From: intuitionist1@gmail.com   
      
   On Friday, November 2, 2018 at 12:23:29 AM UTC+3, Sabbir Rahman wrote:   
   > > [[Mod. note -- Two points:   
   > > * There's good reason to think that actual black holes in the universe   
   > >   do not have r<0 regions.  That is, r<0 regions are only present if   
   > >   you have *vacuum* Schwarzschild/Kerr solutions, with no matter anywhere   
   > >   in the universe, and in particular, no matter ever present near to   
   > >   or inside the black holes.  If you form black holes by collapsing   
   > >   matter any time after the big bang, I don't think you get r<0 regions.   
   > > * If you form multiple-black-hole spacetimes by collapsing matter any   
   > >   time after the big bang, so far as we know you don't get multi-sheeted   
   > >   Einstein-Rosen spacetimes, either.   
   > > -- jt]]   
   >   
   > It seems that you have in mind specifically astrophysical black hole   
   > candidates, which are clearly extremely complex objects.   
   >   
   > If we restrict ourselves instead to tiny (i.e. sub-elementary-particle)   
   > scales, and assume that _only_ particles of class A and B initially exist,   
   > and that they have very tiny masses, and also that they collapse to form   
   > very tiny naked ring singularities (i.e leading to the fast Kerr solution   
   > specifically), then that would be a more useful scenario to consider.   
   >   
   > The ring singularity would then bound a disc connecting the two regions,   
   > and there would be no problem with particles crossing the disc from one   
   > sheet to the other in that case (such trajectories are known to exist and   
   > they do not cause the wormhole to pinch shut before the particles can   
   > cross). Applying the argument I made earlier there would have to be   
   > particles of all four types present in each region even before the   
   > singularity forms.   
   >   
   > It is not even necessary to have multiple black holes or multiple ring   
   > singularities. As soon as a single ring singularity forms, particles of   
   > class A and B will cross from r>0 to r<0 and so in r<0 there will be   
   > particles present of all four classes, and once again we would need to   
   > extend GR to a bimetric theory on a double-sheeted spacetime.   
   >   
   > To avoid this conclusion you could try to discard  the r<0 region, but   
   > this is a mathematically valid part of the solution which can be reached   
   > by the particles present, so I see no justification for doing so.   
   >   
   > If you have a reference to a (classical) proof that r<0 regions cannot   
   > physically form in any of the Kerr-type solutions, I would appreciate it if   
   > you could mention them here.   
   >   
   > [[Mod. note --   
   > I don't know of any way to *form* a sub-elementary-particle--sized   
   > black hole (if it wasn't already present in the big bang).  That is,   
   > I don't know of any formation scenario for collapsing that small an   
   > amount of mass-energy to form a black hole.  And if you did collapse   
   > it, I don't see any reason why it would form a naked ring singularity,   
   > i.e., why you'd have |J|/M^2 > 1.  (If you had that much angular   
   > momentum, wouldn't the "centrifugal barrier" prevent collapse?)   
      
   I am not sure why the smallness of the mass would matter - if there are   
   enough mutually attractive particles around they will eventually condense   
   to form black holes. Note that there are a growing number of papers in the   
   literature that use Kerr or Kerr-Newman black holes as models of elementary   
   particles. For example there are many papers by Alexander Burinskii from   
   1984 (or perhaps even earlier) to the present; the paper "Kerr-Newman   
   solution as a Dirac particle" by Arcos & Pereira is particularly detailed.   
   [The reason I consider the Kerr solutions rather than the Kerr-Newman   
   solutions is because I showed in my own 2005 paper "Classical   
   electrodynamics from the motion of a relativistic fluid" that the apparent   
   charge of a black hole is due to the motion of the fluid into them]. Note   
   that the ring singularity has a spinorial structure, and that these   
   Kerr-Newman solutions have a gyromagnetic ratio of 2, making the   
   identification with the electron particularly enticing.   
      
   Please note that my argument above actually holds for any Kerr-type   
   solution  with and r<0 region into which particles can pass from the r>0   
   region, but I focus on the fast Kerr example here as it means that we do   
   not have to think about event horizons etc. But your question about how   
   they can accumulate sufficient angular momentum  is an  interesting one,   
   particularly as this would need to be the case if they are to be identified   
   with elementary particles such as the electron.   
      
   You will hopefully recall the discussion we had on the Schwarzschild black   
   hole, and my argument that as soon as the black hole forms, there must be   
   a branching of the spacetime such that the mass that caused the formation   
   of the black hole gets 'split off' from the exterior, such that the mass   
   of the black hole, and the singularity associated with it (towards which   
   subsequently infalling matter falls) live in two separate branches - so the   
   mass of the black hole remains stable after formation.   
      
   Well, the same argument holds for Kerr black holes - as soon as the black   
   hole forms, there is a topological bifurcations and the matter splits off   
   into a separate branch of spacetime. [The reason again is that the matter   
   responsible for the collapse does not experience the black hole metric,   
   whereas the subsequently infalling matter does, and so there must be two   
   independent metrics present].   
      
   The two spacetime branches are connected at the outer event horizon, and I   
   for the fast Kerr solution to form there would need to be some mechanism by   
   which the mattering collapsing towards the black hole continues to transfer   
   angular momentum to the branch containing the matter responsible for the   
   collapse. That is, although the infalling matter cannot actually enter the   
   branch containing the mass of the black hole, it can still transfer angular   
   momentum to it at the outer EH which is the locus of intersection of the   
   two branches. Once the angular momentum is high enough i.e. at |J|=M^2, the   
   horizons approach disappear (though the matter branch must still remain   
   connected at the 'locus of disappearance' of the EH). Once the horizons   
   disappear, there is no futher transfer of angular momentum and so the   
   result fast Kerr solution will be stable and have fixed mass and angular   
   momentum thereafter.   
      
   > In general, if you do form a black hole by collapsing "stuff"   
   > (mass-energy), I don't think you get an r<0 region.  Rather, the   
   > "stuff" replaces the r<0 region (see, e.g., the diagram in MTW box   
   > 33.2 section G.1, with the text there stating that the r<0 region   
   > "gets fully replaced by the interior of the star that collapsed to   
   > form the black hole" (this statement doesn't depend on the size of   
   > the "star" in question, at least so long as we're dealing with the   
   > classical Einstein equations).   
      
   For the reasons mentioned above and in the earlier thread, I think that the   
      
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