7. Relativity and Cosmology

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Setup

\(Clear["Global`*"]; 
Off[General::spell]; 
Off[General::spell1]; 
    \)

Needs["Calculus`VectorAnalysis`"]

Needs["Graphics`PlotField3D`"]

<< "D:\Mma\Inits\ch7init.m"

Clear["Global`*"];

7.2 Problems

7.2.1  Special Relativity Problems

Problem 2:  Two particle collision

A particle of rest mass m1 collides with a stationary particle of mass m2.  The collision produces two particles with masses m3 and m4.  The threshold kinetic energy in the laboratory is the energy just sufficient to make the center-of-mass emergy equal to the sum of the rest masses of the particles in the final state.

Part (a)    Write the total four-momentum vectors for the initial particles in the laboratory frame and in the center-of-mass frame.

pL1  =  {E1, p1, 0,  0}

pL2  =  {m2, 0, 0, 0}

pLT = pL1+pL2

pCM1  =  {ECM1, kCM1, 0,  0}

pCM2  =  {ECM2, -kCM1, 0, 0}

pCMT = pCM1+pCM2

Part (b)    Consider the scalar invariant p.η.p, where p is the total momentum.  Evaluate this invariant in the laboratory and center-of-mass frames and equate the expressions to solve for the threshold energy.  Express the results in terms of the kinetic energy of the particle with mass m1.

eq1 = pLT.η.pLT == pCMT.η.pCMT

eq2 = ECM1+ECM2 == m3+m4

{eq3,eq4} = {E1==T1+m1,p1==Sqrt[E1^2-m1^2]}

sol = Solve[{eq1,eq2,eq3,eq4},{T1},{E1,p1,ECM1}]//Flatten