In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light . Notations commonly used are
v ≈ c
\beta ≈ 1
\gamma\gg1
\gamma
\beta=v/c
c
The energy of an ultrarelativistic particle is almost completely due to its kinetic energy
Ek=(\gamma-1)mc2
E=\gammamc2 ≈ pc
p=\gammamv
This can result from holding the mass fixed and increasing the kinetic energy to very large values or by holding the energy fixed and shrinking the mass to very small values which also imply a very large
\gamma
Below are few ultrarelativistic approximations when
\beta ≈ 1
w
1-\beta ≈
1 | |
2\gamma2 |
w ≈ ln(2\gamma)
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.
The opposite case is a so-called classical particle, where its speed is much smaller than . Its kinetic energy can be approximated by first term of the
\gamma
Ek=(\gamma-1)mc2=
1 | |
2 |
mv2+\left[
3 | |
8 |
m
v4 | |
c2 |
+...+mc2
(2n)! | |
22n(n!)2 |
v2n | |
c2n |
+...\right]