Hi Sean,
It was nice meeting you at the foundations conference in Munich. It was also really nice reading your essay. I really liked you analysis, and the subject matter is very interesting to me. I have a couple of comments which I'd like to hear your thoughts on.
The first one is mostly a remark: you mentioned a few times that you want to think of the cosmological constant as what drives cosmic expansion. This is exactly what motivated my PhD thesis, and at one point I picked up Eddington's 'Expanding Universe' and found that it was really the idea he had in mind, too. Throughout the 1920s, guys like him and Weyl had been thinking about cosmic expansion in terms of this 'cosmical repulsion' due to Lambda; and in 1924, Weyl even wrote a dialogue between 'Petrus' and 'Paulus' (Saints Peter and Paul), who presented Einstein's views and his own, respectively. The dialogue presents an obvious reaction to the postcard that he got from Einstein in 1923 (dated something like a week prior to Einstein posting his retraction to what he originally wrote about Friedman's paper, which he now called "correct" and "clarifying"), where he famously said "If there is no quasi-static world, then away with the cosmological term." The interesting thing in Weyl's paper, though, is that he didn't know what the hell Einstein was talking about, because he had never heard of Friedman's "correct" and "clarifying" paper, so when he received the post card he clearly though Einstein was saying he wanted to go back to special relativity, which obviously makes no sense if the Universe is expanding. 'Paulus' aka Weyl replied to 'Petrus' aka Einstein at this point in the dialogue, by saying something like, 'but all the spiral nebulae are receding, so the Universe is probably expanding, so dS space-time is the best bet that we know of'.
So anyway, guys like Weyl and Eddington had this idea of a 'cosmical repulsion' that drives expansion in their minds throughout the 1920s. And in Eddington's 'Expanding Universe' (1933), he did not hide the fact that he was really really upset that Einstein, after pretty much remaining silent on the issue of expansion, and not saying anything to anyone about Friedman's paper throughout the 1920s (everyone else learned about it in like 1930 or 1931, after Hubble confirmed the expansion with his redshift-distance relation in 1929), chose Friedman's solution with no cosmological constant (in the Einstein-de Sitter 1932 paper), where the cosmical repulsion can play no fundamental role in explaining the cosmical expansion. The reason is that near to the big bang, the cosmical repulsion is negligible, so the expansion is, to begin with, always less than it was, with Lambda driving accelerated expansion later on.
I do think there is a way around this last problem, which is what I wrote about in my last essay. There are more details about all of this there, if you're interested.
Okay, that's point one, and my comment's already getting long. I think point two is more interesting, though, so I'm still going to post it. It has to do with the motivation for the dS hyperboloid, which you presented in an interesting way in your essay. I wanted to discuss and alternative approach, though, which I think makes fewer assumptions, and provides a really neat way of thinking about this cool geometry that you've presented such an interesting analysis of.
So: what's the one aspect of nature that seems to be physically the most important? Symmetry, right? Symmetry that can be broken, but which is fundamental in nature. So, to begin I want just to assume that reality has an underlying metric, and that it has maximal symmetry. The only other thing I want to require, since relativity indicates that it's a property of nature, is Lorentzian signature of that metric. I don't want to assume that from the outset, though, but will only pick the solution with this property out of the list of possibilities derived from the assumption of maximal symmetry.
The maximally symmetric spaces can all be recovered as constant curvature hypersurfaces of Euclidean space, so begin by writing down 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] This can be re-written in just four coordinates by arbitrarily solving for one of the five, as in [math]x_0=\pm\sqrt{\alpha^2-\sum_{i=1}^{4}x_{i}^2 },[/math] and the result can be neatly written as [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". We can require this x to be real, since we began in E^5, but there's actually no reason that x_0 can't be imaginary, since it's fictitious anyway; i.e., the 4D maxiamlly symmetric spaces are still described in terms of a real Euclidean basis. From this line-element, we can pretty much just read off 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] The eigenvalues of the tensor are all positive except one, which is given by [math]\lambda=\frac{\alpha^2}{\alpha^2-\mathbf{x}^2}.[/math]
The closed sphere is described by setting alpha^2>0 and alpha^2>x^2; hyperbolic space, by setting 0>alpha^2 and x real; and dS space, by setting alpha^2>0 and x^2>alpha^2. If alpha=0, the metric space is degenerate.
So it turns out that just in case alpha^2 is positive (by the way, alpha^2=1/Lambda), and only when x^2>alpha^2 (dS space) does the metric have Lorentzian signature. The two points that are of further interest are that Lambda>0 is what we actually observe anyway, and that this space (actually, all of them) satisfies the vacuum Einstein equation R_ab=Lambda*g_ab.
I think it's pretty cool that with no reference to GR, and with no prior requirement that the metric should be Lorentzian, but only by requiring maximal symmetry to begin with, and an embedding in Euclidean space, it can be shown that dS space is the only one that does have a Lorentzian metric, and the space satisfies the vacuum Einstein equation.
I'm interested to know what you think of all this, because I think there is a lot of overlap in what we're considering.
Best of luck in the contest--and really, a great job done on the essay!
Daryl