Ultrarelativistic limit

In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light .
The expression for the relativistic energy of a particle with rest mass and momentum is given by
The energy of an ultrarelativistic particle is almost completely due to its momentum (), and thus can be approximated by . This can result from holding the mass fixed and increasing to very large values (the usual case); or by holding the energy fixed and shrinking the mass 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 is assumed. Or, similarly, in the limit where the Lorentz factor is very large ().
Expression including mass value.
While it is possible to use the approximation formula_2, this neglects all information of the mass. In some cases, even with formula_3, 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 formula_4 (and the ultrarelativistic limit formula_5). 
Without loss of generality, the same can be shown including the appropriate formula_6 terms.
The generic expression formula_9 can be Taylor expanded, giving:
Using just the first two terms, this can be substituted into the above expression (with formula_11 acting as formula_12), as:
Ultrarelativistic approximations.
Below are some ultrarelativistic approximations in units with . The rapidity is denoted :
Accuracy of the approximation.
For calculations of the energy of a particle, the relative error of the ultrarelativistic limit for a speed is about %, and for it is just %. For particles such as neutrinos, whose (Lorentz factor) are usually above ( practically indistinguishable from ), the approximation is essentially exact.
Other limits.
The opposite case () is a so-called classical particle, where its speed is much smaller than and so its energy can be approximated by .