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 ≫a(0) (a(0) is the MOND constant). The energy-loss rate, E, is many orders smaller than those derived in the literature for theories with no extra scale. Modification to GR, which underlies E, enters only beyond the MOND radius of the particle: r(M)=(Gp/ca(0))(1/2). The spectral cutoff, entering E quadratically, is thus r(M)(-1), not k(dB)=p/ℏ. Thus, E is smaller than published rates, which use k(dB), by a factor ∼(r(M)k(dB))(2)≈10(39)(cp/3×10(11)  Gev)(3). Losses are important only beyond D(loss)≈qℓ(M), where q is a dimensionless factor, and ℓ(M)=c(2)/a(0) is the MOND length, which is ≈2π times the Hubble distance.