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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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3 Answers3

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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 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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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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