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 (scheinbar daher weil der Richtungswechsel hier sofort geschieht; wird die Richtung nicht instantan geändert sondern zuerst horizontal gebremst und dann vertikal beschleunigt hat das Photon während dieser Phase im beschleunigten Bezugssystem eine schnellere Geschwindigkeit als im unbeschleunigten Inertialsystem. 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)
Bild
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