From: rockbrentwood@gmail.com   
      
   Corrections and additions:   
      
   On Tuesday, January 4, 2022 at 7:37:11 AM UTC-6, Rock Brentwood wrote:   
   > This (meaning: the whole affair that this posting was a small part of)   
   > has come to my attention, only now, because the twins involved in it   
   > have died together on the same day yesterday.   
      
   One died on January 3, the other late in December.   
      
   The key point here   
   > 1A. What Is "Zero Size"?   
      
   is that the metric does not determine whether something is squashed down   
   to a single point.   
      
   Dimension is not a metric property, but a topological property. So,   
   regardless of whether the spatial part of the metric is 0 or not,   
   3-dimensions means 3-dimensions.   
      
   The key indicator is where the light cones land. If they land on the t =   
   0 surface at finite radius, then it's 3D and can't be regarded as a   
   single point, or "squashed down to a single point" regardless of what   
   the metric says.   
      
   If they splay out to infinity at t = 0 (and they don't unless A(t) is at   
   least quadratic in t, but anything faster than quadratic is ruled out on   
   the weak energy principle), then it may be okay to identify all the   
   points at t = 0 with a single point. But not otherwise.   
      
   > 1B. What Is The "Singularity"?   
   > As t -> 0, if A(t) -> 0, then the metric reduces to rank 1 ... and the   
   > inverse metric (up to conformal factor) to rank 3. This is actually the   
   > characteristic property of a Newton-Cartan geometry. In effect, light   
   > speed goes to infinity at time 0; the light cones become flattened out   
   > with the t = 0 surface being an envelope of them all.   
      
   The simplest way to make this regular is to treat the metric and its inverse as   
   independent objects, subject to a constraint. The most direct way to do this is   
   to note that the inverse metric g^{mn} generally only appears in combination   
   with root(|g|) for the Einstein-Hilbert action:   
      S = integral g^{mn} root(|g|) R^r_{mrn} d^4 x   
   , so it could be replaced by a tensor density N^{mn} = root(|g|) g^{mn} and   
   rewritten as   
      S = integral N^{mn} R^r_{mrn} d^4 x   
   subject to the constraint   
      N^{mr} g_{rn} = root(|g|) delta^m_n   
      
   More generally, a conformal degree of freedom can be entirely split out from   
   the inverse metric, by writing:   
      S = integral |L|^{1/4} N^{mn} R^r_{mrn} d^4 x   
   with the constraint:   
      N^{mr} g_{rn} = L delta^m_n.   
      
   For the above FRW metric, correspondingly, we would have   
   g_{mn} dx^m dx^n = dt^2 - A(t) (dx^2 + dy^2 + dz^2)   
   and   
   N_{mn} X_m X_n = X^2 + Y^2 + Z^2 - A(t) T^2   
   with   
   L = -A(t)^2   
      
   This provides a route for a smooth passage through a Newton-Cartan metric   
   at t = 0, with the singularity in the scale factor separated out into L. And,   
   now you   
   have a way to talk about the passage from (+---) to (++++) at t = 0, and to   
   address   
   the issue of "junction conditions" for pairing off the two sectors across the   
   t = 0 interface.   
      
   --- SoupGate-Win32 v1.05   
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