Ultrarelativistic limit

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

Paul Dirac 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 ( $γ ≫ 1$ ).

While it is possible to use the approximation $E=pc$ , this neglects all information of the mass; in some cases, even with $p\gg m$ , 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 $c=1$ (and the ultrarelativistic limit $pc\gg mc^{2}$ ). Without loss of generality, the same can be showed including the appropriate $c$ terms.

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