Lorentz transformation

Saburo Higuchi 2001.11.22

Needs["Graphics`Graphics`"]

B[v_] := (1/Sqrt[1 - v^2]) {{1, v}, {v, 1}}

H[u_] := {{Cosh[u], Sinh[u]}, {Sinh[u], Cosh[u]}}

B[v]

H[u]

B[Tanh[u]]

PowerExpand[Simplify[B[Tanh[u]]]]

B3[v_] := 1/Sqrt[1 - v^2] {{1, 0, v}, {0, 1, 0}, {v, 0, 1}}

H3[u_] := {{Cosh[u], 0, Sinh[u]}, {0, 1, 0}, {Sinh[u], 0, 
          Cosh[u]}}

LightCone = 
      Plot[{t, -t}, {t, -10, 10}, 
        PlotStyle -> {RGBColor[1, 1, 0]}, AspectRatio -> 1, 
        AxesLabel -> {t, z}]

LightCone3D = 
      ParametricPlot3D[{t,  t Cos[q], t  Sin[q]}, {t, 0, 10}, {q, 0, 
          2  Pi}, AxesLabel -> {t, y, z}]

DisplayTogether[
      LightCone, 
Graphics[{PointSize[0.02], 
          Point[{0.5, 0}], 
          Point[{0.5, 0.5}]}], 
ParametricPlot[{t, 
          t}, {t, -1, 1}], 
ParametricPlot[{t, 
          0.5  t}, {t, -1, 1}], 
ParametricPlot[{t, 
          0}, {t, -1, 
          1}], 
ParametricPlot[{Sqrt[0.2 + z^2], 
          z}, {z, -1, 
          1}], 
PlotRange -> {{-1, 1}, {-1, 
            1}}, AspectRatio -> 1, Axes -> False, 
      Frame -> 
        True, 
FrameLabel -> {t, z}]

v = 0.2; 
    DisplayTogether[
      LightCone, 
Graphics[{PointSize[0.02], 
          Point[B[v] . {0.5, 0}], 
          Point[B[v] . {0.5, 0.5}]}], 
ParametricPlot[
        B[v] . {t, t}, {t, -1, 1}], 
ParametricPlot[
        B[v] . {t, 0.5  t}, {t, -1, 
          1}], 
ParametricPlot[
        B[v] . {t, 0}, {t, -1, 1}], 
ParametricPlot[
        B[v] . {Sqrt[0.2 + z^2], z}, {z, -1, 
          1}], 
PlotRange -> {{-1, 1}, {-1, 
            1}}, AspectRatio -> 1, Axes -> False, 
      Frame -> True, FrameLabel -> {t, z}]