Time Dilation Calculator (Velocity)
Calculates the dilated (coordinate) time experienced by a stationary observer when a moving clock ticks off a given proper time at relativistic speeds. Essential for understanding special relativity and GPS corrections.
About this calculator
Special relativity predicts that a moving clock runs slower than a stationary one — a phenomenon called time dilation. If a traveler measures a proper time interval τ on their own clock, a stationary observer measures a longer coordinate time t given by: t = τ / √(1 − v²/c²), or equivalently t = γτ, where γ = 1/√(1 − v²/c²) is the Lorentz factor. Here v is the relative velocity between the frames and c ≈ 2.998 × 10⁸ m/s. Proper time τ is always the shortest time interval — the one measured by the clock moving with the object. As v approaches c, γ → ∞, and the stationary observer measures an ever-longer elapsed time for each tick of the moving clock. At everyday speeds γ is indistinguishable from 1, but GPS satellites must correct for a time dilation of ~7 μs/day due to their orbital velocity.
How to use
A muon produced in the upper atmosphere has a proper lifetime of τ = 2.2 μs and travels at v = 0.99c. Step 1: compute β = 0.99, β² = 0.9801. Step 2: √(1 − 0.9801) = √0.0199 ≈ 0.1411. Step 3: t = 2.2 μs / 0.1411 ≈ 15.6 μs. In Earth's frame the muon lives 15.6 μs, long enough to travel ~4.6 km and reach the surface — a result confirmed experimentally. Enter properTime = 2.2 (μs), velocity = 0.99c to reproduce this calculation.
Frequently asked questions
How does velocity-based time dilation work in special relativity?
When an object moves at velocity v relative to an observer, its internal clock — and all physical processes — tick more slowly as seen by that observer. The ratio of dilated time to proper time is the Lorentz factor γ = 1/√(1 − v²/c²). This is not an optical illusion or a measurement artifact; the time dilation is real and symmetric from each frame's perspective. It has been confirmed experimentally with atomic clocks on aircraft, subatomic particles in accelerators, and corrections built into the GPS satellite network.
What is the difference between proper time and dilated time in special relativity?
Proper time τ is the time elapsed on a clock that travels with the moving object — it is the shortest possible elapsed time between two events on that worldline. Dilated (coordinate) time t is the longer time interval measured by a stationary observer watching that moving clock. The relationship is t = γτ, so the stationary observer always measures more elapsed time than the traveler's clock shows. Only the traveler can measure proper time for their own journey; all other observers measure a larger value depending on their relative velocity.
Why do GPS satellites need to correct for time dilation due to velocity?
GPS satellites orbit at about 14,000 km/h, which is fast enough that special relativistic time dilation slows their onboard clocks by approximately 7 microseconds per day relative to clocks on Earth's surface. There is also a competing general relativistic effect from the weaker gravity at altitude that speeds the clocks up by about 45 μs/day. The net effect is a clock gain of roughly 38 μs/day. Without correcting for both effects, GPS position errors would accumulate at roughly 10 km per day, making the system useless for navigation. These corrections are one of the most practical everyday confirmations of Einstein's relativity.