Are there any mathematically defined 'spaces' that we could talk about the universe 'expanding' into?
If I have a two dimensional locally Euclidean surface (a balloon for example) it should be easy to show that it's deformation over time can be embedded in a three dimensional Euclidean space.
Are there any metrics or spaces that can be used to embed a locally Lorentzian space curved by and evolving according to General Relativity?
Maybe we can throw a bone to those of us who need to have a 'space' that space is expanding into.
"Maybe we can throw a bone to those of us who need to have a 'space' that space is expanding into."
How about this?
In the simplest of terms, Einsteinian gravity isn't gravity. It's that the presence of matter causes the space around the object to "curve", which causes things moving in the vicinity to track into the object. We can diagram this (and do) with pretty grid pictures, so think of the grid.
But, of course, if space is space, there is not really a grid.
But the particle physicists are out there frantically searching for the gravitron. Which is curious. Why do you need a PARTICLE to do something, when Einstein tells us that gravity is, essentially, caused by the presence of matter warping the grid (I am being crude, but don't quibble, because it's not half bad for plain language).
So, there are plenty of folks who, rightly, say that there's a fundamental conflict between particle theory and Einstein, and get exercised about it too. (The exercised part is the part where I start saying "Whoa!" because it means that people are starting to treat this stuff like a religion, and adding emotion to it. Which is not the best thing to do if you want to keep your mind flexible enough to be able to fiddle with new theories. No, we should not CARE is Newton or Einstein was right, nor Darwin or Lamarck either. But that's a separate philosophical issue.)
To try and unify these apparently conflicting things, consider this: the gravitron IS the grid. Emanating from matter, as a sort of "energy" - call it gravitational energy (again, give me a break, the language of medieval tribes was not designed for this stuff) - what the gravitrons do is MAKE the grid. They're the thing that is out there "warping" around in the presence of matter, and emanating from it.
Alright, so go back to that physics textbook with the funny 3-d diagram that's trying to show gravity "warping" space.
Now just imagine that the space really is just a void. Utterly empty and nothing. Unless there is matter there throwing out gravitrons to MAKE a grid which then affects other matter that comes within the grid.
Gravity would then move at the speed of light, because it would be a particle, and the grid itself would shape itself at that speed, as the particles make the grid.
But where there's really NOTHING, no matter. There are no gravitrons (or other particles, obviously), and therefore, there's no grid. That's really nothing. There's nothing to organize it. Once any THING is there, there's a particle...and therefore relationship, and once there's matter of any sort, there are gravitrons and a grid, and it's not "Nothing" anymore.
So, "space time" is the warped grid of all of these gravitrons, making a saddleback or whatever the current theory of the shape of the universe says, and popping and buzzing along out of nothing, maybe, just because there's a chunk of something about.
But where there's not something there's nothign, and no grid either.
It's our language that wants to call nothing "something", and include the something that's nothing in the universe. So, fine, consider everything outside the saddleback as "the universe", but without rules, because there's nothing to behave out there, and nothing can get out of the saddleback, because the saddleback is the structure made by the gravitrons which are the grid.
Nothing goes on forever, and we can't see the end of it.
It is peopled by things like the dream you will have next Thursday. (Does that exist yet?)
But the three-dimensional space is unnecessary. Gauss pointed out that the surface of a sphere is two-dimensional; all properties of the surface can be obtained without any reference to a third dimension.