ULTRARELATIVISTIC LIMIT

In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light $c$ , such that its total energy $E 2 = m 2 c 4 + p 2 c 2$ is almost completely due to its momentum ( $p c \gg m c^2$ ), and thus can be approximated by $E = p c$ .

In general, the ultrarelativistic limit of an expression is the resulting simplified expression when $p c \gg m c^2$ is assumed. Or, similarly, in the limit where the Lorentz factor is very large ( $\gamma \gg 1$ ).

Goodness of the approximation

For calculations of the energy of a particle, the relative error of the ultrarelativistic limit for a speed $v = 0.95 c$ is about 10%, and for $v = 0.99 c$ it is just 2%. For particles such as neutrinos, whose γ are usually above 10⁶ ( $v$ very close to c), the approximation is essentially exact.

Other limits

The opposite case is a so-called classical particle, where its speed is much smaller than $c$ and so its energy can be approximated by $E = m c^2 + \frac{p^2}{2m}$ .

ULTRARELATIVISTIC LIMIT

Goodness of the approximation

Other limits

See also