From: intuitionist1@gmail.com   
      
   On Thursday, June 14, 2018 at 9:25:29 AM UTC+3, Gregor Scholten wrote:   
   > Sabbir Rahman wrote:   
   >   
   > >> And this argument is wrong. As we have seen, the metric for r > R is   
   > >>   
   > >> S_M(r,t)   
   > >>   
   > >> where R is the radial coordinate of the boundary (surface) of the matter   
   > >> distribution ("rigid" structure + dust shells in your case) and S_M is   
   > >> the Schwarzschild metric for the mass M = M(R,t). No matter if R < 2GM   
   > >> or R > 2GM. For r < R, the metric is   
   > >>   
   > >> R(r,t) = S_{M(r,t)}(r,t)   
   > >>   
   > >> with S_{M(r,t)} being the Schwarzschild metric for the mass M(r,t) <= M   
   > >> that is contained in r.   
   > >>   
   > >> If a shell of the matter discribution that is at r + dr at time t   
   > >> encounters a more inner shell at r at time t + dt, then the matter   
   > >> distribution is at r changed from M(r,t) to M(r,t+dt), changing the   
   > >> metric at r from   
   > >>   
   > >> R(r,t) = S_{M(r,t)}(r,t)   
   > >>   
   > >> to   
   > >>   
   > >> R(r,t+dt) = S_{M(r,d+dt)}(r,t+dt)   
   > >>   
   > >> Where we have to note that after the formation of a black hole, and   
   > >> already a little earlier, that case cannot occur that a shell is static   
   > >> (like the rigid structure in your scenario).   
   > >>   
   > >> At every r and t (apart from the singularity), the metric is smooth (no   
   > >> abrupt change between r and r + dr or t and t + dt), so there is a   
   > >> physical link everywhere.   
   > >   
   > > Okay, here is the problem I have with this. Any infalling matter outside   
   > > the dust cloud experiences the full mass M of the interior   
   >   
   > Unless it has already entered yet the interior, i.e. the region r < R   
   > where R is the radial coordinate of the surface of the dust cloud.   
   > Before entering the interior of the dust clould, i.e. as long as the   
   > infalling matter is at some r > R, it experiences the full mass M = M(R)   
   > of the dust cloud and therefore the Schwarzschild metric S_{M(R)}{r,t),   
   > but as soon as the infalling matter has fallen through the surface of   
   > the dus clould and is currently located at some r < R, it is no longer   
   > experiencing the full mass M(R), but the lower mass M(r), and therefore   
   > the Schwarzschild metric S_{M(r)}(r,t) for this lower mass.   
   >   
   > > and thus on   
   > > crossing the event horizon is doomed to hit the singularity after some   
   > > period of time that depends upon M.   
   > >   
   > > Your analysis above for the matter in the interior is fine and I have no   
   > > issues with it. HOWEVER, there is a priori no singularity there, and   
   > > neither does there necessarily ever have to be.   
   >   
   > Assumed the infalling matter crosses the event horizont *after* the   
   > black hole has formed, i.e. when the radial coordinate R of the boundary   
   > of the dust cloud is already < rs. Then the infalling matter first   
   > crosses the event horizon at r = rs and then, a little later, crosses   
   > the dust cloud boundary at r = R < rs and enters the interior of the   
   > dust cloud. Before crossing the event horizon at rs and as well after   
   > crossing the event horizon and before crossing the dust cloud surface at   
   > R, the infalling matter feels the Schwarzschild metric S_{M(R)}(r,t) for   
   > the full mass M = M(R). After crossing the dust cloud boundary at R, the   
   > infalling matter no longers feels that Schwarzschild metric, but instead   
   > the Schwarzschild metric S_{M(r)}(r,t) for the lower mass M(r) < M(R),   
   > where r < R is the radial current radial coordinate of the infalling   
   > matter. So, the infalling matter at r < R feels the same metric like the   
   > particles of the dust cloud at r.   
   >   
   > Nevertheless, the infalling matter is doomed to hit the singularity,   
   > just like the particles of the dust cloud itself are. After the dust   
   > cloud is inside its own Schwarzschild radius, no repulsive forces can   
   > support it against gravitational collaps any more. As we have seen, any   
   > support against gravitational collaps that is active short before the   
   > formation of the black hole (= short before the radial coordinate R of   
   > the boundary of the dust cloud deceeds the Schwarzschild radius rs) is   
   > very unstable and destroyed by the lowest disturbance.   
      
   It is true that a dust cloud will continue to collapse (I have   
   already explained why the standard picture is incorrect in this   
   case, and what the correct picture is). However, this will not be   
   true for a generic interior matter configuration. [I also explained   
   this in more detail in a previous response that appears still to   
   be with the moderators].   
      
   So if what you were saying were true in general (i.e. not specifically   
   for a dust cloud - where as it happens it is not true either, but   
   anyway...), then it would in principle allow a particle to fall   
   into a black hole, and then subsequently escape from it when it   
   finds itself in a region that is no longer inside its own Schwarzschild   
   radius. Clearly this cannot be the case.   
      
   > And of course, the time the infalling matter takes to hit the   
   > singularity is a little longer than in the case that the dust cloud has   
   > already collapsed to the singularity so that R = 0.   
   >   
   > > Thus, we _do_ have two incompatible metrics here - the Schwarzschild black   
   > > hole metric for a black hole of mass M, and the metric R(r,t) that you   
   > > have described above, which is indeed Schwarzschild, but always for some   
   > > mass M(R) < M for r < 2GM.   
   >   
   > The Schwarzschild black hole metric for a black hole of mass M applies   
   > only outside the dust cloud, in the region r > R, that contains the   
   > regions R < r < rs and r > rs. Inside the dust clould, only the   
   > Schwarzschild metric for the lower mass M(r) < M applies. And at the   
   > boundary R, both metrics match.   
      
   This latter statement is incorrect for r < rs for the reasons I   
   have already given. A topological bifurcation must take place at   
   r=rs  when the black hole forms.   
      
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