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"Is this just an accident
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or is it part of a big picture?"
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It turns out,
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it's part of a big picture
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and all of general relativity is
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based on this one great equivalence of two quantities
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which are very different attributes.
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Why should the amount by which you're attracted to the Earth
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be also a measure of how much you hate acceleration?
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Two different features, right?
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But they happen to be the same.
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Anyway, what physicists do