  We show that a surface term should be added to the Einstein-Hilbert action in
order to properly describe quantum transitions occurring around a black hole.
The introduction of this boundary term has been advocated by Teitelboim and
collaborators and it has been used in the computation of the black hole
entropy. Here, we use it to compute the gravitational corrections to the
transition amplitudes giving rise to Hawking radiation. This surface term
implies that the probability to emit a particle is given by $e^{- \Delta A/4}$
where $\Delta A$ is the change in the area of the black hole horizon induced by
the emission. Its inclusion at the level of the amplitudes therefore relates
quantum black hole radiation to the first law of black hole dynamics. In both
cases indeed, the term expressing the change in area directly results from the
same boundary term introduced for the same reason: to obtain a well defined
action principle.
