Ultrarelativistic limit

In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light $c$. 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 $$p = \gamma m v$$ 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 $E$ fixed and shrinking the mass $m$ 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).

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: $d ≈ e^{aτ}/(2a)$, 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: $E_{CM} ≈ √2E_{1}E_{2}$ 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 $v = 0.95c$ is about $10$%, and for $v = 0.99c$ 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 ($v ≪ c$) is a so-called classical particle, where its speed is much smaller than $c$. Its kinetic energy can be approximated by first term of the $$\gamma$$ binomial series:


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