# Length Contraction

The length of an object measured in a reference frame that is moving with respect to the object is always less than the proper length. This effect is known as length contraction.

• Length contraction only takes place along the direction of motion.

The proper length, Lp, of an object is the length of the object measured by someone at rest relative to the object.

Derivation:

Consider a spaceship traveling with a speed v from one star to another. There are two observers: one on Earth and the other in the spaceship. The observer at rest on Earth and also assumed to be at rest with respect to the two stars measures the distance between the stars to be Lp. According to this observer, the time it takes the spaceship to complete the voyage is ∆t= Lp/v. Because of time dilation, the space traveler using the spaceship clock measures a smaller time of travel: ∆tp= ∆t/γ. The space traveler claims to be at rest and sees the destination star moving towards the spaceship with speed v. Because the spaceship reaches the star in the time ∆tp, the traveler concludes that the distance L between the stars is shorter than Lp. The distance measured by the space traveler is L = v∆tp = v(∆t/γ).

Because Lp = v∆t, we see that
Lp= γL

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