Linear stability of the new relativistic theory of modified Newtonian
  dynamics
  We have recently proposed a simple relativistic theory which reduces to
Modified Newtonian Dynamics (MOND) for the weak-field quasistatic situations
applied to galaxies, and to cosmological behaviour as in the $\Lambda$CDM
model, yielding a realistic cosmology in line with observations. A key
requirement of any such model is that Minkowski space is stable against linear
perturbations. We expand the theory action to 2nd order in perturbations on a
Minkowski background and show that it leads to healthy dispersion relations
involving propagating massive modes in the vector and the scalar sector. We use
Hamiltonian methods to eliminate constraints present, demonstrate that the
massive modes have positive Hamiltonian and show that a non-propagating mode
with a linear time dependence may have negative Hamiltonian for wavenumbers $k<
k_*$ and positive otherwise. The scale $k_*$ is estimated to be around
$\lesssim \mathrm{Mpc}^{-1}$ so that the low momenta instability may only play
a role on cosmological scales.
