From: tjroberts137@sbcglobal.net   
      
   On 7/10/19 1:56 AM, rockbrentwood@gmail.com wrote:   
   > On Tuesday, June 25, 2019 at 2:33:20 AM UTC-5, Tom Roberts wrote:   
   >> Moreover [sic], in General Relativity   
   >   
   > Nobody was talking about General Relativity, except you. This has   
   > nothing to do with what you're replying to and is therefore   
   > irrelevant.   
      
   But IMHO it is relevant, as the only physical justification for SR is as   
   the local limit of GR.   
      
   > In any case, the generalization to General Relativity is that the 0-G   
   > motion (i.e. the free fall motion) is the one that has the largest   
   > time   
      
   This is just not true. As I said before, there can be two twins in orbit   
   around a mass that have different elapsed proper times between meetings.   
   BOTH twins are in freefall (zero proper acceleration).   
      
   >> each with ZERO proper acceleration, have different elapsed proper   
   >> times between meetings: put one twin in circular orbit around a   
   >> mass, put the other in a highly elliptical orbit around the same   
   >> mass, arrange for their orbital periods to have a ratio that is a   
   >> rational number, and orient them so the orbits periodically   
   >> intersect.   
   >   
   > ... and the time difference is 0 (at least to the order alpha^2 =   
   > 1/c^4, if not exactly), for orbits with the same mean radii,   
   > independent of shape.   
      
   Only for complete orbits with equal mean radii. But given an elliptical   
   twin, they can meet after partial orbits, and there's no need for their   
   orbits to have the same mean radii.   
      
   Arrange things so the elliptical twin is always outside the circular   
   twin's orbit between meetings, and it is clear that the former will have   
   the larger elapsed proper time between meetings.   
      
   	(Elliptical twin moves slower relative to the mass's   
   	 locally inertial frame => larger proper time.   
   	 Elliptical twin is at higher gravitational potential   
   	 => larger proper time. This is speaking loosely.)   
      
   Arrange things so the elliptical twin is always inside the circular   
   twin's orbit between meetings, and it is clear that the former will have   
   the smaller elapsed proper time between meetings.   
      
   	(Elliptical twin moves faster relative to the mass's   
   	 locally inertial frame => smaller proper time.   
   	 Elliptical twin is at lower gravitational potential   
   	 => smaller proper time. This is speaking loosely.)   
      
   > [...] it is a bit misleading on your part to be bringing it up here   
   > at all in a discussion about the Twin Paradox in SPECIAL relativity.   
      
   I also posted a description in flat spacetime in which both twins   
   accelerate, which can be configured so that the twin with the larger   
   proper acceleration can have EITHER the larger or the smaller elapsed   
   proper time between meetings.   
      
   Bottom line: it is the PATHS that matter, not the acceleration.   
   Acceleration does not even enter into the calculation, and is important   
   only insofar as it affects the paths.   
      
   Tom Roberts   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)