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Gravitational radiation and angular momentum flux from a slow rotating dynamical black hole

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 constructing the reference frame in terms of the compatible constant spinors and the compatibility of the coupling leading order Newman-Penrose equations. By using the slow rotation and small-tide approximation for a spinning black hole, we chose the horizon cross-section is spherical symmetric. It turns out the flux formula is rather simple and can be compared with the known results. Directly from the energy flux formula of the slow rotating 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. Thus a slow rotating dynamical horizon will asymptotically approaches an isolated horizon during late time.