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Prof. Legolasov
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In case your ball is infinitely lighter than the black hole, the answer is infinity. You can never be sure it is not coming back. But in reality your ball has a finite mass which can not be neglected. Its mass is to be added to the black hole's mass M, therefore increasing its size. An outside observer will see that his ball got sucked into the black hole approximately when the ball reaches this (increased) gravitational radius.

(Based on my comment above)

UPDATE (CAUTION): This answer actually might turn out to be wrong, as suggested by two (independent) people in the comments below. I don't understand any of their argumentation so far, but there is a real possibility that I am wrong (and in this case I want to understand why). Please do not consider my answer to be an absolute truth until this issue is resolved.

Guys, please try to back up your opinion with some references. Maybe somebody who understands this subject better than me should comment.

P.S. The "increased" gravitational radius should be understood only as an analogy which gives a reasonable approximation to the time $t$ when a ball can be considered to be sucked into the black hole. An actual GR-based calculation here requires a solution of the two-body problem, which, as far as I know, is not solved in GR and can only be calculated numerically.


To @Timaeus and @Nathaniel: I will try to reformulate my answer a little. If our ball is massive, then it must have a gravitational radius. We can think of it as of a small black hole.

As I mentioned above, we do not know how two black holes would interact when their horizons intersect. But when the distance between two black holes is large enough and the mass of the ball is much less than the mass of our astrophysical black hole, we could approximate this behaviour with a ball's geodesic worldline in the gravitational field of the black hole.

BUT: this only works when the distance between the black hole and the ball is large.

So I have proposed a heuristic method of calculating the maximal time that we need to wait in order to determine that our ball does not come back. I take the $r_i$ which is the (unphysical) increased gravitational radius of the total black hole:

$$ r _i > r. $$

And I see this radius as some kind of a threshold on distance. When our ball reaches $r _i$ it already passes the point of no return, since the horizons begin to merge and this process (I believe) is irreversible.

Now Mr. @Timaeus can draw his beloved spacetime diagram and see that it takes finite time for the ball to reach $r _i$, turn on its drive and thrust its way back to the space station.

This is how I arrive at my initial conclustion: it takes finite time to become sure that the ball is not coming back.

As a special case I would consider a massless probe ball which by definition does not increase the gravitational radius: $r _i = r$. In this case the answer becomes infinite.

In case your ball is infinitely lighter than the black hole, the answer is infinity. You can never be sure it is not coming back. But in reality your ball has a finite mass which can not be neglected. Its mass is to be added to the black hole's mass M, therefore increasing its size. An outside observer will see that his ball got sucked into the black hole approximately when the ball reaches this (increased) gravitational radius.

(Based on my comment above)

UPDATE (CAUTION): This answer actually might turn out to be wrong, as suggested by two (independent) people in the comments below. I don't understand any of their argumentation so far, but there is a real possibility that I am wrong (and in this case I want to understand why). Please do not consider my answer to be an absolute truth until this issue is resolved.

Guys, please try to back up your opinion with some references. Maybe somebody who understands this subject better than me should comment.

P.S. The "increased" gravitational radius should be understood only as an analogy which gives a reasonable approximation to the time $t$ when a ball can be considered to be sucked into the black hole. An actual GR-based calculation here requires a solution of the two-body problem, which, as far as I know, is not solved in GR and can only be calculated numerically.

In case your ball is infinitely lighter than the black hole, the answer is infinity. You can never be sure it is not coming back. But in reality your ball has a finite mass which can not be neglected. Its mass is to be added to the black hole's mass M, therefore increasing its size. An outside observer will see that his ball got sucked into the black hole approximately when the ball reaches this (increased) gravitational radius.

(Based on my comment above)

UPDATE (CAUTION): This answer actually might turn out to be wrong, as suggested by two (independent) people in the comments below. I don't understand any of their argumentation so far, but there is a real possibility that I am wrong (and in this case I want to understand why). Please do not consider my answer to be an absolute truth until this issue is resolved.

Guys, please try to back up your opinion with some references. Maybe somebody who understands this subject better than me should comment.

P.S. The "increased" gravitational radius should be understood only as an analogy which gives a reasonable approximation to the time $t$ when a ball can be considered to be sucked into the black hole. An actual GR-based calculation here requires a solution of the two-body problem, which, as far as I know, is not solved in GR and can only be calculated numerically.


