Please forgive me if these are dumb questions. Wikipedia is wonderful, but it tends (unsurprisingly) not to address the questions I have, and so the SDMB is my only place to ask these things.

In the standard two-dimensional ball-on-a-blanket model used to illustrate how gravity bends spacetime, is it understood that this effect is happening in every direction at once equally? In other words, a ball on a blanket in the three-D world on earth bends the blanket downward, towards the center of gravity in the center of the earth. But this illustration appears to show the effect of gravity bending spacetime away from the center of gravity, and what’s more, it only shows this effect taking place on one plane – as if the whole thing was still a ball on a blanket on earth. Wouldn’t the bending of spacetime be distributed equally around the sphere? (Ignoring for the moment any other sources of gravity nearby.)

When an object is in space, is it “in” that space, or is it displacing that space? Is there spacetime only where mass isn’t?

If just one celestial object has an appreciable warping effect on spacetime, then back at the singularity when all the mass in the universe was in one place, was it sort of bending space inward so far, that that’s why there wasn’t any (space)?

If space began at the singularity and expanded outward in every direction at once, then it is a spherical thing with an ever-increasing diameter, right? Is the expansion at the edges of space pulling in some way on the rest of space? If the expansion (inflation) of space at the edges is driving the expansion of space within the sphere, then could the ever-increasing diameter explain the fact that expansion is speeding up?

I think you are justifiably confused by the picture that you linked to. It is quite misleading, muddling the 2D analogy with 3D reality. You are correct - the bending of spacetime is pretty much impossible for a human brain to visualize, but it is 3D-symmetrical around the center of the earth.

Spacetime is everywhere, aside perhaps from undefined behavior at singularities. I guess that’s why they call it a “continuum”.

John Wheeler’s aphorism on GR:
“Mass tells space-time how to curve, and space-time tells mass how to move.”

Note that when cosmologists talk about the “diameter” of the universe, they are talking about the observable universe, not the entire universe. The “edge” of the observable universe is only a boundary in the sense that it’s the furthest that we can (in principle) ever see. It does not represents a physical boundary - the universe may continue indefinitely beyond. The universe has no edge in the sense that you describe. Again, it’s almost impossible to visualize, but the universe is not expanding “into” anything preexisting. It is the space itself that is expanding. And the only sense in which there is a sphere is in our arbitrary demarcation of the observable universe, the spherical region of the expanding universe centered on us that is all that we can ever see.

The process of metric expansion itself is determined by the total mass and energy in the universe, including the mysterious “dark energy”. I think I should leave that to somebody else with more expertise to describe precisely.

I wouldn’t take the rubber sheet analogy too seriously, it is an analogy that illustrates a few limited points, but can be seriously misleading.

General relativistic spacetime is probably best philosophically treated like Newtonian space-time (e.g. a background), however it is also a background that has much more structure and that extra structure dynamically influences and is dynamically influenced by the mass-energy within in it.

Initially expansion in the Universe depends on initial conditions (for which we do not or cannot know the reasons for) and the decelerating effect of the radiation and matter within it. However in the last approximately third of the Universe’s history it is thought that the predominant factor is mysterious dark energy which has an accelerating effect on expansion.

One important point about curvature in relativity is that all that actually matters is what’s called intrinsic curvature: You measure sets of distances and/or angles, and find that the relationship between those distances and/or angles is not the same as it would be in flat space. Ergo, the space you’re in is not flat… but that’s based entirely on measurements made within the space. There need not be any other dimension into which the space is curved (what’s called “extrinsic curvature”), and even if there is, there’s no way for us to know it.

Thanks, great answers so far. I understand that there is no pre-existing “something” for space to be expanding into; in fact, I’ve never actually heard a cosmologist refer to the circumference or diameter of space as such. I’m just assuming that an essentially regular expansion in all directions from a beginning point would naturally result in a spherical leading edge, because a regular expansion in every direction wouldn’t make a cube or whatever.

What I don’t understand is, why does gravity bend space away from itself, rather than towards? (at least, according to the picture)

Here is something else I wonder about: If, at the singularity, there were an absolutely huge but still finite amount of mass in one place; and if that mass was flung outward from a central point; then at some point the amount of mass outside the halfway point would be greater than the amount of mass inside it; would that preponderance of mass then exert a greater gravitational pull away from the center, than the remaining mass was pulling in? (hope that makes sense)

But, in any event, you are still incorrectly imagining an “explosion” expanding as a sphere in pre-existing space. The universe does not work that way. The big bang happened everywhere in space, and everywhere in space looks the same (modulo random fluctuations and their consequences). There is no “center”, there is no “expanding sphere”.

The expansionary “impetus” is the initial conditions at the big bang. From there, the mass-energy of the universe always works to slow or reverse the expansion. Later, when the universe is larger, dark energy becomes significant, working (for arcane technical reasons) in the opposite direction to mass-energy to accelerate the expansion.

The rubber sheet analogy sort of annoys me, because the demonstration requires gravity to work - so whilst it is a great visual demonstration of the effects of the gravity phenomenon, it doesn’t have any explanatory power.

I think in many ways it’s better to visualise gravity warping spacetime as if it’s distorting the lines on a sheet of graph paper - an object moves along what it perceives to be a straight line, but follows a curved path in the broader view.

I think the biggest problem with the rubber-sheet analogy is that it’s a dead end: It’s not enough to do anything with on its own, and nor does it lead in any way to any other model that is enough. In particular, it has no numbers to it, and none of the models which do have numbers build upon it in any way.

Take two groups of students, teach one of them the rubber-sheet model and the other nothing at all. Both groups will be able to solve the same set of problems in GR: That is to say, none of them. Take another two groups, teach one of them both the rubber-sheet model and a mathematical model, and the other group just the mathematical model, and they’ll again both be able to solve the same problems. In no case does the rubber-sheet model help with much of anything.

Let me correct myself here. On reflection, I think the word “continuum” probably refers more to the mathematically unified treatment of space and time as space-time in relativity.

No, you had it right the first time. Physicists also sometimes refer to lower-dimensionality systems as being a “continuum” (provided, of course, that they are in fact continuous). Though the phrase “spacetime continuum” is probably so common because you’ve got to call it something, and most of the other relevant words are already used up (you couldn’t call it, say, “the spacetime space”). It’s not usually particularly necessary to call attention to the fact that spacetime is continuous.

It’s a very crude analogy - the one good point I can say about is it crudely illustrates the focusing effect of (and hence attractive nature of the gravity of) normal matter.

One example of place where the analogy hopelessly fails is that in general relativity that true vacuums (as opposed to lambda-vacuums etc) are Ricci flat and hence the gravity outside a massive body in bog-standard general relativity is described by Weyl curvature. However 2D and 3D surfaces do not have Weyl curvature, so the 2D rubber sheet should have no curvature at points where there is no mass. This can be seen in 2+1D general relativity where there are no gravitational effects in the vacuum outside of a body.

I kind’ve think Sean Carroll tries to oversimplify on this one: the dynamics in a spatially homogenous and isotropic spacetime are only expand, contract or do nothing and “do nothing” can be ruled out as a realistic possibility with some very simple arguments. Those dynamics are governed by the stress-energy of the matter, radiation, fields, etc in the spacetime and in such a spacetime the stress-energy can be reduced to just the energy density and pressure of those things. So whether such a spacetime expands or contracts and whether it is expanding at an accelerating rate, etc is down to energy density, pressure and boundary conditions.