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This article https://www.quantamagazine.org/why-black-hole-interiors-grow-forever-20181206/ describes how black hole interiors grow forever although the exterior volume of the black hole does not appear to change.

The article imagines a black hole as a funnel, where "The funnel gets deeper and deeper, so that infalling things never quite reach the mysterious singularity at the bottom."

I'm trying to reconcile this with my understanding that an observer who enters the event horizon will have all its future world-lines converge at the singularity. As I understand, inside the event horizon space becomes like a waterfall, drawn downwards faster than the speed of light, so that any observer will be inexorably drawn towards the singularity.

Under that reasoning it seems like an observer would never actually come closer to the singularity because all space in front of it is also falling.

This is just an intuitive picture, and I fully expect this intuition should break down.

Do observers ever reach the singularity? Or does the eternally growing interior of the black hold prevent this from happening? If so, how do we reconcile this with the fact that all the observer's world-lines must converge at the singularity?

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The article is simply wrong when it says "The funnel gets deeper and deeper, so that infalling things never quite reach the mysterious singularity at the bottom." In reality, the time from crossing the horizon to reaching the singularity is largely independent of when you cross the horizon.

I'll try to explain intuitively where this idea of ever-increasing volume comes from, but this may be tough going if you aren't familiar with special or general relativity.

Suppose you're sitting outside the event horizon of a black hole (never mind how you avoid falling in; perhaps you're in a rocket ship, or perhaps there's a Dyson sphere supporting you or something). You toss an object of negligible mass into the hole, then you wait a short time, then you toss a second object in (an exact duplicate of the first toss, but at a later time).

The spacetime events of the two tosses are "timelike separated", meaning one happens objectively after the other. The events of the two objects reaching any other (smaller) distance from the event horizon are likewise timelike separated. If you plot the objects' worldlines (paths in spacetime), they have the same shape, because you threw the objects identically, but they are offset from each other in time.

That's true before they cross the event horizon. The worldlines after they cross the horizon are still the same shape, but they are spacelike separated, meaning they are "side by side" with neither one objectively earlier or later in time than the other.

You can extend this to a large number of people located all around the hole (all at the same height) continually tossing objects into it. The events of the tosses now form a cylinder in spacetime, or rather a "hypercylinder" whose cross section is a sphere instead of a circle. The lengthwise direction on this hypercylinder is the time direction. The spacetime locations of the tossed objects shortly after they cross the horizon also form a cylinder, but now the lengthwise direction is a spatial direction: all of the tossed objects are "side by side" on this cylinder and they all head for the singularity "in parallel" in some sense.

If you start tossing the objects shortly after the hole initially forms from collapsing matter, then the longer you keep doing it, the longer this internal cylinder ends up being. That's essentially why they say that the "interior volume" increases with time.

This doesn't mean that objects tossed in later are farther away from the singularity. All of the objects are almost exactly the same distance from the singularity. If they were stopwatches that started ticking at 0 when they were tossed, they'd all show almost exactly the same elapsed time just before hitting the singularity. The cylinder is not a space through which they have to pass to get to the singularity. It isn't even a space they could in principle explore, if they were equipped with thrusters; they can only reach a portion of it in their limited time before hitting the singularity, and the size of that portion is almost exactly independent of when they were tossed in.

I think the interest in growing cylindrical black hole interiors started with "How big is a black hole?" by Christodoulou and Rovelli arXiv:1411.2854. Technically, what they did is define the interior volume to be the maximum volume of any surface bounded by a sphere outside the hole. That maximal surface ends up having approximately the same shape as the cylinder I defined above.

Personally I find it very difficult to take this whole thing seriously, because the expanding cylinder appears to have no physical significance whatsoever. In every measurable sense, black holes are extremely stable objects. They don't get larger in any measurable sense when they aren't eating matter. This includes even measurements performed inside the event horizon.

The maximal-surface definition of volume fails to give a sensible result in cosmology. The universe appears to be spatially flat, meaning the volume of a region bounded by a sphere of surface area $4πr^2$ is most naturally taken to be $\frac43 πr^3$. For example if you want to estimate the number of galaxies inside the sphere, you multiply the average density of galaxies by that volume. But the maximal spacetime volume enclosed by the sphere is not that spatially flat region; it's a curved surface with a larger volume that has no physical interpretation that I know of.

Christodoulou and Rovelli argue that the maximal-surface volume is a natural notion of enclosed volume, but this seems to be based entirely on the fact that it behaves reasonably in the very simplest case (flat spacetime). It already fails to behave reasonably in FLRW cosmology, which is almost as simple, and it fails spectacularly in the case of a black hole.

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  • $\begingroup$ "The lengthwise direction on this hypercylinder is the time direction" - By the exterior clock, which in no frame can show time larger than the age of the universe. So, as the universe gets older, the black hole interior becomes larger. While the $t$ coordinate is spatial there, it is limited by the age of the universe. Essentially the same idea as in your answer, just a bit differently phrased +1 $\endgroup$
    – safesphere
    Dec 4, 2020 at 8:35
  • $\begingroup$ @ChiralAnomaly Yeah, the question definitely shouldn't have been closed. All I meant by spacelike separation of the worldlines is that if you plot them in ingoing coordinates they're related by $t$ translation, which is spatial inside the horizon. That's true only if they don't do something to break the symmetry like accelerating to meet each other. The fact that the interior is spatially homogeneous (at late times) is a global property, but it's relevant because it means that all objects see an isomorphic region of the interior, not a region whose size depends on when they fell in. $\endgroup$
    – benrg
    Dec 4, 2020 at 20:40
  • $\begingroup$ @benrg Ok, now I understand what you meant about the worldlines. I deleted my previous comments because they're obsolete. $\endgroup$
    – Chiral Anomaly
    Dec 5, 2020 at 0:12

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