Time dilation is the relativistic effect where time passes slower in stronger gravitational fields or at higher velocities.Time dilation is an effect of relativity theory in which time intervals stretch in a moving reference frame or in a gravitational field compared to a stationary observer. In special relativity, time slows down by a factor of γ = 1/√(1 − v²/c²), where v is the speed of motion and c is the speed of light. Gravitational time dilation is predicted by general relativity: clocks in a stronger gravitational potential run slower. Both effects have been precisely measured and they add together.
The foundations were laid in the works of Hendrik Lorentz (1892, 1895) and Henri Poincaré (1905), who introduced 'local time' and coordinate transformations. In 1905, Albert Einstein showed that time dilation is a direct consequence of two postulates: the equivalence of inertial frames and the constancy of the speed of light. Experimental confirmation came from the observation of atmospheric muons (1941): the lifetime of fast muons was extended, and they reached the surface. Later, the effect was tested with atomic clocks on airplanes (Hafele–Keating, 1971) and is used daily in GPS.
Time dilation preserves the invariance of proper time dτ² = dt² − dx²/c² in flat spacetime. In a moving system, worldlines maximize proper time, leading to the 'twin paradox.' Gravitational time dilation arises from the metric: the component g₀₀ = (1 + 2φ/c²) determines the rate of clocks. The equivalence principle connects both types: acceleration is indistinguishable from gravity, so clocks in an accelerating rocket also slow down.
GPS satellites travel at tremendous speeds and are farther from Earth, where gravity is weaker. Without relativistic corrections, their clocks would diverge from Earth clocks by 38 microseconds per day, and navigation would be off by kilometers daily.
On GPS satellites, time runs faster by 45 microseconds per day due to weaker gravity (they are farther from Earth), but slower by 7 microseconds due to speed; the net correction is 38 microseconds, without which the navigation error would reach tens of kilometers after one day.
| P | variable | probability of survival or decay — probability |
| t | variable | time — time second |
| α | variable | power-law exponent — exponent |
| t_ff | variable | free-fall time — time second |
| G | variable | gravitational constant — gravitational constant cubic metre per kilogram per second squared |
| ρ | variable | mass density — mass density kilogram per cubic metre |
| Δf | variable | Change in clock frequency — frequency change hertz |
| f | variable | Clock frequency at reference height — frequency hertz |
| g | variable | Local gravitational acceleration — gravitational acceleration metre per second squared |
| Δh | variable | Height difference — height difference metre |
| c | constant | Speed of light in vacuum — speed of light = 299792458 metre per second |
| \Delta t | variable | time delay between images — time delay second |
| D_{\Delta t} | variable | time delay distance — distance metre |
| \Delta\phi | variable | difference in lensing potential — lensing potential difference |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |
| D_{\Delta t} | variable | Time delay distance — time delay distance metre |
| z_l | variable | Redshift of the lens — redshift |
| D_l | variable | Angular diameter distance to the lens — angular diameter distance metre |
| D_s | variable | Angular diameter distance to the source — angular diameter distance metre |
| D_{ls} | variable | Angular diameter distance between lens and source — angular diameter distance metre |
| Δt | variable | time delay between two images or signals — time delay second |
| D_Δt | variable | time-delay distance, a combination of angular diameter distances in the lens system — time delay distance metre |
| Δτ | variable | difference in the Fermat potential between two light paths — fermat potential difference |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |










