On the comoving distance as an arc-length in four dimensions |
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Authors: | Boudewijn F. Roukema |
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Affiliation: | ;1Inter-University Centre for Astronomy and Astrophysics Post Bag 4, Ganeshkhind, Pune, 411 007, India;2DARC/LUTH, Observatoire de Paris–Meudon, 5, place Jules Janssen, F-92195 Meudon Cedex, France |
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Abstract: | The inner product provides a conceptually and algorithmically simple method for calculating the comoving distance between two cosmological objects given their redshifts, right ascension and declination, and arbitrary constant curvature. The key to this is that just as a distance between two points 'on' the surface of the ordinary 2-sphere 2 is simply an arc-length (angle multiplied by radius) in ordinary Euclidean 3-space ℰ3, the distance between two points 'on' a 3-sphere 3 (a 3-hyperboloid ℋ3) is simply an 'arc-length' in Euclidean 4-space ℰ4 (Minkowski 4-space ℳ4), i.e. an 'hyper-angle' multiplied by the curvature radius of the 3-sphere (3-hyperboloid). |
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Keywords: | cosmology: observations cosmology: theory |
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