# Books/papers with this version of the FLRW metric?

I'm looking for references/books/papers/articles which discuss this specific form of the Robertson-Walker (RW) metric (or see the cartesian version (4) below) : $$\tag{1} ds^2 = dt^2 - \frac{a^2(t)}{(1 + k \, r^2 /4)^2}\big(dr^2 + r^2 (d\vartheta^2 + \sin^2 {\vartheta} \; d\varphi^2)\big).$$ I have about 10 books on General Relativity, classics like Misner-Thorne-Wheeler (MTW), Weinberg, Schutz, Ohanian, Landau-Lifchitz (LL), Wald, ... and only two of them present metric (1) as an exercice, without discussion : MTW and LL. They are all working with other versions of the RW metric : $$\tag{2} ds^2 = dt^2 - a^2(t) \Big( \frac{1}{1 - k \, \tilde{r}^2} \; d\tilde{r}^2 + \tilde{r}^2 (d\vartheta^2 + \sin^2 {\vartheta} \; d\varphi^2) \Big),$$ or $$\tag{3} ds^2 = dt^2 - a^2(t) (d\chi^2 + \mathcal{S}_k^2(\chi) \, (d\vartheta^2 + \sin^2 {\vartheta} \; d\varphi^2)),$$ where $$\mathcal{S}_k(\chi) = \sinh{\chi}, \qquad \text{if k = -1,}$$ $$\mathcal{S}_k(\chi) = \chi, \qquad \text{if k = 0,}$$ $$\mathcal{S}_k(\chi) = \sin{\chi}, \qquad \text{if k = 1.}$$ Yet, metric (1) has several advantages : its space part is conformal to the euclidean metric, and most calculations are much easier with cartesian coordinates : $$\tag{4} ds^2 = dt^2 - \frac{a^2(t)}{\big(1 + \frac{k}{4} \, (x^2 + y^2 + z^2) \big)^2}(dx^2 + dy^2 + dz^2).$$ Metric (1) has even a nice symmetry under "radial inversion" (if $k \ne 0$) : $$\tag{5} r \, \Rightarrow \, r' = \frac{4}{r},$$ and may be compared a bit with the Schwarzschild metric expressed in its "isotropic" version.

So anyone has suggestions of books/papers that discuss version (1) or (4) ? I'm looking for more discussion on the range of the radial coordinate $r$ in version (1), especially for $k = -1$ (there's a coordinate singularity at $r = 2$, and yet the metric (1) is still valid for $r > 2$).

• Are you thinking this might be like the maximally extended Schwarzschild metric, where the region $r\gt 2$ corresponds to an unexpected second copy of spacetime? If so I don't think this is the case, or at least I have never heard of this. I think the region $r>2$ is just physically meaningless. It's an interesting thing to think about though. – John Rennie Sep 8 '16 at 15:05
• @JohnRennie, yes ! This is it ! The second copy may have several interpretations ! A simple copy of $r < 2$ (redundancy of the radial coordinate), or really a second disconnected open universe ! – Cham Sep 8 '16 at 15:08
• If the second disconnected open universe make sense, then this is very interesting (while not very important, I admit) : it gives a nice example of the possibility to have several disjoint (i.e. "parallel") universes in classical General Relativity, described by a single metric. I never saw this before, so this is why I'm so curious about this apparent possibility with the "extended" RW metric. All other cases I've seen are wormholes (i.e. connected universes). – Cham Sep 8 '16 at 15:13
• It's just that given how well known the maximally extended Schwarzschild metric is, if the RW metric could be interpreted in this way I would expect it to be in all the GR textbooks. But it isn't. – John Rennie Sep 8 '16 at 15:30
• @JohnRennie, let me add that the maximally extended Schwarzschild metric you probably reffering to isn't exactly of the same type we're discussing here. The "extension" of the Schwarzschild metric which is closer to the RW extension I'm discussing here is found with the "isotropic" formulation of the Schwarzschild metric. Not the Kruskall-Szekeres version (KS). The wormhole isn't the same in the KS and the isotropic extensions. See for example this paper : arxiv.org/pdf/0901.0215.pdf (only the first half is appropriate for our discussion) – Cham Sep 8 '16 at 15:45