From: g.scholten@gmx.de   
      
   Sabbir Rahman 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.   
      
   In other words: you consider the case that the radial coordinate R of   
   the surface/boundary of the matter configuration currently matches the   
   Schwarzschild radius rs = 2GM of that matter configuration, where M is   
   the total mass of that configuration. So, the black hole has *not* just   
   formed, it is rather currently *forming*. More precisely, we have to   
   distinguish two scenarios:   
      
   (1) The matter configuration has always been as it is now, the radial   
   coordinate R of the boundary has always been equal to 2GM, there has   
   been no collaps before, no time when R was > 2GM. So, the black hole has   
   never formed, and is not forming currently, but is on the edge between   
   formed and not formed. Such a configuration would be highly unstable,   
   though, there would a highly unstable equlibrium between gravity and   
   repulsive forces - the slightest disturbance would destroy the   
   equlibrium and cause gravity to become dominant so that the   
   configuration would collaps. Therefore, such a scenario would be highly   
   unrealistic.   
      
   (2) The matter configuration was collapsing before, so that R was > 2GM   
   before. In the moment when the black hole is forming, i.e. R is exactly   
   2GM (as already emphasized, the black hole has *not* just formed in that   
   moment, it *is* actually *forming*), gravity and repulsive forces could   
   balance each other if we assume the repulsive forces are arbitrary   
   strong, but the preceding collaps unavoidably makes up a disturbance to   
   this balance, so that the repulsive forces are dominated by gravity and   
   fail to support against further collaps. Even if we assume that the   
   repulsive forces slow down the collaps when R is approaching 2GM from   
   above, the collaps will always make up some at least slight disturbance,   
   preventing an equlibrium to establish. So, in the moment when R equals   
   2GM, R is currently decreasing, even if the velocity of decreasing is   
   very low. So, R unavoidably becomes < 2GM, and as soon as that happens,   
   the black hole has formed, and a further collaps of the matter   
   configuration cannot be avoided.   
      
      
   > 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.   
      
   It has not caused the black to form, it currently is causing 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.   
      
   And because of this, the black hole has actually not yet formed (it   
   rather is currently forming), and the gravitational forces on that shell   
   can just be balanced by repulsive forces. But due to the preceding   
   collaps, that made the radial coordinate R of the shell decrease from >   
   2GM to 2GM, there is some at least slight disturbance that destroys the   
   balance, so that the gravitational forces dominate the repulsive forces   
   and the collaps continues.   
      
      
   > 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)   
      
   However, it can only just resist, i.e. the mechanism can just balance   
   the gravitational forces, and the slightest disturbance makes it fail to   
   balance. So, in fact, it cannot resist.   
      
      
   > 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   
      
   Even if we assume that the balance of repulsive forces and gravity would   
   be stable at R = 2GM, so that the case that the black hole would have   
   formed is never accomplished and the black hole will forever stay in the   
   state of being currently forming, but never achieves the state of being   
   formed, there is no coexistence of metrics. Inside the region r < R =   
   2GM, there is only one metric, namely the metric S_{M(r)}(r,t) which is   
   the Schwarzschild metric for the mass M(r) inside the radial coordinate r.   
      
   And a particle that enters the region r < R = 2GM from outside, i.e.   
   from r > R = 2 GM, experiences nothing but the metric S_{M(r)}(r,t). And   
   of course, this would in principle allow such a particle to escape from   
   the region r < R and travel back to r = R = 2GM. The particle could not   
   travel into the region r > R, just as the repulsive forces that support   
   the shell at r = R against gravitational collaps cannot become strong   
   enough the make that shell expand again, but like the shell at r = R,   
   the particle could stay at r = R.   
      
   Imagine a rocket with thrusters: when the rocket is at r = R = 2GM, the   
   thrusters can just balance the gravitational attraction of the matter   
   configuration, their thrust can not dominate them.   
      
   And there is nothing in contradiction to the statement that a particle   
      
   [continued in next message]   
      
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