Physics/Option Modules/Turning Points in Physics/Time Dilation
Physics · Turning Points in Physics · A2 Level

Time Dilation

If the speed of light really is the same for every observer, no matter how they're moving, then time itself cannot be fixed. Two perfectly good clocks, side by side and identical in every way, will disagree about how much time has passed — not because either one is faulty, but because time dilation is built into the structure of space and time itself.

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A moving clock runs slow

Special relativity rests on the fact that the speed of light in a vacuum, $c$, is measured as the same value by every observer in every inertial frame, regardless of how fast the observer or the light source is moving. Accepting this one strange experimental fact has a direct consequence for time: a moving clock, observed from a frame in which it is moving, is measured to run slow compared with an identical clock at rest in the observer's own frame. This effect is called time dilation, and it applies to every physical process that marks the passage of time, not just mechanical clocks — a beating heart, a radioactive decay, a vibrating atom all run slow in exactly the same way when observed from a frame in which they are moving.

To make this precise, physicists distinguish between two different times measured for the same pair of events. Proper time, $t_0$, is the time interval between two events measured in the frame in which those two events happen at the same place — for instance, a clock's own reading of the interval between two of its ticks, measured by an observer travelling along with that clock. Any other observer, in a frame moving relative to that clock, necessarily measures a longer time interval, $t$, between the same two events.

The Lorentz factor

The proper time and the dilated time measured by a moving observer are related by:

$t = \gamma t_0$
where $\gamma$ is the Lorentz factor, the single quantity that controls the size of every relativistic effect in this module:
$\gamma = \dfrac{1}{\sqrt{1 - v^2/c^2}}$
and $v$ is the relative speed between the two frames.

Since $v$ is always less than $c$ for anything with mass, the ratio $v^2/c^2$ always lies between 0 and 1, which makes the quantity under the square root always lie between 0 and 1 too — and dividing 1 by a number between 0 and 1 always gives a result of 1 or more. So $\gamma \geq 1$ always, which means $t \geq t_0$ always: every other frame measures a longer, dilated time than the proper time measured in the clock's own rest frame, never a shorter one. At everyday speeds, $v \ll c$, so $v^2/c^2$ is minute, $\gamma$ is indistinguishably close to 1, and time dilation is utterly unnoticeable — a passenger jet or even an orbiting satellite dilates time by only a tiny fraction of a second over years of travel. Only as $v$ becomes a significant fraction of $c$ does $\gamma$ grow large enough to matter in any measurable way.

Worked exampleTime dilation at a given fraction of c
1
A spacecraft's onboard clock measures a journey to last exactly $1.00$ hour of proper time, while travelling at $v = 0.80c$ relative to Earth. How long does an observer on Earth measure the journey to take?
2
Find the Lorentz factor: $\gamma = \dfrac{1}{\sqrt{1-v^2/c^2}} = \dfrac{1}{\sqrt{1-0.80^2}} = \dfrac{1}{\sqrt{1-0.64}} = \dfrac{1}{\sqrt{0.36}} = \dfrac{1}{0.6} \approx 1.667$.
3
Apply $t = \gamma t_0 = 1.667 \times 1.00\,\text{hour} \approx 1.67\,\text{hours}$.
The Earth-based observer measures the journey taking noticeably longer — about 40 minutes longer — than the astronaut's own onboard clock records, purely as a consequence of the astronaut's clock moving relative to Earth.
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Graph of the Lorentz factor gamma on the vertical axis against speed v as a fraction of the speed of light c on the horizontal axis, running from 0 to 1. A single smooth curve starts at gamma equals 1 when v/c equals 0, stays close to gamma equals 1 for most of the range up to about v/c equals 0.6, then curves upward increasingly steeply, rising sharply as v/c approaches 1, with a dashed vertical asymptote line drawn at v/c equals 1 that the curve approaches but never touches.
The Lorentz factor γ stays close to 1 at everyday speeds, then rises sharply and without bound as v approaches c.
Reading the graph

This curve is the shape that underlies every relativistic effect covered across this module — time dilation, length contraction, relativistic momentum and relativistic energy all scale directly with this same $\gamma$. Below about $v=0.5c$, $\gamma$ is still under 1.2, so relativistic corrections stay modest; beyond about $v=0.9c$, $\gamma$ climbs past 2 and keeps accelerating upward without any limit as $v\to c$.

