The calculations are complicated and error-prone, but software is available to perform the symbolic manipulations by computer (see, for example, Hartle, 2003). We can form the contraction of this to get the Ricci tensor and the scaler curvature. The Riemann curvature tensor is a complicated expression involving derivatives of the metric tensor: All the triangles are congruent (of equal shape and size) in the Poincaré metric. Escher to create some of his remarkable images. The image at the head of this article is the triangular hyperbolic tiling that inspired the artist M. Moreover, the angular deficit (the amount by which the sum of the three angles falls short of ) determines the area of the triangle. Indeed, the entire disk can be foliated by a set of such lines.įor any triangle, the sum of the angles is strictly less than. In the right panel, there are seven geodesics, all parallel and all converging to the same point on the absolute. ![]() 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. 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. In this geometry, Euclid’s fifth postulate no longer holds. Poincaré was presenting a model for non-Euclidean geometry. Right: a set of parallel geodesics, all converging to the same point on the absolute. 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. It turns out that the geodesics in Poincaré’s metric are the circular arcs contained within the disk which intersect the absolute orthogonally. The citizens of the disk could not reach the bounding circle - known as the absolute - in a finite time. The radius of the “unit disk“ with this metric is given by integrating from the centre to a point on the boundary of the disk:īut this integral diverges as, confirming for the disk-dwellers that their universe is infinite in extent. (In relativity, the parameter is the proper time.) The distance is given byįrom this, we can derive quantities known as Christoffel symbols,Īnd write down equations for the geodesics In differential geometry, the metric tensor contains all that is needed to determine shortest paths, or geodesics. The geometry of Poincaré’s disk is encapsulated in the metric giving the distance increment. 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. 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. 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. 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. 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. ![]() Henri Poincaré described a beautiful geometric model with some intriguing properties.
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