zonde said:
It's sometimes such a challenge to talk with you PeterDonis. You can turn on it's head such a simple thing that I am at loss how to explain your mistake.
Perhaps I'm misunderstanding the question you're asking. Let me step back for a bit and try to describe things without phrasing it as an answer to a specific question.
The maximally extended Schwarzschild spacetime has a total of *four* regions. The best way to globally visualize this spacetime and its four regions is using a Kruskal diagram, as seen for example here:
http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates
Region I is the "normal" part of spacetime we're used to, the exterior region that's outside any horizons. Region II is the interior of the black hole. Region III is a *second* exterior region; and region IV is the interior of the white hole.
The standard "river model" covers regions I and II; that is, it views space as flowing inwards towards the black hole. However, note that in this model, there is *no* white hole. More precisely, the white hole portion of the maximally extended spacetime, region IV, is not covered by the standard river model; so it makes no sense within that model to talk about objects falling towards the white hole. Anything that falls inward will eventually fall into the *black* hole, region II.
There is also a second possible "river model", which is obtained by using outgoing Painleve coordinates instead of ingoing Painleve coordinates. This second "river model" covers regions IV and I; that is, it views space as flowing outwards from the white hole. In this model, we can talk about objects moving towards the white hole; but they can't possibly reach the white hole because its horizon is moving inwards at the speed of light.
Now if we look at the full extended spacetime, as shown on the Kruskal diagram, we can see that an observer in region I can move inward, at speeds approaching the speed of light; this corresponds to moving on a worldline that is tilted to the left at an angle approaching 45 degrees. Such an observer, if he were way down in the lower right corner of the diagram, might want to think of himself as moving towards the white hole. However, he will never reach the white hole; he will never reach region IV. Instead, he will eventually cross the black hole horizon and enter region II.
Also, if we look at the full extended spacetime, we can see that there are timelike worldlines that leave region IV, enter region I, and then leave region I and enter region II. Some of these worldlines will be geodesics, i.e., the worldlines of freely falling objects. (The Wikipedia page doesn't show any of these worldlines, but some of the figures in MTW do.) We can use either one of the two "river models" to describe what happens to objects that follow these worldlines:
- The standard river model will view the object as rising away from the black hole (like a ball thrown upwards), coming to rest, then falling back in and entering the black hole; but this model can't show where the object ultimately came from, because it ultimately came from the white hole, and the white hole isn't covered by the standard river model.
- The second river model will view the object as coming out of the white hole, rising upwards, coming to rest, then falling back down; but this model can't show where the object ultimately goes to, because it ultimately goes into the black hole, and the black hole isn't covered by the second river model.
But note that in *both* cases, the object starts by moving upward, then comes to rest, then falls back down; this shows that gravity is attractive throughout the spacetime. There is no region where anything is "repelled" by either the white hole or the black hole. Furthermore, the change in the object's motion, since it is freely falling, is entirely due to the change in river velocity along its trajectory; this is true regardless of which river model you use to describe its motion. This is why I said there is no "additional" acceleration, over and above that produced by the river.
(Remember that even though the second river model has the river flowing outwards, its velocity decreases as you go outwards. An object that comes to rest at a finite height is moving at *less* than the Newtonian "escape velocity", so it is moving *inward* relative to the river.)
Does this help any?
[Edit: I should probably also add that there are other worldlines in the maximally extended spacetime that are also relevant:
- There is a set of worldlines that starts from spatial infinity in the infinite past, and falls inward at exactly the Newtonian "escape velocity". This set of worldlines covers regions I and II, and these worldlines are used to construct the frame field of ingoing Painleve observers, which underlies the standard river model.
- There is a set of worldlines that starts at the white hole singularity and moves outward at exactly the Newtonian "escape velocity", eventually reaching spatial infinity in the infinite future. This set of worldlines covers regions IV and I, and these worldlines are used to construct the frame field of outgoing Painleve observers, which underlies the second river model.
It's a good exercise to work through how observers following these worldlines would describe the objects following the worldlines I described above, the ones that rise upward, come to rest, and then fall back in. This may help to reduce some of the confusion.]
