Hawking Radiation as tunneling and the unified first law of thermodynamics for a class of dynamical black holes

An analysis of relations between the tunneling rate and the unified first law of thermodynamics at the trapping horizons of two kinds of spherically symmetric dynamical black holes is investigated. The first kind is the Vaidya-Bardeen black hole, the tunneling rate $\Gamma \sim e^{\triangle S}$ can be obtained naturally from the unified first law at the apparent horizon, which holds the form $dE_{H}=TdS+WdV$. Another is the McVittie solution, the action of the radial null geodesic of the outgoing particles does not always has a pole at the apparent horizon, while the ingoing mode always has one. The solution of the ingoing mode of the radiation can be mathematically reduced to the case in the FRW universe smoothly. However as a black hole, the physical meaning is unclear and even puzzling.