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How about somebody riding with the particle?
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If you're riding with the particle
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your 0 component
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is m0·c, is m·c.
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There is no velocity for you, and you have no momentum,
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as the particle has no momentum as seen by the particle,
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so the components of the vector P
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become m·c and 0.
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What's the length square of this vector?
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Well, it's the square of this guy and case closed;
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there's nothing here to subtract from here.
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That's a quick way to verify