# Ultrarelativistic limit

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

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/√1 − *v*^{2}/*c*^{2} is very large (*γ* ≫ 1).[1]

## Expression including mass value

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 showed including the appropriate terms.

Derivation |
---|

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: |

## Ultrarelativistic approximations

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

- 1 −
*v*≈ ^{1}⁄_{2γ2} *E*−*p*=*E*(1 −*v*) ≈ ^{m2}⁄_{2E}= ^{m}⁄_{2γ}*φ*≈ ln(2*γ*)- Motion with constant proper acceleration:
*d*≈*e*^{aτ}/(2*a*), where d is the distance traveled,*a*=*dφ*/*dτ*is proper acceleration (with*aτ*≫ 1), τ is proper time, and travel starts at rest and without changing direction of acceleration (see proper acceleration for more details). - Fixed target collision with ultrarelativistic motion of the center of mass:
*E*_{CM}≈ √2*E*_{1}*E*_{2}} where*E*_{1}and*E*_{2}are energies of the particle and the target respectively (so*E*_{1}≫*E*_{2}), and*E*_{CM}is energy in the center of mass frame.

## Accuracy 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 (Lorentz factor) are usually above 10^{6} (v practically indistinguishable from c), the approximation is essentially exact.

## Other limits

The opposite case (*pc* ≪ *mc*^{2}) is a so-called **classical particle**, where its speed is much smaller than c and so its energy can be approximated by *E* = *mc*^{2} + ^{p2}⁄_{2m}.

## Notes

## References

- Dieckmann, M. E. (2005). "Particle simulation of an ultrarelativistic two-stream instability".
*Phys. Rev. Lett*.**94**(15): 155001. Bibcode:2005PhRvL..94o5001D. doi:10.1103/PhysRevLett.94.155001. PMID 15904153.