Gravitational potential and non-relativistic Lagrangian in modified gravity with varying G

Abstract

Abstract We have recently shown that the baryonic Tully–Fisher (BTF) and Faber–Jackson (BFJ) relations imply that the gravitational ‘constant’ G in the force law vary with acceleration a as 1/a. Here we derive the converse from first principles. First we obtain the gravitational potential for all accelerations and we formulate the Lagrangian for the central-force problem. Then action minimization implies the BTF/BFJ relations in the deep MOND limit as well as weak-field Weyl gravity in the Newtonian limit. The results show how we can properly formulate a non-relativistic conformal theory of modified dynamics that reduces to MOND in its low acceleration limit and to Weyl gravity in the opposite limit. An unavoidable conclusion is that a0, the transitional acceleration in modified dynamics, does not have a cosmological origin and it may not even be constant among galaxies and galaxy clusters.