From: intuitionist1@gmail.com   
      
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   On Tuesday, June 12, 2018 at 10:43:37 AM UTC+3, Gregor Scholten wrote:   
   > Sabbir Rahman wrote:   
   >   
   > >> I just read again the chapters 32.4 and 32.5 of MTW, and in fact, they   
   > >> write what I already explained: all the particles inside the collapsing   
   > >> dust cloud fall simultanously towards the point r = 0, there is no   
   > >> "encountering" of the outermost shell of particles and the more inner   
   > >> shells.   
   > >>   
   > >> In 32.4, they consider a dust cloud with zero pressure and argue that   
   > >> the metric inside the cloud equals the Friedmann metric of a contracting=   
   > >   
   > >> universe. And like there is no encountering of shells of galaxies in a   
   > >> contracting universe, there is no encountering of shells of dust particle=   
   > > s.   
   > >>   
   > >> In 32.5, they consider a collapsing star with pressure gradient. They   
   > >> write that the surface of the star is no longer free-falling due to the   
   > >> forces caused by the pressure gradient, but that qualitatively seen the   
   > >> process of collaps is like in the case of zero pressure.   
   > >>   
   > >> So, I am really wondering what makes you think that the standard picture   
   > >> would be that the outer shell of dust particles encounters the more   
   > >> inner shells and sweep them along with it?   
   > >   
   > > The solutions given in in MTW are incorrect, but the reason why is,   
   > > admittedly, not at all obvious.   
   >   
   > Obvios or not, please explain why you think that the solutions given in   
   > in MTW are incorrect. In the following, you do not do that, you just   
   > consider a different picture.   
      
   It turns out that the dust cloud is a very special case where it happens   
   to be particularly to understand immediately why the standard solution   
   is incorrect. I introduced the rigid framework scenario to derive an   
   important preliminary result which can then be applied to the dust cloud   
   case to show where the problem arises.   
      
   > And you still did not answer my questions what makes you think that the   
   > standard picture would be that the outer shell of dust particles   
   > encounters the more inner shells and sweep them along with it?   
      
   Again, I introduced the rigid framework scenario as it makes it easier   
   to explain why I say this about the standard picture of spherical   
   collapse.   
      
   > > I think with hindsight that rather than considering a freely falling   
   > > dust cloud, it would be somewhat easier to consider an (admittedly   
   > > rather artificial) mass distribution which happens to have almost   
   > > the same mass density as the dust cloud throughout the interior but   
   > > which is NOT allowed to collapse by keeping the mass fixed in place   
   > > through mechanical means of some kind.   
   > >   
   > > To be more concrete, let us consider an approximately spherically   
   > > symmetric, rigid, solid (e.g. metallic), spherical lattice of mass   
   > > M-dm (i.e. just less than the mass M of the dustcloud we have been   
   > > considering) and radius r = 2M-dr (i.e. slightly less than the   
   > > radius of the initial dust cloud).   
   > >   
   > > Then this rigid structure will have approximately the same average   
   > > density as the dust cloud we started with, albeit with very slight   
   > > less mass and of slightly smaller radius. Let us assume that it   
   > > does not have enough mass within any radius to form a black hole.   
   > >   
   > > Now, at some distance outside this rigid structure let there be an   
   > > infalling thick shell of dust of mass precisely dm just outside   
   > > r=2M (the thickness should be finite but not too large).   
   > >   
   > > Just outside that shell, let there be a large number of concentric   
   > > infalling shells of dust each of tiny (or infinitesimal) mass. I   
   > > hope that this scenario is clear.   
   > >   
   > > Clearly this is what is going to happen:   
   > >   
   > > After a short time, a black hole will be formed as soon as the   
   > > entirety of the first collapsing dust shell has crossed the   
   > > Schwarzschild radius at r=2M. Note that the entire rigid lattice   
   > > in the interior remains fixed, so that the metric R(r,t) in the   
   > > interior basically remains fixed throughout that interior. The   
   > > lattice will be totally oblivious to the fact that a black hole has   
   > > just formed.   
   >   
   > This is impossible. As soon as the black hole has formed, the rigid   
   > lattice which is now inside the black hole starts to collaps. To resist   
   > against the gravitational compression and remain fixed, its atoms had to   
   > travel on spacelike wordlines, i.e. to exceed the speed of light in a   
   > local frame.   
      
   There is no 'gravitational compression' (=="abrupt change of   
   metric") -  the rigid frame does not experience anything different than   
   it did before the black hole formed. Indeed how could it, as nothing of   
   any significance has happened locally (or will even happen in the   
   future, given that the dust shells are of such tiny mass) that could   
   cause such a sudden change in the metric.   
      
   > In addition, after the first dust shell has entered the region r < 2M,   
   > the metric R(r,t) in that region is no longer determined by the matter   
   > from the lattice alone, but also influenced by the matter of the dust   
   > shell. The infinitesimal mass contribution dm of the dust shell yields a   
   > infinitesimal change in the metric R(r,t).   
      
   You ask me why I think the dust particles sweep the inner shells along   
   with  them - and yet here you are insisting that they be swept inwards   
   by that very same infinitesimal dust shell. I actualy agree with you -   
   the infinitesimal dust shell can only make an infinitesimal change in   
   the metric. So R(r,t) basically does not change in any noticeable way.   
   For the entire rigid framework however to be somehow forced to collapse   
   into a singularity would require an enormous change in the metric -   
   increasing larger as we get closer to the singularity. The infinitesimal   
   collapsing dust cloud simply cannot effect that change and indeed such a   
   change cannot physically happen.   
      
   > > After another short time, the second shell of dust will cross the   
   > > Schwarzschild surface. From Birkhoff's theorem, this shell of dust   
   > > will follow the geodesics of the Schwarzschild interior solution -   
   > > with metric S(r,t) -   
   >   
   > Birkhoff's theorem tells us that at the outermost shell, the metric has   
   > to equal the external Schwarzschild metric, and by this, the interior   
   > Schwarzschild metric. However, you are not considerung the outermost   
   > shell here, but the the second shell seen from the innermost one.   
      
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