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H. I. Ringermacher (General Electric Research Ctr.), L. R. Mead (U. of Southern Mississippi)
We describe an exact 5-D solution of Einstein's equations based upon a hyperbolic axisymmetric Poincaré 3-space, (dx2+dy2+dz2)/ (1+B(x+y))2, that appears to represent the geometry due to dark matter alone underlying the structure of spiral galaxies. The triaxial 3-space, (dx2+dy2+dz2) /(1+B(x+y+z))2, is also a solution. The observed Euclidean 3-space and the hyperbolic space of the dark matter do not have a common center so that care must be taken in calculating proper length. The 4th and 5th dimensions are proper time and cosmological time (epoch), respectively. Geodesic motion yields observed flat rotation curves of spirals. From these, we obtain "isochrones": the locus of stellar positions at fixed proper time and epoch in their motion along geodesics. Isochrones are of the form r(\phi)=A/ln[B tan(\phi/2N)], where B(0.05-3.0) and N(2-64) are shape constants, while A scales. The isochrones yield the structures of the Hubble spiral galaxy classes including barred and grand designs. Polar ring solutions are found including angled rings, as well as warped "integral" and "banana" rings. Other galactic problems are also resolved. The similarity of rotation curves for grand spirals and bars, first pointed out by Rubin, is explained by a duality transformation of the parameters describing grands and bars in the rotation curves. The 75% abundance of spirals arises from the dimensional symmetry of the Poincaré 3-space: there are 3 axisymmetric coordinate permutations (x+y, x+z, y+z) and only 1 triaxial (x+y+z), amounting to geometric "equipartition". The solution correctly represents the known effects of dark matter in galaxies and thereby suggests that this geometry provides the skeletal stabilizing structure underlying the dynamics attributed to dark matter into which baryonic matter may be added.
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Bulletin of the American Astronomical Society, 37 #4
© 2005. The American Astronomical Soceity.