To a distant observer, is the trajectory of an object falling toward a black hole always reversible?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.