Time dilation

Imagine a sprinter and a referee standing together. They compare watches and set them to the same time, so they agree the watches are running in sync.

Now the sprinter runs past the referee at a speed close to the speed of light (say 0.90c). The referee says: “To me, it looks like your watch is running slowly.”

But the sprinter disagrees. From the sprinter’s point of view, their own watch is running normally. Instead, the sprinter claims that the referee’s clock (and in fact everything happening in the stadium ) seems to be happening more slowly.

Key idea: different inertial observers disagree about time intervals when they move relative to each other.

Time dilation equations:

Formula:

where is the Lorentz factor, a useful number to determine first when doing calculations:

Formula:

Definitions:

• proper time (measured in the frame where the clock/event is at rest)

• observed time (measured by an observer who sees that clock/event moving at speed v)

Example calculation: pendulum on a train moving near the speed of light

A train moves at v = 0.92c past a station platform. A pendulum inside the train has a time period of 1.20 s. Determine the time period according to: (a) a passenger on the train, (b) an observer on the platform.

Step 1: Calculate the Lorentz factor

Step 2: Decide which time is proper

The pendulum is at rest in the train, so the passenger measures the proper time:

Step 3: Apply time dilation, for someone on the platform everything on the train is moving in slow motion so appears to take longer. Since is bigger than 1, multiply the proper time to make it bigger.

Proper time is measured in the frame where the clock/event is at rest. Observed time t is

measured by an observer who sees that clock/event moving.

Quick examples (pairs):

• Sprinter’s watch (one tick):
  Proper time : sprinter times one tick on their own watch.
  Dilated time : referee times that same tick on the sprinter’s watch as taking longer as he runs past.

• Decay half-life (moving particles):
  Proper time : half-life measured when the particles are at rest in a lab.
  Dilated time : longer half-life measured by an observer who sees the particles moving fast.

• Particle travelling between two detectors (A → B):
  Proper time : time between arrival at A and arrival at B in the particle’s rest frame.
  Dilated time : time between detector signals measured in the lab frame.

• Beam of particles passing one detector (front then back):
  Proper time : time between front and back passing the detector measured by the lab detector clock. Dilated time : longer time measured in the beam frame.

What evidence is there that time dilation is real?

Cosmic rays high in the atmosphere produce unstable particles (such as muons). They travel towards the ground at speeds close to c. Without time dilation, most should decay before reaching a lower detector — but experiments show many more survive than classical physics predicts.

Lets look at what should happen if we don’t consider time dilation

Worked evidence example (muons): Two detectors are separated by . Muons travel at . Muon half-life at rest is .

Prediction without special relativity (no time dilation)

Step 1: Work out the travel time in the lab (detector) frame

Step 2: Number of half-lives and expected fraction reaching the lower detector

Expected percentage reaching the second detector .

Prediction including time dilation

Step 1: Calculate the Lorentz factor

Step 2: Dilated half-life

Step 3: New expected fraction reaching the lower detector

Conclusion: time dilation explains why far more fast-moving particles are detected at lower altitude than classical decay would predict.