In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light c.
The energy of an ultrarelativistic particle is almost completely due to its momentum (pc >> mc2), 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 >> mc2 is assumed. Or, similarly, in the limit where the Lorentz factor ? = 1/ is very large (? >> 1).
While it is possible to use the approximation , this neglects all information of the mass. In some cases, even with , the mass may not be ignored, as in the derivation of neutrino oscillation. A simple way to retain this mass information is using a Taylor expansion rather than a simple limit. The following derivation assumes (and the ultrarelativistic limit ). Without loss of generality, the same can be shown including the appropriate terms.
The generic expression can be Taylor expanded, giving:
Using just the first two terms, this can be substituted into the above expression (with acting as ), as:
Below are some ultrarelativistic approximations in units with c = 1. The rapidity is denoted ?:
For calculations of the energy of a particle, the relative error of the ultrarelativistic limit for a speed v = 0.95c is about 10%, and for v = 0.99c it is just 2%. For particles such as neutrinos, whose ? (Lorentz factor) are usually above 106 (v practically indistinguishable from c), the approximation is essentially exact.
The opposite case (pc mc2) is a so-called classical particle, where its speed is much smaller than c and so its energy can be approximated by E = mc2 + .