That is at odds with our previous blog post where a collapse will slow down in the global Schwarzschild coordinates, and a "frozen star" will be the end result.
The mathematical question is if there exists a manifold with a reasonable metric which, furthermore, fulfills the Einstein equation.
If the star would continue collapsing after an infinite time of the global Schwarzschild time has passed, is it really possible to extend the spacetime manifold in such a way? Does the manifold after that just contain the internals of the star, or could it also contain the exterior space? What would it mean that the exterior, asymptotically a Minkowski space, would exist after an infinite time has passed?
This reminds us of nonstandard models of arithmetic, where an outside observer sees the ordering of natural numbers as N + Z + Z ..., where N denotes the usual ordering and Z denotes the ordering of negative and positive integers. For an outside observer, natural numbers continue after an infinite number of them have been counted.
If the manifold after an infinite global time just contains the internals of the star, is that a reasonable solution? Why did the internals of the star survive an infinite global time but the surrounding Minkowski space did not?
An astronaut falling into the forming black hole will pass the event horizon at the infinite global Schwarzschild time. What if, at the same time, a ray of light was moving outward from inside the star, and the astronaut meets that ray of light at the infinite time? What is the path of the light after that? It cannot go to the non-existent surrounding space. Does it turn back? But would the sharp turn violate the principle that the metric has to be smooth?
There is no principle in general relativity which forces the proper time of a particle to move forward infinitely long. The proper time of a photon does not progress at all. Why should we require that the proper time of a falling astronaut should extend past the moment where he meets the event horizon? Penrose and Hawking should present grounds for extending the proper time of the astronaut.
A collapsing star and the Kruskal-Szekeres coordinates
Let us use the Kruskal-Szekeres coordinates which have been extended to include the interior of an event horizon. The coordinates come with the Schwarzschild metric outside the star. Inside the star, we have a metric which is between a flat Minkowski space and the Schwarzschild metric.
Let us assume that a spherically symmetric star collapses. Let us assume that the last layer of atoms passes the Schwarzschild radius at an event (spacetime point) x and no interior layer of atoms has yet passed its respective Schwarzschild radius. That is, the event horizon forms at x.
Suppose that a ray of light is rising upwards from the star and comes to x. Since the Schwarzschild metric inside the event horizon dictates that every ray of light goes towards the center, our ray must turn abruptly and start descending back into the collapsing star.
Does that mean the metric is not smooth at x? In a smooth metric, all rays of light should go along smooth paths and there should be no abrupt turns.
Is it possible to glue together a Schwarzscild metric and the interior metric without introducing singularities?
Folklore says that, using the coordinates of a freely falling observer, the spacetime is smooth at the event horizon. Suppose that x happens at the center of a falling spaceship. People in the spaceship will not observe any sharp turn of the ray of light. Rather, they will think the ray is still going upward after x.
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