A local Lagrangian for MOND as modified inertia. (arXiv:1904.07321v1 [gr-qc])

<a href="http://arxiv.org/find/gr-qc/1/au:+Costa_R/0/1/0/all/0/1">Renato Costa</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Franzmann_G/0/1/0/all/0/1">Guilherme Franzmann</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Pereira_J/0/1/0/all/0/1">Jonas P. Pereira</a>

We propose a local Lagrangian for a point particle where its inertia part is

modified in the regime of small accelerations and its potential energy is kept

intact. This Lagrangian is such that for the standard gravitational central

force it recovers the deep MOdified Newtonian Dynamics (MOND) (accelerations

$ll a_0approx 10^{-10}$m,s$^{-2}$) equations of motion in the case of a

circular orbit. We test the stability of the equations in the gravitational

scenario by slightly perturbing them. The perturbations turn on higher

derivative terms which are unstable in the deep MOND regime and have timescales

larger than 1-3 billion years. Thus, there always are regions in the outskirts

of galaxies where instabilities are irrelevant. For intermediate MOND regimes,

instability timescales should be smaller than 1 billion years. We interpret

what this could mean astrophysically. We also present ways to probe our

approach and describe some of its subtleties, especially related to the strong

equivalence principle (violated in general) and how in some cases it could

overcome Ostrogradsky’s instabilities (with naturally occurring piecewise

Lagrangians). Our main conclusion is that our MOND-like proposal constitutes a

possible recipe where Ostrogradsky instabilities could be `tamed’ (when their

timescales are larger than the age of the universe and for some piecewise

Lagrangians), besides being a falsifiable approach in various contexts. This is

relevant to start addressing practical ways to separate MOND as modified

gravity and as modified inertia.

We propose a local Lagrangian for a point particle where its inertia part is

modified in the regime of small accelerations and its potential energy is kept

intact. This Lagrangian is such that for the standard gravitational central

force it recovers the deep MOdified Newtonian Dynamics (MOND) (accelerations

$ll a_0approx 10^{-10}$m,s$^{-2}$) equations of motion in the case of a

circular orbit. We test the stability of the equations in the gravitational

scenario by slightly perturbing them. The perturbations turn on higher

derivative terms which are unstable in the deep MOND regime and have timescales

larger than 1-3 billion years. Thus, there always are regions in the outskirts

of galaxies where instabilities are irrelevant. For intermediate MOND regimes,

instability timescales should be smaller than 1 billion years. We interpret

what this could mean astrophysically. We also present ways to probe our

approach and describe some of its subtleties, especially related to the strong

equivalence principle (violated in general) and how in some cases it could

overcome Ostrogradsky’s instabilities (with naturally occurring piecewise

Lagrangians). Our main conclusion is that our MOND-like proposal constitutes a

possible recipe where Ostrogradsky instabilities could be `tamed’ (when their

timescales are larger than the age of the universe and for some piecewise

Lagrangians), besides being a falsifiable approach in various contexts. This is

relevant to start addressing practical ways to separate MOND as modified

gravity and as modified inertia.

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