28.5 Relativistic Momentum

Relativistic momentum extends classical momentum to handle objects moving at speeds close to the speed of light. Without this correction, momentum calculations break down at high velocities, which matters for everything from particle accelerators to cosmic ray physics.

Relativistic Momentum

Formula for Relativistic Momentum

The relativistic momentum formula looks similar to the classical one, but with an extra factor that accounts for relativistic effects:

$p = \gamma mv$

• $p$ = relativistic momentum
• $m$ = the object's rest mass (also called invariant mass)
• $v$ = the object's velocity relative to the observer
• $\gamma$ = the Lorentz factor, defined as:

$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$

where $c$ is the speed of light (approximately $3 \times 10^8$ m/s).

The Lorentz factor is what makes relativistic momentum behave so differently from classical momentum. At everyday speeds, $\gamma$ is essentially 1, so $p = \gamma mv$ just reduces to $p = mv$. But as $v$ approaches $c$, $\gamma$ grows without bound. This means an object's momentum increases far more steeply than you'd expect from the classical formula, and it approaches infinity as $v \to c$. That's why no object with mass can actually reach the speed of light: you'd need infinite energy to push its momentum that high.

Conservation of momentum still holds in relativity, but you must use the relativistic formula when speeds are significant fractions of $c$.

Significance of Rest Mass

Rest mass (invariant mass) is the mass of an object measured in its own rest frame, where it's stationary relative to the observer. This value doesn't change no matter how fast the object moves or who's observing it.

You may encounter the older idea of "relativistic mass," which suggests that mass itself increases with speed. This concept is considered outdated and tends to cause confusion. Modern physics treats rest mass as the only meaningful mass. The effects that "relativistic mass" tried to capture are already built into the Lorentz factor $\gamma$.

All the key relativistic equations, including $E = mc^2$ (where $m$ is rest mass), work cleanly with rest mass. Sticking to rest mass keeps your calculations consistent and avoids unnecessary complications.

Classical vs. Relativistic Momentum

Classical momentum is simply $p = mv$, and it works perfectly well for objects moving at ordinary speeds. Relativistic momentum, $p = \gamma mv$, adds the Lorentz factor to account for speeds approaching $c$.

Here's how they compare at different speeds:

• At $v = 0.01c$: $\gamma \approx 1.00005$. Classical and relativistic momentum are virtually identical.
• At $v = 0.5c$: $\gamma \approx 1.15$. Relativistic momentum is about 15% higher than the classical prediction.
• At $v = 0.9c$: $\gamma \approx 2.29$. Relativistic momentum is more than double the classical value.
• At $v = 0.99c$: $\gamma \approx 7.09$. The classical formula is now wildly inaccurate.

The pattern is clear: at low speeds ($v \ll c$), $\gamma \approx 1$ and the two formulas agree. As speed climbs toward $c$, relativistic momentum pulls sharply ahead of the classical prediction because $\gamma$ grows rapidly. Classical momentum is a low-speed approximation that breaks down when velocities become a significant fraction of the speed of light.

Relativistic Effects and Reference Frames

Momentum measurements depend on the observer's reference frame, and special relativity introduces effects that matter when comparing measurements between frames moving at high relative velocities:

• Time dilation: A moving clock runs slower compared to a stationary one. This affects how you measure velocity (and therefore momentum) from different frames.
• Length contraction: Objects are measured to be shorter along their direction of motion. This also influences how different observers calculate momentum.

These effects are already baked into the Lorentz factor, so using $p = \gamma mv$ automatically accounts for them.

Relativistic momentum is also tightly connected to relativistic energy. Both quantities increase with velocity and both approach infinity as $v \to c$. The full energy-momentum relation, $E^2 = (pc)^2 + (mc^2)^2$, ties them together and is one of the most important results in special relativity.