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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?

Qmechanic
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Pardeep J
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    See "https://en.wikipedia.org/wiki/Gullstrand%E2%80%93Painlev%C3%A9_coordinates". – Cinaed Simson May 07 '20 at 19:54
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    It's not exactly clear what you're asking here. The GP time coord is the proper time of a free-falling observer. – PM 2Ring May 07 '20 at 20:30
  • Some related questions, although they aren't specifically about Gullstrand-Painlevé: https://physics.stackexchange.com/q/470705/123208 & https://physics.stackexchange.com/q/53334/123208 – PM 2Ring May 07 '20 at 20:33
  • From what I understand, any observer in the coordinate system that doesn't freefall right alongside that clock will have a proper time that differs from that of the clock due to differences in kinetic and/or gravitational time dilation, so its $\tau$ will differ from $t'$. – Gumby The Green Mar 21 '22 at 10:13

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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.

Charles Francis
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