I understand that special relativity tells us that our perception of the passage of time can change depending on reference frames. But why does time pass the rate at which it does in my reference frame and is this a constant rate?
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Frame of reference dictates the "relative" rate of time passage. Rate of time passage within a frame can not be properly defined because then it becomes a question "rate of passage with respect to what"? Every observer has its own sense of rate of time passage which is always the "normal" rate, or 1 (as ACuriousMind hinted in his comment). There is no way to sense the changed sense, because, sense itself is changed, depending upon frame of reference.
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