In some sense a black hole is always an approximation unless we are the ones falling in. It takes infinite time for us to observe anything cross the event horizon. But we do see it slow down then become redshifted to invisibility fairly quickly. It’s just that information is never technically lost to us, only to the precision of our measuring apparatus, in finite time. It never crosses from our perspective.
This is because gravity causes time to happen more slowly relative to an external observer (the person in that field sees far sources experiencing faster time instead). The event horizon is the point at which this time to an external observer reaches 0. A mathematical asymptote.
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