Here is a good question about black holes that is often asked, but very rarely answered satisfactorily. I have found a solution that satisfies me. The solution came from a discussion with Reinhard Prix, wherein he corrected a mistaken impression I had about general relativity. (I don’t know if he ever appreciated the philosophical significance of my enlightenment. We’re still discussing it…)
The question: given that, from the point of view of an outside observer, a body falling into a black hole never reaches the event horizon, how can it be said that the black hole has gained the body’s mass?
Before answering the question, it is helpful to discuss its background in more detail.
When Alice falls into the black hole, no outside observer ever observes her crossing the event horizon. Light from Alice (hanging above the event horizon) is visible to the observer forever. Her pocket watch is soon seen to slow almost to a halt, she is red-shifted practically out of view. In principle, however, she is always visible.
From Alice’s point of view, she passes the radius that she previously considered to be the black hole’s event horizon in a short amount of time. She also experiences many other wonderful things, and dies. But those are stories for another time.
Physical statements must always refer to observation.
This principle, championed by the best minds since Newton, distinguishes physics from mathematics. It also subjects physical statements to testing by observation (at least in principle) and thereby distinguishes them from hocus-pocus. Together with the policy of actually testing statements with real observations, it forms the essence of physical science.
On close scrutiny, however, the Alice effect (with some other commonly accepted properties of black holes) seems to be at odds with observation.
To an outside observer, a body such as Alice’s dropped into a black hole never passes the event horizon. So, to that observer, the mass of the body would never be added to that of the black hole.
How then can a black hole ever grow?
A similar problem: if matter in a region of space is almost dense enough to be a black hole, the above Alice effect almost happens, so a body dropped into the region is observed from the outside to reach it only after an immense time, so in effect, the region can never be seen to become a bona fide black hole.
How then can a black hole come to exist at all?
I asked these questions of a lot of people, including professional physicists, without ever getting a satisfactory answer. (But I got an interesting variety of reactions.)
The problem is resolvable, but only with more information and reflection. In the following will appear:
It is easy to make an incorrect conclusion about the Alice effect. That is, because Alice is observed to continue to move (albeit ever slower), she would continue to respond to stimuli from the outside (albeit ever slower).
However, there is a particular time, in the time of an outside observer, beyond which any stimulus sent to Alice never gets a response. A radar wave sent after this point to bounce off Alice never returns at all.
This detail can be seen clearly in spacetime diagrams for the Schwarzschild solution in Eddington-Finklestein coordinates (see especially d’Invierno’s book, p. 220). These coordinates have the neat property of rendering paths of light that head directly into the black hole as straight lines.
To draw these diagrams, two of the spatial coordinates are removed: the diagram below shows time vs. radius from the black hole’s center. Furthermore, some of the details that usually appear in texts are removed for conciseness, and an illustration of a radar signal bouncing off Alice is added.
So although an outside observer always sees light coming from Alice, past this point, the observer can no longer interact with Alice, or perform an experiment with Alice.
This suggests a revision to the notion of observation: if ‘observation’, is taken to mean an optical interaction, then the paradox is resolved. (Earlier in the discussion it was tacitly assumed that ‘observation’, meant an optical viewing.)
In this revised view, the light the observer sees coming from Alice beyond this point is merely an image of the increasingly distant past. Beyond this point, Alice can be said to be no longer in the observer’s interactive universe. The region just outside the event horizon, is inside the observer’s interactive universe—and experiments do not indicate she is there. In this sense, it is reasonable to say she is inside the black hole.
It is interesting that the standard interpretation of quantum mechanics also involves interactive observations. Although classical mechanics and relativity are usually presented in terms of non-interactive observations, in this extreme case in general relativity, non-interactive observation leads to a paradox.
Anil Zenginoglu pointed out to me that there is an aspect of un-knowability involved in Alice’s fall into the black hole: There is no point at which an outside observer can be absolutely certain that on her way down, Alice will not become acquainted with aliens who have a powerful rocket, and, with them, come blasting back out of the black hole.
Of course, the observer can calculate the vanishingly small interval of Alice’s time remaining for her to complete the social graces necessary to procure such a lift, and thereby free themselves from lingering hope.