From: tjroberts137@sbcglobal.net   
      
   On 7/4/19 2:42 PM, Mike Fontenot wrote:   
   > On 7/4/19 1:19 AM, Tom Roberts wrote:   
   >> it is straightforward to construct a twin scenario in which the   
   >> older twin experiences less acceleration than the younger twin, or   
   >> more acceleration than the younger twin. (Both of these are   
   >> discussed earlier in this thread.)   
   >   
   > I don't know what you're referring to there.  Please elaborate.   
      
   On 6/26/2019 I posted:   
   Arrange two turntables to have their centers at rest in some inertial   
   frame, and arrange them so their rims are tangent. On each rim place a   
   clock, and arrange their rotation rates to be a rational number, such   
   that the clocks periodically meet. By carefully choosing the rotation   
   rates and the turntables' radii, you can arrange for the clock with the   
   SMALLER acceleration to have a larger speed relative to the inertial   
   frame, and thus accumulate less proper time between meetings.   
      
   That is contrary to the notion that "acceleration is all that matters".   
      
   >> So relying on acceleration to "explain" the twin paradox can lead   
   >> you astray.   
   >   
   > Show me exactly WHERE I've made a mistake in my description of the   
   > use of the Minkowski diagram in resolving the twin "paradox".   
      
   I don't think you made a mistake in the description, but rather in the   
   interpretation. There simply is more involved than just the   
   acceleration. The entire path is involved, and acceleration comes in   
   only insofar as it affects the paths of the twins/clocks. After all, the   
   acceleration does NOT enter into the calculation, but the paths do:   
      
            ElapsedProperTime = sqrt( \integral g_ij dx^i dx^j )   
        where the integral is taken over the path through spacetime   
        and the {g_ij} are the metric components relative to the   
        coordinates {x^i}; here the metric signature is +---.   
      
   >> But by considering the paths of both twins, you will never be   
   >> wrong.   
   >>   
   >> Ultimately, the twin paradox displays the basic geometrical fact   
   >> that different paths between a given pair of points can have   
   >> different path lengths.   
   >   
   > Using only the proper time of each twin says NOTHING about the   
   > correspondence of the ages of the two twins when they are separated,   
      
   Of course not. That is not an observable: what you get for comparing   
   their ages when they are separated depends on your method of comparison.   
      
   	All observables are invariants (as are all measurements).   
   	That comparison is explicitly coordinate dependent.   
      
   > THAT'S what the twin "paradox" is all about.   
      
   Not really. The usual "paradox" is that they have different ages when   
   reunited. That is an observable.   
      
   I still say what I said before: for determining the age of friends on   
   earth "right now", no sensible spacefaring pilot would use whatever   
   locally inertial frame they happen to be at rest in, they would use   
   EARTH'S inertial frame. Because ages calculated using your CADO approach   
   do not behave sensibly (i.e. as ages ought to behave: uniform and   
   monotonic); a sensible pilot knows that pushing their throttle does not   
   really change anybody else's age. Moreover, the only time the question   
   of earthbound people's ages would come up is when communicating with   
   earth, and for that the earth's frame must be used.   
      
   	(Compared to travel at speeds approaching c, the earth's   
   	 orbital speed is negligible, so earth can be considered   
   	 to be at rest in an inertial frame.)   
      
   Tom Roberts   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)