From: fortunati.luigi@gmail.com   
      
   Luigi Fortunati giovedì 26/05/2022 alle ore 04:56:55 ha scritto:   
   > Richard Livingston martedì 24/05/2022 alle ore 18:05:03 ha scritto:   
   >>> For example, if I put two clocks at the maximum distance on the floor,   
   >>> they remain synchronized and, therefore, horizontally there is no   
   >>> acceleration (there is quiet).   
   >>>   
   >>> If, on the other hand, I put a clock on the floor and one on the   
   >>> ceiling, they don't stay synchronized and, therefore, there is   
   >>> acceleration vertically (there is no quiet).   
   >>   
   >> According to GR, yes.   
   >>   
   >> Rich L.   
   >   
   > In the free-falling elevator, does the clock on the floor stay   
   > synchronized with the clock on its ceiling?   
   >   
   > Luigi   
   >   
   > [[Mod. note -- Yes, apart from tidal effects.  -- jt]]   
      
   This is exactly the point!   
      
   Do the clocks on the floor of the free-falling elevator stay   
   synchronized with those on the ceiling in the presence of the tides or   
   not?   
      
   [[Mod. note -- The short non-mathematical answer is that the clocks in   
   (at rest with respect to) the free-falling elevator stay synchronized   
   to within a tidal-field tolerance.   
      
   A more precise answer is both longer and involves a bit of mathematics:   
   Consider a pair of clocks A and B.  A is stationary with respect to the   
   elevator (whether the elevator is free-falling or not), say sitting on   
   the elevator floor.  B starts out right next to A, synchronized with A.   
   Now we quickly raise B a vertical distance y, wait some fixed time interval   
   (much longer than the raising time), then quickly lower B back to be next   
   to A again.  Then we measure the difference D between A and B's clock   
   readings, i.e., we measure the desynchronization between A and B.   
   	[Notice that A and B are stationary next to each other   
   	when we make this measurement, so the measurement is   
   	unambiguous and doesn't depend on the exact time when   
   	we make the measurement.]   
      
   Now we (gedanken) repeat this experiment for many different values of y,   
   and investigate how D varies with y.  In particular, let's write D as a   
   Taylor series in y:   
     D(y) = D0 + D1*y + (1/2!)*D2*y^2 + (1/3!)*D3*y^3 + ...   
      
   If y=0, then we never moved clock B, so D(y=0) must be 0.  Hence the   
   power-series coefficient D0 must be 0.   
      
   The Taylor-series coefficient D1 measures that part of the clock   
   de-synchronization which is linear in (i.e., proportional to) the height y.   
   According to general relativity, D1 is proportional to the Newtonian   
   "little g" in the elevator, so D1=0 if the elevator is in free-fall.   
      
   According to general relativity, the Taylor-series coefficients D2, D3, ...   
   are determined by the tidal fields.  Unless there's something rather unusual   
   about the tidal fields, D2, D3, ... are all non-zero whether the elevator   
   is in free-fall or not.   
      
   So the more precise answer to your question is: according to general   
   relativity, if (and only if) the elevator is in free-fall then D1=0,   
   i.e., the clocks stay synchronized to 1st (linear) order in their   
   height difference y.  Unless there's something very unusual about the   
   local tidal fields, the clocks are desynchronized at 2nd and higher   
   order in y.   
      
   (The immediately previous paragraph is precisely what we mean by the   
   phrase "the clocks stay synchronized apart from tidal effects".  That   
   is, "tidal effects" means the D2, D3, ... terms in the Taylor series,   
   so saying that the clocks stay synchronized apart from tidal effects   
   means precisely that they stay synchronized in the D1 (linear) term.   
   -- jt]]   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)