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 $E_{k}=(\gamma -1)mc^{2}$ . The total energy can also be approximated as $E=\gamma mc^{2}\approx pc$ 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 $E$ 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). ^{[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: $d \approx e aτ /(2 a)$ , 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: $E CM \approx \sqrt 2 E 1 E 2$ 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 $106$ ( $v$ 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)mc^{2}={\frac {1}{2}}mv^{2}+\left[{\frac {3}{8}}m{\frac {v^{4}}{c^{2}}}+...+mc^{2}{\frac {(2n)!}{2^{2n}(n!)^{2}}}{\frac {v^{2n}}{c^{2n}}}+...\right]

References

Fahrer / Team für die Suche eingeben: