The Dark Matter Problem
General Relativistic Galactic Rotation Curves
in a Friedman Dust Universe
with Einstein's Lambda
James G. Gilson
Keywords: Cosmology, Dust Universe, Dark Energy, Dark matter
In this paper, the general relativistic replacement for the Newtonian inverse square law of gravitation is obtained from the Friedman Cosmology equations. This version of the inverse square law is shown to contain information about the amount of dark energy mass contained in a specific region through a mass term $M_\Lambda^-$ dependent on Einstein's Lambda and, importantly for this paper, it also contains information about the amount of dark matter mass in the same region through a term $M_P^+$. This work derives from the Dust Universe Model which gives a complete cosmological description of the movement and evolution of the astrophysical space substratum which as usual is represented by a spatially uniform or constant mass density distribution at zero pressure. Thus definite spatial regions of the substratum can only be regarded as holding regions for un clumped mass, as primitive galaxies might be described. Consequently, to describe actual galaxies that have condensed from such a region, the more general solution of Einstien's Field eqtions involving the pressure term is needed to explain clumping and the resultant galactic form. The general relativist version of the inverse square law is written in a form applicable to the case of bound circular orbiting about a spherically symmetric central gravitational spatially distributed source force. Thus the behaviour of masses cycling within or outside the source region can be analysed. The formula for the galactic rotation curves for stars rotating within or outside the source region is obtained. A very simple galactic model is used consisting of just two components, the halo and the bulge with all visible orbiting stars, The conclusion is that the pressure term from general relativity and in the consequent Friedman equations is adequate to explain the constancy of the function of rotational velocity as a function of orbital distance from the centre of gravity starting at the massive core of the galaxy. A simple and parameter adaptable computer program using Mathematica has been constructed to display diagrams of galactic rotation curves. This program is available for downloading.
