, gamma = 1/√(1 - β²), where β is the speed as a fraction of the speed of light. Coordinate time is the proper time multiplied by gamma, so at beta 0.600 gamma is 1.250000 and one year on the moving clock is 1.250 yr in the frame watching it. At beta 0.99 gamma is 7.088812.
What is the difference between proper time and coordinate time?: Proper time is what a clock reads along its own path, between two events that happen in the same place for that clock. Coordinate time is what the frame watching it measures between those same two events, using clocks at rest in that frame. The conversion runs one way, coordinate = gamma times proper, which is why this tool asks which of the two you entered and labels both in the answer.
Why does gamma come out as exactly 1 at ordinary speeds?: Because 1/sqrt(1 - v^2/c^2) evaluated left to right in double precision throws the answer away. At 30 m/s, beta squared is about 1.0e-14, and subtracting that from 1 leaves only a couple of its significant figures; at 1 m/s the subtraction returns exactly 1, and gamma with it. This page computes gamma minus 1 through expm1 and log1p instead, so 30 m/s comes back as 1 + 5.007e-15, a gap of 158.007 ns over a Julian year of 31,557,600 seconds.
Does this include the gravitational time dilation that GPS corrects for?: No. This is special relativity between inertial frames, so it accounts for relative speed and nothing else. A satellite moving at 3,874 m/s runs 7.214 µs a day slow from its speed alone, which this tool works out, but gravitational time dilation runs the other way, is larger, and is not computed here.
How fast is 0.6 c in metres per second?: 179,875,474.8 m/s. c is exactly 299,792,458 m/s by definition, because the metre has been defined from it since 1983, so turning a fraction of c into m/s adds no measurement error of its own. The tool prints that figure beside the answer whichever unit you type the speed in.
What happens at or above the speed of light?: The speed is refused with the reason, rather than printed as an infinity. At exactly c the formula divides by zero, and above c there is no inertial frame to convert between. Every speed a double can tell apart from c is accepted, including one a hair under it: at beta 0.9999999999 gamma is 70710.675195.

Exact special-relativistic arithmetic: gamma is 1/√(1 - β²) for one inertial frame moving at constant velocity relative to another, and the page labels which interval is the proper time and which is the coordinate time, because the conversion runs one way only. A speed at or above c is rejected rather than returned as infinity. Gravity and accelerating frames are outside this model and are not approximated here.