Relativistic momentum ($\mathbf{p}=\gamma m\mathbf{v}$)

For a massive particle of invariant rest mass $m$ and velocity $\mathbf{v}$ in an inertial frame,Together these form the four-momentum and satisfy $E^2-p^2{c}^2=m^2{c}^4$. At low speed, $\mathbf{p}\simeq m\mathbf{v}$; as $v\to c$ for fixed nonzero rest mass, the momentum grows without bound. The expression $\gamma m\mathbf{v}$ is not used for a massless photon; its momentum magnitude is $E/c$.

The relativistic expression also sets magnetic focusing scales. In a quadrupole magnet with $B_y=Gx$, a nearly axial charged particle has $d{p}_x/dt\simeq -q{v}_{z}Gx$ from the Lorentz force. In the small-angle approximation, $x^{\prime }=dx/dz\simeq p_x/p_z$. With constant leading longitudinal momentum $p_z$,Thus the focusing coefficient uses relativistic longitudinal momentum, rather than an uncorrected $m{v}_z$. The opposite transverse plane has the opposite focusing sign for an ideal quadrupole magnet.