From: g.scholten@gmx.de   
      
   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.   
      
   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.   
      
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