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I think (pure intuition) that every object that falls towards a massive center, in the condition of free fall, should always be able to return to the starting point. Even if this massive center is a black hole. I always liked this idea.

I also think that this should be a principle or a postulate of gravitational potential energy.

It's said that what crosses the events horizon never returns, but it's also said that for a distant observer, this time of crossing is infinite.

I would like to know:

To a distant observer, can an object that falls toward a black hole always return to the starting point?

Isn' t that a good case for hollow black holes?

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  • $\begingroup$ What do you mean by starting point? $\endgroup$ – JMac Jul 20 at 1:15
  • $\begingroup$ @JMac - Starting point is the place where the free fall trajectory started. $\endgroup$ – João Bosco Jul 20 at 1:19
  • $\begingroup$ What is a “hollow black hole”? $\endgroup$ – G. Smith Jul 20 at 1:25
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    $\begingroup$ Why do you think that pure intuition should provide any insight into black holes? $\endgroup$ – G. Smith Jul 20 at 1:26
  • $\begingroup$ @G.Smith - Hollow black hole in my imagination can be a region of space determined by the Schwarzschild radius that contains nothing inside (not even time or space) if nothing crosses events horizon from the point of view of the distant observer, $\endgroup$ – João Bosco Jul 20 at 1:40
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GR doesn't have observers who can observe things at a distance, so this type of reasoning about "from the point of view of a distant observer" is a bad conceptual trap to fall into. Observers can only receive signals (such as light rays) from distant objects.

Logically connected to this is the fact that GR doesn't have a notion of simultaneity for distant events, so it doesn't make sense to talk about whether an object has passed through the event horizon "now" according to some distant observer.

But basically, the answer to your question is no.

I think (pure intuition) that every object that falls towards a massive center, in the condition of free fall, should always be able to return to the starting point.

This may sound appealing, but it isn't true. A black hole spacetime is divided into an exterior region and an interior region. Once an object has passed into the interior region, it can never get back into the exterior region.

If you want to talk about observers, then suppose the observer knows how to predict the motion of the infalling object, e.g., they know that it started at rest from a certain exterior point and then underwent free fall. Then there is a time on the observer's clock when they know that they will never be able to receive any more signals from the object, if the observer stays outside the horizon. This is the time at which there is no intersection of the following regions: (1) the observer's future light cone, (2) the exterior of the black hole, and (3) the future light cone of the object (inferred because we assume we can predict its motion).

Personally, I find it extremely difficult to reason about this sort of thing unless I draw a type of diagram called a Penrose diagram. I have a simple, nonmathematical explanation of Penrose diagrams in this book: http://www.lightandmatter.com/poets/ . See section 11.5.

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To a distant observer, can an object that falls toward a black hole always return to the starting point?

In the frame of the external observer the infalling observer is always outside the horizon since it takes an infinite amount of coordinate time to fall in. Therefore he could in principle always decide to fly back, given it has an appropriate propulsion system.

In practice it depends on the time it would take the infalling observer to turn on its rocket, since it crosses the horizon in finite proper time, let's say at τ=1, if it hasn't turned on its rocket at, say, τ=0.999, it will not have enough proper time left to turn it on before it is already too late.

In the frame of the outside observer that moment gets stretched infinitely long, but for the infalling object it is a very short period, so if the outside observer sees the infalling object frozen on the horizon without it having it's rocket turned on he will know that the worldline of the object will most probably end in the black hole.

In other words: if the outside observer observes the infalling observer's clock frozen 1 second before τ=1, but knows that it would take 2 seconds of proper time to turn on the engine, he knows that the infalling observer will not make it back.

Isn' t that a good case for hollow black holes?

No, why should it, most infalling matter does not have a rocket attached to it so it freely falls in. In the frame of the outside observer the infalling material asymptotically slows down before it hits the horizon, but the material that was already inside the star before it collapsed is still inside it.

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  • $\begingroup$ In the frame of the external observer No, GR doesn't have global frames of reference. There is no such thing as the frame of a distant observer, observing something falling into a black hole. See physics.stackexchange.com/questions/458854/… $\endgroup$ – Ben Crowell Jul 20 at 12:02
  • $\begingroup$ the infalling observer is always outside the horizon ... Therefore he could in principle always decide to fly back, No, your repeated use of "always" is expressing the concept that there is a notion of simultaneity between the distant observer and the infalling observer. There is no such notion in GR. $\endgroup$ – Ben Crowell Jul 20 at 12:03
  • $\begingroup$ It is a known fact that in the frame of the far away observer the infalling object never crosses the horizon due to gravitational time dilatation, I can quote you a bunch of sources if you don't believe me. There are hypersurfaces of constant t, and if you choose your t in a way that it is the time of the far away observer you can tell that the infalling object is frozen at the horizon. There are of course some caveats when you compare the case with special relativity (although you could also construct problems on Ehrenfest's rotating disc in SR already), but that's another story for itself. $\endgroup$ – Yukterez Jul 20 at 17:21
  • $\begingroup$ The fact that gravitational time dilation is real and not only a question of light travel time can be seen when you place a stationary observer close to the horizon and let him send signals to the external observer, if both are stationary with respect to each other the light travel time is the same for all the signals, so if the received timespan between the signals is larger you know the gravitational time dilation is real (although not symmetric like in the kinematic case, so you have to combine the relative and the absolute components of the effect in GR). $\endgroup$ – Yukterez Jul 20 at 17:44
  • $\begingroup$ In a, admittedly not very realisitic Gedankenexperiment you can also allow the infalling observer to carry a small wormhole or antitelephone which is connected to the external observer, then the outside observer would also see the infalling observer frozen outside of the horizon when he looks through the wormhole, while the infalling observer would see the outside observer in normal time when he looks through the wormhole (in the frame of a radial freefaller infalling with the escape velocity the gravitational and kinematic time dilation component relative to the bookkeeper cancel out). $\endgroup$ – Yukterez Jul 20 at 18:11
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Bodies falling toward a black hole CAN return to their starting point provided they don't venture too close to the black hole. You can watch this happening in a fascinating speeded -up time lapse clip available on the internet, showing a swarm of stars orbiting the supermassive black hole in Sagittarius. The innermost of these stars describes quite small ellipses around the black hole, and it is fascinating to watch it rapidly speed up as it approaches the perigee of its orbit, in close proximity to the black hole, and go zooming off again to slow down as it approaches apogee. If I remember rightly, the sequence was taken over a timespan of 18 years, during which the innermost star performs several orbits of the black hole.

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