One way of proving the positive energy theorem in general relativity is to express the energy in terms of the asymptotic values of a spinor field on a hypersurface. Given its asymptotic values near infinity, the spinor field is then propagated to the rest of the hypersurface by being assumed to satisfy some differential equation, chosen in such a way that the energy can be expressed as an integral of an everywhere positive function over the hypersurface. The main problem is then to prove the existence of solutions to the differential equation. Furthermore, no physical interpretation of this auxiliary spinor field on the entire hypersurface is known. With the same technique, Ludvigsen and Vickers gave a proof of a special case of the Penrose inequality, which states that the total energy is bounded from below by the square root of the area of a convex trapped surface. In this paper we investigate the role of the auxiliary spinor field in their argument. It is shown that the spinor field can in fact be completely eliminated in their proof. This not only simplifies the proof but also implies that no existence theorem is needed and there is no problem of having a spinor field without clear physical meaning. The possibility of doing the same simplification for the positive energy theorem is briefly discussed.