On Tuesday, March 20, 2018 at 7:09:40 AM UTC-5, lydiamarie...@gmail.com wrote:   
   > When translation generators are added in, the invariants have to   
   > re-stated in terms of coordinate differences or differentials.   
      
   Actually, if the translation generators are deformed with variable   
   coefficients, then things will become a bit more complicated. For   
   instance, with your deformation Px -> -a X, you no longer have Lie_{Px}   
   dx = 0, but instead, Lie_{Px} dx = d(Px x) = -da, which is not zero if a   
   is not a constant.   
      
   Instead, the original invariants (s = t + uv, rr = xx + yy + zz + 2wtu +   
   vwuu, etc.) would have to be deformed -- and in such a way that the   
   combination vrr - wss would then actually give you a metric for a   
   hyperbolic or spherical geometry, or something similar when q is   
   non-zero.   
      
   > And now, the note, itself: For the reasons about to be cited, the very   
   > existence of the Cosmic Microwave Background (CMB) should be regarded   
   > as direct proof that the universe must be: * signature changing, with   
   > an initial null hypersurface,...   
      
   The take-home, as I understand this remark, is that the initial   
   hypersurface is *not* spacelike, then, as is normally assumed for   
   signature change geometries.   
      
   A point of interest. Einstein-Bose condensates exhibit much of the same   
   mathematics used to describe signature change (and more generally:   
   spacetime geometry) -- thus leading to a notion of "emergent   
   spacetimes". This is actually being used to study in the lab (in an   
   analog fashion) the very issues you're raising here!   
      
   > 	*	geodesically complete near its initial hypersurface,   
      
   This is highly non-trivial, despite its simplicity! It would be nice to   
   see the math worked out on this in detail just to see what it all looks   
   like.   
      
   > 	*	causally connected across spacelike separations, and   
   >         *       a non-trivial self-consistency requirement that may   
   >         suppress backwards propagation along the past light cone!   
      
   The constraints may be more severe than you're leading on to. I was   
   originally going to post a followup to the picture I sent you noting the   
   difficulty of getting any kind of sensible boundary conditions with the   
   wave operator (w(XX + YY + ZZ) - vTT, with w = 1, v = t using your   
   conventions) as t -> 0, since that would be the Poisson equation.   
      
   Thanks for putting it out on Google+ and noting the correction. I should   
   actually be using the Hodge-deRham operator, which (if you normalize v =   
   4t) would give you an *extra* term yielding the equation 1(XX + YY + ZZ)   
   Phi - 4tTT Phi + 6T Phi = 0. So, it's actually the heat equation, for t   
   ~ 0!   
      
   But this still raises an interesting question: what are the boundary   
   conditions for this? Since t goes like tau^2 for t > 0 (the null   
   geodesics are parabolic), then you actually have a function Phi =   
   Phi(tau) that satisfies a boundary condition Phi(-tau) = Phi(tau),   
   Phi'(0) = 0, Phi''(0) given by the heat equation @ t = 0 ... and a   
   similar boundary condition for t < 0, with t = -sigma^2.   
      
   Whether the boundary conditions rule out a non-triival solution for Phi   
   at t = 0 or not, there will still probably be some kind of restriction   
   ... born of self-consistency ... on any past-directed propagation along   
   the light cone!   
      
   > Inflation is known, by Mansouri, to be a consequence arising by a   
   > junction condition for interfacing a Euclidean and Lorentzian   
   > geometry across a null hypersurface.   
      
   Mansouri's work may not be as well known as it should be. This bears   
   citing a few references, too; just to see how the arguments work out   
   (e.g. the derivation of inflation solely from the condition that the   
   initial hypersurface be null).   
      
   There are references on the Euclidean wormhole I am aware of, but it   
   might be a good idea to see them here, too.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)