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 11.1 illustrates this concept. To an observer on 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

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

The distance between the same two events—the 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 $L_0\text{,}$ because the points at which the muon is produced and decays are stationary relative to Earth. To the muon, Earth, the air, and the clouds are moving, and so the distance $L$ it sees is not the proper length.

Figure 11.10 (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. Earth, the air, and the 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

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

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

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

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 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.

Example 11.2 Calculating Length Contraction: The Distance Between Stars Contracts When You Travel at High Velocity

Suppose an astronaut, such as the twin discussed in Simultaneity and Time Dilation, travels so fast that $\gamma =\text{30}\text{.}\text{00}\text{.}$ (a) She travels from Earth to the nearest star system, Alpha Centauri, 4.300 light years (ly) away as measured by an Earth-bound observer. How far apart are Earth and Alpha Centauri as measured by the astronaut? (b) In terms of $\mathrm{c,}$ what is her velocity relative to Earth? You may neglect the motion of Earth relative to the sun. (See Figure 11.11.)

Figure 11.11 (a) The Earth-bound observer measures the proper distance between Earth and Alpha Centauri. (b) The astronaut observes a length contraction, since Earth and Alpha Centauri move relative to her ship. She can travel this shorter distance in a smaller time (her proper time) without exceeding the speed of light.

Strategy

First note that a light year (ly) is a convenient unit of distance on an astronomical scale—it is the distance light travels in a year. For part (a), note that the 4.300 ly distance between Alpha Centauri and Earth is the proper distance $L_0\text{,}$ because it is measured by an Earth-bound observer to whom both stars are approximately stationary. To the astronaut, Earth and Alpha Centauri are moving by at the same velocity, and so the distance between them is the contracted length $L\text{.}$ In part (b), we are given $\gamma \text{,}$ and so we can find $v$ by rearranging the definition of $\gamma$ to express $v$ in terms of $c$.

Solution for (a)

1. Identify the knowns. $L_0-\mathrm{4.300\; ly;}$ $\gamma =\text{30}\text{.}\text{00}$
2. Identify the unknown. $L$
3. Choose the appropriate equation.
   
    
   11.25 $$L=\frac{L_0}{\gamma }$$
4. Rearrange the equation to solve for the unknown.
   11.26 $$\begin{array}{lll}L& =& \frac{L_0}{\gamma }\\ & =& \frac{\mathrm{4.300\; ly}}{\text{30.00}}\\ & =& \text{0.1433 ly}\end{array}$$

Solution for (b)

1. Identify the known. $\gamma =\text{30}\text{.}\text{00}$
2. Identify the unknown. $v$ in terms of $c$
3. Choose the appropriate equation.
   11.27 $$\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
4. Rearrange the equation to solve for the unknown.
   11.28 $$\begin{array}{lll}\gamma & =& \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\\ \text{30.00}& =& \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\end{array}$$

   Squaring both sides of the equation and rearranging terms gives

   11.29 $$\text{900}\text{.}0=\frac{1}{1-\frac{v^2}{c^2}}$$

   so that

   11.30 $$1-\frac{v^2}{c^2}=\frac{1}{\text{900}\text{.}0}$$

   and

   11.31 $$\frac{v^2}{c^2}=1-\frac{1}{\text{900}\text{.}0}=0\text{.}\text{99888}\text{.}\text{.}\text{.}.$$

   Taking the square root, we find

   11.32 $$\frac{v}{c}=0\text{.}\text{99944},$$

   which is rearranged to produce a value for the velocity.

   11.33 $$v=0.9994c$$

Discussion

First, remember that you should not round off calculations until the final result is obtained, or you could get erroneous results. This is especially true for special relativity calculations, where the differences might only be revealed after several decimal places. The relativistic effect is large here $\text{(}\gamma =30.00\text{),}$ and we see that $v$ is approaching, not equaling, the speed of light. Since the distance as measured by the astronaut is so much smaller, the astronaut can travel it in much less time in her frame.

People could be sent very large distances, perhaps 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 Earth they know forever. Even if they returned, thousands to millions of years would have passed on 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 Relativistic 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 ($vc$) 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 11.12.) 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 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 11.12 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.