A Finite but Unbounded Universe

Henri Poincaré described a beautiful geometric model with some intriguing properties. He envisioned a circular disk in the Euclidean plane, where distances were distorted to give it geometric properties quite different from those of Euclid’s Elements. He supposed that the temperature varied linearly from a fixed value at the centre of the disk to absolute zero on the boundary, and that lengths varied in proportion to the temperature.

Poincaré argued that creatures, whom we will call disk-dwellers, living on the disk would be unable to detect that the disk was finite, and would regard it as infinitely large, with geometric properties quite different from the Euclidean plane. Thus, for two paths that we — with our external Euclidean perspective — would regard as equal in length, a disk-dweller would require more steps to cover the path near the boundary than that closer to the centre. To move from one point to another, the disk-dweller could follow the shortest path only if he or she travelled on a circular arc rather than along a Euclidean straight line.

The Metric

The geometry of Poincaré’s disk is encapsulated in the metric giving the distance increment. In differential geometry, the metric tensor {g_{\mu\nu}} contains all that is needed to determine shortest paths, or geodesics. The distance {ds} is given by

\displaystyle ds^2 = g_{\mu\nu} dx^\mu dx^\nu \,.

From this, we can derive quantities known as Christoffel symbols,

\displaystyle {\Gamma^\lambda}_{\mu\nu} := \textstyle{\frac{1}{2}} g^{\lambda\sigma} \left[ \partial_\mu g_{\sigma\nu} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu} \right] \,,

and write down equations for the geodesics

\displaystyle \frac{d^2 x^\lambda}{d\tau^2} + {\Gamma^\lambda}_{\mu\nu} \frac{d x^\mu}{d\tau}\frac{d x^\nu}{d\tau} = 0 \,.

(In relativity, the parameter {\tau} is the proper time.)

For the Poincaré disk, the metric is

\displaystyle ds^2 = \frac{4(dx^2 + dy^2)}{(1-(x^2+y^2))^2}

so the metric tensor is

\displaystyle g_{\mu\nu} = \begin{bmatrix} \frac{4}{(1-(x^2+y^2))^2} & 0 \\ 0 & \frac{4}{(1-(x^2+y^2))^2} \end{bmatrix}

The radius of the “unit disk“ with this metric is given by integrating {ds} from the centre {(0,0)} to a point {(1,0)} on the boundary of the disk:

\displaystyle \int_0^x \frac{4\ dx}{(1-x^2)^2} = \left[ \log\frac{1+x}{1-x} +\frac{2x}{1-x^2} \right] \,.

But this integral diverges as {x\rightarrow 1}, confirming for the disk-dwellers that their universe is infinite in extent. The citizens of the disk could not reach the bounding circle — known as the absolute — in a finite time.

Hyperbolic Geometry

It turns out that the geodesics in Poincaré’s metric are the circular arcs contained within the disk which intersect the absolute orthogonally. Also included are all the straight lines through the centre of the disk; these are the only geodesics that are Euclidean straight lines, but the disk-dwellers would regard all the geodesics as straight lines.

Left: two parallel lines. Right: a set of parallel geodesics, all converging to the same point on the absolute.

Poincaré was presenting a model for non-Euclidean geometry. In this geometry, Euclid’s fifth postulate no longer holds. For any “straight line” and a point not on it, there is an infinity of lines through the point and parallel to the given line. The two arcs in the Figure (left panel) are parallel: both are of infinite length (in the Poincaré metric) and they have no common point. In the right panel, there are seven geodesics, all parallel and all converging to the same point on the absolute. Indeed, the entire disk can be foliated by a set of such lines.

For any triangle, the sum of the angles is strictly less than {180^\circ}. Moreover, the angular deficit (the amount by which the sum of the three angles falls short of {2\pi}) determines the area of the triangle. The image at the head of this article [credit to Wikimedia Commons] is the {(6,4,2)} triangular hyperbolic tiling that inspired the artist M. C. Escher to create some of his remarkable images. All the triangles are congruent (of equal shape and size) in the Poincaré metric.

The Riemann curvature tensor is a complicated expression involving derivatives of the metric tensor:

\displaystyle {R^\lambda}_{\mu\nu\sigma} = \partial_\nu {\Gamma^{\lambda}}_{\mu\sigma} - \partial_\sigma {\Gamma^{\lambda}}_{\mu\nu}  + {\Gamma^{\eta}}_{\mu\sigma} {\Gamma^{\lambda}}_{\eta\mu} - {\Gamma^{\eta}}_{\mu\nu} {\Gamma^{\lambda}}_{\eta\sigma} \,.

We can form the contraction of this to get the Ricci tensor { {R}_{\mu\nu} = {R^\lambda}_{\mu\lambda\nu}} and the scaler curvature {R = g^{\mu\nu} R_{\mu\nu}}. The calculations are complicated and error-prone, but software is available to perform the symbolic manipulations by computer (see, for example, Hartle, 2003). It turns out that the scalar curvature for the Poincaré disk has constant value {-2}. The Gaussian curvature is related to this: {K = \frac{1}{2}R} so the Gaussian curvature of the disk is {-1} (this was the motivation to include the factor 4 in the definition of the metric).

Poincaré’s disk model is one of several models of hyperbolic geometry. Others include Poincaré’s half-plane model [see earlier blog posts] and the Cayley-Klein model. Eugenio Beltrami used these models to show that hyperbolic geometry was consistent if Euclidean geometry was. Henri Poincaré rediscovered (and popularised) the disk and half-plane models years later.

Sources

{\bullet} Hartle, James B., 2003: Gravity: an Introduction to Einstein’s General Relativity. Addison Wesley, San Francisco.

{\bullet} Sossinsky, A. B., 2012: Geometries. American Mathematical Society. ISBN: 978-0-8218-7571-1.


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