The geometry of open universes

Here is the Robertson–Walker metric found for the closed universe, with $r = \sin\psi/\sqrt k$:

Guth's proposal is to keep this formula and simply allow $k < 0$. He compares it with a rule like "today's stock price equals yesterday's plus the change", which works whether the change is positive or negative.

But he warns that this trick does not always work. For $k < 0$, $\sqrt k$ is imaginary, so $r = \sin\psi/\sqrt k$ makes no sense with a real angle. Worse, in the $\psi$ form,

a negative $k$ with real $\psi$ makes every $ds^2$ negative. That space would still be homogeneous and isotropic, but distances must be positive (positive definite), so it is not acceptable. The $r$ form is the "magical" one in which flipping the sign of $k$ just works.

Why it works is shown by letting the angle become imaginary as well, $\psi = i\chi$. The sine of an imaginary number is a hyperbolic sine, $\sin(i\chi) = i\sinh\chi$, so with $k = -|k|$:

Both sign problems cancel: $d\psi^2$ becomes $-d\chi^2$, $\sin^2\psi$ becomes $-\sinh^2\chi$, and $a^2/k$ becomes $-a^2/|k|$, so the metric is positive again and $r$ stays positive. (Guth argues this in words and promises to show that $r$ stays positive; the formula above makes it explicit.)

What makes a metric acceptable for cosmology? Not just "no contradictions" or "the right limits", but the two properties we observe: homogeneity and isotropy. Isotropy about the origin is clear, since the angular part is an ordinary sphere. Homogeneity is hidden, because the coordinates single out an origin.

For the closed universe Guth gives a recipe for proving that every point is equivalent to the origin. Take an arbitrary point $(r_0, \theta_0, \phi_0)$, and:

1. compute its four-dimensional coordinates $(x_0, y_0, z_0, w_0)$ on the 3-sphere;
2. apply a four-dimensional rotation that carries this point to the "North Pole" $(0, 0, 0, R)$, which is the origin of the coordinates;
3. convert the rotated coordinates back to $r', \theta', \phi'$.

A rotation does not change distances on a sphere, so the metric keeps exactly the same form in the primed coordinates, with the chosen point now at the origin. Guth admits he has never carried out the algebra in full, but expects each step to work equally well for negative $k$, so the open universe is homogeneous too.

He also quotes a theorem without proof (proved, for example, in Steven Weinberg's Gravitation and Cosmology, as Guth notes on p. 19 of the transcript): every homogeneous, isotropic three-dimensional space can be written in Robertson–Walker form with suitable coordinates. Closed, flat and open universes are the only possibilities.

The three cases differ sharply in size.

• Closed ($k > 0$). The metric is singular where $kr^2 = 1$, i.e. at the equator $\psi = \pi/2$. There $r$ reaches its largest value $1/\sqrt k$, then decreases again as $\psi \to \pi$. Each value of $r$ labels two places, one in each "hemisphere". The total volume is finite.
• Flat ($k = 0$). Ordinary Euclidean space in polar coordinates; $r$ runs to infinity.
• Open ($k < 0$). Nothing in the metric is singular, so $r$ can grow without limit. But $r$ is a coordinate, not a distance. Integrating the metric along a radial line gives the physical (proper) distance from the origin:

This grows without limit, so the open universe is infinite. (In the lecture Guth wonders aloud whether it grows faster than linearly; in fact it grows more slowly, like a logarithm of $r$. Turned around, the circumference of a sphere grows exponentially with its proper radius.)

The two-dimensional version of the open metric is exactly the Gauss–Bolyai–Lobachevsky geometry of Lecture 10. Klein's coordinates for it look complicated because they were chosen to show other things; in Robertson–Walker coordinates, physicists find it straightforward.

• The Robertson–Walker metric in its $r$ form works for any sign of $k$; the $\psi$ form would give negative distances for $k < 0$ unless $\psi$ is imaginary, which turns sines into hyperbolic sines.
• Homogeneity can be shown by rotating the embedding sphere (closed case) and carries over to $k < 0$; every homogeneous, isotropic space has Robertson–Walker form (a theorem Guth quotes).
• Closed universes are finite ($r \le 1/\sqrt k$, double-valued); flat and open universes are infinite, with proper radius $a\sinh^{-1}(\sqrt{-k}\,r)/\sqrt{-k}$ in the open case.