Mach's principle and the dynamics of galaxies: v2 Nov 2012

A basic model for the dynamics of galaxies is given which explains the anomalous rotation curves and the predominant spiral structure without the need for dark matter. The model is based on a metric inspired Mach's principle but which may not satisfy Einstein's equations. 85A05; 85A15, 85A40, 83F05, 83C40, 83C57 an interim approach. It uses a metric which is obtained perturbing a standard metric to make it model a specific formulation of Mach's principle. The perturbation stops the metric from satisfying Einstein's equations and therefore the approach adopted here is not fully relativistic. We hope in the future to find a metric which does satisfy Einstein's equations and which gives a similar dynamic. This paper can be seen as motivation for the search for this metric. Mach's principle (21) is the principle that the local concept of inertial frame is with the distribution of the matter in the universe. The principle is often stated using the wording determined by rather than correlated with. However this formulation begs the question of how this determination takes place and we prefer a less contentious statement. It is well known that solutions to Einstein's equations need not satisfy Mach's principle, see eg Rindler (27). One particular solution which breaks Mach's principle and which is germane to the main subject matter of this paper is the Kerr solution for the metric near a rotating mass in an otherwise empty universe. In the Kerr universe there is only one central mass and no other mass to determine the inertia of the central mass. If the solution were Machian then the inertial frame at every other point would rotate with the central mass and the solution would be a simple coordinate transformation of the Schwarzschild