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Aug28-08, 05:26 AM
PF Gold
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Based on Friedmann’s equation, if k=0, the energy density of the universe would have to be equal to the critical density and imply that the universe would expand forever, albeit at an ever-decreasing rate, i.e.

[tex]\Omega = \frac {\rho}{\rho_c} = \frac {8\pi G \rho}{3H^2} = 1[/tex]

Current models, inclusive of matter, cold dark matter and dark energy, suggest that present-day [tex][\Omega][/tex] is very close to 1, such that [k=0], which in-turn is said to support a spatially flat universe. However, this only gives a mathematical rationale for the definition of spatial curvature without necessarily helping with any visualisation. Of course, we can visualise geometrical 2-D flatness in terms of the angles of a triangle adding up to 180 degrees, but mapping this onto a physical 3-D concept of space appears to be more problematic.

The attachment `space.jpg` shows an equilateral triangle that is assumed to enclose a vast area of space at 2 different points in time. As a possible futile attempt to consider 3-D space curvature, I have included a triangular pyramid to parallel the 2-D concept of flatness. The problems of simultaneity are ignored and it is simply assumed that the length of the sides and angles are relative to an inertial observer and space-like at each time instance. There are a number issues raised for clarification

- It is recognised that the example is intrinsically flawed in that space is a 3-D concept that cannot really be visualised in terms of a flat 2-D triangle.

- Spatial flatness only requires the angles of this triangle to add up to 180 degrees at any single instance in time, i.e. t1 or t2? However, in the real universe, we would have to first establish a spatially flat reference triangle at [t1], which we might assume remains geometrically flat at [t2]?

- Of course, we have now introduced the effects of time and the possibility of ‘distortion’ due to spacetime curvature (?). However, while each corner of the equilateral triangle might travel along curved spacetime in an expanding universe (?) this does not seem to implicitly change its spatial flatness, i.e. spatial curvature and spacetime curvature can be independent of each other?

- However, if we run time backwards, does space within our triangle always remain flat? If not, what ‘point’ in the universe would this space be curving around in a homogeneous model without any centre of gravity?

- I realise that last point may be meaningless in that the curvature of 3-D space may have no meaningful centre or axis. However, if perpendiculars could be drawn to any tangents of curvature in a higher dimension would these perpendiculars converge to a ‘point’ and what meaning might be inferred on this point?

- Finally, is it possible that the curvature of space has no physical meaning, i.e. it can only be inferred from the motion of some object moving through this space?

I realise that the actual answers to some of these issues may be buried in GR maths or Riemann n-D manifolds, but was trying to focus on whether there was any simple approach that might help visualise the basic concepts. My initial conclusion is that space curvature is harder to visualise than spacetime curvature. However, would welcome any clarification of the issues raised in the first 3 posts. Thanks
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