From: intuitionist1@gmail.com   
      
   On Friday, June 15, 2018 at 9:41:49 AM UTC+3, Gregor Scholten wrote:   
   >   
   > You are wrong, it is true for a generic interior matter configuration,   
   > too. If M is the mass of the generic interior matter configuration and   
   > its boundary (surface) is located at radial coordinate R < 2GM (so that   
   > the black hole has already formed), then the gravitational collaps of   
   > the matter configuration is unavoidable.   
      
   To try to avoid the possiblity of just going to go round in circles at   
   this point, I will try once again to explain to you why your argument is   
   wrong, but this time I will try to make my argument a little more   
   precise. In particular I will try to be a bit more careful when   
   discussing the infinitesimals than I was before, as this is key to   
   sorting the truth from falsehood.   
      
   So, let us consider a generic spherically symmetric matter configuration   
   in which a black hole has just formed at radius R=2M, and without loss   
   of generality, let us assume that this is the smallest radius at which a   
   black hole forms in that configuration, so that *all* of the matter   
   inside R=2M is outside of its Schwarzschild radius, and can indeed be a   
   _long_ distance outside of its Schwarzschild radius, relatively speaking   
   at least for r strictly less than 2M.   
      
   Let us ignore the matter outside of R=2M for now (we could suppose for   
   convenience that there is none, without changing the basic argument),   
   and let us focus on the infinitesimal shell of matter that has just   
   caused the black hole to form.   
      
   I think before in this thread I may have been a bit sloppy and said (or   
   at least implied) that this infinitesimal shell of matter is inside its   
   Schwarzschild radius when the black hole forms.   
      
   In actual fact, if we are more careful, we will realise that _none_ of   
   that infinitesimal shell lies inside its Schwarzschild radius. Indeed   
   only particles _outside_ that shell feel the effect of the full mass M.   
   That infinitesimal shell itself only feels the effect of the mass M-dm,   
   where dm is the infinitesimal mass of that shell.   
      
   Now it is a KNOWN FACT that, precisely because the matter at R<2M -   
   including importantly the infinitesimal outer shell discussed above -   
   lies outside its own Schwarzschild radius, it _can_ in principle   
   indefinitely resist (by applying thrusters, because of interatomic   
   forces, or by whatever other mechanism), falling into its own   
   Schwarzschild radius. The fact that it can do this even in principle,   
   means that you are wrong, and that collapse of the interior is _not_   
   inevitable. Therefore both the interior metric R(r,t) and the   
   Schwarzschild black hole metric S(r,t) for the full mass M must coexist,   
   and therefore a topological bifurcation MUST occur when the black hole   
   forms.   
      
   In particular, although that infinitesimal shell of matter just happened   
   to be sufficient to create the black hole, it a priori has insufficient   
   influence to be able to make any of the matter at r<2M collapse inside   
   its Schwarzschild radius.   
      
   It is true that for matter infinitesimally close to R=2M it may be able   
   to make an infinitesimal change to the metric, but this infinitesimal   
   change is _not_ in general going to be enough to cause that subsequent   
   infinitesimal shell to collapse inside its own Schwarzschild radius.   
      
   The argument that you are making requires that this infinitesimal shell   
   will _always_ make a sufficiently large [infinitesimal] change in the   
   metric to force the subsequent inner shell fall inside its Schwarzschild   
   radius. But this is simply not the case.   
      
   Giving anecdotal evidence for specific cases that an infinitesimal shell   
   can trigger a larger scale collapse does _not_ imply that the collapse   
   of each infinitesimal shell encountered is _necessarily_ into its own   
   Schwarzschild radius. (It might be, but definitely does not have to be).   
      
   > Let's assume for example that the matter configuration is "rigid" in the   
   > sense that there are repulsive interatomic forces that support against   
   > gravitational collaps. Those repulsive forces fail to support at least   
   > after R has become < 2GM.   
      
   This seems to be a statement of faith on your part, not a statement of   
   fact.   
      
   > Let's imagine the generic interior matter configuration splitted into   
   > shells again. For the outermost shell at r = R(t0) < 2GM, the   
   > Schwarzschild metric S_M(R(t0),t0) for the full mass M = M(R(t0))   
   > applies at time t = t0. Due to that, the outermost shell collapses   
   > unavoidably.   
      
   At t0, the outermost shell only experiences M-dm, where dm is the   
   infinitesimal mass of that shell. If you think carefully about it,   
   _none_ of that infinitesimal shell experiences the full mass, and so   
   actually _none_ of it is inside its Schwarzschild radius, and _none_ of   
   it collapses unavoidably. Any further collapse is in principle   
   avoidable. [That is not to say however that any matter outside that   
   infinitesimal shell that subsequently crosses R=2M will not unavoidably   
   collapse into the singularity. Indeed it must, by Birkhoff's theorem].   
      
   Please note the general principle that I derived earlier that you can   
   never have more than mass M inside a radius of 2M. In actual fact, I   
   should have made a slightly stronger statement that holds - and this is   
   very important here when we start talking about infinitesimals - and   
   that is that the for radius r < 2m, the mass inside that radius is   
   always LESS THAN m. Equivalently, during spherical collapse, NO MASS CAN   
   EVER BE INSIDE ITS OWN SCHWARZSCHILD RADIUS.   
      
   > If the next inner shell at R(t0) - dr does not collaps,   
   > too, at time t0, the outermost shell encounters it at time t0 + dt.   
   > After that, the Schwarzschild metric S_M(R(t0) - dr, t0 + dt) for the   
   > full mass M applies at the next inner shell at R(t0) - dr, and by this,   
   > that next inner shell is unavoidably forced to collaps, too. This shell   
   > then encounters the third shell, and so on. So, shell by shell is forced   
   > to collaps, yielding all shells collapsing in the end.   
      
   No, this is not _necessarily_ the case. The collapse is avoidable in   
   principle throughout the interior and for every shell for which r<2M.   
      
   > Or let's use another picture. Let's split the generic interior matter   
   > configuration into an inner part with its boundary R_inner, let's say   
   > R_inner = R(t0)/2, and an outer part R_inner < r < R(t0). The mass   
   > M_inner of the inner part is much lower than the mass M of the total   
   > configuration (for uniform density and without taking spatial curvature   
   > into account, M_inner = M/8 would apply), so that we can assume that   
   > R_inner > 2GM_inner and the inner part on its own would therfore not   
   > make up a black hole. So, the inner part on its own would not be caused   
      
   [continued in next message]   
      
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