From an outsider's perspective, how can a black hole grow if nothing ever crosses the event horizon?

Due to time dilation, an outside observer never sees a falling object actually cross the event horizon. I'm not referring to the optical illusion of red-shifted light making objects appear to fade away before crossing the event horizon. I'm referring to the very real effect time dilation has on objects near the event horizon, causing them to slow down to a halt from the frame of reference of an outside observer.

This outside observer can see, however, that the black hole has a non-zero mass and volume. How can this be if from the frame of reference of the outside observer, nothing has ever crossed the event horizon and been integrated into the black hole?

In other words, how can an outside observer say that a black hole has a non-zero mass and volume, but also, nothing has ever crossed its event horizon and contributed to said mass and volume?

First off, there's no real time dilation effect. In coordinate systems that cover the event horizon (Kruskal-Szekeres, Eddington-Finkelstein, Gullstrand-Painlevé, Kerr-Schild), objects fall through it without so much as a hiccup. The only real slowing-down effect is from redshift (as seen from outside).

Suppose you watch from the side as someone lobs a basketball into a black hole, against a background of stars. The basketball has mass, so even when it's far from the black hole, it gravitationally lenses the stars behind it. As it approaches the black hole, that lensing will overlap/merge with the lensing of the black hole, so the shape of the black hole will seem to have a slight bulge in the direction of the basketball. As more time passes, the basketball will appear to flatten against the horizon of the black hole (while redshifting into invisibility), and the bulge will spread around so that the combined system appears spherical again, but with a slightly larger mass.

Technically, you never see the object fall through and you never see a "true" black hole, but it makes no practical difference, because the difference between what you see and the ideal limit decreases exponentially with time. The time constant is comparable to the light crossing time of the black hole, which is around 1 day for a huge black hole like M87's, and less than a millisecond for a stellar-mass black hole. So after even one year, never mind billions, you'll have no hope of distinguishing the frozen star from a "true" black hole.

• So, purely theoretically, from the perspective of a faraway observer, any black hole that formed a finite time ago is just a bunch of highly redshifted objects spread very close to the event horizon, but not under it. Right? Sep 30, 2023 at 14:29
• @Ruslan That's pretty much the take away I'm getting as well.
– user52416
Sep 30, 2023 at 15:57
• @Ruslan: I believe this is one proposed solution to the black hole information paradox but it's not known for certain to be true. Oct 1, 2023 at 12:16
• Actually, now I realize that as the conglomerate of the objects gets more mass, its Schwarzschild radius grows, so newer objects will appear on larger distances from the center. Then I suppose the whole system should be filled with matter during its growth, not just have a thin layer above the event horizon. Oct 1, 2023 at 20:57
• @benrg I would like to shift to the point of reference of the infalling object. I keep hearing conflicting information about whether or not the universe speeds up behind them; again, not referring to optical illusions, but real effects of time dilation. If it does, do they actually experience crossing the event horizon before the end of time? If it doesn't, wouldn't that conflict with the events experienced by the outside observer?
– user52416
Oct 2, 2023 at 13:08

Due to time dilation, an outside observer never sees a falling object actually cross the event horizon.

This is correct, but perhaps not quite in the way that you visualize it.

It's trivially true, in a literal sense, that an outside observer can never see anything cross the event horizon, since this would require seeing inside the event horizon, and no light (or other information) can escape from there to the outside. The most the outside observer could hope to see, even in principle, would be the object hitting the event horizon and disappearing (but they can't even quite see that, for the reasons discussed below).

It's also true that, the closer the falling object gets to the event horizon, the longer it will take for any light it emits to get away from the black hole (and the less of it will make it out at all, and the more redshifted the light that does escape will be). In fact, as the object approaches the event horizon, the time it takes for light emitted by it to escape the black hole tends to infinity.

However, trying to interpret and visualize this as:

[an] effect time dilation has on objects near the event horizon, causing them to slow down to a halt from the frame of reference of an outside observer

is probably not the best way to think about it. A better way of looking at what's happening is that time dilation is causing light (and other signals) to take longer to escape the black hole, and thus the outside observer is seeing things that happened a longer time ago in the past.

