jordankonisky said:
The impact of a bowling ball on the shape of a stretched gridded sheet is often used as a tool to help some of us to better visualize how the presence of a mass influences the curvature of spacetime? However, while this kind of diagram is helpful, I think that it is misleading because it depicts the perturbation of only a single slice of the space-time continuum. In real fact, the massive object is totally enmeshed in a space-time continuum. Rather, I visualize a raisin enmeshed in a bread loaf in which the loaf represents the full continuum of space-time and no part of the raisin exists, indeed can exist, outside the space-time continuum. Am I thinking about this in the right way?
Raisins (representing events) embedded in a loaf (representing space) is a common popularization of how to visualize space. But it's really oriented to expaining "curved space" and not "curved space-time". Which is common to most popularizations, due to the abstractness of curved space-time.
As far as "expanding space" goes, one interesting paper that I think makes some good points is "Expanding space, the root of all evil?"
<<Link>>
My summary of the paper would be that the concept of expanding space has its detractors, and can be and frequently is mis-used and leads to notable misconceptions (many of which we try to straighten out here on PF, with varying degrees of success). But it has supporters, as well, amongst them I would include the authors of this paper, who attempt to present the idea of "expanding space" in a way that they claim is less likely to cause confusion.
As far as curvature goes, most discussions of curvature really only visualize the curvature of space, and not space-time. The "rubber sheet" would need to be not a spatial rubber sheet, but a space-time diagram, for the analogy to represent space-time curvature. Unfortunately, the concept of a space-time diagram seems to be hard to get people to utilize, for reasons that I don't really understand.
There's another analogy to curved space (and not space-time) due to Einstien that I think has some merit, the "heated slab" approach. See for instance
, A Einstein, "Relativity, the special and general theory"
<<link>>.
I believe that the expanding slab idea has some limitations as to what it can model, but it serves as a conceptual model that may help people who are extremely used to envisioning only Euclidean geometries. A generalization of the approach may be general enough to handle small sections of actual space-time, for instance in Straumann, "Reflections on Gravity"
<<link>>. In the generalization, one includes the idea that gravity affects clocks as well as rulers.
While the idea is helpful and probably less prone to misconceptions than the rubber sheet analogy, it does have some limits, though I haven't seen these limts discussed in the literature. For instance, the relativity of simultaneity is a basic feature of special relativity that is frequently not understood is not well-represented by this kind of model, which tries to "tweak" familiar notions of geometry into a more general form. Another concern is handling global topology.
I was going to comment more on curved space-time geometry as opposed to just spatial geometry, but, I think the post has already gone on long enough , and possibly too long.
One final note. Popularizations are not a replacement for learning the full theory - which, ultimately, involves a lot of math. They do the best they can to present some features of the theory in a way that's understandable without the math, but they are not the full theory. Frequently people criticize the full theory by criticizing the popularizations (because the populariations are all they understand). Well, the popularizations ARE flawed, in many cases. The flaws can be minimized by applying them only to the situations for which they are intended and suitable for, but one should not expect a full understanding based only on popularizations :(.