Dear George:
Thanks a lot for continuing to respond to me here. I appreciate your comments, and hope you do find the discussion interesting.
" " "Note that if the family of observers are rotating there are no simultaneous space sections for them." You mean there is no spacelike surface describing the space they are "actually" rotating through, as cosmic time progresses? If there's an enduring three-dimensional space (or EBU with boundary), and the events of space-time form the map of events that occur in that space (or boundary) as it evolves, then however warped their description may be in the proper frame, there have to be spacelike surfaces that contain the events that happened simultaneously, at the same value of absolute time." There are spacetimes where no such surfaces exist. That is a key discovery of Godel in his 1949 paper on the Godel universe, which is a specific example."
The fact that there are spacetimes like Goedel's representing rotating families of observers where no such surfaces exist, does not mean that every description of a rotating family of observers has to have closed timelike curves. Just because one solution satisfies A and B, does not mean that every solution satisfying A has to satisfy B. As a specific example along the lines of my fourth response above, on Sep. 14, 2012 @ 04:53, please consider the de Sitter cosmological model that's described by the line-element ds^2=-dT^2+cosh(aT)^2/a^2*dOmega3^2, in which the cosmological 3-sphere contracts from cosmic T=-infty to T=0 and then expands afterwards. Since a 3-sphere is a 2-sphere plus a 1-sphere with a twist, we can populate it with a homogeneous distribution of particles that all move in the same direction with the same coordinate velocity, viz. as moving through that twisted 1-sphere dimension always with the same velocity in passing T. This space-time, which clearly has no real closed timelike curves, can therefore be described with respect to this family of rotating observers, from whose perspective the slices of constant T will be described locally as a set of spacelike hypersurfaces that are not synchronous.
"Well the idea of broken symmetry is very important. You have a theory with a symmetry, and that characterises the dynamics in general. Then specific solutions break the symmetry - but they still inherit many important properties from the theory with the full symmetry." Sure. I agree with you that this is one---evidently useful---way of looking at it. My point was that if the Universe itself should then be said to break the symmetry, then the symmetry may not be a fundamental law, but could be due to a property of the Universe. I actually do have something specific in mind here that might help to make my point more clear. It requires entertaining the possibility that coordinates do have immediate metrical meaning, which may be opposed to general covariance as a fundamental principle, but seems valid from the point-of-view of cosmology and its implication that there are actually preferred time lines and spatial sections.
Please consider, if you will, the possibility that space and duration do have some fundamental order, so that emergent space-time is consequently ordered as well, as any set of space-time coordinates would scale relative to these basic coordinates. One possibility is that these should be coordinated real lines; but that would only be at all realistic, according to the success of relativity theory, if the coordinated space had Lorentzian metrical form. To assume that, however, is to already assume the symmetry about which I'd like to gain some insight. The next simplest possibility for a basic maximally symmetric space relative to which space-time coordinates should scale, is a sphere. So let us consider the real 4-spheres with induced metric, [math]ds^2=\sum_{\mu=0}^{4}{dx_i}^2,[/math] [math]\sum_{\mu=0}^{4}{x_{\mu}}^2=\alpha^2.[/math]
Here, although the metric of the embedding space is written explicitly with Euclidean form, we'll demand only that the coordinates that describe the sphere are real. Therefore, an implicit assumption that I'm making here is that the most natural coordination of real lines is the Cartesian one.
Now, it's easy to show that there are four distinct real 4-dimensional hypersurfaces with this induced metric, which are easily differentiated from one another by arbitrarily solving for x_0, [math]x_0=\pm\sqrt{\alpha^2-\sum_{i=1}^{4}{x_i}^2},[/math] so that each "sphere" may be described instead, as a 4-dimensional real Riemannian manifold, in a real Cartesian coordinate basis, with metric, [math]ds^2=d\mathbf{x}^2+\frac{(\mathbf{x}\cdot{d\mathbf{x}})^2}{\alpha^2-\mathbf{x}^2},[/math] where [math]\mathbf{x}=(x_1,x_2,x_3,x_4)[/math] is a real vector, and alpha is the sphere's *intrinsic* `radius of curvature'. For, in doing so, we may allow the coordinate x_0, describing the extra dimension of the embedding space, to be imaginary, so that the embedding space need not be Euclidean, but may instead be pseudo-Euclidean, or Minkowskian,---without affecting the requirement that the coordinates of the maximally symmetric space must be real, by definition.
Now, from this line-element it's straightforward to write down the corresponding elements of the metric tensor in this basis, [math]g_{ij}=\frac{1}{\alpha^2-\mathbf{x}^2}\left\{\begin{array}{ll}
\alpha^2-(\mathbf{x}^2-{x_i}^2), & i=j \\
x_ix_j, & i\neq{j}
\end{array}\right.,[/math] which always has 3 positive eigenvalues, along with [math]\lambda=\frac{\alpha^2}{\alpha^2-\mathbf{x}^2}.[/math] Therefore, when [math]0\leq\mathbf{x}^2\leq\alpha^2[/math] and alpha^2 is positive (so that x_0 is actually real), the closed 4-sphere has positive-definite metric tensor. Otherwise, the line-element describes one of three distinct, maximally symmetric real open hypersurfaces of 5-dimensional Minkowski space, depending on the sphere's `radius of curvature', alpha:---when [math]\alpha\in\mathbb{R}{\backslash}0,[/math] [math]\mathbf{x}^2>\alpha^2,[/math] and the metric tensor is Lorentzian (lambda is negative), the open `sphere', which may be described as a timelike hyperboloid of one sheet in 5D Minkowski space, is de Sitter space; when alpha=0, the spherical hypersurface is a degenerate null-cone in Minkowski space; and when [math]\alpha\in{i}\mathbb{R}\backslash0,[/math] the metric tensor is positive-definite, with lambda>0 for all [math]\mathbf{x}\in\mathbb{R}^4[/math], and this open *anti*-de Sitter `sphere', can be described as a spacelike hyperboloid of two sheets in 5D Minkowski space.
The point of this derivation is to show a way that a maximally symmetric *real* metric space with *implicit* Lorentzian signature---through which space-time might derive both its order and metrical structure---may be derived from first principles, i.e. without assuming Lorentzian signature a priori. Incidentally, in this derivation the Loretzian signature occurs only in the case of a positive cosmological constant, when alpha^2 is positive, which is something that all the cosmological evidence also points to.
It's also very helpful to see that, purely for reasons of mathematical consistency, it's sloppy in a sense to refer to closed spherical space as the positive curvature analogue of hyperbolic space. Actually, if we don't differentiate between "Riemannian" and "pseudo-Riemannian" metrics a priori based on their signature, it's apparently more consistent to describe de Sitter space as the positive curvature analogue of hyperbolic space, and the closed sphere as a maximally symmetric hypersurface of either of those real spaces, or of Euclidean space.
Daryl
