A new approach to the central gravitational field in vacuum space / Implementation of General Relativity in finite space to the MOND theory

We know, from the publication of the Schwarzschild metric (1916) and the Birkhoff’s theorem (1923) that a central symmetric gravitational field in infinite vacuum space is automatically static in the framework of General Relativity. This result has intrigued many researchers. Indeed, Lemaitre’s works seemed to suggest that the dynamic behavior of a metric could be inherent to General Relativity. We have searched for a method allowing getting round the Birkhoff’s theorem, and we have finally found the following methodology: For defining the metric of a central gravitational field in vacuum, we have started from an undefined metric written in tortoise coordinates, instead of starting from an undefined metric written in the coordinates used by Schwarzschild. In tortoise coordinates, the solution obtained by canceling the Ricci tensor depends on two arbitrary functions, the first one being a function of (t+r), and the second one being a function of ( t−r) (t being the temporal coordinate multiplied by the speed light, and r being the radial coordinate). In a first approach, it appears that some convenient transformations (if well justified) of coordinates lead to the classical Schwarzschild metric, whatever arbitrary functions used. However, we have found that if we choose for the arbitrary functions a triangular signal, the situation changes dramatically: in this case, the coordinates t and r permute their temporal/spatial role over time; the metric is broken down in four different versions which replace each other cyclically over time. Consequently, it becomes impossible to find a general transformation of coordinates allowing recovering the Schwarzschild metric. This new metric describes an area in which the orbital velocity of a massive particle does not depend on the radial distance; this result allows recovering the Tully-Fisher law without any dark matter; we find also that the Riemannian space-time curvature is oscillatory. So, we came up with our methodology to find, in the framework of General Relativity, a new solution for the central gravitational field in vacuum space, which is considered currently as impossible; more, this solution could be physical, as it explains the Tully-Fisher law without any dark matter. By combining this new metric and the classical Schwarzschild metric, it becomes possible to justify approximately in a galaxy the velocity profile of stars found in the MOND theory, without any dark matter, in the framework of General Relativity