Does a black hole have to have a point-like singularity at its core?

[Asympotically_fat]

@wolfpup

What the poster is struggling with, as am I, is some kind of intuitive explanation for the apparent paradox between the “frozen in time” consequence of infinite time dilation and the fact that according to everything we know both empirically and theoretically, matter does fall into black holes in a pretty straightforward Newtonian fashion, creating powerful accretion disks in the process, and black holes become more massive accordingly.

My attempts to gain an intuitive non-mathematical insight into this apparent paradox may or may not be sound, so I invite appropriate criticism. They have centered around the idea that this apparent freezing due to infinite time dilation is merely an artifact that arises from the use of inappropriate coordinates and implicit but incorrect assumptions about simultaneity. The Schwarzschild solution to the Einstein field equations for a non-rotating black hole (and the equivalent Kerr solution for a rotating one) are valid as one approaches the event horizon and can be used to show the aforementioned switching of the time dimension with the radial spatial dimension beyond the EH, but it’s ill-behaved at the EH itself, where it blows up and appears to show the time dilation factor tending to infinity as the distance to the EH tends to zero. However under the appropriate geometries and coordinate systems the spacetime curvature at the EH can be shown to be well-behaved and finite.

There seem to be multiple different ways of trying to grasp this intuitively. One way is that from the perspective of an external observer, time dilation really does freeze the infalling object – in space. But the Schwarzchild coordinate system that leads to this conclusion also leads to the conclusion that space itself is rapidly flowing into the black hole, exceeding the speed of light beyond the EH (which space is allowed to do, as distinct from objects in it), where time and the spatial dimension pointing to the singularity have correspondingly switched places and the singularity becomes the object’s future.

At no point would you actually see the intrepid astronaut frozen forever at the EH (he’d be red-shifted out of detectability anyway), but more importantly, I’m suggesting that the idea that the astronaut would forever be at rest at the EH as seen by an external observer seems to stem from a misapplication of the Schwarzchild solution.

The easiest way to see is from a Penrose diagram like the below:

We can ignore the bottom (white hole interior) and left (parallel universe) regions of the diagram(s) as they are features of the extended solution.

In a Penrose diagram light always has worldlines at 45 degree angles to the vertical axis, and the worldlines of objects moving at less than the speed of light are always at an angle that is less than 45 degrees to the vertical axis.

It should be easy to see that nowhere on the BH singularity (upper zig-zag) is causally connected to the whole of the BH exterior region (right region), which means an observer hitting the singularity cannot see the entire future of the Universe played out before they hit the singularity.

The misapprehension that you are able see the entire future of the Universe before you hit the singularity comes from taking Schwarzschild coordinates (the dotted lines in the BH exterior region. NB as illustrated the lines do not mark out equal amounts of coordinate time or distance) too literally. Schwarzschild coordinates map all the events that lie on the event horizon to the same time coordinate* as timelike future infinity (i[sup]+[/sup]), to which all events in BH exterior region are causally connected and in the past (therefore if you make it to i[sup]+[/sup] you are able to see every event that happened in the BH exterior region).

*Strictly speaking they don’t because the Schwarzschild time coordinate goes to infinity at both the event horizon and at i[sup]+[/sup], so the problem is more from making faulty assumptions by incorrectly taking limits when the coordinates are singular.