From: goldenfieldquaternions@gmail.com   
      
   On Tuesday, March 27, 2018 at 2:28:09 PM UTC-6, rockbr...@gmail.com wrote:   
   > On Wednesday, March 21, 2018 at 7:01:43 AM UTC-5, Lawrence Crowell wrote:   
   >> I will get the question out of the way. Why is this presuming the   
   >> universe is a non-Riemannian manifold or geometry? General relativity   
   >> is physics based on Riemannian differential geometry.   
      
   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.   
      
   LC   
      
   >   
   > Actually, General Relativity is based primarily on [in order of   
   > importance] (1) the equivalence principle, (2) the continuity equations   
   > that embody conservation laws for mass, energy and momentum (and angular   
   > momentum and moment) and (3) the assumption that the geometry is locally   
   > 3+1 -- conservation laws that (as Lydia pointed out) are abruptly   
   > violated on any "singularities" in a solution, thereby undermining one   
   > of the chief premises used to establish the theory! For this reason, any   
   > solution with a singularity has to be excluded from   
   > consideration. Therefore, a no-go theorem that essentially makes   
   > singularities inevitable can only be taken as a proof by contradiction   
   > that the basic assumption (of the geometry being Riemannian) is false.   
   >   
   > Also, on a technical note, the theory can only be founded properly on a   
   > Riemann-*Cartan* geometry, since Riemannian geometries do not support   
   > the infrastructure required to embody non-natural objects (such as a   
   > Lorentz bundle or SL(2,C) bundle -- which you need even to express the   
   > idea of a spinor, never mind expressing anything involving them!) You   
   > need also the 3-currents for angular momentum and moment.   
   >   
   > So, (1), (2) and (3) together mandate a Riemann-Cartan geometry, not a   
   > Riemannian geometry; and General Relativity (where but for the   
   > historical accident of having predated the formulation of Riemann-Cartan   
   > geometry by about 10 years) is actually founded on a Riemann-Cartan   
   > geometry. The only real choice is not between geometry types, then, but   
   > whether to adopt the Einstein-Hilbert Lagrangian (which, when rewritten   
   > in intrinsic form in a Riemannian-Cartan geometry is very awkward, since   
   > it requires adding extra terms to subtract out the contorsion from the   
   > native connection so as to get the Levi-Civita connection) or the   
   > Riemann-Cartan "native" form of the Einstein-Hilbert Lagrangian (which   
   > uses the Riemann-Cartan's native R scalar, rather than the R based on   
   > the Levi-Civita connection). That gives you one of 2 theories:   
   > Einstein-Hilbert(-in-Riemann-Cartan-geometry-form) or   
   > Einstein-Cartan. The empirical differences between the two are very   
   > small (but could affect early cosmology, as Trautman has pointed out!),   
   > so the matter is not yet decided. Exterior solutions will not see any   
   > difference, since both torsion and contorsion cannot propagate outside   
   > matter in Einstein-Cartan gravity (nor in any of a large range of other   
   > Lagrangians built on a Riemann-Cartan geometry) ... by a simple counting   
   > argument on number of degrees of freedom for the spin tensor and torsion   
   > tensor.   
   >   
   > Note that assumptions (1) and (2) do not mandate a Riemannian geometry   
   > nor one that is pseudo-Riemanninan. For instance, Newton's law of   
   > gravity falls within this mould but is described with a (most decidedly   
   > *non*-[pseudo-]Riemannian) Newton-Cartan geometry. The only qualifier to   
   > that is that it can be embedded in a 4+1 geometry as I, myself, have   
   > pointed out here on a previous occasion, such that the geodesic law   
   > directly embodies the Newtonian law of gravity. In fact, you can easily   
   > find such a metric by taking the Schwarzschild metric and substituting   
   > the Lydia's invariant (ds = dt + v du) in for the proper time and 1/c^2   
   > for v, and make it applicable to all motion under the influence of a   
   > potential G(r) by substituting -GM/r by G(r). Move all the terms for the   
   > proper time over to the same side as the line element, cancel out the   
   > dt^2 terms, and rescale so that the metric has the asymptotic form   
   > dx^2 + dy^2 + dz^2 + ... = 0 (as v -> 0). Note what extra terms crop up in   
   > the 4+1 metric where G is present.   
   >   
   >> The CMB occurred relatively late in the evolution of the universe,   
   >> here late being compared to inflationary cosmology etc.   
   >   
   > Actually, the CMB occurred at the *end* of the radiation dominant   
   > phase. The radiation dominant phase existed (as far as anyone can   
   > determine) at all earlier times before that.   
      
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