Relativistic MOND from modified energetics

We begin to investigate the question of what modifications in the energy-momentum tensor can yield the correct MOND regime. As a starting study, we refrain from insisting on an action principle and focus exclusively on the equations of motion. The present work, despite the absence of an explicit action functional, can be regarded to extend Milgrom's modified inertia approach to relativistic domain. Our results show that a proper MOND limit arises if the energy-momentum tensor is modified to involve the determinant of the metric tensor in reference to the flat metric, where the latter is dynamically generated as in the gravitational Higgs mechanism. This modified energy-momentum tensor is conserved in both Newtonian and MONDian regimes.