  One interesting proposal to solve the black hole information loss paradox
without modifying either general relativity or quantum field theory, is the
soft hair, a diffeomorphism charge that records the anisotropic radiation in
the asymptotic region. This proposal, however, has been challenged, given that
away from the source the soft hair behaves as a coordinate transformation that
forms an Abelian group, thus unable to store any information. To maintain the
spirit of the soft hair but circumvent these obstacles, we consider Hawking
radiation as a probe sensitive to the entire history of the black hole
evaporation, where the soft hairs on the horizon are induced by the absorption
of a null anisotropic flow, generalizing the shockwave considered by Hawking et
al. To do so we introduce two different time-dependent extensions of the
diffeomorphism associated with the soft hair, where one is the backreaction of
the anisotropic null flow, and the other is a coordinate transformation that
produces the Unruh effect and a Doppler shift to the Hawking spectrum.
Together, they form an exact BMS charge generator on the entire manifold that
allows the nonperturbative analysis of the black hole horizon, whose surface
gravity, i.e. the Hawking temperature, is found to be modified. The
modification depends on an exponential average of the anisotropy of the null
flow with a decay rate of 4M, suggesting the emergence of a new 2-D degree of
freedom on the horizon, which could be a way out of the information loss
paradox.
