  We discuss the behaviour of massive modes near a horizon based on a study of
the dispersion relation and wave packet simulations of the Klein-Gordon
equation. We point out an apparent paradox between two (in principle
equivalent) pictures of black hole evaporation through Hawking radiation. In
the picture in which the evaporation is due to the emission of positive-energy
modes, one immediately obtains a threshold for the emission of massive
particles. In the picture in which the evaporation is due to the absorption of
negative-energy modes, such a threshold apparently does not exist. We resolve
this paradox by tracing the evolution of the positive-energy massive modes with
an energy below the threshold. These are seen to be emitted and move away from
the black hole horizon, but they bounce back at a "red horizon" and are
re-absorbed by the black hole, thus compensating exactly for the difference
between the two pictures. For astrophysical black holes, the consequences are
curious but do not affect the terrestrial constraints on observing Hawking
radiation. For analogue gravity systems with massive modes, however, the
consequences are crucial and rather surprising.
