Astroparticle Physics

3.3 Lorentz Transformation 45energies and momenta are obtained by a Lorentz transformation.IfEand p are energy and momentum in the centerof-masssystem and if the laboratory system moves with thevelocity β relative to p ‖ , the transformed quantities E ∗ andp‖ ∗ in this system are calculated to be (compare Fig. 3.5)( E∗)( )( )γ −γβ E=,p−γβ γ⊥ ∗ p = p ⊥ . (3.50)‖p ∗ ‖The transverse momentum component is not affected by thistransformation. Instead of using the matrix notation, (3.50)can be written asE ∗ = γE− γβp ‖ ,p‖ ∗ =−γβE + γp (3.51)‖ .For β = 0 and correspondingly γ = 1 one trivially obtainsE ∗ = E and p‖ ∗ = p ‖ .Fig. 3.5Illustration of a LorentztransformationA particle of energy E = γ 2 m 0 , seen from a systemwhich moves with β 1 relative to the particle parallel to themomentum p, gets in this system the energyE ∗ = γ 1 E − γ 1 β 1 p ‖√γ1 2= γ 1 γ 2 m 0 − γ − 11√(γ 2 m 0 )γ 2 − m 2 01√= γ 1 γ 2 m 0 − m 0 γ1 √γ 2 − 1 2 2 − 1 . (3.52)If γ 1 = γ 2 = γ (for a system that moves along with aparticle) one naturally obtainsE ∗ = γ 2 m 0 − m 0 (γ 2 − 1) = m 0 .