Flat Space, Curved Spacetime?

One of the first visualisation of spacetime curvature we often come across is the idea of a heavy ball on a rubber sheet. Of course, this image has its limitations in that it is just illustrating the curvature of 2D-space, i.e. time is not really being addressed.

So we will move quickly onto the centre image of a sphere with a triangle on it, highlighting that the angles of this triangle don’t add up to 180 degrees within this curved geometry. In addition, this visualisation also suggests that parallel lines in some geometries may converge. Possibly, the next step leads us naturally to the balloon analogy of cosmology, alluding to not only curvature, but also the expansion of the universe. However, we should note the caveat in this visualisation, which states that only the surface of the balloon, not its volume, be included in the model.

So what frame of reference is the surface of the balloon universe said to be curving around?

Often, this analogy is accompanied with an almost subliminal message that it might also explain how a finite universe is boundless, i.e. the surface of a sphere has no edge. However, it is not clear whether anybody can really justify this position, i.e.

Is it really saying that setting off in any direction only leads back to the start?

This analogy is also somewhat contradicted by the apparently spatial flatness of the ΛCDM concordance model , which we shall discuss in a later section. Of course, most presentations of this nature are normally focusing on the curvature resulting from the proximity to some large mass-density, which does not necessarily reflect the assumptions of a low mass density homogeneous universe, which may or may not have a centre of gravity. Of course, if we accept that mass is just a form of energy, i.e. E=mc², then any localised concentrations of energy density, resulting from small-scale perturbations within a large-scale homogeneous universe, may have what appears to be fairly extreme gravitational effects, such as black holes. However, it might be worth trying to clarify what might be understood as spacetime curvature as opposed to spatial curvature.