From: tjroberts137@sbcglobal.net   
      
   On 9/30/18 1:15 AM, richalivingston@gmail.com wrote:   
   > On Wednesday, September 26, 2018 at 1:42:05 PM UTC-5, Tom Roberts wrote: ...   
   >> [description of] a light clock with a light pulse bouncing between two   
   >> mirrors. [...] Then it is straightforward to see that the trajectory of   
   >> the light pulse relative to the lab is a series of straight lines with   
   >> corners at the successive locations of the mirrors' centers when it   
   >> bounces. It is quite clear that the rate of bouncing measured in the lab   
   >> depends ONLY on the size of the clock and how fast the mirrors move   
   >> relative to the lab (i.e. how far apart the corners/bounces are); the   
   >> mirrors' acceleration DOES NOT MATTER (i.e. it does not matter how they get   
   >> to successive positions of the bounces/corners). This is just basic   
   >> geometry, and if your simulation does not show this then it is wrong.   
   >   
   > Unless I am misinterpreting your argument, I believe the last statements   
   > about acceleration not affecting the period of the clocks is incorrect.   
      
   For a centrifuge with a fixed radius, the speed of the mirrors relative to the   
   lab is directly related to their acceleration, making it easy to confuse the   
   dependence.   
      
   That said, a more careful analysis shows my statements are indeed wrong:   
      
   Consider two centrifuges with vertical axes and different radii, but the same   
   tangential speed relative to the lab; they have different accelerations. Let   
   each have a light clock with mirrors separated vertically (along the axis of   
   rotation). Plot the bounce points (in 3-d) in the lab for each centrifuge, and   
   project them onto a single horizontal plane -- each centrifuge's points all lie   
   on a circle, and the tangential speed of the centrifuge is related to the arc   
   of   
   the circle between the projected points (which is equal for the two   
   centrifuges). Since the radii are different, the chords of the circle between   
   projected points are different, and Pythagoras' theorem (applied to the chord   
   and the vertical separation of the mirrors) implies the light path lengths are   
   different -- the two light clocks tick at (slightly) different rates.   
      
   This leads us to a rather different lesson: in relativity, clocks are   
   considered   
   to be pointlike [#]. These light clocks are pointlike, and behave as relativity   
   predicts, ONLY when their internal light path is much smaller than their   
   acceleration between bounces (i.e. only when the two chords above differ by a   
   negligible amount). They do, of course, behave as relativity predicts for all   
   INERTIAL motions, even when they are not really pointlike.   
      
   	[#] E.g. atomic clocks are pointlike compared to GPS orbits.   
      
   In retrospect, this should have been obvious, especially for a light clock in a   
   centrifuge oriented with its mirrors along the radius (i.e. at different   
   "heights" in the equivalent "gravitational field").   
      
   > -Consider the two mirrors to be at the same radius in the centrifuge, but   
   > displaced along the axis of the centrifuge so the light pulses are   
   > describing a zig-zag along a cylinder at that radius.   
      
   NO! In the lab, the points of the bounces are all on that cylinder, but the   
   light rays follow STRAIGHT LINES between the bounce points. You sort of talk   
   around this, but it OUGHT to be clear that the light follows straight-line   
   paths   
   between bounces (i.e. on the faces of a many-sided polygon, not a cylinder).   
      
   > [... description of what appears to be similar, using very different words]   
      
   Bottom line: clocks must be pointlike or their behavior can differ from what   
   relativity predicts.   
      
   Tom Roberts   
      
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