Suppose F ~ ~ := 1 − − 2 M ( v ) r , then the Lagrangian for null radial geodesics of the "advanced(/ingoing)" Vaidya spacetime Eq(7) is L = − − F ~ ~ v ˙ ˙ 2 + 2 v ˙ ˙ r ˙ ˙ , which has an ingoing solution v ˙ ˙ = 0 and an outgoing solution r ˙ ˙ = F ~ ~ 2 v ˙ ˙ in accordance with the definition of v in Eq(4). Now, we can construct a complex null tetrad which is adapted to the ingoing null radial geodesics and employ the Newman–Penrose formalism for perform a full analysis of the Vaidya spacetime. Such an ingoing adapted tetrad can be set up as l a = ( 1 , F ~ ~ 2 , 0 , 0 ) , n a = ( 0 , − − 1 , 0 , 0 ) , m a = 1 2 r ( 0 , 0 , 1 , i csc ⁡ ⁡ θ θ ) , and the dual basis covectors are therefore l a = ( − − F ~ ~ 2 , 1 , 0 , 0 ) , n a = ( − − 1 , 0 , 0 , 0 ) , m a = r 2 ( 0 , 0 , 1 , sin ⁡ ⁡ θ θ ) .