2. General Physics

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2.4 Problems

2.4.5  Modern Physics

Problem 2:  Relativistic Collision

\(p1 = m1\ γ1\ {1, 0, 0, v1}; 
p2 = m2\ {1, 0, 0, 0}; \)

p3 = m3 γ3 {1, 0, 0, v3};

η = DiagonalMatrix[{1, -1, -1, -1}];

γrule = 
      Join[Solve[{p1 . η . p1 == (m1^2)}, γ1]
            [[2]], 
        Solve[{p3 . η . p3 == (m3^2)}, γ3][[2
          ]]]

\(iRule = {
        1\/\@\(\(-1\) + v_\^2\) :> 1\/\(I\ \@\(1 - v\^2\)\)}; 

    γrule = γrule //. iRule\)

vrule = 
      Solve[{p1 . η . p1 == (m1^2), p3 . η . p3 == (m3^2)}, {v1, 
            v3}][[4]]

eq1 = Simplify[Thread[p1 + p2 == p3] /. γrule]

sol1 = 
      Simplify[PowerExpand[
          Simplify[
            PowerExpand[
              Solve[eq1[[1]], {m3}]
                    [[1]] /. vrule]]]]

sol2 = 
      Simplify[Solve[
              eq1[[4]] /. sol1 /. 
                γrule, {v3}][[1
            ]] /. {(Sqrt[1 - (v1^2)]) -> (1/γ)1}]

p1 + p2 == p3 //. Join[sol1, sol2]