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.

7 pages, minor corrections to the original version. Accepted for publication in The European Physical Journal C

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• Scalar field dark matter in clusters of galaxies
• Gravitational lensing with $f(\chi )={\chi }^{3/2}$ gravity in accordance with astrophysical observations
• Scale-Invariant Dynamics of Galaxies, MOND, Dark Matter, and the Dwarf Spheroidals
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• Strong gravitational lensing in $f\left(\chi \right)={\chi }^{3/2}$ gravity
• Brane-$f(R)$ gravity and dark matter
• Atemporality from Conservation Laws of Physics in Lorentzian-Euclidean Black Hole
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• Thermodynamics and phase transitions of galactic clustering in higher-order Modified Gravity
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• Galaxy Rotation Curves via Conformal Factors
• MOND as the weak field limit of an extended metric theory of gravity with torsion
• A relativistic description of MOND using the Palatini formalism in an extended metric theory of gravity
• A simple model for explaining Galaxy Rotation Curves
• Modified Regge calculus as an explanation of dark energy
• The massive elliptical galaxy NGC 4649 from the perspective of extended gravity
• Astrophysical constraints and insights on extended relativistic gravity
• Effective $f(R)$ actions for modified Loop Quantum Cosmologies via order reduction
• Isothermal distributions in MONDian gravity as a simple unifying explanation for the ubiquitous $\rho \propto r^{-3}$ density profiles in tenuous stellar halos
• A Phase Space Diagram for Gravity
• A Critique of Covariant Emergent Gravity
• Gravitational anomalies signaling the breakdown of classical gravity
• Wide binaries as a critical test for Gravity theories
• Flat Galactic Rotation Curves from Geometry in Weyl Gravity
• Extended models of gravity in SNIa cosmological data using genetic algorithms
• Extending cosmology: the metric approach
• Mach like principle from conserved charges
• Gravity Rainbow Effects on Higher Curvature Modification of R2 inflation
• Dynamics of clusters of galaxies with extended $f(\chi )$ gravity
• Modified Newtonian Dynamics as an entropic force
• Stellar structure of quark stars in a modified Starobinsky gravity
• Equivalent Gravities and Equivalence Principle: Foundations and experimental implications
• Recovering the fundamental plane of galaxies by $f(R)$ gravity
• General Rule for Boundary conditions from the Action Principle
• Modified Newtonian dynamics and hadronic scales