28.3 Length Contraction

Learning Objectives

By the end of this section, you will be able to:

• Describe proper length.
• Calculate length contraction.
• Explain why we don’t notice these effects at everyday scales.

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 1 in Section 28.2 illustrates this concept. To an observer on the Earth, the muon travels at $0.950c$ for $7.05\mu \mathrm{s}$ from the time it is produced until it decays. Thus it travels a distance

$$L_0=v\mathrm{\Delta }t=(0.950)(3.00\times {\text{10}}^8\text{m/s})(7.05\times {\text{10}}^{-6}\text{s})=2.01\text{km}$$

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

$$L=v\mathrm{\Delta }{t}_0=(0\text{.}\text{950})(3\text{.}\text{00}\times {\text{10}}^8\text{m/s})(2\text{.}\text{20}\times {\text{10}}^{-6}\text{s})=0\text{.627 km}.$$

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.

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

$$v=\frac{L_0}{\mathrm{\Delta }t}.$$

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

$$v=\frac{L}{\mathrm{\Delta }{t}_0}.$$

The moving observer travels with the muon and therefore observes the proper time $\mathrm{\Delta }{t}_0$. The two velocities are identical; thus,

$$\frac{L_0}{\mathrm{\Delta }t}=\frac{L}{\mathrm{\Delta }{t}_0}.$$

We know that $\mathrm{\Delta }t=\gamma \mathrm{\Delta }{t}_0$. Substituting this equation into the relationship above gives

$$L=\frac{L_0}{\gamma }.$$

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 length $L$ 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 () 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.

Summary

• All observers agree upon relative speed.
• Distance depends on an observer’s motion. Proper length $L_0$ is the distance between two points measured by an observer who is at rest relative to both of the points. Earth-bound observers measure proper length when measuring the distance between two points that are stationary relative to the Earth.
• Length contraction $L$ is the shortening of the measured length of an object moving relative to the observer’s frame:

  $${\text{L=L}}_0\sqrt{1-\frac{v^2}{c^2}}=\frac{L_0}{\gamma }.$$

Conceptual Questions

Problems & Exercises