62.5: Momentum

In Newton's classical mechanics, momentum is $\mathbf{p}=m \mathbf{v}$. Under special relativity, this is modified to be

$$\mathbf{p}=\gamma m \mathbf{v} .$$

Relativistically, it is this definition of momentum that is conserved. Newton's second law in the form $\mathbf{F}=m \mathbf{a}$ is no longer valid under special relativity, but Newton's original form $\mathbf{F}=d \mathbf{p} / d t$ is still valid, using this definition of momentum $\mathbf{p}$.

Notice that as $v \rightarrow c$, we have $\gamma \rightarrow \infty$ (by Eq. 62.3.2), and so momentum $p \rightarrow \infty$. As a body goes faster, its momentum increases in such a way that it becomes increasingly difficult to make it go even faster. This means that it is not possible for a body to move faster than the speed of light in vacuum, $c$.