  A four-dimensional asymptotic expansion scheme is used to study the next
order effects of the nonlinearity near a spinning dynamical black hole. The
angular momentum flux and energy flux formula are then obtained by asymptotic
expansion and the compatibility of the coupling Newman-Penrose equations. After
constructing the reference frame in terms of the compatible constant spinors,
the energy-momentum flux is derived and it is related to the black hole area
growth. Directly from the flux formula of the spinning dynamical horizon, we
find that the physically reasonable condition on the positivity of the
gravitational energy flux yields that the shear will monotonically decrease
with time.
