92 Relativistic Addition of Velocities

Imagine paddling a kayak down a fast-moving river. Even if you stop paddling, the river current continues to carry you downstream. If you paddle forward, your speed relative to the river increases, and your speed relative to the riverbank becomes even greater.

This familiar situation illustrates one of the most common ideas in classical mechanics: velocities add together. Your motion relative to the shore depends on both your velocity through the water and the velocity of the water itself.

For objects moving at ordinary speeds, this simple rule works extremely well. It accurately predicts the motion of cars on highways, airplanes flying through moving air, boats crossing rivers, and people walking on moving sidewalks.

In previous chapters, however, we discovered that measurements of space and time change when objects move at speeds close to the speed of light. Because velocity depends on both distance and time, it should not be surprising that the rules for combining velocities must also change.

Einstein's second postulate states that every observer measures light traveling at the same speed in a vacuum, regardless of the motion of the source or the observer. If velocities simply added together, this postulate could not be true. Special relativity therefore replaces the familiar rule of velocity addition with a new equation that is consistent with the constant speed of light.

In this section, we first review the classical method of adding velocities before introducing the relativistic equation. We then apply these ideas to one of the most important observational tools in modern astronomy and medicine: the Doppler effect, which allows scientists to determine the motion of distant objects by measuring changes in the wavelength or frequency of light.

Classical Velocity Addition

To simplify the discussion, we will consider motion in only one dimension. In this case, positive values represent motion in one direction and negative values represent motion in the opposite direction.

Suppose a child rides on a sled moving at a constant speed of

relative to an observer standing on the ground. While riding on the sled, the child throws a snowball at a speed of

relative to the sled.

We define the following quantities:

• [latex]v[/latex]: velocity of the sled relative to the ground.
• [latex]u'[/latex]: velocity of the snowball relative to the sled.
• [latex]u[/latex]: velocity of the snowball relative to the observer on the ground.

For one-dimensional motion, the classical rule is remarkably simple:

If the child throws the snowball forward, both velocities point in the same direction.

The snowball moves faster relative to the ground because it already shares the sled's forward motion before being thrown.

Now suppose the child throws the snowball backward with the same speed relative to the sled. In this case, the snowball's velocity relative to the sled is negative.

The velocity measured by the observer on the ground becomes

The negative sign indicates that the snowball moves in the direction opposite the sled's motion. Although the sled continues moving forward, the snowball was thrown backward with enough speed to move backward relative to the observer on the ground.

Classical velocity addition successfully describes virtually every situation encountered in daily life because our speeds are extremely small compared with the speed of light.

Relativistic Velocity Addition

Classical velocity addition works remarkably well for everyday situations because the speeds involved are extremely small compared with the speed of light. However, Einstein's second postulate of special relativity states that every observer measures light traveling through a vacuum at the same speed, [latex]c[/latex], regardless of the motion of the source or the observer.

This experimental fact immediately reveals a problem with the classical equation. Imagine a car traveling along a highway at night while its headlights shine forward.

If classical velocity addition were always valid, an observer standing beside the road would measure the speed of the light as

where [latex]v[/latex] is the speed of the car.

Experiments show that this prediction is incorrect. The driver measures the headlights moving away at the speed of light, [latex]c[/latex], and the observer standing beside the road also measures exactly the same speed, [latex]c[/latex]. No matter how fast the car moves, both observers obtain the same value.

Because the speed of light is invariant, the classical equation must be replaced by a new relationship whenever objects move at speeds that are a significant fraction of the speed of light.

Notice the important difference between the classical and relativistic equations. The denominator contains the additional factor

which becomes important only when both velocities are comparable to the speed of light.

When the speeds involved are much smaller than [latex]c[/latex], the quantity

is extremely small. In that case, the denominator is almost equal to 1, and the equation simplifies to the familiar classical result:

This explains why classical mechanics is an excellent approximation for ordinary transportation, sports, and engineering applications, even though it is not fundamentally correct at relativistic speeds.

Example 92.1: Why Everyone Measures the Same Speed of Light

A spaceship travels directly toward Earth at a speed of

[latex]v=0.500c.[/latex]

The crew transmits a laser signal toward Earth. Observers on the spaceship measure the light leaving the spacecraft at the speed

[latex]u'=c.[/latex]

What speed does an observer on Earth measure for the approaching light?

