rjbeery said:
At some point the mass would be enough such that their gravitational forces would compress them together; perhaps it would first form a star, but at some point traditional GR states that a black hole would form, agreed?
This is basically one of the alternatives I gave in my last post: at some point the self-gravity of the assemblage of golf balls is non-negligible, so they gradually transition from an assemblage of golf balls to a single big sphere of matter, something like a planet. At some much further point, if we keep on adding mass, yes, the mass of the object will become larger than the mass limit on white dwarfs or neutron stars or any other stable structure, and at that point, it will collapse to a black hole.
rjbeery said:
Let's call that time T_blackhole.
Time in what chart? Remember that the Schwarzschild chart does not cover the black hole or its horizon, so you can't describe the process of black hole formation in this chart. You have to use some other chart--for example, Painleve or Eddington-Finkelstein. In those charts, yes, you can assign some particular coordinate time ##T_b## to the formation of the black hole.
rjbeery said:
Do you agree that GR suggests all golf balls approaching the newly-formed event horizon after T_blackhole should be visible at time T?
The same question as I asked above applies here (time in what chart?), but I'll assume we're using a chart that meets the requirement I gave above, that you can assign a particular finite time ##T_b## to the formation of the hole.
Let's first consider a simpler case than yours, where a single object falls into the hole, and the object's mass is negligible--it does not change the hole's mass in any way. (If we wanted to be somewhat rigorous, we could consider this as a limiting case of the infall of an object of mass ##m## as ##m \rightarrow 0##.) In this case, yes, light rays from the object will be visible at any time ##T > T_b## in the region outside the hole's horizon. As ##T## increases, the light visible at that time will have been emitted by the object increasingly close to its crossing the horizon. (Technically, this light will also be increasingly redshifted, so we have to assume that the light is being observed by idealized detectors that can detect light of arbitrarily long wavelength. We also have to assume that the light is continuously emitted, i.e., we are ignoring the quantum nature of light.)
However, in a chart in which we can assign a finite time ##T_b## to the hole's formation, we can also assign a finite time ##T_c## to the infalling object crossing the horizon. How does that work? Simple: in this chart, light emitted by the object as ##T \rightarrow T_c## gets received by an observer far away at a time ##T \rightarrow \infty##, because of the time delay (in this chart) caused by the hole's gravity. (This is the same general kind of effect that has been measured in the much weaker field of the Sun as the Shapiro time delay of radar signals.) Light emitted at exactly ##T = T_c## by the object never gets to the faraway observer: it stays at the horizon forever. And light emitted by the object at ##T > T_c## hits the singularity, though at a later time than the object itself.
All of the above assumed that the mass of the hole never changed once formed, so that the horizon never grew. Now consider a case still somewhat simpler than yours, where a single golf ball falls in, but it has non-negligible mass ##m##, so it does increase the size of the hole as it falls in. Suppose you are sitting somewhere well away from the hole, watching the golf ball fall past you. Then, if the hole's mass, as you measure it, is ##M## at some instant just before the golf ball of mass ##m## falls past you, then just after the golf ball passes you, you will measure the hole's mass to be ##M + m##. That is, you measure the hole's mass increase as soon as the golf ball's mass ##m## is closer to the hole's horizon than you are. (Note that this is a general rule not limited to black holes; it applies to any massive object that gains mass by things falling into it.)
In other words, in the case of a hole into which mass is falling, the "mass of the hole" is not just a function of time; it's a function of time and radius. If you are at radius ##R## (note that the radial coordinate in the chart I'm using is the same as in the Schwarzschild chart; only the time coordinate is different), then the mass that you attribute to the hole is a function ##F(T, R)##; if a golf ball of mass ##m## passes you at time ##T##, then ##F(T - \epsilon, R) = M## and ##F(T + \epsilon, R) = M + m##. Someone at a different radius ##R'## would see that particular golf ball pass at some different time ##T'##, and would observe the mass increase from ##M## to ##M + m## at that time.
Now consider an observer who is "hovering" just above the horizon an instant before the golf ball we've been discussing falls past him. The mass function ##F## at his radius ##R = 2M + \delta r## will switch values from ##M## to ##M + m## at some time ##T_h##; i.e., ##F(T_h - \epsilon, 2M + \delta r) = M## and ##F(T_h + \epsilon, 2M + \delta r) = M + m##. But suppose that ##\delta r## is just equal to the increase in horizon radius due to the mass ##m## falling in; then this observer, who was just outside the horizon at ##T < T_h##, will be exactly on the horizon at ##T = T_h##--and at that instant, which is the instant the golf ball falls past him, he will no longer be able to maintain altitude, but will start falling inward towards the singularity.
Now, what about light from this observer, and from the golf ball that falls in? Take the latter first. If its mass were negligible, then the horizon radius would not increase when it falls in, so it would reach the horizon at some time ##T > T_h##; but since the horizon radius does increase, because the golf ball's mass is not negligible, the ball reaches the horizon at ##T = T_h##; what happens is that, at some time ##T_g##, a short time before ##T_h##, the horizon starts growing from radius ##2M## to radius ##2M + \delta r = 2(M + m)##, such that it reaches radius ##2(M + m)## just at ##T_h##, which is the instant that the golf ball falls to that same radius. To a distant observer, it will look as though the ball's light redshifts a little faster than it would have if the hole's mass had not increased. (We're assuming a classical BH that never evaporates, btw.) That is, at some time ##T##, the frequency of light being received by a distant observer from the ball will be a bit less than it would have been if the hole's mass had not increased (because the last light visible from the ball is emitted just before ##T_h##, instead of just before some time greater than ##T_h##).
Light from the observer that was hovering at radius ##2M + \delta r = 2(M + m)## was redshifted too, as seen by a distant observer; but before the golf ball fell in, that redshift was constant. However, when the horizon starts growing, at time ##T_g##, the redshift of the light emitted by this observer starts increasing; and by time ##T_h##, when that observer is at the horizon, the light he emits can no longer escape. So a distant observer will, at some time ##T > T_g## (the exact time will depend on how much time delay the hole's mass imposes on outgoing light), see the hovering observer's redshift start to increase, and it will keep increasing forever, since light the hoverer emits as ##T \rightarrow T_h## will be received by an observer at infinity at ##T \rightarrow \infty##, just as any other light emitted by an object falling through the horizon.
![golfball_black_holes_zps339d1899[1].png](https://www.physicsforums.com/attachments/golfball_black_holes_zps339d1899-1-png.73838/ "golfball_black_holes_zps339d1899[1].png")
