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On the one hand, black holes are said to be able to grow by acquiring mass. In order to see the event horizon’s radius increase, you first have to see mass fall into it.

But on the other hand, I’m led to believe that you can never see any mass cross the event horizon; it will instead look like it gets asymptotically closer but never reaches it.

It would seem that the effect (black hole increasing in size) must come before the cause (mass falling in).

What resolves this apparent contradiction? How can an outside observer observe a black hole growing?

Timwi
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The problem is this statement: "In order to see the event horizon’s radius increase, you first have to see mass fall into it." The truth is that you can never see a black hole's event horizon from the outside, because it is always in the future. A black hole twists time and space round so that on the event horizon the whole of an observer's future time (his future light cone) points into the hole. You can no more see the event horizon than you can see tomorrow. What you can see from the outside is the effect of the black hole's gravity on light rays passing through the space surrounding the event horizon, in the past.

When you look directly towards a black hole, the rays of light you see come from the distant past, from all the way back to when the black hole was formed. It only looks black because a few seconds of outgoing light from the last few seconds of an object's fall gets stretched out over millions of years. It has been struggling to climb out of the gravity well for all this time. If you spread the energy from a few seconds of light rays out over millions of years, it looks dark.

When mass falls into the hole, the hole gets bigger, the gravity outside the hole gets stronger, and there is more distortion of passing light rays further out from the hole. The region where light rays get distorted appears bigger. And the 'black circle' you see in the middle (containing spread out light from the distant past) gets bigger. But you are not seeing the horizon itself - only light that came uncomfortably close and managed to escape.

Nullius in Verba
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    I think your answer needlessly confuses the notion of a black hole with what you can see. It's the event horizon that fundamentally defines a black hole. – StephenG - Help Ukraine Oct 02 '22 at 16:14
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    But the questioner is asking about what you can see. The contradiction claimed is between needing to see mass fall into the event horizon for it to get bigger, and never being able to see things fall into the event horizon because they appear to slow down as they approach. They can't both be true. The question is which statement is wrong and why? – Nullius in Verba Oct 02 '22 at 16:29
  • Thank you for your answer. May I ask for clarification: Are you saying that it is possible to see the black hole grow (i.e.: see its gravitational effect get stronger) while still seeing an object “stuck” near the event horizon? And we are supposed to assume that the stuck object is just a remnant of the light it gave off, but the object itself can be safely assumed to have crossed the event horizon even if we don’t see that happening? – Timwi Oct 03 '22 at 19:11
  • Note that “never in the past” does not equate “always in the future”. Parts of the event horizon will have spacelike separation from a typical observer. – TimRias Oct 03 '22 at 21:34
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I will suggest a method of thinking about this which is not very rigorous but is easy to grasp.

The Schwarzchild Radius is given by :

$$R_S=\frac{2GM} {c^2}$$

Now as the mass falling in approaches the black hole's event horizon (before reaching $R_S$) it will reach a point where the total mass of the black hole and the falling mass is such that a new event horizon is formed at :

$$R_S+\delta r=\frac {2G(M+\delta m)}{c^2}$$

That is the new event horizon forms by "reaching out" to the falling matter as they merge.

So the falling matter never has to cross the original event horizon, it simply forms part of a new body with a larger event horizon and only needs to get close enough to do that. We don't have to wait for an infinite time, it happens in a finite time.

StephenG - Help Ukraine
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  • So the outside observer does eventually see the mass “fall in” by seeing it swallowed up by the growing event horizon? So the thing about asymptotic approach isn’t actually correct because it assumes a constant event horizon? So in reality all mass is eventually observed to be swallowed up by the event horizon? – Timwi Oct 02 '22 at 14:50
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    Keep in mind that the Schwarzchild model of a black hole is a very special case of a solution to the Einstein Field Equations that has no spin, no charge and is considered to always have existed and does not change. Real models of black holes are quite a lot more complex and I'm just offering a simplistic model for intuitive purposes only. They do grow and we see them grow (e.g. black hole mergers have been detected). – StephenG - Help Ukraine Oct 02 '22 at 14:58
  • Thank you for the explanation!! – Timwi Oct 02 '22 at 15:07
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    You will also never see the object cross the new horizon. – TimRias Oct 02 '22 at 16:14
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But on the other hand, I’m led to believe that you can never see any mass cross the event horizon; it will instead look like it gets asymptotically closer but never reaches it.

Yes, it is possible. You just need really big masses involved for that. In 2015, astronomers were witnesses of such event for the first time, see First evidence of black holes merger. Since then they detected about 100 such mergers within the LIGO gravitational wave observatory. The GW190521, three years ago, was the most heavy one.

JanG
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