Consider the standard picture of gravitational collapse to form a   
   Schwarzschild black hole of a spherically-symmetric dust cloud of   
   total mass M, made of identical particles that have zero size,   
   nonzero mass, and that do not interact with anything.   
      
   The dust particles are initially at rest, and are distributed such   
   that the dust can can be considered continuous with a definite outer   
   boundary. Outside the dust cloud is vacuum, and with the spherical   
   symmetry we can apply Birkhoff's theorem and conclude that the   
   region outside the outer boundary is isometric to some region of   
   Schwarzschild spacetime.   
      
   Since the dust particles don't interact with anything, each one   
   follows a geodesic in the metric. Since the metric is continuous,   
   the particles at the outer boundary are in a region with a metric   
   that is very nearly Schwarzschild, and thus the geodesics they   
   follow are all infalling. This region with a metric that is nearly   
   Schwarzschild extends inside the outer boundary for some distance,   
   and all the particles there are infalling (perhaps at different   
   rates -- details don't matter). Since the outer boundary is infalling,   
   the entire dust cloud must be getting smaller (collapsing). The   
   specific details of this collapse are not important; it is sufficient   
   to know that the dust cloud is collapsing.   
      
   According to standard picture, nothing special happens when the   
   outer boundary passes r=2M. That is, the above description still   
   holds and the cloud simply keeps collapsing, all the way down to   
   r=0, where a curvature singularity forms.   
      
   However, there seems to be a problem with this standard picture of   
   spherical collapse, and I will try to explain my concern.   
      
   Let's say that the metric inside the collapsing matter is R(r,t),   
   and let's say that the Schwarzschild interior is S(r,t). At the   
   moment that the black hole forms at t=t0, say, it is true that the   
   boundary conditions require that R(r=2M,t=t0)=S(r=2M,t=t0). However   
   in general it will obviously not be the case that R(r,t)=S(r,t) for   
   r<2m or t>t0 as R(r,t) is some non-black hole metric and S(r,t) is   
   the Schwarzschild interior metric.   
      
   Usually it is assumed that there is nothing outside the dust cloud,   
   and that it has mass M, so that the black hole forms when the   
   outermost shell reaches r=2M. This outermost shell follows the   
   geodesics assuming that the metric is S(r,t).   
      
   Now, in the standard picture, because the outer shell of matter is   
   part of the Schwarzschild interior, and _also_ part of the dust   
   cloud, it is known from Birkhoff's theorem that it must end up in   
   a singularity, and moreover because there can only be one metric   
   on the interior submanifold, it must sweep all of the dust particles   
   inside the cloud with it into the singularity.   
      
   The problem is, however, that the particles in the dust cloud live   
   in a manifold with metric R(r,t), and the particles in the boundary   
   live in a manifold with metric S(r,t). According to the standard   
   picture, as this outer shell of particles collapses towards the   
   singularity, all of the dust particles it encounters along the way   
   must become swept along with it. But this means that the particles   
   that had been experiencing/following the geodesics of the metric   
   R(r,t) must now start experiencing the metric of the Schwarzschild   
   interior S(r,t).   
      
   It is claimed that the particles in the dust cloud follow a geodesic   
   path (i.e. according to R(r,t)) that is very close to S(r,t).   
   However, this is _only_ true for the particles very close to the   
   boundary of the dust cloud, because R(2M-dr,t0+dt) is very close   
   to S(2M-dr,t0+dt) there. It is _not_ true for particles further   
   inside the dust cloud where R(r<2M,t>t0) != S(r<2M,t>t0). The two   
   metrics are finitely different whenever we move a finite distance   
   either radially inwards or in the temporal future from the event   
   horizon after the time of formation of the black hole.   
      
   The standard claim that the outer boundary 'sweeps' up the particles   
   in the interior of the dust cloud to follow the Schwarzschild   
   geodesics implies that by some mechanism as yet to be explained,   
   the metric R(r<2M,t>t0) experienced by the particles inside the   
   dust cloud instantaneously start experiencing the new Schwarzschild   
   metric S(r<2M,t>t0). This is a _finite_ and _discontinuous_ change   
   in the metric from R(r,t) -> S(r,t).   
      
   Now, the only way that the metric can change in this way is if there   
   is some matter that causes it to do so, but the infinitesimal shell   
   of matter that is doing the 'sweeping' only has an infinitesimal   
   mass, and this is insufficient to cause the finite change in the   
   metric that is required in the interior of the dust cloud. One might   
   try to argue that as the matter is swept along, the mass inside   
   this inwardly sweeping shell becomes finite, but this cannot be the   
   case, because that shell would then have infinite density, resulting   
   in a singularity. The boundary also cannot somehow 'push' particles   
   along with it because (i) this is a dust cloud and so the particles   
   cannot 'push', and (ii) even if it were not a dust cloud, the   
   particles would not be able to accelerate the swept particles   
   instantaneously to the speed of light, which is the speed at which   
   the boundary is collapsing.   
      
   So, it seems to me that the standard picture cannot be correct   
   because there is no physical mechanism by which the finite change   
   in the metric that would be required in the interior of the dust   
   cloud can be effected.   
      
   The problem can be traced back to the _assumption_ that the outer   
   shell of particles in the matter cloud follows the Schwarzschild   
   interior geodesics and sweeps through the interior of the dust   
   cloud. In actual fact, all particles of the dust cloud continue to   
   follow the geodesics of the dust cloud. Only particles outside the   
   dust cloud (which are not initially postulated to exist, but there   
   is no problem allowing them to be present), will follow the   
   Schwarzschild interior geodesics upon crossing the event horizon.   
      
   These particles cannot 'sweep' through the interior of the dust   
   cloud because of the incompatibility of the metrics already described,   
   and thus the only possibility for both metrics R(r,t) and S(r,t)   
   to be simultaneously present is if a bifurcation or 'branching' of   
   spacetime takes place at the event horizon precisely at the time   
   the black hole is formed. The particles inside the dust cloud will   
   continue to experience the metric R(r,t) for t>t0, and the particles   
   which fall across the BH event horizon after t=t0 will experience   
   the metric S(r,t).   
      
   I hope that this is clear. (Please let me know if it is not).   
      
   Some further issues I have with the standard picture are the fact   
   that the Schwarzschild singularity lies in the temporal future of   
   the event horizon, and so it seems incorrect to think of any kind   
   of 'radial' sweeping taking place. Since the singularity is reached   
      
   [continued in next message]   
      
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