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The GP coordinates version of Sch. metric uses a new time coordinate, t', which is the time read from a free-falling clock that was dropped from rest at large r and falls past the event we're measuring as it occurs.

I accept that coordinate time, so t' in this case, and proper time, τ, are different and I follow the mathematics needed to calculate t'. But I can't seem to understand conceptually how they are two different times?

Please can somebody enlighten me?

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I am not 100% clear about what you are asking, but proper time is the local time of a particular observer, whereas coordinate time is defined globally, so they cannot be the same. They should coincide at the position of the clock.

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