From: rockbrentwood@gmail.com   
      
   [[Mod. note -- I have rewrapped overly-long lines.  -- jt]]   
      
   On Saturday, May 7, 2022 at 5:38:42 AM UTC-5, Luigi Fortunati wrote:   
   > A free-falling brick is an inertial frame?   
      
   If it's not rotating. Otherwise, the free-falling object is in a   
   rotating frame, if the frame is attached to and associated with the   
   object, itself.  If it's not rotating, then it is locally inertial.   
   The distinction between the two rests on Newton's bucket thought   
   experiment.   
      
   > Are a pair of free-falling half-bricks an inertial frame or are they   
   > two distinct inertial frame?   
      
   The curvature of space-time is precisely the warping of a field of   
   locally inertial free-fall frames whereby those that are initially   
   at rest with respect to one another start to accelerate with respect   
   to each other. If you display them as worldlines in a 4-dimensional   
   graph (or in a 3-dimensional graph, where one of the spatial   
   dimensions is suppressed, for the benefit of the unlucky few who   
   are visually impaired to see in 4 dimensions) then you'll see the   
   worldlines for locally inertial free-fall frames - initially parallel   
   in a time-like direction - starting to curve into one another -   
   hence the "falling" action associated with gravity. In this sense,   
   the gravity one feels and experiences is actually a warping in time,   
   first and foremost, rather than a warping in space. The actual   
   contraction is quantified and accounted for in the Raychaudhuri   
   equation, which is closely related to the "geodesic deviation   
   equation".   
      
   In a flat space-time spatially separated locally inertial frames,   
   initially at rest with respect to one another, remain at rest; and   
   so can be said to comprise the different locations of a global   
   inertial frame.   
      
   All of the foregoing applies independent of paradigm - to *both*   
   relativistic *and* non-relativistic theory; so it is neither a   
   construct nor innovation of "general relativity", but rather one   
   which first fully emerged at the onset of general relativity and   
   so has been (falsely) associated with it as a characteristic feature   
   of it. It is a general feature of any theory of gravity that respects   
   the Equivalence Principle.   
      
   In fact, both Newtonian gravity and Einsteinian gravity (specifically:   
   the Schwarzschild solution) can be unified as a one-parameter family   
   of geometries, that are warped versions of the 5-dimensional Bargmann   
   Geometry, via the line element + constraint:   
      
   dx^2 + dy^2 + dz^2 + 2 dt du + a du^2 - 2 V dt^2 + 2aV/(1 + 2aV) dr^2 = 0   
      
   where r = root(x^2 + y^2 + z^2), dr = (x dx + y dy + z dz)/r and   
   V = -GM/r is the potential of a gravitating body of mass M located   
   at r = 0.   
      
   The extra, u, coordinate is the non-relativistic limit of (s - t)   
   c^2, as c goes to infinity, where s is proper time. This has meaning   
   ... which also (by the way) shows that such things as "time dilation"   
   and "twin paradox" are *also* rooted in non-relativistic theory in   
   disguised form as u, and are not features specific to Relativity!   
   The u coordinate shows up, physically, as negative the action per   
   unit mass for an inertial particle.   
      
   When a = 0, this is Newtonian gravity, and it can be generalized   
   by having V be the total potential for all gravitating bodies,   
   rather than just for one. The geodesics for this geometry are the   
   orbits of Newtonian gravity. Since V is a function of the coordinates   
   and velocities of individual bodies, rather than a bona fide field   
   quantity, it is very tempting to try and quantize this geometry   
   directly in quantum *mechanics*. But for the fact that you still   
   have the self-energy and self-force problems to deal with (in starker   
   form, in fact) you'd almost have a full-fledged *geometric* quantum   
   theory for Newtonian gravity - one in which space-time itself is   
   quantized. But the whole "quantizing field theory as mechanics"   
   strategy has these same issues roadblocks, here, as did Feynmann   
   and Wheeler's attempt to do the same with electromagnetic theory   
   in the 1940's.   
      
   The case where a = 0 and V = 0 is the Bargmann geometry, which is   
   the natural geometric arena for non-relativistic theory.   
      
   When a > 0, this is the Schwarzschild solution in which the proper   
   time is given as s = t + a u, and in which c = root(1/a) is an   
   invariant speed (i.e. "light speed").   
      
   The term "dx^2 + dy^2 + dz^2" is the legacy of Euclidean geometry;   
   while "2 dt du" is the legacy of *Galileo*'s principle of Relativity,   
   which is where space and time *actually* became unified into the   
   chrono-geometry of spacetime. The secret eloping of the two, however,   
   went largely unnoticed until it was fully consummated by the addition   
   of the Poincare' term "a du^2", which turns this into a geometry   
   for Minkowski space. The warping of time associated with Newtonian   
   gravity is in the "-2V dt^2" term, while the warping of space,   
   itself, associated with General Relativity is limited to the   
   substantially smaller "2aV/(1 + 2aV) dr^2" term.   
      
   The only effects of paradigm, here, are those limited to the "Special   
   Relativity" term a du^2 and the "General Relativity" term   
   2aV/(1 + 2aV) dr^2 (so that whatever "relativistic corrections" there   
   are, which are made to trajectories have to come from these terms).   
   The main thrust of gravity - and the essential background behind your   
   query - resides with the "Newtonian Gravity" term -2 V dt^2.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)