Topic
Eigenplane
About: Eigenplane is a research topic. Over the lifetime, 2 publications have been published within this topic receiving 3 citations.
Papers
TL;DR: In this paper, the authors present a method for numerical falsification of topological lensing hypotheses without requiring any particular spherical 3-manifold by finding the two eigenplanes of the matrix M.
Abstract: Three-dimensional catalogues of objects at cosmological distances can potentially yield candidate topologically lensed pairs of sets of objects, which would be a sign of the global topology of the Universe. In the spherical case, a necessary condition, which does not exist for either null or negative curvature, can be used to falsify such hypotheses, without needing to loop through a list of individual spherical 3-manifolds. This condition is that the isometry between the two sets of objects must be a root of the identity isometry in the covering space S^3. This enables numerical falsification of topological lensing hypotheses without needing to assume any particular spherical 3-manifold. By embedding S^3 in R^4, this condition can be expressed as the requirement that M^n = I for an integer n, where M is the matrix representation of the hypothesised lensing isometry and I is the identity. Moreover, this test becomes even simpler with the requirement that the two rotation angles, theta, phi, corresponding to the given isometry, satisfy 2\pi / \theta, 2\pi / \phi \in Z. The calculation of this test involves finding the two eigenplanes of the matrix M. A GNU General Public Licence numerical package, called eigenplane, is made available at this http URL for finding the rotation angles and eigenplanes of an arbitrary isometry M of S^3.
1 citations
TL;DR: In this paper, a necessary condition for falsification of topologically lensed pairs of sets of objects is presented. This condition is that the isometry between the two sets of points of objects must be a root of the identity isometry in the covering space S 3, which enables numerical falsification without requiring any particular spherical 3-manifold.
Abstract: Three-dimensional catalogues of objects at cosmological distances can potentially yield candidate topologically lensed pairs of sets of objects, which would be a sign of the global topology of the Universe. In the spherical case (i.e. if curvature is positive), a necessary condition, which does not exist for either null or negative curvature, can be used to falsify such hypotheses, without needing to loop through a list of individual spherical 3-manifolds. This condition is that the isometry between the two sets of objects must be a root of the identity isometry in the covering space S 3 . This enables numerical falsification of topological lensing hypotheses without needing to assume any particular spherical 3-manifold. By embedding S 3 in euclidean 4-space, R 4 , this condition can be expressed as the requirement that M n = / for an integer n, where M is the matrix representation of the hypothesised topological lensing isometry and I is the identity. Moreover, this test becomes even simpler with the requirement that the two rotation angles, θ,Φ, corresponding to the given isometry, satisfy 2π θ, 2π Φ e Z. The calculation of this test involves finding the two eigenplanes of the matrix M. A GNU General Public Licence numerical package, called eigenplane. is made available for finding the rotation angles and eigenplanes of an arbitrary isometry M of S 3 .
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2005 | 1 |
| 2004 | 1 |