Sagnac Effekt

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Yukterez
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Sagnac Effekt

Beitragvon Yukterez » Mo 16. Nov 2015, 01:21

Auf einem Quadrat von x = 1 Lichtsekunde Seitenlänge läuft ein Läufer mit der Geschwindigkeit v=c/2 dessen Seiten entlang, und jeweils ein Photon mit c in die selbe und ein weiteres in die entgegengesetzte Richtung.

Im linken Bild sehen wir die Situation im Ruhesystem des Quadrats, und im rechten Bild im System des bewegten Läufers mit Fokus auf die Differenzgeschwindigkeiten relativ zur Strecke:

Bild

Code: Alles auswählen

plot1 = Manipulate[
 
  c = 1; v = c/2;
 
  L = Line[{{0, 0}, {0, 1}, {1, 1}, {1, 0}, {0, 0}}];
 
 
  P1a = {PointSize[Large], Black, Point[{1 - v t, 1}]};
  T1a = Text[StyleForm["v", FontSize -> 9],
    {1 - v t, 1 + 0.04}];
 
  P2a = {PointSize[Large], Red, Point[{1 - c t, 1}]};
  T2a = Text[StyleForm["c", FontSize -> 9],
    {1 - c t, 1 + 0.04}];
 
  P3a = {PointSize[Large], Blue, Point[{1, 1 - c t}]};
  t4 = Text[StyleForm["c", FontSize -> 9],
    {1 + 0.038, 1 - c t}];
 
  P1b = {PointSize[Large], Black, Point[{1 - v 1 - v (t - 1), 1}]};
 
  P2b = {PointSize[Large], Red, Point[{0, 1 - c (t - 1)}]};
  T2b = Text[StyleForm["c", FontSize -> 9],
    {0 - 0.04, 1 - c (t - 1)}];
 
  P3b = {PointSize[Large], Blue, Point[{1 - c (t - 1), 1 - c 1}]};
  T3b = Text[StyleForm["c", FontSize -> 9],
    {1 - c (t - 1), 1 - c 1 - 0.04}];
 
  P1c = {PointSize[Large], Black,
    Point[{1 - v 1 - v 1, 1 - v (t - 2)}]};
  T1c = Text[StyleForm["v", FontSize -> 9],
    {0 - 0.04, 1 - v (t - 2)}];
 
  P2c = {PointSize[Large], Red, Point[{1 - c 1 + c (t - 2), 1 - c 1}]};
  T2c = Text[StyleForm["c", FontSize -> 9],
    {1 - c 1 + c (t - 2), 1 - c 1 - 0.04}];
 
  P3c = {PointSize[Large], Blue,
    Point[{1 - c 1, 1 - c 1 + c (t - 2)}]};
  T3c = Text[StyleForm["c", FontSize -> 9],
    {0 - 0.04, 1 - c 1 + c (t - 2)}];
 
  P1d = {PointSize[Large], Black,
    Point[{1 - v 1 - v 1, 1 - v (t - 2)}]};
  P2d = {PointSize[Large], Red, Point[{1 - c 1 + c (t - 2), 1 - c 1}]};
 
  P1e = {PointSize[Large], Black,
    Point[{1 - v 1 - v 1, 1 - v 1 - v (t - 3)}]};
  P2e = {PointSize[Large], Red,
    Point[{1 - c 1 + c 1, 1 - c 1 + c (t - 3)}]};
 
  P1f = {PointSize[Large], Black,
    Point[{1 - v 1 - v 1 + v (t - 4), 1 - v 1 - v 1}]};
  P2f = {PointSize[Large], Red,
    Point[{1 - c 1 + c 1 - c (t - 4), 1 - c 1 + c 1}]};
 
  P1g = {PointSize[Large], Black,
    Point[{1 - v 1 - v 1 + v 1 + v (t - 5), 1 - v 1 - v 1}]};
  P2g = {PointSize[Large], Red,
    Point[{1 - c 1 + c 1 - c 1, 1 - c 1 + c 1 - c (t - 5)}]};
 
  P1h = {PointSize[Large], Black,
    Point[{1 - v 1 - v 1 + v 1 + v 1, 1 - v 1 - v 1 + v (t - 6)}]};
  P2h = {PointSize[Large], Red,
    Point[{1 - c 1 + c 1 - c 1 + c (t - 6), 1 - c 1 + c 1 - c 1}]};
 
  P1i = {PointSize[Large], Black,
    Point[{1 - v 1 - v 1 + v 1 + v 1,
      1 - v 1 - v 1 + v 1 + v (t - 7)}]};
  P2i = {PointSize[Large], Red,
    Point[{1 - c 1 + c 1 - c 1 + c 1,
      1 - c 1 + c 1 - c 1 + c (t - 7)}]};
 
