In static spherically symmetric perfect fluid solutions of Einstein field equations the central pressure diverges when the compactness parameter $\alpha~(\equiv r_{S}/R)$ reaches some critical value $\alpha_{c}\leq 8/9$. This fact is commonly understood as the birth of central gravitational singularity and event horizon. What is first, the event horizon or the singularity? How to understand the initial event horizon at the star center, which have a zero surface area?
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What is first, the event horizon or the singularity?
The horizon forms first. Every event on the horizon has the singularity in its causal future. No event anywhere in spacetime has any part of the singularity in its causal past. So unambiguously the horizon is first.
How to understand the initial event horizon at the star center, which have a zero surface area?
It is not a particularly different interpretation from the rest of the horizon. It is simply the first event at which light does not escape to future null infinity.
Dale
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It depends. For a faraway observer a black hole takes an infinite time to form. The innermost particles collapse to a small black hole first. The mass of a sphere increases with $R^3$, while the Schwarzschild radius increases linearly with $M$, the mass of the hole. So if the whole sphere lies still outside its associated Schwarzschild radius, every sphere inside lies outside its SR also. So only when the whole sphere lies within the associated SR, the black hole will have formed.
The density of the sphere of (non-rotating) dust is not constant throughout the sphere and the center will first develop an associated SR. As the compactness parameter shows. The radius will then grow to the radius associated with the total mass. So the radius grows from the inside to the outside. The matter collapses towards one point (as assumed in GR, but in reality particles are not pointlike, but a soft and smoothened, continuous singularity can still develop), and at the moment 8/9 of the mass finds itself behind the SR it still needs some time to collapse to a point. So, event horizon first, and then the singularity.
Seen rom the outside the 1/9 left outside the horizon will never reach the hole and seems to be frozen in time above the horizon.
I have read that some doubt black holes actually never develop at all. I can't find it though and I'm not sure what they mean.
This holds for all continuous non-rotating spherical masses. Up to the Planck mass. There are no processes in nature though that develop a black hole when their mass is smaller than about 1.4 solar masses, but in principle a small spherical, non-rotating distribution of mass (say without charges, which normally stop collapse) can collapse to a BH. With a minimum though of the Planck mass.
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Great discussion on this question.
It seems to me, though, that the event horizon can never form, in principle.
If we take as a working hypothesis that an event horizon does form, it will have to form initially around some point. This point will have to be the density maximum of the immediate vicinity.
The initial event horizon around this initial point would have to have an extremely tiny initial radius.
One of two things would then happen (given the assumption that it formed at all). Either (1) it would evaporate so quickly due to Hawking radiation that it could not grow virtually at all in that short time, or (2) it would "consume" whatever was in its immediate vicinity, but after that, anything else (like the next quark over, say) would be so far away from its tiny radius that such growth would appear to take infinite time to happen, so effectively not happen.