Experimental evidence: cosmic ray muons

Muons are unstable subatomic particles, similar to electrons but far heavier, that decay with a mean (rest-frame) lifetime of about $\tau_0 = 2.2\,\mu\text{s}$. Cosmic rays striking the upper atmosphere, roughly $15\,\text{km}$ above the Earth's surface, produce large numbers of muons travelling downward at speeds close to $c$ — typically around $v = 0.995c$. Whether these muons survive long enough to reach ground-level detectors turns out to be a direct, quantitative test of time dilation.

Worked exampleWhy cosmic ray muons reach the ground
1
Ignoring relativity entirely: at $v = 0.995c \approx 2.985\times10^8\,\text{m}\,\text{s}^{-1}$, the time to cross $15\,\text{km} = 1.5\times10^4\,\text{m}$ is $t = \dfrac{d}{v} = \dfrac{1.5\times10^4}{2.985\times10^8} \approx 5.03\times10^{-5}\,\text{s} = 50.3\,\mu\text{s}$.
This is over 22 times the muon's mean lifetime of $2.2\,\mu\text{s}$. Classically, so few muons should survive that many lifetimes that essentially none should reach the ground.
2
Relativity changes this by dilating the muon's decay clock, as measured from the Earth's frame. First find $\gamma$ at $v=0.995c$: $\gamma = \dfrac{1}{\sqrt{1-0.995^2}} = \dfrac{1}{\sqrt{1-0.990025}} = \dfrac{1}{\sqrt{0.009975}} \approx \dfrac{1}{0.09987} \approx 10.0$.
3
The muon's mean lifetime, as measured from the Earth's frame, is dilated: $t = \gamma t_0 = 10.0 \times 2.2\,\mu\text{s} = 22\,\mu\text{s}$.
This is the muon's decay "clock" running slow, as seen from Earth — ten times slower than its own proper lifetime.
4
The $50.3\,\mu\text{s}$ crossing time found in step 1 is now only about $50.3/22 \approx 2.3$ dilated mean lifetimes, rather than $22.9$ rest-frame lifetimes. A survival fraction after $2.3$ lifetimes, using $N=N_0 e^{-t/\tau}$, is $e^{-2.3} \approx 0.10$, or about 10% — a small but very much non-negligible, easily detectable fraction, compared with the utterly negligible survival predicted without relativity.
This dramatic difference between the classical and relativistic predictions, and the fact that muons genuinely are detected reaching the ground in large numbers, is one of the most direct, everyday pieces of experimental evidence for time dilation.
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Looking ahead

The muon result can equally well be explained from the muon's own point of view, using length contraction rather than time dilation — the next lesson picks this up and shows both explanations give exactly the same physical outcome.

Key takeaways

  • Time dilation: a clock (or any process marking time) observed from a frame in which it is moving is measured to run slow compared with an identical process at rest in the observer's own frame.
  • Proper time $t_0$ is the time between two events measured in the frame where both events happen at the same place — for example, a clock's own reading of its own ticks.
  • The dilated time measured in any other frame is $t = \gamma t_0$, where $\gamma = 1/\sqrt{1-v^2/c^2}$ is the Lorentz factor and $v$ is the relative speed between the frames.
  • Since $v < c$ always, $\gamma \geq 1$ always, so $t \geq t_0$ always — every other frame measures time running equal to or slower than the proper time, never faster.
  • At $v=0.80c$, $\gamma\approx1.67$: a 1.00 hour proper-time journey is measured as 1.67 hours from Earth.
  • Cosmic ray muons ($\tau_0=2.2\,\mu\text{s}$, $v\approx0.995c$, $\gamma\approx10.0$) reach sea level in large numbers because their decay clock runs ten times slower in the Earth's frame, giving roughly 10% survival over the ~50 μs crossing time rather than the classically predicted near-zero survival.
  • $\gamma$ stays close to 1 for most everyday and even fairly high speeds, only rising sharply and without bound as $v$ approaches $c$.
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