  Graphics[
   
   Piecewise[{
     
     {{L, P1a, P2a, P3a, T1a, T2a, t4}, t <= 1},
     {{L, P1b, P2b, P3b, T1a, T2b, T3b}, t > 1 && t <= 2},
     {{L, P1c, P2c, P3c, T1c, T2c, T3c}, t > 2 && t <= 8/3},
     {{L, P1d, P2d}, t > 8/3 && t <= 3},
     {{L, P1e, P2e}, t > 3 && t <= 4},
     {{L, P1f, P2f}, t > 4 && t <= 5},
     {{L, P1g, P2g}, t > 5 && t <= 6},
     {{L, P1h, P2h}, t > 6 && t <= 7},
     {{L, P1i, P2i}, t > 7 && t <= 8}
     
     }], PlotRange -> {{-0.25, 1.25}, {-0.1, 1.1}}, Frame -> True
   ],
  {t, 0, 8/3}];
 
plot2 = Manipulate[
 
  c = 1; v = c/2; γ = Sqrt[1 - v^2];
 
  t1 = γ/(c + v);
  t2 = 1/Sqrt[c^2 - v^2];
  t3 = γ/v;
  t4 = t3 + 1/2/Sqrt[c^2 - v^2];
  t5 = 8/3 γ;
 
  La = Line[{{0, 0}, {0, 1}, {1 γ, 1}, {1 γ, 0}, {0,
      0}}];
  Ld = Line[{{0, 1 - γ}, {0, 1}, {1, 1}, {1, 1 - γ}, {0,
       1 - γ}}];
  Le = Line[{{0, 0}, {0, 1}, {1, 1}, {1, 0}, {0, 0}}];
  Lf = Line[{{0, 0}, {0, 1}, {1, 1}, {1, 0}, {0, 0}}];
  Lg = Line[{{0, 0}, {0, 1}, {1, 1}, {1, 0}, {0, 0}}];
  Lh = Line[{{0, 0}, {0, 1}, {1, 1}, {1, 0}, {0, 0}}];
  Li = Line[{{0, 0}, {0, 1}, {1, 1}, {1, 0}, {0, 0}}];
 
  P1a = {PointSize[Large], Black, Point[{1 γ - v t, 1}]};
  T1a = Text[StyleForm["v", FontSize -> 9],
    {1 γ - v t, 1 + 0.04}];
 
  P2a = {PointSize[Large], Red, Point[{1 γ - (c + v) t, 1}]};
  T2a = Text[StyleForm["c+v", FontSize -> 9],
    {1 γ - (c + v) t, 1 + 0.04}];
 
  P3a = {PointSize[Large], Blue,
    Point[{1 γ, 1 - Sqrt[c^2 - v^2] t}]};
  T3a = Text[
    StyleForm[
     "\!\(\*SqrtBox[\(\*SuperscriptBox[\(c\), \(2\)] - \
\*SuperscriptBox[\(v\), \(2\)]\)]\)", FontSize -> 9],
    {1 γ + 0.1, 1 - Sqrt[c^2 - v^2] t}];
 
  P1b = {PointSize[Large], Black, Point[{1 γ - v t, 1}]};
  T1b = Text[StyleForm["v", FontSize -> 9],
    {1 γ - v t, 1 + 0.04}];
 
  P2b = {PointSize[Large], Red,
    Point[{0, 1 - Sqrt[c^2 - v^2] (t - t1)}]};
  T2b = Text[
    StyleForm[
     "\!\(\*SqrtBox[\(\*SuperscriptBox[\(c\), \(2\)] - \
\*SuperscriptBox[\(v\), \(2\)]\)]\)", FontSize -> 9],
    {0 - 0.1, 1 - Sqrt[c^2 - v^2] (t - t1)}];
 
  P3b = {PointSize[Large], Blue,
    Point[{1 γ, 1 - Sqrt[c^2 - v^2] t}]};
  T3b = Text[
    StyleForm[
     "\!\(\*SqrtBox[\(\*SuperscriptBox[\(c\), \(2\)] - \
\*SuperscriptBox[\(v\), \(2\)]\)]\)", FontSize -> 9],
    {1 γ + 0.1, 1 - Sqrt[c^2 - v^2] t}];
 
  P1c = {PointSize[Large], Black, Point[{1 γ - v t, 1}]};
  T1c = Text[StyleForm["v", FontSize -> 9],
    {1 γ - v t, 1 + 0.04}];
 
  P2c = {PointSize[Large], Red,
    Point[{0, 1 - Sqrt[c^2 - v^2] (t - t1)}]};
  T2c = Text[
    StyleForm[
     "\!\(\*SqrtBox[\(\*SuperscriptBox[\(c\), \(2\)] - \
\*SuperscriptBox[\(v\), \(2\)]\)]\)", FontSize -> 9],
    {0 - 0.1, 1 - Sqrt[c^2 - v^2] (t - t1)}];
 
