Time Dilation — Physica

Equivalent forms

\Delta t = \Delta t_0 / \sqrt{1-\beta^2}

\Delta t \approx \Delta t_0 (1 + v^2/(2c^2))

Two postulates and a Pythagorean triangle yield the entire structure of relativistic time.

Symbol	Name	SI unit	Dimension	Range
$\Delta t$	Dilated timeoutput Time interval measured by stationary observer	s	T	0 – 10000000000
$\Delta t_0$	Proper time Time measured by clock at rest in its own frame	s	T	0 – 10000000000
$v$	Relative velocity Speed of moving frame relative to observer	m/s	L·T⁻¹	0 – 299792457
$c$	Speed of light Speed of light in vacuum	m/s	L·T⁻¹	299792458 – 299792458

Unit systems

Symbol	SI	CGS	Imperial
$\Delta t$	s	—	—
$\Delta t_0$	s	—	—
$v$	m/s	cm/s	ft/s
$c$	m/s	cm/s	ft/s

Where it holds

Exact in special relativity for inertial frames. General relativity adds gravitational time dilation

\Delta t

_grav

= \Delta t_{0}/\surd (1 - 2GM/(rc^{2}))

. For GPS, both effects matter

(\sim 38 \mu s/day

net).

Dimensional analysis

$[\Delta t] = T = T$ / dimensionless $\checkmark (\gamma is$ dimensionless)

Discovery

Albert Einstein · 1905

In his special relativity paper, Einstein derived time dilation directly from the postulate that c is invariant. Demonstrated experimentally by Bailey et al. (1977) measuring muon lifetimes at 0.9994c in the CERN storage ring — agreement to 0.2%.

Try this

Why do GPS satellites' clocks tick faster than yours?

A clock moving at speed v relative to you ticks slower by the Lorentz factor γ. How much slower, and when does this matter for everyday technology?

Research status: stable

Real-world applications

GPS — without relativistic corrections, positions would drift $\sim 10\,\mathrm{km/day}$ .
Particle physics — long muon flight paths in storage rings (g-2 experiment).
Cosmic ray atmosphere physics — $\mu ^{-}$ from 15 km altitude reach surface only $via \gamma \approx 10$ –20 dilation.
Atomic clocks — comparing optical clocks at different heights tests general-relativistic redshift to $10^{-18}$ .

Common misconceptions

Time dilation only happens at relativistic speeds — actually all motion dilates time; it's just minute below $\sim 10$ % c.
The 'slow' clock is the moving one absolutely — no, each observer sees the other's clock as slow (symmetry of inertial frames).
Twin paradox proves SR inconsistent — false; the traveling twin's acceleration breaks the symmetry.

Experimental verification

Hafele-Keating (1971): atomic clocks on commercial jets vs ground showed predicted nanosecond shifts. Muon decay in atmosphere (cosmic ray secondaries reach surface despite

2.2 \mu s

lifetime). GPS routinely corrects

for \sim 7 \mu s/day SR

slowdown +

\sim 45 \mu s/day GR

speedup. Direct measurement in Bailey 1977 to 0.2%.

Derivation

Consider a light clock: photon bouncing between mirrors separated by L.

In the rest frame, period

= 2L/c

.

In a frame where clock moves at v, photon traces a longer diagonal path of length

2\surd (L^{2} + (v\Delta t/2)^{2}) = c\cdot \Delta t

.

Solving gives

\Delta t = (2L/c)/\sqrt{1-v^{2}/c^{2}} = \gamma \Delta t_{0}

.

Limiting cases

v \to 0

⟶

\Delta t \to \Delta t_{0}

Stationary clocks agree (Galilean limit).

v \to c

⟶

\Delta t \to \infty

Light-speed clock would never tick; massive objects can't reach c —

\gamma

diverges.

v ≪ c⟶

\Delta t \approx \Delta t_{0}(1 + \tfrac{1}{2}\beta ^{2})

Correction is tiny at everyday speeds:

\Delta /\Delta t_{0} \sim 10^{-15} at 1000\,\mathrm{km/h}

.

What if…

What if you could travel at 0.9999c to Alpha Centauri (4.37 ly)?

$\gamma \approx 70.7$ . Ship time $= 4.37 / 0.9999 / 70.7 \approx 22$ days. Earth time $\sim 4.4$ years. Twin paradox in action.

What if c were 100 m/s?

Walking would be relativistic. Clocks on every bike would visibly desynchronize; the world would be famously paradoxical and macroscopic technology would have to compensate constantly.

1

Astronaut at 0.6c for 1 year (proper)

Given ·

\Delta t 0:: 31557600
v:: 179875470
c:: 299792458

Find · \Delta t

Steps

$\beta = v/c = 0.6 \to \gamma = 1/\sqrt{1-0.36} = 1/0.8 = 1.25$
$\Delta t = 1.25 \times 1$ year $= 1.25$ years
Astronaut ages 1 year while Earth ages 1.25 years.

Result ·

\Delta t = 1.25

years (Earth time)

2

GPS satellite (v ≈ 3874 m/s)

Given ·

\Delta t 0:: 86400
v:: 3874
c:: 299792458

Find ·

\Delta t - \Delta t_{0}

(per day)

Steps

$\beta ^{2} = (3874/3e_{8})^{2} \approx 1.67e-10$
$\gamma \approx 1 + \tfrac{1}{2}\beta ^{2} \approx 1 + 8.35e-11$
$\Delta = 86400 \times 8.35e-11 \approx 7.22 \mu s/day$ . GR effect (height) gives + $45 \mu s/day$ in opposite sense; $net \sim 38 \mu s/day$ .

Result ·

\approx 7.2 \mu s/day

slower (SR effect only)

Lorentz Factor→Mass-Energy Equivalence→