My books on General Relativity do not tell anything about the following and I would like to clarify this thing.

Consider only the open FLRW universe ; $k = -1$ (also called hyperbolic universe), of the following metric (note : that metric is given in Misner-Thorne-Wheeler on page 722, exe 27.4, also in Landau-Lifchitz, at the end of paragraph 111): $$\tag{1} ds^2 = dt^2 - \frac{a^2(t)}{(1 - r^2 /4)^2}\big(dr^2 + r^2 (d\vartheta^2 + \sin^2 {\vartheta} \; d\varphi^2) \big). $$ Usually, we introduce a new radial coordinate : $$\tag{2} \tilde{r} = \frac{r}{1 - r^2/4}, $$ such that the metric (1) becomes $$\tag{3} ds^2 = dt^2 - a^2(t) \Big( \frac{d\tilde{r}^2}{1 + \tilde{r}^2} + \tilde{r}^2 \, (d\vartheta^2 + \sin^2 {\vartheta} \; d\varphi^2) \Big), $$ or else $$\tag{4} r = 2 \tanh{(\chi/2)} $$ (or $\tilde{r} = \sinh{\chi}$) such that $$\tag{5} ds^2 = dt^2 - a^2(t) \big(d\chi^2 + \sinh^2 {\chi} \, (d\vartheta^2 + \sin^2 {\vartheta} \; d\varphi^2) \big). $$ However, these coordinate transformations bypass a particularity of metric (1) above : it has a coordinate singularity at $r = 2$, and is still valid for $r > 2$ (remember that metric (1) describes an isotropic and homogeneous spacetime, so curvature is regular everywhere : $0 \le r < \infty$. Curvature invariants don't even depend on $r$). Since $\tilde{r}$ should be positive, the transformation (2) is valid only for $r < 2$. Also, the transformation (4) is defined only if $r < 2$.

So the question is simple : What is the part of spacetime described by $r > 2$, according to the metric (1) ?. Is this another open spacetime, "parallel" to the part described by $r < 2$ ? Or do we have to restrict ourselves to $r < 2$ only (why rejecting $r > 2$) ?

Take note that metric (1) is invariant under the radial coordinate reversal : $$\tag{6} r = 4/r', $$ so points of coordinate $r < 2$ could be mapped to points of coordinate $r > 2$ (there's something similar with the Schwarzschild metric written in isotropic coordinates).

Also, the proper radial distance of a point of coordinate $r < 2$ to the observer located at $r = 0$ is easily computed : $$\tag{7} \mathcal{D} = a(t) \ln{\Big(\displaystyle{\frac{2 + r}{2 - r}}\Big)} \equiv 2 \, a(t) \arg\tanh{(r/2)}. $$ (this is simply $\mathcal{D} = a(t) \, \chi$ if you do the transformation (4) above). That distance diverge at $r = 2$, so we can't define the distance to points of $r > 2$.

Am I right in saying that the part of spacetime with $r > 2$ can be interpreted as a disconnected "parallel" open universe, a bit like the second sheet of a 2 sheets hyperboloid ? (see the picture there : http://virtualmathmuseum.org/Surface/hyperboloid2/hyperboloid2.html)

  • $\begingroup$ @JohnRennie, no it is not ! With $k = -1$ and the geometric factor $$\frac{1}{(1 + k r^2 /4)^2},$$ the metric (1) describes an open space (hyperbolic). Check your maths/books and compare with metric (3), which have an opposite sign in front of $k$. I'm sure of this. The closed universe is defined for $k = +1$, so it would give the geometric factor $$\frac{1}{(1 + r^2/4)^2}.$$ The metric (3) then has the factor $$\frac{1}{1 - \tilde{r}^2}$$ for the closed universe. $\endgroup$ – Cham Sep 6 '16 at 22:05
  • $\begingroup$ Also, take note that the metrics (1) and (3) get an hyperbolic $\sinh{\chi}$, when you substitute the coordinate transformation (4) (or $\tilde{r} = \sinh{\chi}$). This then implies the open universe, not the closed one (which need a trigonometric $\sin{\chi}$) ! $\endgroup$ – Cham Sep 6 '16 at 22:14
  • $\begingroup$ Oops, sorry, I read the post in a hurry and misread it. $\endgroup$ – John Rennie Sep 7 '16 at 6:17
  • $\begingroup$ Isn't your metric (1) the Poincare disk, or one of its many varients? $\endgroup$ – John Rennie Sep 7 '16 at 11:17
  • $\begingroup$ @JohnRennie, metric (1) is a variant of the Poincaré "disk" (in 3D + time) if $k = -1$, but I think it isn't important (it's "just" a name !). Most authors are defining the RW metric with metric (3) (metric (1) appears to be less well known), but a simple radial coordinate transformation gives metric (1). Misner-Thorne-Wheeler give metric (1) on page 722 as the "true" RW metric (from the 1935-1936 papers of Robertson and Walker). It has many calculations advantages since it is "isotropic", and its space section is conformal to the euclidian metric ; $d\ell^2 = f(r)(dx^2 + dy^2 + dz^2)$. $\endgroup$ – Cham Sep 7 '16 at 13:00

