Ultrarelativistic limit

Template:Short description In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light Template:Math. Notations commonly used are $v\approx c$ or $\beta \approx 1$ or $\gamma \gg 1$ where $\gamma$ is the Lorentz factor, $\beta =v/c$ and $c$ is the speed of light.

The energy of an ultrarelativistic particle is almost completely due to its kinetic energy $E_k=(\gamma -1)m{c}^2$. The total energy can also be approximated as $E=\gamma m{c}^2\approx pc$ where $p=\gamma mv$ is the Lorentz invariant momentum.

This can result from holding the mass fixed and increasing the kinetic energy to very large values or by holding the energy Template:Math fixed and shrinking the mass Template:Math to very small values which also imply a very large $\gamma$. Particles with a very small mass do not need much energy to travel at a speed close to $c$. 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). Template:Citation needed

Ultrarelativistic approximations

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: Template:Math, where Template:Mvar is the distance traveled, Template:Math is proper acceleration (with Template:Math), Template:Mvar 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: Template:Math where Template:Math and Template:Math are energies of the particle and the target respectively (so Template:Math), and Template:Math 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 Template:Math is about Template:Math%, and for Template:Math it is just Template:Math%. For particles such as neutrinos, whose Template:Mvar (Lorentz factor) are usually above Template:Math (Template:Mvar practically indistinguishable from Template:Mvar), the approximation is essentially exact.

Other limits

The opposite case (Template:Math) is a so-called classical particle, where its speed is much smaller than Template:Mvar. Its kinetic energy can be approximated by first term of the $\gamma$ binomial series:

$E_k=(\gamma -1)m{c}^2=\frac{1}{2}m{v}^2+[\frac{3}{8}m\frac{v^4}{c^2}+...+m{c}^2\frac{(2n)!}{2^{2n}(n!{)}^2}\frac{v^{2n}}{c^{2n}}+...]$

See also

• Relativistic particle
• Classical mechanics
• Special relativity
• Aichelburg–Sexl ultraboost