Relativistic kinematics

Relativistic kinematics relates positions, times, velocities, energy and momentum between inertial frames using special relativity. The Lorentz transformation changes the components while preserving the relevant Minkowski interval or Minkowski norm. It describes which motions and reactions are possible without by itself supplying the forces or reaction probability.

For a particle of rest mass $m$, its four-momentum gives $E^2-p^2{c}^2=m^2{c}^4$. At $v=0.6c$, the Lorentz factor is $1.25$, so $E=1.25m{c}^2$ and $p=0.75mc$. The relation remains the same in another inertial frame, although $E$ and $p$ separately change. Combining this invariant with conservation laws yields a reaction threshold.

The kinematic minimum energy needed to produce an allowed final state. In the centre-of-momentum frame, the threshold for freely separated massive final particles is the sum of their rest energies, with no relative kinetic energy. Conservation of total energy and momentum then determines the corresponding laboratory threshold. Charges and other selection rules must also permit the reaction; the mass bound alone does not guarantee an observable transition.