On Wednesday, March 28, 2018 at 1:38:11 AM UTC-5, Lawrence Crowell wrote:   
   > What you lay out below is in greater detail than what I said. I suppose   
   > I should have said Riemannian geometry is the mathematics general   
   > relativity is based on. Riemann-Cartan geometry is a technical   
   > modification. The equivalence principle is a physical statement for the   
   > connections used in GR, in particular the geodesic equation.   
      
   The equivalence principle, itself, is paradigm-neutral, and applies   
   across the board both to GR and non-relativistic gravity (whose   
   underlying also has geodesics, but is not Riemannian). Sometimes what   
   people refer to by the statement is a mixture of (1) and (3); (1) being   
   that the motion of matter is determined by its initial position and   
   velocity independent of its constituency and (3) being that the   
   spacetime is locally Minkowski.   
      
   I put (2) (the continuity equation) up ahead of (3) intentionally. If   
   you had to choose between violating the continuity equation (and   
   conservation laws) versus relaxing (3), I think most people would give   
   (2) higher priority. This is the situation you face with singularities   
   -- where (2) is most definitely violated! All the more so, when you have   
   no go results which virtually mandate the presence of singularities in   
   your solutions.   
      
   Remember that GR, itself, (its Lagrangian) was essentially *singled out*   
   from other possibilities by the requirement that it yield a conservation   
   law. The moment you allow it to be violated in your solutions, you've   
   undercut the very foundation of the theory itself.   
      
   I think Lydia's point about radiation dominant solutions going   
   asymptotically with a metric like dt^2 - t (dx^2 + dy^2 + dz^2) for t ~   
   0 is the key point. That basically forces you into signature change as t   
   -> 0!   
      
   But it would have helped to see a similar construction for black hole   
   solutions (that makes the "Euclidean Wormhole" solution more   
   transparent).   
      
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