Length Contraction

Figure 28.9 People might describe distances differently, but at relativistic speeds, the distances really are different. (credit: Corey Leopold, Flickr)

Have you ever driven on a road that seems like it goes on forever? If you look ahead, you might say you have about 10 km left to go. Another traveler might say the road ahead looks like it’s about 15 km long. If you both measured the road, however, you would agree. Traveling at everyday speeds, the distance you both measure would be the same. You will read in this section, however, that this is not true at relativistic speeds. Close to the speed of light, distances measured are not the same when measured by different observers.

Proper Length

One thing all observers agree upon is relative speed. Even though clocks measure different elapsed times for the same process, they still agree that relative speed, which is distance divided by elapsed time, is the same. This implies that distance, too, depends on the observer’s relative motion. If two observers see different times, then they must also see different distances for relative speed to be the same to each of them.

The muon discussed in Example 28.1 illustrates this concept. To an observer on the Earth, the muon travels at $0.950c$ from the time it is produced until it decays. Thus it travels a distance

$L_{0}=v\Delta t=(0.950)(3.00 \times 10^{8}\:m/s)(7.05\times 10^{-6}\:s)=2.01\:km$ [Equation 28.16]

relative to the Earth. In the muon’s frame of reference, its lifetime is only $2.20\: \mu s$. It has enough time to travel only

$L=v\Delta t_{0}=(0.950)(3.00\times 10^{8}\:m/s)(2.20\times 10^{-6}\:s)=0.627\:km$ [Equation 28.17]

The distance between the same two events (production and decay of a muon) depends on who measures it and how they are moving relative to it.

The Earth-bound observer measures the proper length $L_{0}$, because the points at which the muon is produced and decays are stationary relative to the Earth. To the muon, the Earth, air, and clouds are moving, and so the distance $L$ it sees is not the proper length.

Figure 28.10 (a) The Earth-bound observer sees the muon travel 2.01 km between clouds. (b) The muon sees itself travel the same path, but only a distance of 0.627 km. The Earth, air, and clouds are moving relative to the muon in its frame, and all appear to have smaller lengths along the direction of travel.

Length Contraction

To develop an equation relating distances measured by different observers, we note that the velocity relative to the Earth-bound observer in our muon example is given by

$v=\frac{L_{0}}{\Delta t}$ [Equation 28.18]

The time relative to the Earth-bound observer is $\Delta t$, since the object being timed is moving relative to this observer. The velocity relative to the moving observer is given by

$v=\frac{L}{\Delta t_{0}}$ [Equation 28.19]

The moving observer travels with the muon and therefore observes the proper time $\Delta t_{0}$. The two velocities are identical; thus,

$\frac{L_{0}}{\Delta t}=\frac{L}{\Delta t_{0}}$ [Equation 28.20]

We know that $\Delta t=\gamma \Delta t_{0}$. Substituting this equation into the relationship above gives

$L=\frac{L_{0}}{\gamma}$ [Equation 28.21]

Substituting for $\gamma$ gives an equation relating the distances measured by different observers.

If we measure the length of anything moving relative to our frame, we find its $L$ to be smaller than the proper length $L_{0}$ that would be measured if the object were stationary. For example, in the muon’s reference frame, the distance between the points where it was produced and where it decayed is shorter. Those points are fixed relative to the Earth but moving relative to the muon. Clouds and other objects are also contracted along the direction of motion in the muon’s reference frame.

People could be sent very large distances (thousands or even millions of light years) and age only a few years on the way if they traveled at extremely high velocities. But, like emigrants of centuries past, they would leave the Earth they know forever. Even if they returned, thousands to millions of years would have passed on the Earth, obliterating most of what now exists. There is also a more serious practical obstacle to traveling at such velocities; immensely greater energies than classical physics predicts would be needed to achieve such high velocities. This will be discussed in Relatavistic Energy.

Why don’t we notice length contraction in everyday life? The distance to the grocery shop does not seem to depend on whether we are moving or not. Examining the equation $L=L_{0}\sqrt{1-\frac{v^{2}}{c^{2}}}$, we see that at low velocities ($v$) the lengths are nearly equal, the classical expectation. But length contraction is real, if not commonly experienced. For example, a charged particle, like an electron, traveling at relativistic velocity has electric field lines that are compressed along the direction of motion as seen by a stationary observer. (See Figure 28.12.)

As the electron passes a detector, such as a coil of wire, its field interacts much more briefly, an effect observed at particle accelerators such as the 3 km long Stanford Linear Accelerator (SLAC). In fact, to an electron traveling down the beam pipe at SLAC, the accelerator and the Earth are all moving by and are length contracted. The relativistic effect is so great than the accelerator is only 0.5 m long to the electron. It is actually easier to get the electron beam down the pipe, since the beam does not have to be as precisely aimed to get down a short pipe as it would down one 3 km long. This, again, is an experimental verification of the Special Theory of Relativity.

Figure 28.12 The electric field lines of a high-velocity charged particle are compressed along the direction of motion by length contraction. This produces a different signal when the particle goes through a coil, an experimentally verified effect of length contraction.

Source: Rice University, https://openstax.org/books/college-physics/pages/28-3-length-contraction
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