lgw wrote:Thanks for the responses everyone!

Let me explain more what motivates this thought experiment: I'm poking at the oddity that GR requires that being inside a distant uniform shell of matter can have an extreme effect on local conditions, in stark contrast to classical expectation (after all, classically that would have 0 effect).

Why do you say that? As far as I know, if the observer inside the shell can be considered to have negligible mass, then from what I understanding spacetime is flat inside a spherical shell, see

here. It is true that if you depart from a friend far outside the shell, enter the shell, hang out inside it for a while and then exit and return to your friend, you will have aged less than them, but I don't think this is really an "effect on local conditions" inside the shell, more a matter of how the flat region inside the shell is linked up to the approximately flat region far outside the shell.

lgw wrote:I'm trying to see how to reconcile these, and in fact it doesn't seem far-fetched to me from a GR perspective. As you approach the horizon, the radial and time axes are "twisting" as you move in, an effect that increases steadily as you get close.

That "twisting" is purely a matter of the coordinate system you choose, there is no such twisting effect in the Kruskal-Szekeres coordinate system I mentioned.

lgw wrote:But once past the horizon, now what? With no hyperdense core in the formation of the black hole, it seems reasonable that the situation would actually be uniform inside. The "twist" of the axes has completed, and you're inside a closed universe now.

You can choose surfaces of simultaneity (3D slices through 4D spacetime that represent space at a particular "instant" in some coordinate system, see

relativity of simultaneity if you're not familiar with the idea that simultaneity depends on the choice of coordinate system) such that the spatial curvature is indeed uniform inside the horizon horizon. Just look at the dotted line "F" in the diagram below from p. 528 of the textbook

Gravitation by Misner/Thorne/Wheeler, drawn over the [url=http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates#Qualitative_features_of_the_Kruskal.E2.80.93Szekeres_diagram]Kruskal-Szekeres diagram[/url] which I mentioned in a previous comment (I can't emphasize enough how much getting a feel for Kruskal-Szkekeres diagrams will help your intuitions about black holes!) If you look at the corresponding

embedding diagram for F to the right, it's a universe which is closed along one "short" axis but extends forever along the other "long" axis, like an infinite cylinder (like all embedding diagrams, one of the three spatial coordinates is kept fixed so we can represent the curvature of space on a 2D slice through 3D space where the other two dimensions are allowed to vary).

F is a hyperbola bounded by the two black hole event horizons in the Kruskal-Szkeres diagram, so if you pick different hyperbolas with the same boundary, they represent different simultaneity surfaces where the closed spatial dimension has different sizes--and ones below F have a larger size for the closed dimension, and any above F would have a smaller size, and any infalling observer will pass through each surface in order from bottom to top. So, an observer inside the horizon can be seen as living in a universe where space is "collapsing" along one spatial axis ("one spatial axis" in the embedding diagram anyway, in terms of the curvature of 3D space I believe each simultaneity surface is a hypercylinder which is shrinking in

two different directions, while remaining infinitely extended in the third). In this picture the singularity is a future event, experienced by everyone living in this hypercylinder, where the closed dimension(s) shrink to zero length, similar to the Big Crunch in cosmology (where the universe would shrink to zero along

all spatial dimensions).

lgw wrote:To further explain why I think this, lets consider another example: take all the contents of our observable universe, and contract all large scale structure about 10-fold, moving the clusters of galaxies closer until it all fit into a sphere of about 4.6 Gly in radius, with appropriately higher CMBR temperature and so on (or we could imagine the contents looking just as they did when the universe was somewhat younger, if you prefer). Per the guess in the Schwarzschild radius article in Wikipedia, we now have a black hole.

That claim in the wiki article has a "citation needed" so you shouldn't put too much trust in it. As pointed out in

this entry from the

usenet physics FAQ, in an expanding universe you can no longer use the Schwarzschild radius to determine whether a given collection of matter will become a black hole:

Sometimes people find it hard to understand why the Big Bang is not a black hole. After all, the density of matter in the first fraction of a second was much higher than that found in any star, and dense matter is supposed to curve spacetime strongly. At sufficient density there must be matter contained within a region smaller than the Schwarzschild radius for its mass. Nevertheless, the Big Bang manages to avoid being trapped inside a black hole of its own making and paradoxically the space near the singularity is actually flat rather than curving tightly. How can this be?

The short answer is that the Big Bang gets away with it because it is expanding rapidly near the beginning and the rate of expansion is slowing down. Space can be flat even when spacetime is not. Spacetime's curvature can come from the temporal parts of the spacetime metric which measures the deceleration of the expansion of the universe. So the total curvature of spacetime is related to the density of matter, but there is a contribution to curvature from the expansion as well as from any curvature of space. The Schwarzschild solution of the gravitational equations is static and demonstrates the limits placed on a static spherical body before it must collapse to a black hole. The Schwarzschild limit does not apply to rapidly expanding matter.

lgw wrote:Now, one even weirder idea I'll throw out here: we've discussed in other threads the idea that that the sum of universal entropy must increase in the time direction in which the universe is seen as expanding (and CMBR temperature falling), and thus "the future" will always seem to be in the expanding-universe direction. If we accept that for the sake of argument (or argue about in in the old thread), then we'd have a universe that instead had a singularity in its past, and a cosmological constant much like our own - definitely measureable, but inexplicable if one considered only local conditions. Sound like a familiar universe?

I don't understand, what would be inexplicable? Our universe already has a singularity in its past, that's what the Big Bang is (of course quantum gravity might change that, but I'm just talking about what's predicted by general relativity).