Gravitational Cherenkov Losses in Theories Based on Modified Newtonian Dynamics

Survival of high-energy cosmic rays (HECRs) against gravitational Cherenkov losses is shown not to cast strong constraints on modified Newtonian dynamics (MOND) theories that are compatible with general relativity (GR): theories that coincide with GR for accelerations $\gg a_0$ ($a_0$ is the MOND constant). The energy-loss rate, $\dot{E}$, is many orders smaller than those derived in the literature for theories with no extra scale. Modification to GR, which underlies $\dot{E}$, enters only beyond the MOND radius of the particle: $r_M=(Gp/c{a}_0{)}^{1/2}$. The spectral cutoff, entering $\dot{E}$ quadratically, is thus $r_M^{-1}$, not $k_{\mathrm{dB}}=p/\hslash$. Thus, $\dot{E}$ is smaller than published rates, which use $k_{\mathrm{dB}}$, by a factor $\sim (r_M{k}_{\mathrm{dB}}{)}^2\approx {10}^{39}(cp/3\times {10}^{11}\mathrm{Gev}{)}^3$. Losses are important only beyond $D_{\mathrm{loss}}\approx q{\ell }_M$, where $q$ is a dimensionless factor, and ${\ell }_M=c^2/a_0$ is the MOND length, which is $\approx 2\pi$ times the Hubble distance.