91 Length Contraction

In the previous section, we learned that different observers can measure different time intervals for the same event. If time depends on the observer, what about distance?

Our everyday experience suggests that distances are fixed. If two people measure the length of a road, a hospital hallway, or the distance between two cities, they obtain the same value as long as they use accurate measuring tools. Any disagreement is usually attributed to measurement error rather than to the observers themselves.

Special relativity reveals that this intuition is only approximately correct. At speeds much smaller than the speed of light, all observers measure nearly identical distances. At relativistic speeds, however, the measured distance between two points depends on the observer's state of motion.

This result is a direct consequence of time dilation. Since all observers agree on the relative speed between two objects, and speed is defined as distance divided by time, differences in measured time require corresponding differences in measured distance.

Proper Length

Recall that velocity is defined as

Observers moving relative to one another measure different elapsed times because of time dilation. Nevertheless, they still agree on the relative speed of the moving object. Therefore, the distance associated with the motion cannot be the same for every observer.

This relationship can be illustrated using the muon discussed in the previous chapter.

From the perspective of an observer on Earth, the muon travels at

and survives for

The Earth-bound observer therefore calculates the distance traveled as

From the muon's own reference frame, however, the situation is different. The muon measures its proper lifetime to be only

During that shorter time interval, it can travel only

Both calculations describe the same physical events—the creation of the muon and its decay—yet the measured distance between those events is different.

This demonstrates an important consequence of special relativity: distance, like time, depends on the observer's frame of reference.

In the muon example, the points where the muon is created and where it decays are fixed relative to Earth. Therefore, the observer on Earth measures the proper length.

From the muon's perspective, however, the Earth, the atmosphere, and the clouds move rapidly past it. Because those locations are moving relative to the muon, the distance between them is no longer the proper length.

Length Contraction

In the previous section, we introduced the concept of proper length—the distance measured by an observer who is at rest relative to the two points being measured. We now develop the mathematical relationship between the proper length and the length measured by an observer who is moving relative to those points.

We continue using the example of the relativistic muon. Both the observer on Earth and the observer traveling with the muon agree on one important quantity: the muon's speed relative to Earth.

For the Earth-bound observer, the distance between the muon's creation and decay is the proper length, so the speed is

where

• [latex]L_0[/latex] is the proper length measured on Earth.
• [latex]\Delta t[/latex] is the dilated time measured by the Earth-bound observer.

From the muon's perspective, the Earth, atmosphere, and clouds move past it. The muon measures the shorter distance [latex]L[/latex] during its own proper lifetime [latex]\Delta t_0[/latex], giving

Although the two observers disagree about the measured distance and elapsed time, they agree on the relative speed. Therefore, the two expressions for velocity must be equal:

From the previous chapter, we know that time dilation relates the two time intervals through

Substituting this expression into the velocity equation gives

The proper time appears on both sides of the equation and cancels, leaving

Finally, substituting the definition of the Lorentz factor,

produces the equation for relativistic length contraction:

Because the square-root term is always less than or equal to one, the measured length is always less than or equal to the proper length.

When an object is at rest relative to an observer, its speed is

so the equation becomes

In other words, there is no length contraction when there is no relative motion.

As the speed approaches the speed of light, the quantity

becomes increasingly small, causing the measured length to shrink dramatically. This is why, from the muon's perspective, the atmosphere is much thinner than it appears to observers on Earth. The muon reaches Earth's surface not because it travels faster than light, but because the distance it must cross is contracted in its own frame of reference.

Example 91.1: Calculating Length Contraction—The Distance Between Stars Shrinks at Relativistic Speeds

One of the most surprising predictions of special relativity is that distances measured along the direction of motion become shorter for observers moving relative to those distances.

Suppose an astronaut, like the traveling twin discussed in the previous section, travels at such a high speed that her Lorentz factor is

[latex]\gamma=30.00.[/latex]

The nearest star system, Alpha Centauri, is 4.300 light-years (ly) from Earth as measured by observers on Earth.

Determine:

1. The distance between Earth and Alpha Centauri measured by the astronaut.
2. The astronaut's speed as a fraction of the speed of light.

Strategy

A light-year (ly) is the distance that light travels in one year. It is a unit of distance—not time—and is especially convenient for describing astronomical distances.

For part (a), the Earth and Alpha Centauri are essentially stationary relative to observers on Earth, so the measured distance

[latex]L_0=4.300\ \text{ly}[/latex]

is the proper length.

