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 or or where is the Lorentz factor, and is the speed of light.
The energy of an ultrarelativistic particle is almost completely due to its kinetic energy . The total energy can also be approximated as where 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 . 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
Ultrarelativistic approximations
Below are few ultrarelativistic approximations when . The rapidity is denoted :
- 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 ≈ √2E1E2 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.
Accuracy of the approximation
For calculations of the energy of a particle, the
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 binomial series: