28.4

The Relativity of Length: Length Contraction

Because of time dilation, observers moving at a constant velocity relative to each other measure different time intervals between two events. For instance, Example 2 in the previous section illustrates that a trip from earth to Alpha Centauri at a speed of v = 0.95c takes 4.5 years according to a clock on earth, but only 1.4 years according to a clock in the rocket. These two times differ by the factor . Since the times for the trip are different, one might ask whether the observers measure different distances between earth and Alpha Centauri. The answer, according to special relativity, is yes. After all, both the earth-based observer and the rocket passenger agree that the relative speed between the rocket and earth is v = 0.95c. Since speed is distance divided by time and the time is different for the two observers, it follows that the distances must also be different, if the relative speed is to be the same for both individuals. Thus, the earth observer determines the distance to Alpha Centauri to be light-years. On the other hand, a passenger aboard the rocket finds the distance is only light-years. The passenger, measuring the shorter time, also measures the shorter distance. This shortening of the distance between two points is one example of a phenomenon known as length contraction.
The relation between the distances measured by two observers in relative motion at a constant velocity can be obtained with the aid of Figure 28-6. Part a of the drawing shows the situation from the point of view of the earth-based observer. This person measures the time of the trip to be , the distance to be L0, and the relative speed of the rocket to be . Part b of the drawing presents the point of view of the passenger, for whom the rocket is at rest, and the earth and Alpha Centauri appear to move by at a speed v. The passenger determines the distance of the trip to be L, the time to be , and the relative speed to be . Since the relative speed computed by the passenger equals that computed by the earth-based observer, it follows that . Using this result and the time-dilation equation, Equation 28.1, we obtain the following relation between L and L0:

| (28.2)

The length L0 is called the proper length; it is the length (or distance) between two points as measured by an observer at rest with respect to them. Since v is less than c, the term is less than 1, and L is less than L0. It is important to note that this length contraction occurs only along the direction of the motion. Those dimensions that are perpendicular to the motion are not shortened, as the next example discusses.

Figure 28-6 (a) As measured by an observer on the earth, the distance to Alpha Centauri is L0, and the time required to make the trip is .(b) According to the passenger on the spacecraft, the earth and Alpha Centauri move with speed v relative to the craft. The passenger measures the distance and time of the trip to be L and , respectively, both quantities being less than those in part a.

| Example 4 | | | The Contraction of a Spacecraft

An astronaut, using a meter stick that is at rest relative to a cylindrical spacecraft, measures the length and diameter of the spacecraft to be 82 and 21 m, respectively. The spacecraft moves with a constant speed of v = 0.95c relative to the earth, as in Figure 28-6. What are the dimensions of the spacecraft, as measured by an observer on earth?
Reasoning The length of 82 m is a proper length L0, since it is measured using a meter stick that is at rest relative to the spacecraft. The length L measured by the observer on earth can be determined from the length-contraction formula, Equation 28.2. On the other hand, the diameter of the spacecraft is perpendicular to the motion, so the earth observer does not measure any change in the diameter.
Solution The length L of the spacecraft, as measured by the observer on earth, is

Both the astronaut and the observer on earth measure the same value for the diameter of the spacecraft: . Figure 28-6a shows the size of the spacecraft as measured by the earth observer, and part b shows the size measured by the astronaut.

Problem solving insight The proper length L0 is always larger than the contracted length L.

Need more practice?

Interactive LearningWare28.1
After a long journey, an astronaut is heading home at a speed of 0.850c relative to earth. Her instruments tell her that she has to go. (a) Based on her clock, how much time (in hours) will it take her to travel this distance? (b) According to a stationary observer on earth, how far is she from the planet? (c) What does this observer calculate for the time (in hours) it will take for her to arrive?
Related Homework: Problem 10

When dealing with relativistic effects we need to distinguish carefully between the criteria for the proper time interval and the proper length. The proper time interval between two events is the time interval measured by an observer who is at rest relative to the events and sees them occurring at the same place. All other moving inertial observers will measure a larger value for this time interval. The proper length L0 of an object is the length measured by an observer who is at rest with respect to the object. All other moving inertial observers will measure a shorter value for this length. The observer who measures the proper time interval may not be the same one who measures the proper length. For instance, Figure 28-6 shows that the astronaut measures the proper time interval for the trip between earth and Alpha Centauri, whereas the earth-based observer measures the proper length (or distance) L0 for the trip.
It should be emphasized that the word “proper” in the phrases proper time and proper length does not mean that these quantities are the correct or preferred quantities in any absolute sense. If this were so, the observer measuring these quantities would be using a preferred reference frame for making the measurement, a situation that is prohibited by the relativity postulate. According to this postulate, there is no preferred inertial reference frame. When two observers are moving relative to each other at a constant velocity, each measures the other person’s clock to run more slowly than his own, and each measures the other person’s length, along that person’s motion, to be contracted.

Check Your Understanding 2

The drawing shows an object that has the shape of a square when it is at rest in an inertial frame. When this square moves, length contraction changes its shape. In each of the following three situations the speed is the same, and the velocity is in the plane of the square, directed parallel to (a) side AB, (b) diagonal AC, or (c) side AD. In which situation is the area of the object the least? (The answer is given at the end of the book.)

Background: Length contraction occurs only along the direction of the motion. Dimensions perpendicular to the motion are not contracted.
For similar questions (including calculational couterparts), consult Self-Assessment Test 28.1, which is described next.

Test your understanding of the material in Sections 28.1, 28.2, 28.3 and 28.4 :

· | Events and Inertial Reference Frames
· | The Postulates of Special Relativity
· | The Relativity ofTime:Time Dilation
· | The Relativity of Length: Length Contraction

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