Time Dilation Calculator

What time dilation means

Two people with identical stopwatches will not agree on how much time passed if one of them is moving. The moving clock reads less. That gap is time dilation, and it comes from the geometry of spacetime rather than from a fault in the watch.

Special relativity ties the size of the effect to speed alone. A clock on a jet, a satellite or a particle in an accelerator runs slow relative to the ground clock by a factor set by v divided by c, the speed of light. At walking pace the factor differs from 1 by about one part in 10^17, which is why nobody notices until the clock is atomic or the speed is a decent fraction of c.

One quantity shows up in every version of the arithmetic: the Lorentz factor, written gamma. It converts between proper time, the reading on the clock that makes the trip, and coordinate time, the reading on the clock that stays home. The calculator above returns that factor, the moving clock reading and the gap between the two clocks.

The Lorentz factor and the formula

The formula is gamma = 1 / sqrt(1 - v squared / c squared), and the moving clock reads t divided by gamma. Nothing else enters: no mass, no distance, no direction. The speed of light is exact by definition, and NIST's CODATA entry lists c as 299,792,458 metres per second with zero uncertainty, which is the value used here.

Two limits are worth memorising. As speed goes to zero, gamma goes to 1 and the clocks agree. As speed goes to c, gamma grows without bound, so no finite push gets an object with mass to c. At 0.99c gamma is 7.089; at 0.999c it is 22.366.

Because gamma depends on speed squared, the effect stays tiny far from c and then climbs steeply. A probe at 0.1c has gamma 1.005, so one year of Earth time equals 363.4 days on the probe, a gap of 1.83 days. Push the same probe to 0.9c and one Earth year becomes 159.2 days on board.

Worked example: five years at 0.8c

A ship leaves at 0.8c and returns when the ground clock has recorded 5.00 years. Gamma at 0.8c is 1.667, so the ship clock records 5.00 / 1.667 = 3.00 years. The traveller comes home 2.00 years younger than the twin who stayed put.

Run the same factor in the other direction and the arithmetic holds. One year on a ship at 0.9c matches 2.294 years on the ground, and one year at 0.99c matches 7.089 years. Those are the numbers behind the twin paradox, and they are symmetric in the sense that each observer measures the other clock as the slow one during the steady cruise.

The turnaround is what breaks the symmetry. The travelling twin feels acceleration, changes inertial frame and comes back to a single clock comparison that both sides can read, and that clock is the one that lost time. The steady-speed formula above covers the cruise legs; the acceleration itself contributes a separate term that this tool does not model.

Dilation across the speed range

The table makes the shape of the curve obvious. Below 0.5c the correction is small enough to ignore for anything short of precision timing. Between 0.8c and 0.99c gamma more than quadruples, which is why fiction reaches for those speeds when it wants a noticeable age gap.

Two rows deserve a second look. At 0.8660c gamma is 2.000, the point where the ship clock runs at half speed, so a 10 year mission ages the crew 5 years. At 0.999c a single ship year corresponds to 22.366 Earth years, and the fuel budget for that trip is a separate problem.

Read the last two columns as conversion rates. Multiply Earth years by the fourth column to get ship years, or multiply ship years by the fifth column to get Earth years. The two columns are reciprocals of each other, which is a quick way to check any row.

Everyday speeds and the size of the effect

Time dilation is measurable long before it is dramatic. An airliner at 250 metres per second accumulates about 30 nanoseconds of lag per day, which is far below the drift of a mechanical watch but easy for an atomic clock. The International Space Station at 7.66 km/s loses about 28.2 microseconds per day from speed alone, and the weaker gravity at 400 km adds back only about 3.8 microseconds, leaving a net slowdown of roughly 24 microseconds per day for the crew.

Every figure in that table is computed from the same formula the calculator uses, so you can reproduce any row by entering the speed and 24 hours. The jump from the station to the 0.1c probe is the point: a factor of about 3,900 in speed turns microseconds into minutes.

Why GPS satellites carry the correction

GPS works by timing radio signals that travel at the speed of light, so a clock error of one microsecond becomes a range error of about 300 metres. The satellites orbit fast enough and high enough that relativity cannot be ignored. NIST's atomic clock group puts the speed effect at 7 microseconds per day slower and the weaker gravity at 45 microseconds per day faster, for a net gain of 38 microseconds per day.

Multiply that 38 microseconds by the speed of light and the position error comes to about 11.4 km per day, which matches the roughly 11 km figure in NASA's GNSS training material. The same material notes that satellite clocks are built with a fixed fractional offset of -4.46475 x 10^-10 to cancel the net effect, and that receivers apply a further correction of up to 23 nanoseconds for orbital eccentricity.

None of this needs the traveller to notice anything. The correction is baked into the hardware and the navigation message, and the blue dot on a phone depends on it every second.

What this calculator covers, and what it leaves out

The tool applies special relativity to steady motion: one speed, one elapsed time, one Lorentz factor. It returns the speed as a share of light speed, gamma, the clock reading on the moving object, and the gap between that clock and the one that stayed still. Those four numbers cover most homework, most sanity checks and most arguments about how much a fast trip would age a crew.

Gravitational time dilation is out of scope. A clock deeper in a gravity well runs slow, and the size of that effect depends on the mass of the planet and the height difference, not on speed. GPS needs both terms, which is why the 38 microsecond net in the table above is larger than the 7 microsecond speed term alone.

Two practical notes on the inputs. Speeds must stay below c, and the calculator reports an error at or above it because gamma is undefined there. For speeds in the metres per second range the gap is a fraction of a millisecond, so the tool reports it in microseconds and the gamma field shows the tiny difference.