Einstein gravity extended by a scale covariant scalar field with Bekenstein term and dynamical mass generation

Under carefully chosen assumptions a single general relativistic scalar field is able to induce MOND-like dynamics in the weak field approximation of the Einstein frame (gauge) and to modify the light cone structure accordingly. This is shown by a Lagrangian model formulated in the framework of integrable Weyl geometry. It contains a Bekenstein-type (``aquadratic'') term and a second order term generating additional mass energy for the scalar field. Both are switched on only if the gradient of the scalar field is spacelike and below a MOND-typical threshold, like in the superfluid model of Berezhiani/Khoury. The mass term induces non-negligible energy and pressures of the scalar field and leads to gravitational light deflection compatible with MOND-ian free fall trajectories. In the weak field (Newton-Milgrom) approximation the Bekenstein term implies a deep MOND equation for the scalar field. In this model the external field effect of the MOND approach has to be reconsidered.