Ultrarelativistic limit

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

Max Planck 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 ( $p c \gg m c^2$ ), and thus can be approximated by $E = p c$ . 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 $p c \gg m c^2$ is assumed. Or, similarly, in the limit where the Lorentz factor is very large ( $\gamma \gg 1$ ).^[1] Here are some ultrarelativistic approximations (in units with c=1):

1-v ≈ 1/(2γ²)
E-p = E*(1-v) ≈ m²/(2E) = m/(2γ)
rapidity φ ≈ ln(2γ)
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 ≈ $\sqrt{2E_{1}E_{2}}$ where E₁ and E₂ are energies of the particle and the target respectively (so E₁≫E₂), 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 ( $v$ very close to 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 $E = m c^2 + \frac{p^2}{2m}$ .

References

↑ Dieckmann, ME (April 2005). "Particle simulation of an ultrarelativistic two-stream instability". Physical Review Letters 94 (15): 155001. Bibcode:2005PhRvL..94o5001D. doi:10.1103/PhysRevLett.94.155001. PMID 15904153.

Ultrarelativistic limit

Accuracy of the approximation

Other limits

See also

References