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

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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