Ciro Santilli
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# Lorentz transform consequence: everyone sees the same speed of light | 🗖 nosplit | ↑ parent "Lorentz transformation" | 311

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OK, so let's verify the main desired consequence of the Lorentz transformation: that everyone observes the same speed of light.
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Observers will measure the speed of light by calculating how long it takes the light going towards cross a rod of length laid in the x axis at position .
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TODO image.
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Each observer will observe two events:
• : the light touches the left side of the rod
• : the light touches the right side of the rod
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Supposing that the standing observer measures the speed of light as and that light hits the left side of the rod at time , then he observes the coordinates: $$t1​x1​t2​x2​​=T1=X1=cL​=X1+L​ (11)$$
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Now, if we transform for the moving observer: $$t1′​x1′​t2′​x2′​​=γ(t1​−c2vx1​​)=γ(x1​−vt1​)=γ(t2​−c2vx2​​)=γ(x2​−vt2​)​ (12)$$ and so the moving observer measures the speed of light as: $$c′​=t2′​−t1′​x2′​−x1′​​=(t2​−c2vx2​​)−(t1​−c2vx1​​)(x2​−vt2​)−(x1​−vt1​)​=(t2​−t1​)−c2v​(x2​−x1​)(x2​−x1​)−v(t2​−t1​)​=1−c2v​t2​−t1​x2​−x1​​t2​−t1​x2​−x1​​−v​=1−c2v​cc−v​=cc−v​c−v​=c​ (13)$$
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