318 6 Special Theory of Relativity
Inverse transformations
cpx=γ(cpx′+βE′) (6.41)
cpy=cp′y (6.42)cpz=cp′z (6.43)E=γ(E
′
−βcp′x∗) (6.44)
E′^2 −c^2 p′^2 =E^2 −c^2 p^2 =m^20 =Invariant (6.45)
γ′=γγ 0 (1−ββ 0 cosθ 0 ) (6.46)
γ 0 =γγ′(1+ββ∗cosθ∗) (6.47)where zeros refer to the particle’s velocity, Lorentz factor and the angle in the
S-system while primes refer to the corresponding quantities in theS′system.
Transformation of angles
tanθ′=
sinθ
γ(
cosθ−ββ 0) (6.48)
tanθ=sinθ′
γc(cosθ′−β/β′)(6.49)
Optical Doppler effect
ν′=γν(1−βcosθ) (6.50)ν=γν′(1+βcosθ∗) (6.51)whereνis the frequency in theS-systemandν′is the frequency in theS′-system,
θandθ′are the corresponding angles,βis the source velocity andγis the corre-
sponding Lorentz factor.
Threshold for particle production
Consider the reaction
m 1 +m 2 →m 3 +m 4 +M (6.52)T(threshold)=1
2 m 2[(m 3 +m 4 +M)^2 −(m 1 +m 2 )^2 ]T(threshold)=(Sum of final masses)^2 −(sum of initial masses)^2
2 ×mass of target particle