Dear Ken,
I can't reasonably expect you to accept, at face value, my claim that in my view maximal symmetry lies at the heart of everything. Or at least I can't reasonably expect you to accept that I'm justified in saying so, when you see my position on time as being opposed to that. So, I thought I should try to explain my reasoning in a post here. I hope you do find the analysis interesting, because it leads to what I think are a couple of very intriguing results.
As I said my reasons lie along the lines of a teleparallel description of gravity that's based on a maximally symmetric background metric, interpreted geometrically, I'll begin by showing what I think that metric has to be. Afterwards, I'll explain how I think the teleparallel aspect of the picture should work.
So, we begin by hypothesising that there is this maximally symmetric geometry that lies at the heart of it all, and the first question is, "What is it?" I think it should be real, so my first instinct is to coordinate a number of real lines. But since space-time has Lorentzian signature, Euclidean space seems like a bad background geometry to use for this purpose, and imposing Lorentzian signature on a "real" metric space by simple definition seems contrived. A more natural definition of a real metric space with maximal symmetry and zero curvature is that it is Euclidean space (and this is anyway the usual definition), and according to *that* definition, Minkowski space is naturally derived through a Wick rotation of one coordinate. From this point of view, Minkowski space isn't real.
Instead, we can consider spherically symmetric spaces with the induced metric, [math]ds^2=\sum_{\mu=0}^{4}dx_{\mu}^2,[/math] [math]\sum_{\mu=0}^{4}x_{\mu}^2 =\alpha^2.[/math]
By demanding only that the four dimensions of the maximally symmetric space itself are real, it's easy (and interesting) to see that this metric actually represents four distinct real 4D spaces, by arbitrarily solving the bottom equation for one coordinate and allowing that it can be real or imaginary; i.e., by writing [math]x_0=\pm\sqrt{\alpha^2-\sum_{i=1}^{4}x_{i}^2 }.[/math]
Then, the metric for these 4D real Riemannian spaces in this Cartesian coordinate basis can be written [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 now the spherically symmetric space's *intrinsic* "radius of curvature". From this line-element it's straightforward to write down the components of the metric tensor in this basis:
[math]g_{ij}=\frac{1}{\alpha^2-\mathbf{x}^2}\cdot[\alpha^2-(\mathbf{x}^2-{x_i}^2)],~\mathrm{if}~i=j,[/math] [math]g_{ij}=\frac{1}{\alpha^2-\mathbf{x}^2}\cdot{x_i}{x_j},~\mathrm{if}~i\neq{j}.[/math]
From here, we can solve the eigenvalue problem; and it turns out that the metric tensor always has three positive eigenvalues, along with [math]\lambda=\frac{\alpha^2}{\alpha^2-\mathbf{x}^2}.[/math]
The four distinct geometries described by the metric are therefore as follows: (i) when [math]\alpha^2\geq\mathbf{x}^2(\geq0)[/math] and alpha^2 is positive (this is the one case where x_0 is actually real), the space is a closed 4-sphere with positive-definite metric tensor; (ii) when [math]\alpha^2\leq\mathbf{x}^2[/math] and alpha is a non-zero real constant, lambda is negative so the *real* metric is Lorentzian; (iii) when alpha=0, lambda=0 so the metric is degenerate and the metric describes a lightlike hypersurface of 5D Minkowski space; and finally, (iv) when alpha is purely imaginary (and non-zero), the metric is positive-definite for all [math]\mathbf{x}\in\mathbb{R}^4.[/math]
In terms of the original embedding, these four geometries are: (i) a closed 4-sphere in 5D Euclidean space; (ii) a hyperboloid of one sheet in 5D Minkowski space; (iii) a light cone in 5D Minkowski space; and (iv) a hyperboloid of two sheets in 5D Minkowski space.
The point of the derivation is that, beginning from the requirement of maximal symmetry and the usual definition of "real space", and maintaining the real basis in an embedded space, it can be shown that the only *real* space with maximal symmetry *and* Lorentzian signature is case (ii), which is de Sitter space. From the point of view of wanting to describe the apparent symmetries of nature, and particularly the Lorentzian symmetry of space-time, as fundamental properties of reality, this result seems significant--especially when it's noted that the cosmological evidence supports a pure cosmological constant which could be essentially geometrical. Then, the vacuum Einstein equation would be [math]R_{ab}=\Lambda{g_{ab}},[/math] which indeed the above metric is a solution of, with [math]\Lambda\equiv\frac{1}{\alpha^2},[/math] although the metric wasn't derived with reference to general relativity; e.g., I allowed for general metrical signature, but instead maintained that the space had to be real. But as I said, only in case (ii) is the metric really relativistic, because that's the only case in which this real space has Lorentzian signature.
I hope you don't mind if I try to explain briefly how I'd make use of this result. I'll do that in another post.