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In existence of mass, spacetime gets warped, this means time speeds up near the mass, so how can we measure the actual rate of flow of time?

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There is no absolute time. Time always passes at the normal rate for you, your clock, and everything in your frame of reference no matter what your conditions. A distant observer considered to be in a different frame of motion, acceleration, and / or gravitational conditions will also perceive their own time perception and clock as running normally. Yet each of you may see time running at a different rate for the other. Each is equally valid, there is no privileged or absolute time.

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