From: intuitionist1@gmail.com   
      
   On Monday, June 11, 2018 at 3:04:19 PM UTC+3, Gregor Scholten wrote:   
   > Sabbir Rahman wrote:   
   >   
   > >> A very informative diagram of the gravitational-collapse process can   
   > >> be found in figure 32.1c of Misner, Thorne, and Wheeler, "Gravitation"   
   > >> (W. H. Freeman, 1973).   
   > >> -- jt]]   
   > >   
   > > [Moderator's note:  There is a new edition of MTW out, at a reasonable   
   > > price to boot.  -P.H.]   
   > >   
   > > Please note that the discussion of spherical collapse in Ch.32 of MTW   
   > > does not address the issues I am raising here (if it did, there would   
   > > have been no point in my raising them as they would already have been   
   > > addressed!).   
   > >   
   > > Throughout that discussion in MTW, it is _assumed_ (without anywhere   
   > > making that assumption explicit presumably because it is taken to be   
   > > 'common sense') that the interior of the star and [the radial inward   
   > > progression of] the Schwarzschild interior refer to the same   
   > > (sub)manifold.   
   >   
   > 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.   
      
   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.   
      
   [[Mod. note -- Unfortunately, the rigid lattice implies the existence   
   of strong compressive forces to support this structure.  Those   
   compressive sources will themselves contribute to the stress-energy   
   tensor (in a rather complicated way).  Your original formulation of   
   a freely falling dust cloud is actually a lot easier to work with.   
   -- jt]]   
      
   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.   
      
   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) - and eventually disappear into the Schwarzschild   
   singularity after some amount of time that depends upon the mass   
   M. The same will happen to each subsequent shell of dust as it   
   crosses the Schwarzschild surface.   
      
   Now, in the standard picture of gravitational collapse, each component   
   of the rigid lattice structure, which lies inside the black hole,   
   must be swept up by the first infalling dust shell of mass dm as   
   it passes that component and eventually end up in the Schwarzchild   
   singularity. This means that the rigid metric R(r,t) must turn into   
   the interior Schwarzschild solution S(r,t) as the thick dust shell   
   (and specifically the outermost infinitesimal shell of that thick   
   shell) sweeps by, taking the rigid lattice with it as it heads   
   towards the Schwarzschild singularity.   
      
   There are two possibilities:   
      
   1) On the one hand, if this _does_ happen, then you need to explain   
   the physical mechanism by which the infinitesimal shell of mass at   
   the outer boundary of the dust cloud managed to effect the   
   transformation of the rigid metric R(r,t) into the Schwarzschild   
   interior metric S(r,t), into which the subsequent dust shells also   
   fall.   
      
   2) On the other hand, if this does _not_ happen, and R(r,t) remains   
   relatively unchanged by the radially inward sweeping shell of mass   
   dm, then there is a problem - we have two metrics, the rigid metric   
   R(r,t), and the metric of the Schwarzschild interior solution S(r,t),   
   both coexisting indefinitely. This means that a topological bifurcation   
   must have taken place when the black hole formed. The matter   
   responsible for the collapse is in the submanifold with metric   
   R(r,t), which is no longer accessible to the exterior infalling   
   matter after the black hole forms. Any subsquently falling matter   
   falls into submanifold with metric S(r,t), which corresponds to the   
   vacuum interior Schwarzschild solution.   
      
   I claim that 1) is not physically possible, and therefore that 2)   
   is correct, and a topological splitting takes place.   
      
   I hope that this argument is clear. Once you have understood it I   
   will (iA) be able to go on to explain why the solutions given in   
   MTW are wrong.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)