  We consider two applications of the factorization of infrared dynamics in QED
and gravity. The first is a redefinition of the Lorentz transformations that
makes them commute with supertranslations. The other is the process of particle
creation near a black hole horizon. For the latter we show that the emission of
soft particles factors out of the S-matrix in the fixed-background
approximation and to leading order in the soft limit. The factorization is
implemented by dressing the incoming and outgoing asymptotic states with clouds
of soft photons and soft gravitons. We find that while the soft photon cloud
has no effect, the soft graviton cloud induces a phase shift in the Bogolyubov
coefficients relating the incoming and outgoing modes. However, the flux of
outgoing particles, given by the absolute value of the Bogolyubov coefficient,
is insensitive to this phase.
