Length contraction

A sprinter and a referee compare two 30 cm rulers to confirm they’re identical.

The sprinter then runs at 0.90c holding the ruler pointing in the direction of motion.

The referee says: “To me, the ruler and the sprinter look narrower in the direction they’re running (contracted along the track).”

The sprinter disagrees. From the sprinter’s point of view, their own ruler is normal length. Instead, the sprinter says the referee, the rulers, the whole stadium and even the track look shorter because the stadium is moving past at 0.90c in the opposite direction.

Key idea: length contraction only happens parallel to the direction of motion (objects don’t get shorter sideways/upwards).

Length contraction equation:

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

Formula:

Definitions:

• proper length (measured in the frame where the object is at rest)

• observed length (measured by an observer who sees the object moving at speed v)

Example calculation: train carriage moving near the speed of light

A train carriage has a proper length = 14.0 m (measured by a passenger). The train passes a station at v = 0.92c.

Find the length of the carriage measured by the platform observer.

Step 1: Calculate the Lorentz factor

Step 2: Decide which length is proper

The carriage is at rest in the train frame, so is the proper length.

Step 3: Apply length contraction

Answer: passenger measures 14.0 m, platform observer measures 5.49 m (contracted along motion).

Proper length is measured in the frame where the object is at rest. Observed length is measured by an observer

who sees that object moving.

Quick examples (pairs):

• Ruler held by sprinter:
  Proper length : sprinter measures their ruler (30 cm).
  Observed length : referee measures the moving ruler (shorter).

• 100 m track:
  Proper length : referee measures the track (100 m).
  Observed length : sprinter measures the moving track (shorter).

• Spaceship length:
  Proper length : astronaut measures the ship.
  Observed length : planet observer measures the ship passing (shorter).

• Distance between detectors:
  Proper length : lab measures detector separation.
  Observed length : particle measures the detector separation (shorter).

• Length of a beam of particles:
  Proper length : beam frame measures its own length.
  Observed length : detector frame measures the moving beam (shorter).

What evidence is there that length contraction is real? We can test relativity using unstable particles from cosmic rays. Classically, many should decay before reaching a lower detector. Special relativity explains the extra survivors.

This time we explain it using length contraction: in the particle’s frame, the distance to the lower detector is shorter, so the particle reaches it sooner and more survive.

Two detectors are separated by = 1.6 km (lab). Muons travel at v = 0.995c. Muon half-life at rest is μ

.

Prediction without special relativity (no length contraction)

Step 1: Travel time in the lab frame

Step 2: Expected fraction reaching detector 2

Expected percentage reaching the second detector .

Prediction including length contraction (muon frame)

Step 1: Lorentz factor

Step 2: Contracted distance in the muon frame

Step 3: Time to reach detector 2 in the muon frame

Step 4: New expected fraction reaching detector 2

Conclusion: in the muon frame the detector separation is shorter, so many more survive to reach the lower detector.