  P3c = {PointSize[Large], Blue,
    Point[{1 γ - (c + v) (t - t2), 1 - Sqrt[c^2 - v^2] t2}]};
  T3c = Text[StyleForm["c+v", FontSize -> 9],
    {1 γ - (c + v) (t - t2), 1 - Sqrt[c^2 - v^2] t2 - 0.04}];
 
  P1d = {PointSize[Large], Black, Point[{0, 1 - v (t - t3)}]};
  T1d = Text[StyleForm["v", FontSize -> 9],
    {0 - 0.03, 1 - v (t - t3)}];
 
  P2d = {PointSize[Large], Red,
    Point[{1/2 + Sqrt[c^2 - v^2] (t - t3), 1 - γ}]};
  T2d = Text[
    StyleForm[
     "\!\(\*SqrtBox[\(\*SuperscriptBox[\(c\), \(2\)] - \
\*SuperscriptBox[\(v\), \(2\)]\)]\)", FontSize -> 9],
    {1/2 + Sqrt[c^2 - v^2] (t - t3), 1 - γ - 0.08}];
 
  P3d = {PointSize[Large], Blue,
    Point[{0, 1 - 1/Sqrt[3] + (c - v) (t - t3)}]};
  T3d = Text[StyleForm["c-v", FontSize -> 9],
    {0 - 0.049, 1 - 1/Sqrt[3] + (c - v) (t - t3)}];
 
  P1e = {PointSize[Large], Black, Point[{0, 1 - v (t - t3)}]};
  T1e = Text[StyleForm["v", FontSize -> 9],
    {0 - 0.03, 1 - v (t - t3)}];
 
  P2e = {PointSize[Large], Red,
    Point[{1, 1 - γ + (c - v) (t - t4)}]};
 
 
  P3e = {PointSize[Large], Blue,
    Point[{0, 1 - 1/Sqrt[3] + (c - v) (t - t3)}]};
  T3e = Text[StyleForm["c-v", FontSize -> 9],
    {0 - 0.049, 1 - 1/Sqrt[3] + (c - v) (t - t3)}];
 
  Graphics[
   
   Piecewise[{
     
     {{La, P1a, P2a, P3a, T1a, T2a, T3a}, t <= t1},
     {{La, P1b, P2b, P3b, T1b, T2b, T3b}, t > t1 && t <= t2},
     {{La, P1c, P2c, P3c, T1c, T2c, T3c}, t > t2 && t <= t3},
     {{Ld, P1d, P2d, P3d, T1d, T2d, T3d}, t > t3 && t <= t4},
     {{Ld, P1e, P2e, P3e, T1e, T3e}, t > t4 && t <= t5}
     
     
     }], PlotRange -> {{-0.25, 1.25}, {-0.1, 1.1}}, Frame -> True
   ],
  {t, 0, t5}];
 
{plot1, plot2}

In dem Moment in dem der Läufer um die Ecke biegt (zu dessen Eigenzeit τ = x/γ/c) wechselt er das Inertialsystem, weshalb die Photonen einen scheinbaren Sprung machen. Auf einem Kreis (im Limit ein unendlichfaches Polygon) findet ein solcher Sprung auf jedem infinitesimalen Teilabschnitt statt:

Bild

Code: Alles auswählen

plot3 = Manipulate[
  Graphics[
   {
    {PointSize[Large], Blue, Point[{
       {+Sin[φ], +Cos[φ]}
       }]},
    {PointSize[Large], Red, Point[{
       {-Cos[φ], -Sin[φ]}
       }]},
   
    {PointSize[Large], Black, Point[{
       {-Cos[φ/4 +
           1/16 (π - 16 ( π/4))], -Sin[φ/4 +
           1/16 (π - 16 ( π/4))]}
       }]},
    Circle[],
    Text[StyleForm["Photon 1", FontSize -> 9],
     {+Sin[φ], 0.07 + Cos[φ]}],
    Text[StyleForm["Photon 2", FontSize -> 9],
     {-Cos[φ], 0.07 - Sin[φ]}],
    Text[StyleForm["Station", FontSize -> 9],
     {-Cos[φ/4 + 1/16 (π - 16 ( π/4))],
      0.07 - Sin[φ/4 + 1/16 (π - 16 ( π/4))]}]
    }
   ,
   Frame -> True,
   PlotRange -> {{-1.3, 1.3}, {-1.3, 1.3}}
   ],
   
   {φ, -π/4, 4 π - π/4}]

Sagnac Calculation, Simon Tyran, Wien (Vienna)
Симон Тыран @ wikipedia | stackexchange | wolfram

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