AcademyRelativity
Academy
Time Dilation
Level 1 - Physics topic page in Relativity.
Principle
A moving clock is measured to run slow compared with a clock at rest with the events.
Notation
\(\Delta\tau\)
proper time between events
\(\mathrm{s}\)
\(\Delta t\)
time interval measured in another inertial frame
\(\mathrm{s}\)
\(v\)
relative speed
\(\mathrm{m\,s^{-1}}\)
\(\beta\)
\(v/c\)
\(\gamma\)
Lorentz factor
\(L\)
light-clock rest height
\(\mathrm{m}\)
\(c\)
speed of light
\(\mathrm{m\,s^{-1}}\)
Method
Derivation 1: Identify proper time
Proper time is measured by a single clock present at both events. That means the events occur at the same place in that clock's frame.
Proper-time condition
\[\Delta x'=0\]
The clock does not move in its own rest frame, so the spatial separation between the two ticks is zero.
Proper time
\[\Delta\tau=\frac{2L}{c}\]
Light travels up and back down the light clock in proper time.
Derivation 2: Light-clock geometry
In the moving frame, light travels a longer diagonal path because the clock moves sideways while light travels upward. The diagonal distance is \(c\Delta t\) and the sideways displacement is \(v\Delta t\).
Vertical leg distance
\[L=c\frac{\Delta\tau}{2}\]
Diagonal geometry
\[\left(c\frac{\Delta t}{2}\right)^2=\left(v\frac{\Delta t}{2}\right)^2+L^2\]
The diagonal hypotenuse is c times half the dilated interval; the horizontal leg is v times half the dilated interval.
Solve for L^2
\[L^2=\frac{c^2(\Delta t)^2}{4}-\frac{v^2(\Delta t)^2}{4}\]
Combine with vertical leg
\[L^2=\frac{c^2(\Delta\tau)^2}{4}\]
Set equal and simplify
\[\frac{c^2(\Delta t)^2}{4}-\frac{v^2(\Delta t)^2}{4}=\frac{c^2(\Delta\tau)^2}{4}\]
Collect terms
\[c^2(\Delta t)^2\left(1-\frac{v^2}{c^2}\right)=c^2(\Delta\tau)^2\]
Solve for time ratio
\[\Delta t=\frac{\Delta\tau}{\sqrt{1-v^2/c^2}}\]
Derivation 3: Lorentz factor form
The dilated interval is larger than proper time by the Lorentz factor.
Lorentz factor definition
\[\gamma=\frac{1}{\sqrt{1-v^2/c^2}}=\frac{1}{\sqrt{1-\beta^2}}\]
Time dilation result
\[\Delta t=\gamma\Delta\tau\]
The moving clock ticks slower: one tick of the moving clock takes longer in the lab frame.
Derivation 4: Use lifetime and distance consistently
For an unstable moving particle, the laboratory sees a longer lifetime. Distance traveled in the lab is \(v\\Delta t\), not \(v\Delta\\tau\).
Lab distance
\[d=v\Delta t=v\gamma\Delta\tau\]
Always use the dilated time interval when calculating distance traveled in the lab frame.
Rules
Time dilation
\[\Delta t=\gamma\Delta\tau\]
Lorentz factor
\[\gamma=\frac{1}{\sqrt{1-\beta^2}}\]
Proper time
\[\Delta\tau=\frac{\Delta t}{\gamma}\]
Examples
Question
A clock moving at
\[0.800c\]
ticks every \[1.00,mumathrm s\]
in its own frame. What interval is measured in the lab?Answer
\[gamma= rac{1}{sqrt{1-0.800^2}}=1.67\]
\[\Delta t=gammaDelta au=1.67,mumathrm s\]
Checks
- The proper time is the smallest time interval between the same two events.
- The clock is never slow in its own rest frame.
- Do not mix the particle's proper lifetime with a laboratory distance unless you transform the interval.
- For \(v\ll c\), \(\gamma\approx1\) and the effect is tiny.