28.4 Relativistic Addition of Velocities

Einstein's special relativity changed how we think about motion at high speeds. At everyday velocities, classical physics works perfectly fine. But once objects start moving at a significant fraction of the speed of light, you need new formulas that account for the fact that nothing can exceed $c$. This section covers the relativistic velocity addition formula and the relativistic Doppler effect.

Relativistic Velocity Addition and Doppler Effect

Relativistic velocity addition formula

In classical physics, you just add velocities together. If you're on a train moving at 50 km/h and you throw a ball forward at 20 km/h, someone on the ground sees the ball moving at 70 km/h. Simple addition. But this breaks down at high speeds because it would let you exceed the speed of light, which relativity forbids.

The relativistic velocity addition formula fixes this:

$u = \frac{v + u'}{1 + \frac{vu'}{c^2}}$

• $v$ is the velocity of the moving reference frame relative to the original frame (e.g., a spaceship's speed as seen from Earth)
• $u'$ is the velocity of an object as measured within that moving frame (e.g., a probe launched from the spaceship)
• $u$ is the velocity of the object as measured from the original frame (e.g., the probe's speed as seen from Earth)
• $c$ is the speed of light in vacuum (approximately $3 \times 10^8$ m/s)

The key feature is the denominator: $1 + \frac{vu'}{c^2}$. At low speeds, $\frac{vu'}{c^2}$ is extremely small, so the denominator is essentially 1, and the formula reduces to plain addition. At high speeds, the denominator grows and prevents the result from ever reaching or exceeding $c$.

Example: A spaceship travels at $v = 0.6c$ relative to Earth. It fires a probe forward at $u' = 0.5c$ relative to itself. Classical addition would give $1.1c$, which is impossible. Using the relativistic formula:

$u = \frac{0.6c + 0.5c}{1 + \frac{(0.6c)(0.5c)}{c^2}} = \frac{1.1c}{1 + 0.3} = \frac{1.1c}{1.3} \approx 0.846c$

The result stays below $c$, as it must.

To solve problems with this formula:

1. Identify which velocity is $v$ (the moving frame) and which is $u'$ (the object within that frame)
2. Substitute the known values into the formula
3. Simplify the denominator first
4. Divide to find $u$

Relativistic effects become significant when velocities exceed roughly 10% of the speed of light (about $3 \times 10^7$ m/s). Real-world situations where this matters include particle accelerators and cosmic ray interactions.

Classical vs. relativistic velocity calculations

Classical velocity addition uses the straightforward formula:

$u = v + u'$

This is based on the Galilean transformation from classical mechanics and works perfectly for everyday speeds. Cars, airplanes, even rockets in low Earth orbit are all slow enough relative to $c$ that the relativistic correction is negligible.

Relativistic velocity addition is needed when speeds become a substantial fraction of $c$. The formula accounts for effects like time dilation (clocks in a fast-moving frame tick slower as seen by a stationary observer) and length contraction (objects appear compressed along their direction of motion at high speeds). These aren't just mathematical quirks; they're real, measured phenomena.

How do you decide which formula to use? Compare the velocities involved to $c$:

• Below ~10% of $c$ (less than about $3 \times 10^7$ m/s): Classical addition is accurate enough. The relativistic correction is tiny.
• Above ~10% of $c$: Use the relativistic formula. The difference between classical and relativistic results becomes measurable and physically meaningful.

Relativistic Doppler effect applications

You're probably familiar with the classical Doppler effect: a siren sounds higher-pitched as an ambulance approaches and lower-pitched as it moves away. The relativistic Doppler effect applies the same idea to light (and other electromagnetic radiation) when the source or observer moves at speeds where relativity matters.

The frequency equation for the relativistic Doppler effect is:

$f = f_0 \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}$

• $f_0$ is the frequency emitted by the source
• $f$ is the frequency measured by the observer
• $v$ is the relative velocity between source and observer

The wavelength equation is:

$\lambda = \lambda_0 \sqrt{\frac{1 - \frac{v}{c}}{1 + \frac{v}{c}}}$

• $\lambda_0$ is the wavelength emitted by the source
• $\lambda$ is the wavelength measured by the observer

Note the sign convention carefully. In these formulas as written above, $v$ is positive when the source is approaching the observer. This gives a higher observed frequency and shorter observed wavelength. Some textbooks use the opposite convention (positive $v$ for recession), so always check which convention your course uses.

Redshift occurs when a source moves away from the observer. The observed wavelength stretches toward the red (longer-wavelength) end of the spectrum. This is how astronomers determine that distant galaxies are receding from us, providing key evidence for the expansion of the universe.

Blueshift occurs when a source moves toward the observer. The observed wavelength compresses toward the blue (shorter-wavelength) end of the spectrum. Nearby galaxies that are gravitationally falling toward the Milky Way, like the Andromeda Galaxy, show blueshift.

To solve Doppler effect problems:

1. Determine whether you need the frequency or wavelength equation
2. Identify the source values ($f_0$ or $\lambda_0$) and the relative velocity $v$
3. Pay close attention to the sign convention for $v$ in your textbook
4. Substitute values and compute the result

Special relativity and spacetime

A few broader concepts tie this all together. Einstein's special relativity rests on two postulates, one of which is that the speed of light is the same in all inertial (non-accelerating) reference frames. This single idea is what forces us to use the relativistic velocity addition formula instead of simple addition.

Minkowski spacetime is the mathematical framework that combines three spatial dimensions with time into a unified four-dimensional structure. In this framework, events are described by four coordinates (three for space, one for time), and the geometry of spacetime explains why time dilation and length contraction occur. Four-velocity extends the concept of velocity into this four-dimensional spacetime, incorporating both spatial motion and the passage of time into a single quantity. These are more advanced topics, but they provide the deeper foundation for everything covered in this section.