From the astronaut's perspective, however, the Earth and the star are moving toward her. Therefore, she measures the contracted length

[latex]L=\frac{L_0}{\gamma}.[/latex]

For part (b), we use the definition of the Lorentz factor to solve for the spacecraft's speed.

Solution (a): Contracted Distance

Step 1: Identify the known quantities.

• Proper length:
  [latex]L_0=4.300\ \text{ly}[/latex]
• Lorentz factor:
  [latex]\gamma=30.00[/latex]

Step 2: Identify the unknown.

[latex]L[/latex]

Step 3: Apply the length-contraction equation.

[latex]L=\frac{L_0}{\gamma}[/latex]

Step 4: Calculate the contracted distance.

[latex]\begin{aligned} L &=\frac{4.300\ \text{ly}}{30.00}\\ &=0.1433\ \text{ly} \end{aligned}[/latex]

Answer (a):

[latex]\boxed{L=0.1433\ \text{ly}}[/latex]

Solution (b): Astronaut's Speed

Step 1: Start with the definition of the Lorentz factor.

[latex]\gamma= \frac{1} {\sqrt{1-\frac{v^2}{c^2}}}[/latex]

Substitute

[latex]\gamma=30.00.[/latex]

[latex]30.00= \frac{1} {\sqrt{1-\frac{v^2}{c^2}}}[/latex]

Step 2: Solve for the speed.

[latex]\begin{aligned} 900 &= \frac{1} {1-\frac{v^2}{c^2}} \\[4pt] 1-\frac{v^2}{c^2} &= \frac{1}{900} \\[4pt] \frac{v^2}{c^2} &= 1-\frac{1}{900} \\ &= 0.99889 \end{aligned}[/latex]

Taking the square root gives

[latex]\frac{v}{c}=0.99944.[/latex]

Therefore, the spacecraft's speed is

[latex]\boxed{v=0.99944c\approx0.9994c.}[/latex]

Discussion

This example illustrates just how dramatic relativistic effects become as an object's speed approaches the speed of light. A Lorentz factor of 30 means that distances measured in the direction of travel shrink by a factor of 30 for the moving astronaut.

Although observers on Earth measure the trip to Alpha Centauri as 4.300 light-years, the astronaut experiences a journey of only 0.1433 light-years. Since both observers agree on the spacecraft's speed, the astronaut also measures a much shorter travel time.

Notice that the spacecraft never exceeds the speed of light. Instead, the combination of time dilation and length contraction allows different observers to measure different distances and travel times while remaining completely consistent with Einstein's postulates.

When performing calculations involving relativistic speeds, it is also important to retain extra significant figures until the final step. Small rounding errors can become noticeable because quantities such as the Lorentz factor change rapidly when speeds are very close to the speed of light.

Healthcare Connection

The same relativistic effects apply to proton therapy and heavy-ion therapy, where charged particles reach relativistic speeds. Engineers must account for length contraction when predicting particle trajectories and energy deposition inside patients, even though patients themselves never experience it.

In principle, special relativity allows travelers moving at speeds extremely close to the speed of light to journey across enormous cosmic distances while experiencing relatively little aging. An astronaut could travel thousands—or even millions—of light-years and age only a few years according to the clocks on board the spacecraft.

From Earth's perspective, however, a very different amount of time would pass. Even if such travelers eventually returned, they would find that thousands or millions of years had elapsed on Earth. Civilizations, continents, ecosystems, and even the night sky itself could be dramatically different. In this sense, relativistic travel provides a way to journey far into Earth's future, but not into its past.

There is, however, an enormous practical obstacle. Accelerating a spacecraft to speeds extremely close to the speed of light requires vastly more energy than predicted by classical mechanics. As an object's speed increases, its relativistic energy grows rapidly, making further acceleration increasingly difficult. We will examine this important limitation in the next chapter on Relativistic Energy.

Why Don't We Notice Length Contraction?

If length contraction is a real physical effect, why doesn't the distance to school, a hospital, or the grocery store appear to change when we walk, drive, or fly?

The answer lies in the speeds involved. The length-contraction equation is

For everyday motion, our speeds are extremely small compared with the speed of light.

Under these conditions, the quantity

is extremely close to 1, making the contracted length almost identical to the proper length.

For example, even a commercial jet traveling at about 250 m/s moves at less than one-millionth of the speed of light. The resulting length contraction is far too small to detect with ordinary measuring instruments.

Although length contraction is negligible in everyday life, it becomes important for particles traveling near the speed of light.

Consider an electron moving through a particle accelerator. To a stationary observer, the electron's electric field is compressed in the direction of motion because of length contraction. This changes how the field interacts with nearby detectors and measuring devices.

