From: tjroberts137@sbcglobal.net   
      
   On 4/12/22 2:21 AM, Mike Fontenot wrote:   
   > On 4/11/22 1:19 PM, Tom Roberts wrote:   
   >> If you are an astronaut in a spaceship that can travel at an   
   >> appreciable fraction of c, and which you maneuver, then such an   
   >> array of clocks is impossible -- each such clock must vary its   
   >> acceleration in concert with yours [...]   
   >   
   > No, that's not correct.   
      
   Yes, it is correct. Finally you say what you are thinking behind the   
   scenes, and I can now explain that you are thinking of an inappropriate   
   analogy.   
      
   > According to the accelerating observer (the AO), whose conclusions   
   > we seek, he and each of his "helper friends" (HF's) undergo EXACTLY   
   > the same (constant) acceleration, as recorded on their   
   > accelerometers.   
      
   This is wrong. You require the helper friends to remain at the same   
   proper distance from the accelerating observer -- i.e. in each   
   successive instantaneously co-moving inertial frame (ICIF) those   
   distances are constant. That is Born rigid motion, and it is well   
   known that HFs lower than the AO must have larger proper accelerations   
   than the AO, and HFs above the AO must have smaller proper   
   accelerations; there is often a limit below which no HF can possibly   
   keep up, so in general the HFs cannot cover the manifold.   
      
       [Note that accelerometers display their proper acceleration.]   
      
   Moreover, whenever the observer maneuvers (changes his proper   
   acceleration), all of the HFs must SIMULTANEOUSLY (in the current ICIF)   
   make corresponding changes to their proper accelerations -- that   
   requires either clairvoyance or detailed pre-planning, because they are   
   separated by spacelike intervals from the AO. As such detailed   
   pre-planning cannot hold for a spacefaring astronaut (who will maneuver   
   his spacecraft based on current observations), I said that this array   
   of HFs is impossible.   
      
   > This is clear by looking at the equivalent scenario in the case of a   
   > constant gravitational field with no accelerations (via the   
   > equivalence principle) ... all of those people are motionless,   
   > unaccelerated, and mutually stationary. And according to them, the   
   > distance between each of them is also constant.   
      
       (I presume you mean a static gravitational field, all   
        those people are located at various altitudes with each   
        4-velocity parallel to the timelike Killing vector, and   
        the field is uniform in that their proper accelerations   
        are all equal. I also presume that "all of those people"   
        are the AO and all the HFs. Note this is not "equivalent"   
        to the above scenario, as I discuss next.)   
      
   This is wrong at several levels.   
   1) The HFs are not "motionless" or "mutually stationary" in any of the   
   AO's ICIFs; this is not Born rigid motion.   
   2) the HFs are not "unaccelerated" -- each has a nonzero proper   
   acceleration (even though they have zero coordinate acceleration [#]).   
   3) the equivalence principle applies only in a region small enough that   
   the curvature of spacetime is negligibly small (compared to measurement   
   accuracy); it CLEARLY does not apply here.   
   4) the distances between the HFs and the AO are not "constant" in any   
   ICIF -- you are confusing constant coordinate difference [#] with   
   constant distance in an ICIF -- the AO and HFs have constant coordinate   
   values (and differences) [#], but not constant distances in any of the   
   ICIFs (as required by your scenario above).   
      
   	[#] Implicitly using coordinates aligned with the static   
   	gravitational field. These are not the coordinates of any   
   	of the ICIFs.   
      
   It seems you have implicitly been thinking of this inappropriate analogy   
   all along. Relativity is more complicated than that (even in flat   
   spacetime).   
      
   > In the acceleration scenario (in the infinite flat spacetime of   
   > special relativity), perpetually-inertial observers, who are   
   > initially stationary with respect to the AO and the HF's, WILL   
   > conclude that the accelerations of the AO and the various HF's, and   
   > their distances apart, DO vary with time.   
      
   But the only way to construct that collection of HFs is for the AO to   
   adhere to a detailed, prearranged plan of accelerations, so each HF can   
   pre-compute their own detailed plan of accelerations. The AO cannot   
   maneuver to, say, land on a discovered planet or orbit a discovered star.   
      
   > But it is not their conclusions that I am interested in.   
      
   But as I keep saying, the conclusions you are interested in are   
   nonsensical and unphysical. And the AO + HFs scenario you have in mind   
   is either impossible, or requires unacceptably rigid adherence to a   
   pre-computed plan (and even then can be limited in scope).   
      
   Tom Roberts   
      
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