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Concept
Ultrarelativistic limit
Summary
In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light c.
The expression for the relativistic energy of a particle with rest mass m and momentum p is given by
The energy of an ultrarelativistic particle is almost completely due to its momentum (pc ≫ mc2), 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 is assumed. Or, similarly, in the limit where the Lorentz factor is very large ().
While it is possible to use the approximation , this neglects all information of the mass. In some cases, even with , 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 (and the ultrarelativistic limit ).
Without loss of generality, the same can be shown including the appropriate terms.
The generic expression can be Taylor expanded, giving:
Using just the first two terms, this can be substituted into the above expression (with acting as ), as:
Below are some ultrarelativistic approximations in units with c = 1. The rapidity is denoted φ:
1 − v ≈
E − p = E(1 − v) ≈ =
φ ≈ ln(2γ)
Motion with constant proper acceleration: d ≈ eaτ/(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: ECM ≈ where E1 and E2 are energies of the particle and the target respectively (so E1 ≫ E2), and ECM is energy in the center of mass frame.
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