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 $a_{-}(t) \; \text{if} \; r > 2$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.