I think I've found a complete and clear understanding of the second universe in the RW metric (1) shown in the question. It is indeed a second sheet of a 2 sheets hyperboloid.

For $k = -1$, the standard RW metric is derived by almost every author as a pseudo-sphere (actually a 2 sheets hyperboloid) of equation $$\tag{1} u^2 - x^2 - y^2 - z^2 = a^2, $$ immersed in a fictious flat space of pseudo-euclidian metric $$\tag{2} d\ell^2 = -\: du^2 + dx^2 + dy^2 + dz^2. $$ Equation (1) is similar to the special relativistic mass shell relation $E^2 - p_x^2 - p_y^2 - p_z^2 = m^2$ (two sheets hyperboloid).

Most authors are using a partial parametrization, describing only half of the hypersurface (1) : \begin{align}\tag{3} u(\chi, \vartheta, \varphi) &= a \cosh{\chi}, \\[6pt] x(\chi, \vartheta, \varphi) &= a \sinh{\chi} \, \sin{\vartheta} \, \cos{\varphi}, \\[6pt] y(\chi, \vartheta, \varphi) &= a \sinh{\chi} \, \sin{\vartheta} \, \sin{\varphi}, \\[6pt] z(\chi, \vartheta, \varphi) &= a \sinh{\chi} \, \cos{\vartheta}. \end{align} As you can see, this parametrization is assuming $u \ge a$, and yet equation (1) also admits the same parametrization with $u \le -\, a$ (changing the sign of $u$ in (3)). When you substitute the parametrization (3) into the flat metric (2), you get the space section of the standard RW metric, with $k = -1$ and $0 < \chi < \infty$ : $$\tag{4} ds^2 = dt^2 - a^2(t) \big( d\chi^2 + \sinh^2 \chi \, (d\vartheta^2 + \sin^2 \vartheta \; d\varphi^2) \big). $$ So this is only half the hyperboloid, which is just equivalent as making a topological choice (i.e. removing the second part of the fictious hypersurface). But then, it is easy to prove that the full Robertson-Walker metric below, with $r < 2$ and $r > 2$, is actually decribing the full hyperboloid (1) (i.e. its two disconnected sheets) : $$\tag{5} ds^2 = dt^2 - \frac{a^2(t)}{(1 - \frac{1}{4} \; r^2)^2} \big( dr^2 + r^2 (d\vartheta^2 + \sin^2 \vartheta \; d\varphi^2) \big). $$ The coordinate singularity at $r = 2$ is simply the gap between both sheets of the double-hyperboloid. The link between coordinates $r$ and $u$ is this ($a$ is considered as a constant here, as in equation (1) and parametrization (3)) : $$\tag{6} r = 2 \, \sqrt{\displaystyle{\frac{u - a}{u + a}}}. $$ It is actually very simple. There is no weirdness, no mystery. The metric (5) is all consistent with equation (1). So as a topological choice, you may really have two disconnected hyperbolic universes (i.e. "parallel" universes), described by the natural extension of metric (5) to all values available of the radial coordinate.

EDIT : I'm adding some funny speculations. In the analysis given above, both sheet-universes are expanding at exactly the same rate (same scale factor $a(t)$ for both hyperbolic sheets).

  1. Could these sheets be a simple example of branes in the context of classical General Relativity ?

  2. Could they be the "reverse" of each other, i.e. one of matter and the other one of anti-matter ?

  3. Could we generalize them to two sheets of different scale factors, expanding at a different rate; $a_{+}(t)$ and $a_{-}(t)$, so they have different amount or type of matter ? What would look like a "generalisation" of the RW metric (5) in this case ?

I guess the answer is "Yes" to 1 and 2. In case of question 3, it may be trivial; just change the scale factor to a position dependant function: $a(t, \, r) = a_{+}(t) \; \text{if} \; r < 2$ and $a_{-}(t) \; \text{if} \; r > 2$.


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