The proper length Lp of an object is the length measured by someone at rest relative to the object. The length of an object measured by someone 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.
Consider a spaceship traveling with a speed v from one star to another. There are two observers: one on the Earth and the other in the spaceship. The observer at rest on the Earth (and also assumed to be at rest with respect to the two stars)
measures the distance between the stars to be the proper length Lp . According to this observer, the time it takes the spaceship to complete the voyage is Δt = Lp /v.
Because of time dilation, the space traveler measures a smaller time of travel by the spaceship clock:
The space traveler claims to be at rest and sees the destination star moving toward the spaceship with speed v. Because the space traveler reaches the star in the time Δtp , he or she concludes that the distance L between the stars is shorter than Lp . This distance measured by the space traveler is L = v Δtp = vΔt/γ
Since Lp = v Δt, so L = Lp / γ = Lp (1 - v2/c 2 ) ½.
We see that :
If an object has a proper length Lp when it is at rest, then when it moves with speed v in a direction parallel to its length, it contracts to the length
L = Lp / γ = Lp (1 - v2/c 2 ) ½.
It is important to emphasize that proper length and proper time are measured in different reference frames. As an example of this point, let us return to the decaying muons( see :muons) moving at speeds close to the speed of light. An observer in the muon reference frame measures the proper lifetime (that is, the time interval Tp), whereas an Earth-based observer measures a dilated lifetime. However, the Earth-based observer measures the proper height (the length Lp) of the mountain in. In the muon reference frame, this height is less than Lp , as the ﬁgure shows. Thus, in the muon frame, length contraction occurs but time dilation does not. In the Earth-based reference frame, time dilation occurs but length contraction does not. Thus, when calculations on the muon are performed in both frames, the effect of “offsetting penalties” is seen, and the outcome of the experiment in one frame is the same as the outcome in the other frame!.
The Relativistic Doppler Effect
Another important consequence of time dilation is the shift in frequency found for light emitted by atoms in motion as opposed to light emitted by atoms at rest. This phenomenon, known as the Doppler effect, was introduced as it concerns to sound waves. In the case of sound, the motion of the source with respect to the medium of propagation can be distinguished from the motion of the observer with respect to the medium. Light waves must be analyzed differently, however, because they require no medium of propagation, and no method exists for distinguishing the motion of a light source from the motion of the observer.
If a light source and an observer approach each other with a relative speed v, the frequency fobs measured by the observer is
fobs = ( √1 + v/c / √1 -v/c) f source
where f source is the frequency of the source measured in its rest frame. Note that this relativistic Doppler shift formula, unlike the Doppler shift formula for sound, depends only on the relative speed v of the source and observer and holds for relative speeds as great as c. As you might expect, the formula predicts that fobs > f source when the source and observer approach each other. We obtain the expression for the case in which the source and observer recede from each other by replacing v with - v.
The most spectacular and dramatic use of the relativistic Doppler effect is the measurement of shifts in the frequency of light emitted by a moving astronomical object such as a galaxy. Spectral lines normally found in the extreme violet region for galaxies at rest with respect to the Earth are shifted by about 100 nm toward the red end of the spectrum for distant galaxies—indicating that these galaxies are receding from us. The American astronomer Edwin Hubble (1889–1953) performed extensive measurements of this red shift to conﬁrm that most galaxies are moving away from us, indicating that the Universe is expanding.
Coming up : Relativistic linear momentum and the
Relativistic form of Newton’s laws.