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Q & A: Zeno at the edge of a black hole

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Most recent answer: 07/10/2011
Q:
In a conversation about black holes with a relative who is a physicist, I understood him to say something like - due to the immense gravity everything slows down and never actually gets to the center of the black hole. If I heard/understood him correctly, this reminded me of Zeno's puzzle, the Achilles. I know that the Greeks did not understand infinity or zero and this puzzle has been long solved. So, I was wondering is there any relation between the two problems mathematically or logically (are physicists missing the concept of zero in a way). Gary
- Gary (age 52)
Belmont, MA, USA
A:
That's a tough question. Let me back up a little to bring other readers on board. Zeno said that if you go halfway, then half of the remaining half, and so on, you'll never get where you're going. The problem with that argument is that it doesn't take into account that each of those smaller steps takes less time. So if you add say (1/2 min + 1/4 min+ 1/8 min....) you end up only taking a minute to go the whole way. 

As a black hole starts to form, you do get a related problem. You can't get away with saying "things are flying toward the horizon of the black hole at such-and-such a rate" to calculate how long it takes to fall in. We have to divide up the time into little slices because of a peculiar effect. As something falls down in gravity, its clocks (and all other measures of local time) slow down, from the point of view of those of us far away. Einstein predicted that effect via very sound arguments, and we measure it frequently. GPS systems would be badly mis-calibrated if they didn't take into account that the satellite clocks (up in the gravitational field) run fast. So in figuring how long it takes to fall into the hole, you have to multiply the time interval of each little time slice by bigger and bigger numbers to allow for that slowing down.

As the object reaches the black hole's horizon (not even the center), the multiplication factor become infinite. Now it's true that some functions that go infinite still have finite integrals (e.g. 1/sqrt(x) near x=0) but not all of them do. This one (the time to fall, as seen by us) doesn't. So it takes something infinitely long to fall in, as seen by us. It turns out that for the black hole even to form, with the process getting slower as it gets closer to forming, also requires an infinite amount of time.

So far, really nothing I said should be a problem. The picture I've given is maybe even less strange than the black hole picture you started with. Now let's look at the same process from the point of view of somebody falling in. They see the black hole form, and the fall into it. They know when they cross the horizon because, looking back, they see our clocks speed up infinitely. The entire future of an increasingly narrow slice of our universe passes before their eyes, if they happen to be looking back.

So they say the hole forms and they fall into it. We say that the hole never even forms. In fact, we say that the stuff that was falling toward the hole ultimately evaporates.

This may strike you as a bit strange.

Mike W.

(published on 07/10/2011)

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