Modified Newton’s gravity in Finsler Space as a possible alternative to dark matter hypothesis

A modified Newton’s gravity is obtained as the weak field approximation of the Einstein’s equation in Finsler space. It is found that a specified Finsler structure makes the modified Newton’s gravity equivalent to the modified Newtonian dynamics (MOND). In the framework of Finsler geometry, the flat rotation curves of spiral galaxies can be deduced naturally without invoking dark matter. PACS numbers: 02.40.-k, 04.25.Nx, 95.35.+d [email protected] [email protected] 1 There are a great variety of observations which show that the rotational velocity curves of all spiral galaxies tend to some constant values[1]. These include the Oort discrepancy in the disk of the Milky Way[2], the velocity dispersions of dwarf Spheroidal galaxies[3], and the flat rotation curves of spiral galaxies[4]. These facts violate sharply the prediction of Newtonian dynamics or Newton’s gravity. The most widely adopted way to resolve these difficulties is the dark matter hypothesis. It is assumed that all visible stars are surrounded by massive nonluminous matters. Though it explains the flat rotation curves of spiral galaxies, the hypothesis has its own weakness. No theory predicts these matters, and they behave in such ad hoc way. There are a lot of possible candidates of dark matter (such as axion, neutrino et al), but none of them satisfactory. Up to now, all of them either undetected or excluded by observation. Because of these troubles induced by dark matter, some models have been built for alternative of the dark matter hypothesis. Their main ideas are to assume that the Newtonian gravity or Newton’s dynamics is invalid in galactic scale. In particular, two models explain well the flat rotational curves successfully without invoking dark matter. One is higher-order gravitational theory[5]. The gravitational potential was supposed[6] of the Yukawa form, φ(r) = − GM r(1 + α) ( 1 + αe0 )