According to the definition of u in Eq(2), one could find that when t increases, the areal radius r would increase as well for the solution u ˙ ˙ = 0 , while r would decrease for the solution r ˙ ˙ = F 2 u ˙ ˙ . Thus, u ˙ ˙ = 0 should be recognized as an outgoing solution while r ˙ ˙ = F 2 u ˙ ˙ serves as an ingoing solution. Now, we can construct a complex null tetrad which is adapted to the outgoing null radial geodesics and employ the Newman–Penrose formalism for perform a full analysis of the outgoing Vaidya spacetime. Such an outgoing adapted tetrad can be set up as l a = ( 0 , 1 , 0 , 0 ) , n a = ( 1 , − − F 2 , 0 , 0 ) , m a = 1 2 r ( 0 , 0 , 1 , i csc ⁡ ⁡ θ θ ) , and the dual basis covectors are therefore l a = ( − − 1 , 0 , 0 , 0 ) , n a = ( − − F 2 , − − 1 , 0 , 0 ) , m a = r 2 ( 0 , 0 , 1 , sin ⁡ ⁡ θ θ ) .