Ultrarelativistic limit

In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light $c$ .

Albert Einstein showed that the expression for the relativistic energy of a particle with rest mass $m$ and momentum $p$ is given by

E^{2}=m^{2}c^{4}+p^{2}c^{2}.

The energy of an ultrarelativistic particle is almost completely due to its momentum ( $pc ≫ mc 2$ ), and thus can be approximated by $E = pc$ . This can result from holding the mass fixed and increasing $p$ to very large values (the usual case); or by holding the energy $E$ fixed and shrinking the mass $m$ to negligible values. 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).

In general, the ultrarelativistic limit of an expression is the resulting simplified expression when $pc ≫ mc 2$ is assumed. Or, similarly, in the limit where the Lorentz factor $γ = 1/ \sqrt 1 - v 2 / c 2$ is very large ( $γ ≫ 1$ ).^[1]

Expression including mass value

While it is possible to use the approximation $E=pc$ , this neglects all information of the mass; in some cases, even with $p\gg m$ , the mass may not be ignored, as in the derivation of neutrino oscillation. A simple way to retain this mass information is using a Taylor expansion rather than a simple limit. The following derivation assumes $c=1$ (and the ultrarelativistic limit $pc\gg mc^{2}$ ). Without loss of generality, the same can be showed including the appropriate $c$ terms.

Derivation
$E=(p^{2}+m^{2})^{1/2}$ $E=p(1+{\frac {m^{2}}{p^{2}}})^{1/2}$ The generic expression $(1+x)^{1/2}$ can be Taylor expanded, giving: $(1+x^{2})^{1/2}=1+{\frac {x^{2}}{2}}-{\frac {x^{4}}{8}}+...$ Using just the first two terms, this can be substituted into the above expression (with ${\frac {m}{p}}$ acting as $x$ ), as: $E=p(1+{\frac {m^{2}}{2p^{2}}})$ $E=p+{\frac {m^{2}}{2p}}$

Derivation

$E=(p^{2}+m^{2})^{1/2}$
$E=p(1+{\frac {m^{2}}{p^{2}}})^{1/2}$

The generic expression $(1+x)^{1/2}$ can be Taylor expanded, giving:

$(1+x^{2})^{1/2}=1+{\frac {x^{2}}{2}}-{\frac {x^{4}}{8}}+...$

Using just the first two terms, this can be substituted into the above expression (with ${\frac {m}{p}}$ acting as $x$ ), as:

$E=p(1+{\frac {m^{2}}{2p^{2}}})$
$E=p+{\frac {m^{2}}{2p}}$

Ultrarelativistic approximations

Below are some ultrarelativistic approximations in units with $c = 1$ . The rapidity is denoted $φ$ :

$1 - v \approx 1 ⁄ 2 γ 2$
$E - p = E (1 - v) \approx m 2 ⁄ 2 E = m ⁄ 2 γ$
$φ \approx ln(2 γ)$
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 is a so-called classical particle, where its speed is much smaller than $c$ and so its energy can be approximated by $E = mc 2 + p 2 ⁄ 2 m$ .

Notes

↑ Dieckmann 2005.

References

Dieckmann, M. E. (2005). "Particle simulation of an ultrarelativistic two-stream instability". Phys. Rev. Lett. 94 (15): 155001. Bibcode:2005PhRvL..94o5001D. PMID 15904153. doi:10.1103/PhysRevLett.94.155001.

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