Structure formation by the Hartree equation

Details Structure formation by the Hartree equation Structure formation by the Hartree equation

Dec 2004

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2004sf2a.conf..687l&link_type=abstract

SF2A-2004: Semaine de l'Astrophysique Francaise, meeting held in Paris, France, June 14-18, 2004. Edited by F. Combes, D. Barret

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The Scale Relativity theory predicts that the formation of structures in gravitational interaction can be described by an Hartree equation or equivalently by a set of coupled Schrodinger and Poisson equations. This system is similar to the one used in quantum gravity (with matter only being quantized not the field), but with the substitution of the ratio of the Planck constant to the mass (hbar/m) by a parameter depending of the system under study. By a change of coordinates we get the hydrodynamical equations of Euler and continuity which make appear a term of potential energy (which is understood as a quantum pressure in the case of Bose-Einstein condensation) which would play here the role of a "dark potential". As applications we show the formation of structures in a medium with a homogeneous density and the formation of a disk from dust around a central star. The linear (equilibrium) solutions will be recalled in both cases and they compare favorably with the observations, but we describe also the phase of non linear dynamical evolution which is thought to converge asymptotically in time towards the equilibrium linear solutions (under current study). In particular the Hartree equation has no singularity of blow-up kind in finite time since the dark potential, which is the manifestation of the fractality of space, prevents from gravitational collapse. We make use of variable in time, with 2D and 3D variations in space numerical codes to solve this problem, by setting as initial conditions either the asymptotic states or localized structures of the soliton type.

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