Time Dilation — Physica

Equivalent forms

\Delta t = \frac{\Delta t_0}{\sqrt{1 - v^2/c^2}}

Time is not universal — motion itself slows the ticking of every physical process.

Symbol	Name	SI unit	Dimension	Range
$Delta_t$	Dilated Timeoutput Time interval measured in the observer frame	s	T	0 – 100
$Delta_t0$	Proper Time Time interval in the clock's rest frame	s	T	0 – 10
$v$	Velocity Relative speed of the clock	m/s	LT^-1	0 – 299792458

Unit systems

Symbol	SI	CGS	Imperial
$Delta_t$	s	s	s
$Delta_t0$	s	s	s
$v$	m/s	cm/s	ft/s

Where it holds

Valid for inertial frames and for any proper time interval. For accelerated clocks, integrate dtau along the worldline.

Dimensional analysis

\Delta t \to [T]

\gamma \cdot \Delta t_{0} \to [1]\cdot [T] = [T]

Both sides [T].

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Discovery

Albert Einstein · 1905

Einstein derived time dilation in his 1905 'On the Electrodynamics of Moving Bodies' paper from the two postulates of special relativity.

Try this

A muon lives 2.2 μs at rest — how far can it travel at 0.99c before decaying?

In the lab frame the muon's lifetime is gamma·Δt0. With gamma ≈ 7.09 and v = 0.99c, compute the lab-frame distance L = v·gamma·Δt0.

Research status: stable

Real-world applications

GPS satellite timing corrections
Muon lifetime extension in cosmic rays
Atomic clock comparisons on aircraft
Particle collider beam lifetimes

Common misconceptions

Time dilation is reciprocal — each observer sees the other's clock as slow
The effect applies to every physical process, not just clocks
The 'twin paradox' is resolved by the traveling twin's acceleration

Experimental verification

Hafele–Keating (1971) atomic-clock flights, muon lifetime in storage rings (CERN, 1977) confirmed dilation to 0.2%, and GPS satellites correct

for \sim 38 \mu s/day

.

Derivation

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

In the rest frame period

is \Delta t_{0} = 2L/c

.

In a frame where the clock moves with velocity v, the photon traverses a diagonal of length

\sqrt(L^2 + (v\Delta t/2)^2)

, giving

c\cdot \Delta t/2 = \sqrt(L^2 + (v\Delta t/2)^2)

.

Solving yields

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

.

Limiting cases

v \to 0

⟶

\Delta t \to \Delta t_{0}

Classical limit: no dilation at rest.

v \to c

⟶

\Delta t \to \infty

A photon's clock would not tick at all.

v = 0.866c

⟶

\Delta t = 2\Delta t_{0}

Moving clocks run at half speed.

What if…

What if the clock is a biological heart?

Heartbeats slow in the observer frame by exactly gamma; the moving person ages slower as measured externally.

What if

v = 0.5c

?

$\gamma \approx 1.155$ ; a 1-second proper interval becomes 1.155 s in the lab.

What if both twins are inertial?

Neither twin ages differently permanently — the asymmetry requires one to accelerate and return.

1

Cosmic-ray muon

Given ·

Delta t0:: 0.0000022
v:: 296794533

Find · Delta_t

Steps

$\gamma = 1/\sqrt{1 - 0.99^2} \approx 7.09$
$\Delta t = \gamma \cdot \Delta t_{0} = 7.09 \times 2.2e-6\,\mathrm{s}$
Distance $= 0.99 \times 299792458 \times 1.56e-5 \approx 4630\,\mathrm{m}$

Result ·

\Delta t = 7.09 \times 2.2e-6\,\mathrm{s} \approx 1.56e-5\,\mathrm{s}

; travels

v\cdot \Delta t \approx 4.63\,\mathrm{km}

.

2

ISS astronaut

Given ·

Delta t0:: 31557600
v:: 7660

Find · Delta_t - Delta_t0

Steps

$\beta = 7660/299792458 \approx 2.556e-5$
$\gamma - 1 \approx \beta ^2/2 \approx 3.27e-10$
$\Delta t - \Delta t_{0} \approx 3.27e-10 \times 31557600 \approx 0.0103\,\mathrm{s}$

Result · Over one year, ISS clock lags Earth

by \sim 0.01\,\mathrm{s}

.

Lorentz Factor→Length Contraction→Relativistic Doppler Effect→