  Instead of using a three dimensional analysis on quasi-local horizons, we
adopt a four dimensional asymptotic expansion analysis to study the next order
contributions from the nonlinearity of general relativity. From the similarity
between null infinity and horizons, the proper reference frames are chosen from
the compatible constant spinors for an observer to measure the energy-momentum
and flux near quasi-local horizons. In particular, we focus on the similarity
of Bondi-Sachs gravitational radiation for the quasi-local horizons and compare
our results to Ashtekar-Kirshnan flux formular. The quasi-local energy momentum
and flux of generic isolated horizons and non-rotating dynamical horizons are
discussed in this paper.
