Existence theorem for solutions of Witten’s equation and nonnegativity of total mass

We prove that given an asymptotically flat (in a very weak sense) initial data set, there always exists a spinor field that satisfies Witten’s equation and that becomes constant at infinity. Thus we fill a gap in Witten’s arguments on the nonnegativity of the total mass of an isolated system, when measured at spatial infinity. We also include a review of Witten’s argument.