Length Contraction and Relativistic Velocity

Length contraction is the effect that an object moving relative to an observer is measured to be shorter along the direction of motion: L = L0 ÷ γ = L0√(1 − v2/c2), where L0 is the proper length measured in the rest frame of the object; velocities add relativistically, u = (v + u’)/(1 + vu’/c2), so that no speed exceeds c. In brief: lengths perpendicular to the motion are unchanged. Length contraction and time dilation are two aspects of the same spacetime geometry: a traveler crossing a distance L0 (measured in the Earth frame) at speed v measures the distance as L0/γ and the time as L0/(γv), in agreement with the time dilation seen from Earth. Contraction is real as a measurement, but an object appears rotated, not just contracted, when photographed.

If clocks run slow in moving frames, lengths must also change to keep the speed of light constant for all observers. This guide derives length contraction, shows its connection with time dilation using the muon and a spacecraft, discusses the relativistic addition of velocities, compares the effects in a table, works through numerical examples and corrects common mistakes.

What is length contraction?

The proper length L0 of an object is its length measured in the frame in which it is at rest. An observer who sees the object moving with speed v along its length, and measures the positions of its two ends at the same time in the observer’s frame, finds the shorter length L = L0 ÷ γ. A meter stick at 0.8c has a length of 0.60 m; at 0.99c, 0.14 m; at 0.9999c, 1.4 cm. Dimensions perpendicular to the motion are not contracted (a derivation using the principle of relativity shows this). The contraction becomes noticeable only at speeds that are a considerable fraction of c. From the object’s own frame, it is the observer’s distances that are contracted. The effect is a consequence of the relativity of simultaneity: observers in different frames do not agree on which positions of the ends are measured “at the same time”.

0 0.5 1.0 0 0.5 1.0 0.6 0.8 v/c L/L0

How is it consistent with time dilation?

Consider a muon traveling at 0.998c down through the atmosphere, from an altitude of 10 km (as measured from the ground). In the Earth frame, the muon’s lifetime is dilated, and it travels 10.4 km before decaying on average. In the muon’s own frame, its lifetime is just 2.2 µs, but the atmosphere moves toward it at 0.998c, and the 10 km thickness of the atmosphere is contracted to 10 ÷ 15.8 = 0.63 km, a distance that the muon can cover in 0.63 km ÷ 0.998c = 2.1 µs. Both viewpoints agree on the physical outcome: the muon reaches the ground. In the same way, a trip to a star 4.0 light-years away at 0.80c takes 5.0 years in the Earth frame, but for the traveler the distance is contracted to 4.0 ÷ 1.667 = 2.4 light-years, which takes 2.4 ÷ 0.80 = 3.0 years, in agreement with the time dilation factor 5.0 ÷ 1.667 = 3.0 years.

How do velocities add?

The relativistic velocity addition formula, for velocities along the same line, is u = (v + u’) ÷ (1 + vu’/c2), where v is the velocity of frame S’ relative to S, u’ the velocity of an object in S’, and u the velocity of the object in S. For velocities much smaller than c, the denominator is close to 1, and u ≈ v + u’. If one of the velocities is c, the result is c, whatever the other. The sum of two speeds below c is always below c. For velocity components perpendicular to the relative motion, the formula is more complicated: uy = u’y ÷ [γ(1 + vu’x/c2)]. The relativistic composition explains the result of Fizeau’s 1851 measurement of the speed of light in moving water, and the aberration of starlight.

Measured length of a one meter rod against its speed 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.25 0.50 0.75 1.00 Speed v / c Measured length L (m) 0.5c: 0.866 m 0.8c: 0.600 m 0.99c: 0.141 m
Measured length of a rod of proper length 1.00 m moving along its length, L = √(1 − v²/c²) × 1.00 m. The contraction is negligible at everyday speeds and only becomes noticeable above a large fraction of the speed of light: 0.866 m at 0.5c, 0.600 m at 0.8c and 0.141 m at 0.99c. The length tends to zero as v approaches c.

Contraction factors compared

Length of a 1.00 m rod moving along its length
Speed v γ Measured length L = 1/γ (m)
0.1c1.0050.995
0.5c1.1550.866
0.8c1.6670.600
0.99c7.090.141
0.9999c70.70.0141

Worked example: rods, spaceships and velocities

  1. Rod. A rod of proper length 2.0 m moves at 0.60c along its length: L = 2.0 ÷ 1.25 = 1.6 m.
  2. Spaceship. A spaceship has a proper length of 100 m and passes the Earth at 0.60c. An observer on Earth measures a length of 100 ÷ 1.25 = 80 m. The time for the ship to pass a fixed point in the Earth frame: 80 m ÷ (0.60 × 3.00 × 108) = 4.4 × 10−7 s.
  3. The same event for the crew. The ship’s crew measures the time for a fixed point on Earth to pass along the whole ship: 100 m ÷ (0.60c) = 5.6 × 10−7 s. The ratio 5.6 ÷ 4.4 = 1.25 = γ: the Earth observer’s time is the proper time (both events at the same point on Earth).
  4. Interstellar trip. At 0.80c, a distance of 4.0 light-years in the Earth frame is contracted to 2.4 light-years for the traveler.
  5. Adding velocities. Two spaceships approach the Earth in opposite directions, each at 0.80c relative to the Earth. The speed of one relative to the other: u = (0.80c + 0.80c) ÷ (1 + 0.64) = 0.976c (in the Galilean case it would be 1.6c).
  6. Probe launched from a ship. A ship moving at 0.90c relative to the Earth launches a probe forward at 0.90c relative to the ship: u = 1.8c ÷ (1 + 0.81) = 0.994c. If the probe is launched backward, with u’ = −0.90c: u = (0.90c − 0.90c) ÷ (1 − 0.81) = 0, so the probe is at rest relative to the Earth.
2.0 m 1.6 m 0.60c 100 m 80 m 0.60c ÷ 1.25 ÷ 1.25

How was length contraction proposed?

George FitzGerald (1889) and Hendrik Lorentz (1892) proposed that moving bodies contract in the direction of motion, in order to explain the null result of the Michelson-Morley experiment, while keeping the ether. Einstein showed in 1905 that contraction follows from the postulates of relativity without any ether, as a property of spacetime itself. James Terrell and Roger Penrose showed in 1959 that a fast-moving object, photographed, does not appear contracted but rotated, since light from different parts takes different times to reach the camera.

Which mistakes are common?

Mistake 1: Using the proper length as the contracted one

L0 is the length in the object’s rest frame, the longest measured length. L = L0/γ is the shorter length in a frame where the object moves.

Mistake 2: Contracting lengths perpendicular to the motion

Only the dimension along the motion changes.

Mistake 3: Thinking that the object is physically compressed

No forces act on the object. The contraction is a feature of measurements of space in different frames.

Mistake 4: Adding velocities by simple addition at high speed

Use the relativistic formula when speeds are a significant fraction of c.