From: rockbrentwood@gmail.com   
      
   On Monday, July 27, 2020 at 5:26:08 PM UTC-5, Luigi Fortunati wrote:   
   > Is it an acceleration with respect to anyone or with respect to a=20   
   > specific reference system?=20   
      
   Newton, in the background to the formulation of his laws, said that all mot=   
   ion was to be referred to a specific frame - putatively one in which the st=   
   ars are "fixed". The laws were formulated in such a way that they hold equa=   
   lly well when taken in reference to any other frame that is in uniform moti=   
   on with respect to this one ... so that one could not determine which was t=   
   he actual stationary one and which ones were moving.   
      
   Galileo (it is NOT well-known here) was Newton's great-grand-math-father in=   
    the Mathematicians' Genealogy. Galileo (eventually) held to the propositio=   
   n that each of these frames stands on equal ground. Newton rejected it by w=   
   ord, but implemented it by deed. In other words,. he tried to have it both =   
   ways, saying one thing and doing another. But it gets the job done: of dist=   
   inguishing a family of frames (out of all those that are possible) as the o=   
   nes that are inertial. They each move at a constant speed in a constant dir=   
   ection with respect to one another.   
      
   The closest modern equivalent of Newton's assertion (and one which revokes =   
   his doctrine of Unknowability on the matter of which frame is the Stationar=   
   y one) is the co-moving frame that is almost literally tied to "fixed stars=   
   " - namely the one given by the CMB: the one which makes it maximally isotr=   
   opic, minimizing all its Doppler shifts.   
      
   The reason Newton had to take this route (though not clearly stated or even=   
    understood by him) is Genidentity. Everything is formulated in the languag=   
   e of spatial geometry. The fundamental object of spatial geometry is the Po=   
   int. A Point defines a location. The concept, however, has no meaning unles=   
   s and until you can say what's to count as the "same" location at two diffe=   
   rent times. Is New York in 2001 the "same place" as New York in 2020? Or is=   
    that "location" somewhere else on the Earth at the same latitude (because =   
   the Earth rotates) or different latitude (because the rotation wobbles) or =   
   different altitude (because the crust fluctuates) or different part of the =   
   Earth's orbit (because the Earth goes around the sun) or different part of =   
   the galaxy (because the solar system orbits the galaxy) or out in intergala=   
   ctic space (because the galaxies move mostly away from each other)? What co=   
   unts as the "same place" at a different time? That's the property of Genide=   
   ntity. And, as you can see, Genidentity is just a back-door way of saying w=   
   hat is Stationary and what is not.   
      
   Without this, you have no Genidentity. Without Genidentity, you have no con=   
   cept of a Location that endures in time or of a Point. Without Point, you h=   
   ave no foundation for Spatial Geometry. Newton's treatise is cast in the la=   
   nguage of spatial geometry, so he needs Point, Location, Genidentity and St=   
   ationarity. Therefore, he had no choice but to refute Galileo's principle, =   
   even if he still tried to have it through the back door by making his laws =   
   invariant under Galilean boosts.   
      
   To fully implement Galileo's principle requires delving deeper than the con=   
   cept of a Point, and deconstructing it into even more fundamental constitue=   
   nts - as a sequence of Point-Instants. The geometry required for this is no=   
   t a geometry at all, but a chrono-geometry: one whose fundamental objects a=   
   re point-instants. That is made necessary by Galileo's principle of Relativ=   
   ity - or by any other principle of Relativity that supersedes it.   
      
   So, it is also the case that the true origin of chrono-geometry (that is: t=   
   he concept of spacetime) lies rooted in Galileo but that it just happens to=   
    also be the case that neither he nor anyone else realized this or that thi=   
   s had to be so until after Relativity was changed from Galilean to Lorentzi=   
   an. In other words, Minkowski didn't marry space and time, nor did Einstein=   
    or Poincare'. They merely ordained the eloping of the two, which took plac=   
   e nearly 300 years before that.   
      
   A chain of Point-instants can be any one-dimensional subspace of this chron=   
   o-geometry. Those are worldlines. Of all the possible ones, a distinguished=   
    subset of them have the property of possessing zero acceleration at each p=   
   oint-instant - one for each direction in space at each speed. So, ultimatel=   
   y, the answer to the question is that an additional structure is imposed on=   
    the chrono-geometry which singles out which of the worldlines are inertial=   
    such that at each point-instant, in   each direction at each speed, passe=   
   s through exactly one such worldline. This structure is embodied by what we=   
    now call an affine connection.   
      
   In Newtonian Physics, all such worldlines represent either motions that are=   
    at a constant speed and direction relative to one another, or an instantan=   
   eous line of points in a snapshot of 3D space at a specific time (a spatial=   
    geodesic). In the relativistic world, in place of the spatial geodesics ar=   
   e the light-like and space-like geodesics.   
      
   Accelerations are taken relative to geodesics. So, for a given motion passi=   
   ng a given point-instant in a given direction and speed, you compare it to =   
   the geodesic possessing those same attributes.   
      
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