Ultrarelativistic limit

Below are few ultrarelativistic approximations when \beta \approx 1. The rapidity is denoted w: : 1 - \beta \approx \frac{1}{2\gamma^2} : w \approx \ln(2 \gamma) • Motion with constant proper acceleration: , where is the distance traveled, is proper acceleration (with ), 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: where and are energies of the particle and the target respectively (so ), and 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 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 . Its kinetic energy can be approximated by first term of the \gamma binomial series: : E_k = (\gamma - 1) m c^2 = \frac{1}{2} m v^2 + \left[\frac{3}{8} m \frac{v^4}{c^2} + ... + m c^2 \frac{(2n)!}{2^{2n}(n!)^2}\frac{v^{2n}}{c^{2n}} + ...\right] == See also ==