In other words, the object falling into the black hole isn't slowing down. As benrg notes, it actually speeds up quickly and falls through the event horizon in finite time. And in doing so it adds mass to the black hole and causes the event horizon to grow bigger.

All that's slowing down (from the outside observer's perspective) is the light that the falling object emits as it's falling in. So the outside observer can in fact "see" the black hole form and grow, at least as much as it's meaningful to speak of seeing a black hole at all. (They obviously cannot receive any light or other information directly from the black hole itself. All they can observe is the infalling mass and the extreme spacetime curvature and its associated phenomena that are consistent with there being a black hole in the middle.) But the outside observer is also seeing (very small amounts of very strongly redshifted) light emitted by objects that fell into the black hole a very long time ago, including some emitted by objects that fell in when the black hole (and specifically its event horizon) was smaller than it is now.

Interpreting these observations in a particular way, the outside observer could claim that they're seeing light coming from inside the black hole's current event horizon (as calculated based on indirect observations). But there's no actual contradiction there, since that light was emitted back when the event horizon didn't extend that far yet!

Of course, in practice, all of this is mostly an exercise in pedantry and deliberately quirky interpretation of observations. What a real outside observer actually sees, as you correctly note, is objects fading away as they fall into the black hole, and the approximate place where they fade away (accounting for the massive optical distortion around the black hole) being approximately where they can calculate that the event horizon should be.

All the rest is just theory, and not particularly fruitful theory either. Sure, you can say that in theory you could sit in a spacecraft orbiting a black hole and see light emitted by an object that fell into it a million years ago, because it's theoretically possible for a photon emitted by the object very close to the event horizon to take a million years to reach you. But in practice you just can't, no matter what kind of incredible sensor technology you may postulate you'd have, because the probability that you'll ever detect a photon emitted during that vanishingly small portion of the object's descent is for all practical purposes zero.

• Why we say the objects become "invisible" due to redshift? It's not just a matter of using a different sensor (i.e. IR) instead of our eyes limited to the visible range?
– Mark
Oct 1, 2023 at 7:20
• @Mark Redshift shifts the frequencies of individual photons, but it also shifts the rate at which you can detect them. When the detection rate is too low, you can no longer form an image. Oct 1, 2023 at 20:06
• So it's similar to the doppler effect in the sense that the passing car's real engine sound (= falling object's real speed) doesn't change but the observer does hear a changing sound (= seeing the object falling slower), and eventually the engine sound just slowly fades away (as does your view on the falling object)? Oct 2, 2023 at 3:26
• " But in practice you just can't, no matter what kind of incredible sensor technology you may postulate you'd have, because the probability that you'll ever detect a photon emitted during that vanishingly small portion of the object's descent is for all practical purposes zero." This is possibly still under selling how fast this happens! (For stellar mass black holes), the expected number of photons emitted by the falling object for the rest of eternity drops below zero in a fraction of a second, and continues dropping exponentially. Oct 2, 2023 at 6:41

Objects that approach the event horizon(from an outsider's perspective),will indeed appear to slow down and become increasingly redshifted as they get closer. As a result, it will take an infinite amount of time (as perceived by the outside observer) for an object to cross the event horizon. This is the essence of the "no hair" theorem for black holes, which states that all the information about an object that falls into a black hole is seemingly lost to an outside observer.

However, from the infalling object's perspective itself, it will indeed cross the event horizon in a finite amount of time. For the infalling object, its journey toward the singularity continues, even though it is never visible to the outside observer after crossing the event horizon.

So, while the outside observer sees objects near the event horizon slowing down and never crossing it (due to time dilation and redshift), the objects themselves do cross the event horizon and contribute to the mass and volume of the black hole as they move closer to the central singularity.

• Would time speed up behind the infalling object before crossing the event horizon due to time dilation?
– user52416
Oct 2, 2023 at 13:12