Strategy

Because both the spacecraft and the light are moving at relativistic speeds, we must use the relativistic velocity addition equation rather than the classical expression.

Solution

Step 1: Identify the known quantities.

[latex]v=0.500c,\qquad u'=c.[/latex]

Step 2: Use the relativistic velocity addition equation.

[latex]u=\frac{v+u'}{1+\frac{vu'}{c^2}}.[/latex]

Step 3: Substitute the known values.

[latex]\begin{aligned} u &=\frac{0.500c+c}{1+\frac{(0.500c)(c)}{c^2}}\\ &=\frac{1.500c}{1+0.500}\\ &=\frac{1.500c}{1.500}\\ &=c \end{aligned}[/latex]

Discussion

Both the astronaut and the observer on Earth measure exactly the same speed for the light:

[latex]\boxed{u=c.}[/latex]

This remarkable result agrees with countless experiments and is one of the defining features of Einstein's Special Theory of Relativity. Unlike ordinary objects, light does not obey the classical rule of adding velocities.

Healthcare Connection

Medical imaging technologies such as laser surgery systems, optical coherence tomography (OCT), pulse oximeters, and fiber-optic endoscopes all rely on the constant speed of light. Their operation assumes that light propagates at the same speed regardless of the motion of the patient or the medical device. This invariance allows these instruments to produce precise and reproducible measurements.

The relativistic velocity equation has another important consequence: no combination of velocities can produce a speed greater than the speed of light. Even when two objects each move at speeds close to [latex]c[/latex], the resulting velocity always remains below this universal limit.

The next example illustrates how relativistic velocity addition differs from the simple symmetry predicted by classical mechanics when ordinary objects—not light—are launched from a moving spacecraft.

Doppler Shift

In the previous section, we learned that the speed of light is the same for every observer, regardless of the motion of the source. However, although the speed remains constant, the wavelength and frequency of the light do change when there is relative motion between the source and the observer.

This phenomenon is known as the Doppler effect. It occurs for all types of waves, including sound, water waves, and electromagnetic radiation.

You have probably experienced the Doppler effect with sound. As an ambulance approaches, its siren sounds higher in pitch. As it passes and moves away, the pitch suddenly becomes lower. The same physical principle applies to light, although instead of hearing a change in pitch, we measure a change in wavelength or frequency.

For light, the relativistic Doppler effect is especially important because astronomical objects often move at speeds that are a significant fraction of the speed of light.

When the source moves away, the observed wavelength becomes longer. This phenomenon is called a red shift because red light has a longer wavelength than blue light.

When the source moves toward the observer, the observed wavelength becomes shorter. This is known as a blue shift.

Because frequency and wavelength are related through

the Doppler shift can also be written in terms of frequency.

Notice that the plus and minus signs are reversed compared with the wavelength equation. This occurs because wavelength and frequency are inversely proportional.

Relativistic Doppler shifts can be measured with great precision. They allow astronomers to determine whether stars and galaxies are moving toward or away from Earth and to estimate their speeds.

Doppler measurements also have many applications at lower speeds. Police radar measures vehicle motion, weather radar detects the movement of rain and storms, and medical Doppler systems measure the motion of blood through the cardiovascular system. At these ordinary speeds, however, the classical Doppler equations are generally sufficient.

Conceptual Questions

10. Explain the meanings of the terms red shift and blue shift in the context of the relativistic Doppler effect.
11. What happens to the relativistic Doppler shift when the relative velocity between the source and the observer is zero? Is this result physically reasonable? Explain.
12. The classical Doppler effect predicts that waves emitted by a source moving away from an observer have longer wavelengths. Does the relativistic Doppler effect predict the same general behavior? Explain.
13. Nearly all galaxies more than about [latex]50\times10^6[/latex] light-years from Earth exhibit a red shift that increases with distance. Assuming the red shift is caused by relative motion, what does this observation imply about the large-scale behavior of the universe? (Hint: On cosmological scales, the expansion of space produces the same observational effect.)

Problems & Exercises

20. A spaceship travels directly toward Earth at a speed of [latex]0.750c[/latex]. It launches a canister at a speed of [latex]0.500c[/latex] relative to the spaceship.
    1. What is the velocity of the canister relative to Earth if it is launched directly toward Earth?
    2. What is the velocity of the canister relative to Earth if it is launched directly away from Earth?
21. Repeat Problem 20 for a spaceship traveling directly away from Earth at [latex]0.750c[/latex].
22. A spaceship approaches Earth at [latex]0.100c[/latex]. A message capsule is sent from Earth toward the spaceship at [latex]0.100c[/latex] relative to Earth. What is the speed of the capsule relative to the spaceship?
23. Suppose the speed of light were only [latex]3000\ \text{m/s}[/latex]. A jet fighter travels toward a stationary ground target at [latex]800\ \text{m/s}[/latex] and fires bullets with a muzzle velocity of [latex]1000\ \text{m/s}[/latex] relative to the jet.
    1. What is the velocity of the bullets relative to the target?
    2. Would relativistic effects be noticeable in everyday life if the speed of light were this small? Explain.
24. A galaxy moves away from Earth at [latex]1000\ \text{km/s}[/latex] and emits light with a wavelength of [latex]656\ \text{nm}[/latex], characteristic of hydrogen.
    1. What wavelength is observed on Earth?
    2. In what region of the electromagnetic spectrum is the observed radiation?
    3. Why can Earth's orbital speed be neglected in this calculation?
25. A space probe travels away from Earth toward the nearest star at [latex]0.250c[/latex]. It transmits radio information at a frequency of [latex]1.00\ \text{GHz}[/latex]. What frequency is received on Earth?
26. Two spaceships move directly toward each other at a relative speed of [latex]0.800c[/latex]. At what speed must a canister be launched from the first spaceship so that it approaches the second spaceship at [latex]0.999c[/latex], as measured by observers on the second spaceship?
27. Two planets move directly toward each other at a relative speed of [latex]0.250c[/latex]. A spaceship launched from the first planet approaches the second planet at [latex]0.750c[/latex], as measured from the second planet. What is the spaceship's velocity relative to the first planet?
28. A missile is launched from one spaceship toward another. It leaves the first spaceship at [latex]0.950c[/latex] and approaches the second spaceship at [latex]0.750c[/latex], as measured from the second spaceship. What is the relative velocity of the two spaceships?
29. One spaceship fires a missile toward another at [latex]0.750c[/latex] relative to the first spaceship. Observers on the second spaceship measure the missile approaching at [latex]0.950c[/latex]. What is the relative velocity of the two spaceships?
30. Near the center of the Milky Way, hydrogen gas moves directly away from Earth while orbiting a black hole. Radiation emitted by the gas has a wavelength of [latex]1875\ \text{nm}[/latex], but it is observed on Earth at [latex]1900\ \text{nm}[/latex]. What is the speed of the gas?
31. A highway patrol officer measures vehicle speed by reflecting radar waves from the vehicle and detecting the Doppler shift. The transmitted radar frequency is [latex]100\ \text{GHz}[/latex], and the reflected signal has a frequency [latex]15.0\ \text{kHz}[/latex] higher than the transmitted signal.What is the velocity of the vehicle? Remember that a reflected radar signal undergoes two Doppler shifts. Do not round intermediate results because the frequency change is very small.
32. Use the relativistic velocity addition equation to prove that a beam of light sent from one observer toward another approaches the second observer at speed [latex]c[/latex] for any relative velocity [latex]v
33. Use the relativistic velocity addition equation to show that a beam of light emitted directly away from another observer moves away from that observer at speed [latex]c[/latex] for any relative velocity [latex]v
34. A galaxy located [latex]12.0\times10^9[/latex] light-years from Earth is moving away from the Milky Way at [latex]0.900c[/latex]. An exploratory probe is sent toward the galaxy.
    1. At what velocity relative to Earth must the probe travel so that it approaches the galaxy at [latex]0.990c[/latex], as measured by observers in that galaxy?
    2. How long does the journey take as measured from Earth? Assume the galaxy continues moving at constant velocity.
    3. After the probe arrives, how long does a radio signal sent back to Earth take to reach us?

    Although such a mission is possible in principle according to special relativity, it is not practical with current technology.