Neutrino Masses and Cosmic Ray Energies below the GZK Cut-Off

In the seesaw hypothesis the small neutrino mass is related to the very large mass of the order 10$^{\mathrm{27}}$ eV (GUT mass), far above the GZK cut-off for cosmic rays. Here we propose that the vacuum of space is a Planck mass plasma of positive and negative Planck mass particles [1]. In this hypothesis Dirac spinors of mass $m$ are composed of gravitationally bound very large positive and negative masses, m$^{\mathrm{\pm }} \quad =$ \textpm (žmc/2G)$^{\mathrm{1/3}}$ ( G, gravitational constant) [2]. For the neutrino mass m $=$ 0.06 eV, one has m$^{\mathrm{\pm }} \quad \approx $ \textpm 2 x 10$^{\mathrm{18}}$ eV, located near the minimum of the ankle for the cosmic ray spectrum. This is just below the GZK cut-off, where the incoming cosmic radiation would resonantly interact with a neutrino dark matter background, showing in the depression of the cosmic ray spectrum at this energy. [1] F. Winterberg, Z. Naturforsch. \textbf{58a}, 231 (2003). [2] H. H\"{o}nl and A. Papapetrou, Z. Phys. \textbf{114}, 478 (1939).