concept

Time dilation

Time dilation is the relativistic effect where time passes slower in stronger gravitational fields or at higher velocities.
18 related articles · 6 formulas · 6 who worked here

Description

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.

History

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.

How it works

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.

In practice

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.

Fun fact

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.

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Formulas

$$P(t) \propto t^{-\alpha}$$
Probability decays as a power law in time.
What is in the formula
Pvariableprobability of survival or decay — probability
tvariabletime — time second
αvariablepower-law exponent — exponent
p(t)\proptot^{-α} · 2509.17163
τ\proptop^{α} · 2607.17359v1
$$t_{\text{ff}} \propto \frac{1}{\sqrt{G \rho}}$$
Free-fall time scales inversely with the square root of density.
What is in the formula
t_ffvariablefree-fall time — time second
Gvariablegravitational constant — gravitational constant cubic metre per kilogram per second squared
ρvariablemass density — mass density kilogram per cubic metre
t\propto\frac{1}{m} · 2512.15393
$$\frac{\Delta f}{f} = \frac{g \Delta h}{c^2}$$
Clock frequency increases with height in a gravitational field.
What is in the formula
ΔfvariableChange in clock frequency — frequency change hertz
fvariableClock frequency at reference height — frequency hertz
gvariableLocal gravitational acceleration — gravitational acceleration metre per second squared
ΔhvariableHeight difference — height difference metre
cconstantSpeed of light in vacuum — speed of light = 299792458 metre per second
\frac{δf}{f}=\frac{gδh}{c^2} · 2604.17157
$$\Delta t = \frac{D_{\Delta t}}{c} \Delta\phi$$
Time delay is proportional to the time delay distance and lensing potential difference.
What is in the formula
\Delta tvariabletime delay between images — time delay second
D_{\Delta t}variabletime delay distance — distance metre
\Delta\phivariabledifference in lensing potential — lensing potential difference
cconstantspeed of light in vacuum — speed of light = 299792458 metre per second
δt_{ij}=\frac{d_{δt}}{c}δφ_{ij} · 2606.03634v1
$$D_{\Delta t} = (1+z_l) \frac{D_l D_s}{D_{ls}}$$
Time delay distance relates angular diameter distances and Hubble constant in lensing.
What is in the formula
D_{\Delta t}variableTime delay distance — time delay distance metre
z_lvariableRedshift of the lens — redshift
D_lvariableAngular diameter distance to the lens — angular diameter distance metre
D_svariableAngular diameter distance to the source — angular diameter distance metre
D_{ls}variableAngular diameter distance between lens and source — angular diameter distance metre
d_{δt}=(1+z_l)\frac{d_ld_s}{d_{ls}}\propto1/h_0 · 2606.22390
$$\Delta t = \frac{D_{\Delta t}}{c} \Delta\tau$$
Time delay from path difference in lensing.
What is in the formula
Δtvariabletime delay between two images or signals — time delay second
D_Δtvariabletime-delay distance, a combination of angular diameter distances in the lens system — time delay distance metre
Δτvariabledifference in the Fermat potential between two light paths — fermat potential difference
cconstantspeed of light in vacuum — speed of light = 299792458 metre per second
δt_{ij}=\frac{d_{δt}}{c}[τ(θ_i)-τ(θ_j)] · 2606.25205

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