Recovering MOND from extended metric theories of gravity

Abstract

We show that the Modified Newtonian Dynamics (MOND) regime can be fully recovered as the weak-field limit of a particular theory of gravity formulated in the metric approach. This is possible when Milgrom's acceleration constant is taken as a fundamental quantity which couples to the theory in a very consistent manner. As a consequence, the scale invariance of the gravitational interaction is naturally broken. In this sense, Newtonian gravity is the weak-field limit of general relativity and MOND is the weak-field limit of that particular extended theory of gravity. We also prove that a Noether's symmetry approach to the problem yields a conserved quantity coherent with this relativistic MONDian extension.

Axiomatic Affine Unification with Large Gravitational Vector Field Yields Vector-Metric Theory of Gravitation, Electromagnetism and Field Description of Mass-Particles

“…(12) n = 4, then we will have 4 G′ . Perhaps it could be shown that these equations yield the results (or reduce to) similar to the ones obtained in the phenomenological theories of Berkenstein's TeVeS [39] and and Milgrom's MOND [41].…”

“…From the expr. (41) follows that written in this form the metric tensor r has units From this follows that the Riemann tensor…”

A Phase Space Diagram for Gravity

“…This regime appears at accelerations lower than the critical value of the MOND a 0 parameter, to reproduce the observed flat rotation curves and Tully-Fisher relation of spiral galaxies. Examples of the above are the modified dynamics approach of MOND, e.g., [1], the Lagrangian MOND schemes of, e.g., [2], covariant MOND formulations, e.g., [3,4], the Tensor-Vector-Scalar formalism of TeVeS of [5], or conformal gravity theories, e.g., [6]. From a cosmological perspective, the qualitative similarity between the early inflationary phase and the current late accelerated expansion phase has been interpreted as evidence for a common physical origin for both, in terms of modified gravity [7].…”

Extending Cosmology: The Metric Approach

“…In order to find an elegant and simple theory of gravity for which a MONDian solution is found, Bernal, Capozziello, Hidalgo & Mendoza [5] used a correct dimensional metric interpretation of Hilbert's gravitational action S f for a point mass source M generating the gravitational field, in such a way that:…”

“…In a series of recent articles, Bernal, Capozziello, Cristofano & de Laurentis [4], Bernal, Capozziello, Hidalgo & Mendoza [5], Carranza et al [10], Hernandez et al [17,18], Mendoza et al [22,23] have shown how relevant the introduction of a new fundamental physical constant a 0 ≈ 10 −10 m/s 2 with dimensions of acceleration is in excellent agreement with different phenomenology at many astrophysical mass and length sizes, from solar-system to extragalactic and cosmological scales. The introduction of the so called Milgrom's acceleration constant a 0 in a description of gravity means that any gravitational field produced by a certain distribution of mass (and hence energy) needs to incorporate the acceleration a 0 together with Newton's gravitational constant G and the speed of light c in the description of gravity.…”