Suppose I'm on a non rotating planet. I have two identical, perfect watches. I synchronize them. Then I throw one of them into the air and catch it. Does the one I throw into the air gain or lose time with respect to the one I was holding?
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Lawrence has the details, but this question can actually be determined exactly without assuming anything about mass distribution, etc. A geodesic (which is what the watch in free-fall follows) in GR has maximal proper time along it. The "stationary" watch, which is actually accelerated, is following some other path and so must experience a shorter proper time. |
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What astounds me is there is considerable quibbling over the nature of the question, but nobody answers it! This is comparatively simple to address. Let us consider the Schwarzschild metric in a weak gravity field $$ ds^2~=~-(1~-~2\phi/c^2)dt^2~+~dr^2~+~r^2d\Omega^2 $$ for $\phi~=~GM/r$ the Newtonian gravity potential. The unit velocity is then $$ 1~=~-(1~-~2\phi/c^2)u_t^2~+~u_r^2~-~\dots $$ where we can consider the motion in the radial direction for simplicity. The derivative of this with respect to the proper time $s$ is then $$ 0~=~-(1~-~2\phi/c^2)u_ta_t~+~u_ra_r. $$ If the gravity potential is zero the solutions are $t~=~g^{-1}sinh(gs)$ $r~=~g^{-1}cosh(gs)$, for $g$ the acceleration parameter. Here $g$ counters the gravitation of the Earth. If the gravity potential is turned on we can the write the time solution solution as $t~=~g^{-1}sinh(gs~+~\gamma)$, which we input into the third equation $$ 0~=~-(1~-~2\phi/c^2)g~cosh(gs)sinh(gs)~+~g~cosh(gs~+~\gamma) sinh(gs~+~\gamma) $$ $$ =~(1~-~2\phi/c^2)\frac{g}{2}sinh(2gs~+~2\gamma) ~+~sinh(2gs) $$ If we consider weak fields, small accelerations and small proper time $s$ we have $$ 0~\simeq~-(1~-~2\phi/c^2)(g^2s~+~g\gamma)~+~g^2s, $$ where $\gamma~\simeq~-2\phi gs/c^2$. The coordinate time is reduced with the turning on of the acceleration. This implies that the watch on the accelerated frame will mark off a shorter interval of time than the watch which is placed on a geodesic motion in the local gravity field with acceleration $g$. This is a gravitational version of the twin paradox. The twin which travels outwards and back is on an accelerated frame, which is a path in spacetime that is non-extremal, or maximal. As a result the proper time marked off is shorter. |
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