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You have to follow this very carefully.
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When you and I crossed,
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remember that was at the origin of coordinates,
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At that instant when our origins touched,
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let us emit a flash of light.
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Okay, maybe when the origins touched, there's a spark,
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the light signal goes out
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and the light signal is detected here, by a light detector.
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That second event,
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detection of the pulse,
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it has a coordinates (x,t) for me;
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it has a coordinates (x',t') for you.