To @Timaeus and @Nathaniel: I will try to reformulate my answer a little. If our ball is massive, then it must have a gravitational radius. We can think of it as of a small black hole.

As I mentioned above, we do not know how two black holes would interact when their horizons intersect. But when the distance between two black holes is large enough and the mass of the ball is much less than the mass of our astrophysical black hole, we could approximate this behaviour with a ball's geodesic worldline in the gravitational field of the black hole.

BUT: this only works when the distance between the black hole and the ball is large.

So I have proposed a heuristic method of calculating the maximal time that we need to wait in order to determine that our ball does not come back. I take the $r_i$ which is the (unphysical) increased gravitational radius of the total black hole:

$$ r _i > r. $$

And I see this radius as some kind of a threshold on distance. When our ball reaches $r _i$ it already passes the point of no return, since the horizons begin to merge and this process (I believe) is irreversible.

Now Mr. @Timaeus can draw his beloved spacetime diagram and see that it takes finite time for the ball to reach $r _i$, turn on its drive and thrust its way back to the space station.

This is how I arrive at my initial conclustion: it takes finite time to become sure that the ball is not coming back.

As a special case I would consider a massless probe ball which by definition does not increase the gravitational radius: $r _i = r$. In this case the answer becomes infinite.

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Prof. Legolasov
  • 16.3k
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In case your ball is infinitely lighter than the black hole, the answer is infinity. You can never be sure it is not coming back. But in reality your ball has a finite mass which can not be neglected. Its mass is to be added to the black hole's mass M, therefore increasing its size. An outside observer will see that his ball got sucked into the black hole approximately when the ball reaches this (increased) gravitational radius.

(Based on my comment above)

UPDATE (CAUTION): This answer actually might turn out to be wrong, as suggested by two (independent) people in the comments below. I don't understand any of their argumentation so far, but there is a real possibility that I am wrong (and in this case I want to understand why). Please do not consider my answer to be an absolute truth until this issue is resolved.

Guys, please try to back up your opinion with some references. Maybe somebody who understands this subject better than me should comment.

P.S. The "increased" gravitational radius should be understood only as an analogy which gives a reasonable approximation to the time $t$ when a ball can be considered to be sucked into the black hole. An actual GR-based calculation here requires a solution of the two-body problem, which, as far as I know, is not solved in GR and can only be calculated numerically.

In case your ball is infinitely lighter than the black hole, the answer is infinity. You can never be sure it is not coming back. But in reality your ball has a finite mass which can not be neglected. Its mass is to be added to the black hole's mass M, therefore increasing its size. An outside observer will see that his ball got sucked into the black hole approximately when the ball reaches this (increased) gravitational radius.

(Based on my comment above)

In case your ball is infinitely lighter than the black hole, the answer is infinity. You can never be sure it is not coming back. But in reality your ball has a finite mass which can not be neglected. Its mass is to be added to the black hole's mass M, therefore increasing its size. An outside observer will see that his ball got sucked into the black hole approximately when the ball reaches this (increased) gravitational radius.

(Based on my comment above)

UPDATE (CAUTION): This answer actually might turn out to be wrong, as suggested by two (independent) people in the comments below. I don't understand any of their argumentation so far, but there is a real possibility that I am wrong (and in this case I want to understand why). Please do not consider my answer to be an absolute truth until this issue is resolved.

Guys, please try to back up your opinion with some references. Maybe somebody who understands this subject better than me should comment.

P.S. The "increased" gravitational radius should be understood only as an analogy which gives a reasonable approximation to the time $t$ when a ball can be considered to be sucked into the black hole. An actual GR-based calculation here requires a solution of the two-body problem, which, as far as I know, is not solved in GR and can only be calculated numerically.

Source Link
Prof. Legolasov
  • 16.3k
  • 2
  • 33
  • 70

In case your ball is infinitely lighter than the black hole, the answer is infinity. You can never be sure it is not coming back. But in reality your ball has a finite mass which can not be neglected. Its mass is to be added to the black hole's mass M, therefore increasing its size. An outside observer will see that his ball got sucked into the black hole approximately when the ball reaches this (increased) gravitational radius.

(Based on my comment above)