This effect has been observed experimentally in particle accelerators such as the Stanford Linear Accelerator (SLAC), whose accelerator tunnel is approximately 3 km long.

From Earth's frame of reference, electrons travel through the full length of the accelerator. From the electron's own frame, however, the accelerator itself is moving at nearly the speed of light and therefore appears dramatically shorter because of length contraction. At a typical operating energy of about 25 GeV, an electron's Lorentz factor γ is roughly 50,000, so in the electron's frame the 3 km accelerator is contracted to only a few centimeters.

Although this seems extraordinary, both descriptions are equally valid because each observer measures space and time within a different inertial reference frame.

The agreement between theoretical predictions and experiments performed at modern particle accelerators provides strong evidence that length contraction is a real consequence of Einstein's Special Theory of Relativity.

Section Summary

• All observers measure the same relative speed between objects, even though they may disagree about the distances traveled and the elapsed times.
• Proper length, denoted by [latex]L_0[/latex], is the distance between two points measured by an observer who is at rest relative to both points. For example, an observer on Earth measures the proper length between two locations that remain fixed on Earth.
• Observers moving relative to an object measure a shorter length along the direction of motion. This phenomenon is called length contraction.

• Length contraction occurs only along the direction of motion. Dimensions perpendicular to the motion are unchanged.
• At everyday speeds, length contraction is far too small to detect. It becomes significant only when an object's speed approaches the speed of light.
• Experimental observations of high-speed particles in accelerators confirm the predictions of relativistic length contraction.

Conceptual Questions

1. To which observer does an object appear longer: an observer moving with the object or an observer moving relative to it? Which observer measures the object's proper length?
2. Relativistic effects such as time dilation and length contraction occur for moving cars and airplanes as well as for particles traveling near the speed of light. Why do these effects seem absent from everyday experience?
3. An astronaut travels at a significant fraction of the speed of light relative to Earth.
   1. Does the astronaut observe their own clocks running more slowly?
   2. How do the astronaut and an observer on Earth each perceive the rate of the other's clocks?
   3. Does the astronaut observe their own spacecraft to be shorter than its proper length?
   4. How does the astronaut measure the distance between stars that lie along the direction of travel?
   5. Do the astronaut and an observer on Earth agree on the spacecraft's speed relative to Earth? Explain.

Problems & Exercises

12. A spaceship has a proper length of 200 m as measured by an observer on board. It passes Earth at a speed of [latex]0.970c[/latex]. What length is measured by an observer on Earth?
13. A sports car has a proper length of 6.0 m. How fast would it need to pass an observer for its measured length to be only 5.5 m?
14. Consider the muon described in Example 90.1.
    1. How far does the muon travel according to an observer on Earth?
    2. How far does it travel according to an observer moving with the muon? Base your calculation on its speed relative to Earth and its proper lifetime.
    3. Verify that the two distances are related by length contraction using [latex]\gamma=3.20[/latex].
15. Suppose the muon in Example 90.1 instead travels at a speed of [latex]0.0500c[/latex].
    1. How long does the muon live according to an observer on Earth?
    2. How far does it travel according to the Earth-bound observer?
    3. What distance does the muon measure in its own frame of reference?
16. Consider the astronaut in Example 91.1, who travels 4.30 ly at a speed of [latex]0.99944c[/latex].
    1. How long does the journey take according to an observer on Earth?
    2. How long does the journey take according to the astronaut?
    3. Verify that the two time intervals are related by time dilation using [latex]\gamma=30.00[/latex].
17. An athlete runs a 100-m race.
    1. How fast would the athlete need to run for the race to appear only 100 yd long in the athlete's frame of reference?
    2. Is your result consistent with the fact that relativistic effects are extremely difficult to observe under ordinary conditions? Explain.
18. Unreasonable Results: An astronaut measures the length of her spaceship to be 25.0 m, while an observer on Earth measures its length to be 100 m.
    1. Calculate the value of [latex]\gamma[/latex] implied by these measurements.
    2. What is unreasonable about the result?
    3. Which assumptions or measurements are inconsistent with the principles of length contraction?
19. Unreasonable Results: A spaceship travels directly toward Earth at a speed of [latex]0.800c[/latex]. An astronaut on board claims that a canister launched from the spaceship moves toward Earth at [latex]1.20c[/latex] relative to Earth.
    1. Using classical velocity addition, calculate the velocity the canister would need to have relative to the spaceship.
    2. What is unreasonable about the result?
    3. Which assumptions are inconsistent with special relativity?