  We study the Ashtekar formulation of linear gravity starting from the ADM
first order action for the non linear theory, linearizing it, and performing a
canonical transformation that coordinatizes the phase space in terms of the
already linearized Ashtekar variables. The results obtained in this way are in
accordance with those obtained through the standard method, in which, after
introducing the Ashtekar variables for the full theory, a linearization around
the flat Abelian connection and its conjugate momentum is performed.
