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What deformations are possible with spacetime?

By 'deformation' I am referring to the kind of change in spacetime caused by the presence of a mass which deforms spacetime sufficiently to deflect photons of light, or in the extreme case of a black hole, 'compress' spacetime into a singularity, or in the case of a rotating binary star system to creates waves comprised of compressed spacetime.

By 'possible', I mean 'theoretically' possible.

Is it a case of what you can do with a room full of jelly, you can also do with spacetime? If this analogy is not good then why is that?

Is it possible to make a hole in spacetime as it is with jelly, in that case creating a volume of vacuum within the jelly?

Pictures of worm holes imply that a volume of spacetime can have a surface. Is it actually possible to create a surface that bounds spacetime?

Must every point in spacetime remain attached to its current neighbours?

In the case of jelly, one could isolate a sphere of jelly and then rotate it relative to the rest of the jelly without creating any kind of gap. Is that possible with space time?

One paper talks about cracks in space-time (following a change of state after the big bang). Is that possible?

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    $\begingroup$ Please define your precise notion of deformation, and what you mean by the deformation being "possible". $\endgroup$ – ACuriousMind Mar 20 '15 at 22:29
  • $\begingroup$ Hi you say is it a case of what you can do with a room full of jelly, you can also do with spacetime? If this analogy is not good then why is that no offence but the analogy you provide is not valid because you can't compare in any meaningful way, jelly and space time. A analogy is good only if it is useful in helping to compare two situations that have some common basis to establish a mental picture. Jelly and spacetime do not share any common characteristics at all,imo, so no analogy exists. You can't make an analogy between an elephant and say, a block of ice. Regards $\endgroup$ – user74893 Mar 21 '15 at 18:09
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You need to be careful about comparing the curvature of spacetime to the deformation of a block of jelly. In particular, in general relativity time is curved as well as space, and this is impossible to represent with the jelly model.

In fact it's just about impossible to give a really good description of spacetime curvature to anyone who doesn't have at least a basic grasp of general relativity. Once you've grasped the maths it's surprisingly straightforward, but it's so unintuitive that I can't think of any analogies that aren't ultimately misleading. So this makes answering your question effectively impossible.

However a few of the points you raise can be addressed. Apart from a few special cases spacetime immediately around any point must look smooth and flat - technically it must look like Minkowski spacetime. So every point must remain attached to its neighbours. This also means you can't cut out a (hyper)sphere of spacetime, rotate it, then put it back.

While you can have a hole in jelly, you can't have a hole in spacetime, because spacetime is by definition everything that exists. There is nothing outside or beyond spacetime. However spacetime can be non-trivially connected, and this is what happens with wormholes.

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Must every point in spacetime remain attached to its current neighbours?

Basically yes, this captures the spirit of allowable deformations.

Some background: If you want to measure distances between points, this is very much in the realm of mathematical analysis. However, it is sometimes useful to be able to discuss "closeness" in a looser (but still rigorous) sense, and this is the realm of topology. Topology deals with things like neighborhoods of points. For example, given two points in a space with a certain topological structure, is it always possible to find disjoint neighborhoods of those points, or might two points be "so close" that any neighborhood of one contains the other.

Mathematically, spacetime is assumed to be a manifold -- a set of points that locally looks like 4-dimensional Euclidean space in terms of topology. That is, around every point is a 4D neighborhood of points that is as well-behaved as the standard $\mathbb{R}^4$ we are more familiar with it. Right away this tells you things shouldn't be too weird on small scales.

Now topological closeness isn't particularly amenable to physical observation; we need something more concrete. Therefore we also endow spacetime with a metric -- a function that, at every point, computes angles between directions as well as infinitesimal distances. Moreover, this function is assumed to vary smoothly from point to point in standard general relativity. Since this is spacetime we're talking about, the metric is smooth (well, to be precise, twice-differentiable) in space as well as time.

This is one case were the rubber sheet imagery actually does work: you can stretch and compress and shear spacetime, but you can't cut it or glue different patches together, as that would induce abrupt changes in the metric (as well as break the topological structure).

Of course, one can always tweak things and modify general relativity so as to allow more violent deformations, but these are just speculative ideas.

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