Ultrarelativistic

In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light $c$ .

Albert Einstein showed that the expression for the relativistic energy of a particle with rest mass $m$ and momentum $p$ is given by

The energy of an ultrarelativistic particle is almost completely due to its momentum ( $pc ≫ mc 2$ ), and thus can be approximated by $E = pc$ . This can result from holding the mass fixed and increasing $p$ to very large values (the usual case); or by holding the energy $E$ fixed and shrinking the mass $m$ to negligible values. The latter is used to derive orbits of massless particles such as the photon from those of massive particles (cf. Kepler problem in general relativity).

In general, the ultrarelativistic limit of an expression is the resulting simplified expression when $pc ≫ mc 2$ is assumed. Or, similarly, in the limit where the Lorentz factor $γ = 1/ \sqrt 1 - v 2 / c 2$ is very large ( $γ ≫ 1$ ).

Below are some ultrarelativistic approximations in units with $c = 1$ . The rapidity is denoted $φ$ :

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 $γ$ (Lorentz factor) are usually above $106$ ( $v$ practically indistinguishable from $c$ ), the approximation is essentially exact.

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