The metric inside the horizon would make matter to move toward the center regardless of how strong a force opposes the movement.
A collapsing symmetric spherical dust ball
General relativity says that the metric outside the ball is the Schwarzschild metric. Outside the event horizon, we can define simultaneous events in a "sensible" way. The standard Schwarzschild coordinates give us a sensible definition of simultaneity.
In these coordinates, the outer surface of the shell moves ever slower, never reaching the Schwarzschild radius R of the collapsing mass M.
The natural spacelike foliation is given by the Schwarzschild standard coordinates t and r.
R
t = ∞ ●
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.
.
/
t = 2 ----------------
/
t = 1 ----------------
/
t = 0 ----------------|
/
surface R
The surface does reach R in a finite proper time of someone standing on the surface. Also, the proper distance of the horizon is finite from a static observer outside the horizon. This implies that the time for surface to descend to the horizon is finite, if measure the time by summing proper time intervals measured by static observers on the path.
But our opinion in this blog is that proper time is nothing more than what a clock shows. And if a clock runs ever slower, it may never progress to a specific time t₀. There is no reason why we should be able to extend proper time beyond that reading of the clock.
Question. Is there a spacelike foliation where the spatial metric is static in the vacuum area, simultaneousness is defined in a "sensible" way, and where the surface would reach R?
For the standard Schwarzschild coordinates the following holds: if a static observer A sends a light signal to a nearby static observer B, and B immediately sends a signal back, then the time coordinate when the signal arrives at B is the average of the time coordinates of when A sent the signal and received the signal.
Suppose then that we have a folio F which in the diagram above contains the dot ● at R, at the external time ∞. Obviously, one cannot define simultaneousness in the above "sensible" way. A static observer A at the dot ● cannot send a signal at all to a static observer B at a slightly larger radius, and vice versa.
The Oppenheimer-Snyder (1939) collapsing dust ball
Inside the collapsing uniform dust ball, Oppenheimer and Snyder (1939) use comoving coordinates and solve the metric in the form above. The coordinate τ measures the proper time of a comoving observer.
The authors move to Schwarzschild-like coordinates which have the metric in the form above. They then match the metric to the static Schwarzschild metric outside the ball:
where r₀ is the Schwarzschild radius of the ball.
In the Schwarzschild-like metric and coordinates t and r, the tangential metric is 1. That is, the circumference around the center of the system, as measured by observers at r is 2 π r. In that sense, we may call an observer at a fixed radius r a "static" observer. It is reasonable to say that an observer at r = 0 is static - our definition agrees with that.
Is the Schwarzschild time t a reasonable global time in the system? So that it would be sensible to say that events having the same coordinate t are "simultaneous" relative to a faraway observer?
In a Schwarzschild-like metric tensor, the non-zero elements are on the diagonal. Observers which are initially static at a certain r and a fixed t will probably agree that the events with the fixed time coordinate t are simultaneous. At least this holds in the case of the Minkowski metric.
Oppenheimer and Snyder calculate the approximate mapping from the Schwarzschild global time t to the proper time τ of comoving observers. When t approaches infinity, the proper time τ approaches a finite value. In the formula above, R_b is the comoving radial coordinate of the surface of the ball, and r₀ is the Schwarzschild radius, and R is the comoving radial coordinate of a dust particle inside the ball.
When t goes to infinity, the formula says that the proper time τ progresses slightly further when R is larger.
Interpreting the Oppenheimer-Snyder result in our Minkowski & newtonian gravity
In our own gravity model, slowdown of clocks is a result of the gravity interaction which gives more inertia to parts of the clock. The stretching of the radial metric is due to the inertia being larger in a radial movement compared to a tangential movement.
In our model the underlying metric of spacetime does not change. The underlying metric is always Minkowski. It is the paths of particles which change.
Let us introduce:
Hypothesis. The global time coordinate in the Schwarzschild-like metric matches the time coordinate of the underlying Minkowski coordinates and metric, where the dust ball is static in the Minkowski coordinates.
Then we have a very nice interpretation for the Oppenheimer-Snyder result. It describes how the dust ball freezes as it collapses.
There is an "event horizon" in the following sense: a photon sent from inside the ball will never get out of the ball. Our event horizon has no strange properties of the one in the standard Schwarzschild solution of general relativity. There is no infinite force at the horizon in our model. And the metric does not switch the roles of the radial coordinate and the time coordinate at the horizon.
The interpretation of the whole collapse process is very mundane in our model: a system of particles moves slower and slower, but never freezes totally. It resembles the "frozen star" hypothesis of general relativity.
The mass-energy concentrates itself close to the Schwarzschild radius
The center of the dust ball falls into an extremely low potential. Its Komar mass is negligible relative to the dust particles at surface of the ball which acquire a lot of kinetic energy, relative to a static coordinate system.
Could this explain why the gravity field of the resulting black hole is spherically symmetric, even if the dust cloud initially was not symmetric?
We need to figure out what happens with spinning dust balls, and how does a merger of two black holes happen.
Conclusions
Our Minkowski & newtonian model makes a collapsing dust ball well behaved. We avoid some strange properties of a standard Schwarzschild black hole.
We need to figure out what happens with rotating black holes, and what happens in a merger of two black holes.
We also need to consider quantum mechanical effects. There are extremely strong forces at the surface of a collapsing dust ball. Are these forces able to materialize new particles from "empty" space?
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