Time Dilation Calculator
Free Time dilation Calculator for relativity. Enter variables to compute results with formulas and detailed steps. See charts, tables, and visual results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
t = tau * gamma = tau / sqrt(1 - v^2/c^2)
Where t = dilated time (measured by stationary observer), tau = proper time (measured by moving clock), gamma = Lorentz factor, v = relative velocity, and c = speed of light. The moving clock always measures less time than the stationary observer.
Worked Examples
Example 1: Cosmic Ray Muon Survival
Problem:Muons created 15 km above Earth move at 0.998c and have a rest-frame half-life of 2.2 microseconds. Can they reach the ground?
Solution:gamma = 1/sqrt(1 - 0.998^2) = 1/sqrt(0.003996) = 15.82\nDilated half-life: 2.2 * 15.82 = 34.8 microseconds\nTravel time at 0.998c over 15 km: 15000 / (0.998 * 3e8) = 50.1 microseconds\nNumber of half-lives: 50.1 / 34.8 = 1.44\nFraction surviving: (1/2)^1.44 = 0.369 = 36.9%\nWithout time dilation: 50.1 / 2.2 = 22.8 half-lives, survival fraction = (1/2)^22.8 = 1.4e-7
Result:With time dilation: 36.9% survive | Without: 0.000014% | Time dilation confirmed by cosmic ray observations
Example 2: Twin Paradox: Trip to Alpha Centauri
Problem:A twin travels to Alpha Centauri (4.37 light-years) at 0.9c. How much does each twin age for the one-way trip?
Solution:gamma = 1/sqrt(1 - 0.81) = 1/sqrt(0.19) = 2.294\nEarth-frame travel time: 4.37 / 0.9 = 4.856 years\nTraveler proper time: 4.856 / 2.294 = 2.117 years\nStay-home twin ages 4.856 years, traveler ages 2.117 years\nAge difference: 4.856 - 2.117 = 2.739 years\nFrom traveler frame: distance contracts to 4.37 / 2.294 = 1.905 light-years\nTravel time: 1.905 / 0.9 = 2.117 years (consistent!)
Result:Earth twin: 4.86 years older | Traveler: 2.12 years older | 2.74 years younger after one-way trip
Frequently Asked Questions
What is time dilation in special relativity?
Time dilation is the phenomenon where time passes at different rates for observers in relative motion. A clock moving relative to a stationary observer ticks more slowly, as measured by that stationary observer. This effect is quantified by the Lorentz factor gamma = 1/sqrt(1 - v^2/c^2): a moving clock records a time interval tau (proper time) while the stationary observer measures a longer interval t = gamma * tau. At 87% of light speed, time runs at half the normal rate (gamma = 2). This is not an illusion or mechanical malfunction but a fundamental feature of spacetime itself. Every physical process, from atomic vibrations to biological aging, is equally affected by time dilation.
How has time dilation been experimentally verified?
Time dilation has been confirmed by numerous experiments with extraordinary precision. The Hafele-Keating experiment (1971) flew cesium atomic clocks around the world on commercial aircraft and measured time differences of hundreds of nanoseconds, matching relativistic predictions. Muons created by cosmic rays in the upper atmosphere survive to reach ground level because their 2.2-microsecond half-life is extended by time dilation at their typical speeds of 0.998c (gamma approximately 15). Particle accelerators routinely observe extended lifetimes of unstable particles. The GPS system must continuously correct for time dilation effects. Most recently, optical lattice clocks have measured time dilation between two clocks separated by just one meter of height difference on Earth surface.
How does time dilation affect GPS satellites?
GPS satellites experience two competing relativistic time effects. First, their orbital velocity of about 3.87 km/s causes special relativistic time dilation that makes their clocks tick about 7 microseconds per day slower than ground clocks. Second, being 20,200 km above Earth in weaker gravity causes general relativistic time dilation (gravitational blueshift) that makes their clocks tick about 45 microseconds per day faster. The net effect is that satellite clocks gain about 38 microseconds per day relative to ground clocks. Without correcting for this, GPS positions would drift by roughly 10 kilometers per day. The correction is applied by setting satellite clock frequencies slightly lower before launch so they match ground clocks when in orbit.
What is proper time and how does it relate to dilated time?
Proper time (tau) is the time measured by a clock that is at rest relative to the observer, or equivalently, the time measured along the worldline of an object in its own rest frame. It is the shortest time interval between two events as measured by any inertial observer, and it is a Lorentz invariant quantity. Dilated time (t) is the time measured by an observer who sees the clock moving, and it is always longer than proper time: t = gamma * tau. The concept of proper time extends to general relativity, where it equals the integral of sqrt(g_mu_nu dx^mu dx^nu) along a worldline, accounting for both velocity and gravitational time dilation. Proper time is the physical aging experienced by an observer along their specific path through spacetime.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy