Relativistic Addition of Velocities
Because Einstein identified the speed of light as a universal speed limit, it can now no longer be strictly true that the ground speed of a person running forward on a fast-moving airplane will just be the sum of the plane's ground speed and the person's speed relative to the plane. This velocity-addition formula, which goes back all the way to Galileo, would in principle allow ground speeds larger than c to be achieved by adding speeds that are individually smaller than c.
As this text discusses, Einstein's Law for Velocity addition is more complicated, but it becomes approximately the same as Galileo's old law when all the speeds involved are low compared to the speed of light.
Figure 28.13 The total velocity of a kayak, like this one on the Deerfield River in Massachusetts, is its velocity relative to the water as well as the water’s velocity relative to the riverbank. (credit: abkfenris, Flickr)
If you have ever seen a kayak move down a fast-moving river, you know that remaining in the same place would be hard. The river current pulls the kayak along. Pushing the oars back against the water can move the kayak forward in the water, but that
only accounts for part of the velocity. The kayak’s motion is an example of classical addition of velocities. In classical physics, velocities add as vectors. The kayak’s velocity is the vector sum of its velocity relative to the water and the water’s
velocity relative to the riverbank.
Classical Velocity Addition
For simplicity, we restrict our consideration of velocity addition to one-dimensional motion. Classically, velocities add like regular numbers in one-dimensional motion. (See Figure 28.14.) Suppose, for example, a girl is riding in a
sled at a speed 1.0 m/s relative to an observer. She throws a snowball first forward, then backward at a speed of 1.5 m/s relative to the sled. We denote direction with plus and minus signs in one dimension; in this example, forward is positive. Let
be the velocity of the sled relative to the Earth,
the velocity of the snowball relative to the Earth-bound observer, and
the velocity of the snowball relative to the sled.
Figure 28.14 Classically, velocities add like ordinary numbers in one-dimensional motion. Here the girl throws a snowball forward and then backward from a sled. The velocity of the sled relative to the Earth is . The velocity of the snowball
relative to the sled is
, while its velocity relative to the Earth is
. Classically,
.
Thus, when the girl throws the snowball forward, . It makes good intuitive sense that the snowball will head towards the Earth-bound observer faster, because it is thrown forward from a moving vehicle. When the girl
throws the snowball backward,
. The minus sign means the snowball moves away from the Earth-bound observer.
Relativistic Velocity Addition
The second postulate of relativity (verified by extensive experimental observation) says that classical velocity addition does not apply to light. Imagine a car traveling at night along a straight road, as in Figure 28.15. If classical
velocity addition applied to light, then the light from the car’s headlights would approach the observer on the sidewalk at a speed . But we know that light will move away from the car at speed
relative to the driver of the car, and
light will move towards the observer on the sidewalk at speed
, too.
Figure 28.15 According to experiment and the second postulate of relativity, light from the car’s headlights moves away from the car at speed and towards the observer on the sidewalk at speed
. Classical velocity addition is not valid.
Relativistic Velocity Addition
Either light is an exception, or the classical velocity addition formula only works at low velocities. The latter is the case. The correct formula for one-dimensional relativistic velocity addition is
where is the relative velocity between two observers,
is the velocity of an object relative to one observer, and
is the velocity relative to the other observer. (For ease of visualization, we often choose to measure
in our reference frame, while someone moving at
relative to us measures
.) Note that the term
becomes very small at low velocities, and
gives a result very
close to classical velocity addition. As before, we see that classical velocity addition is an excellent approximation to the correct relativistic formula for small velocities. No wonder that it seems correct in our experience.
Example 28.3 Showing that the Speed of Light towards an Observer is Constant (in a Vacuum): The Speed of Light is the Speed of Light
Suppose a spaceship heading directly towards the Earth at half the speed of light sends a signal to us on a laser-produced beam of light. Given that the light leaves the ship at speed as observed from the ship, calculate the speed at which
it approaches the Earth.
Figure 28.16
Strategy
Because the light and the spaceship are moving at relativistic speeds, we cannot use simple velocity addition. Instead, we can determine the speed at which the light approaches the Earth using relativistic velocity addition.
Solution
- Identify the knowns.
;
- Identify the unknown.
- Choose the appropriate equation.
- Plug the knowns into the equation.
Discussion
Relativistic velocity addition gives the correct result. Light leaves the ship at speed and approaches the Earth at speed
. The speed of light is independent of the relative motion of source and observer, whether the observer is on
the ship or Earth-bound.
Velocities cannot add to greater than the speed of light, provided that is less than
and
does not exceed
. The following example illustrates that relativistic velocity addition is not as symmetric as classical velocity addition.
Example 28.4 Comparing the Speed of Light towards and away from an Observer: Relativistic Package Delivery
Suppose the spaceship in the previous example is approaching the Earth at half the speed of light and shoots a canister at a speed of . (a) At what velocity will an Earth-bound observer see the canister if it is shot directly towards the Earth?
(b) If it is shot directly away from the Earth? (See Figure 28.17.)
Figure 28.17
Strategy
Because the canister and the spaceship are moving at relativistic speeds, we must determine the speed of the canister by an Earth-bound observer using relativistic velocity addition instead of simple velocity addition.
Solution for (a)
- Identify the knowns.
;
- Identify the unknown.
- Choose the appropriate equation.
- Plug the knowns into the equation.
Solution for (b)
- Identify the knowns.
;
- Identify the unknown.
- Choose the appropriate equation.
- Plug the knowns into the equation.
Discussion
The minus sign indicates velocity away from the Earth (in the opposite direction from ), which means the canister is heading towards the Earth in part (a) and away in part (b), as expected. But relativistic velocities do not add as simply
as they do classically. In part (a), the canister does approach the Earth faster, but not at the simple sum of
.
The total velocity is less than you would get classically. And in part (b), the canister moves away from the Earth at a velocity of , which is faster than the
you would expect classically. The velocities
are not even symmetric. In part (a) the canister moves 0.409
faster than the ship relative to the Earth, whereas in part (b) it moves
slower than the ship.
Doppler Shift
Although the speed of light does not change with relative velocity, the frequencies and wavelengths of light do. First discussed for sound waves, a Doppler shift occurs in any wave when there is relative motion between source and observer.
In the Doppler equation, is the observed wavelength,
is the source wavelength, and
is the relative velocity of the source to the observer. The velocity
is positive for motion away from an observer and negative
for motion toward an observer. In terms of source frequency and observed frequency, this equation can be written
Notice that the – and + signs are different than in the wavelength equation.
Career Connection: Astronomer
If you are interested in a career that requires a knowledge of special relativity, there’s probably no better connection than astronomy. Astronomers must take into account relativistic effects when they calculate distances, times, and speeds of black holes, galaxies, quasars, and all other astronomical objects. To have a career in astronomy, you need at least an undergraduate degree in either physics or astronomy, but a Master’s or doctoral degree is often required. You also need a good background in high-level mathematics.
Example 28.5 Calculating a Doppler Shift: Radio Waves from a Receding Galaxy
Suppose a galaxy is moving away from the Earth at a speed . It emits radio waves with a wavelength of
. What wavelength would we detect on the Earth?
Strategy
Because the galaxy is moving at a relativistic speed, we must determine the Doppler shift of the radio waves using the relativistic Doppler shift instead of the classical Doppler shift.
Solution
- Identify the knowns.
;
- Identify the unknown.
- Choose the appropriate equation.
- Plug the knowns into the equation.
Discussion
Because the galaxy is moving away from the Earth, we expect the wavelengths of radiation it emits to be redshifted. The wavelength we calculated is 1.70 m, which is redshifted from the original wavelength of 0.525 m.
The relativistic Doppler shift is easy to observe. This equation has everyday applications ranging from Doppler-shifted radar velocity measurements of transportation to Doppler-radar storm monitoring. In astronomical observations, the relativistic Doppler shift provides velocity information such as the motion and distance of stars.
Source: Rice University, https://openstax.org/books/college-physics/pages/28-4-relativistic-addition-of-velocities
This work is licensed under a Creative Commons Attribution 4.0 License.