Comment on ‘‘Self-dual teleparallel formulation of general relativity and the positive energy theorem’’ V. Pelykh * Pidstryhach Institute Applied Problems in Mechanics and Mathematics, Ukrainian National Academy of Sciences, 3B Naukova Street, Lviv, 79601, Ukraine (Received 13 July 2004; published 30 November 2005) We give a correct tensor proof of the positive energy problem for the case including momentum on the basis of conditions of the existence of the two-to-one correspondence between the Sen-Witten spinor field and the Sen-Witten orthonormal frame. These conditions were obtained in our previous publications, but the true significance of our works was not estimated properly by Chee, and these were not correctly quoted in his publication. On other hand, the main result of our work is a substantial argument in favor of the geometrical nature of the Sen-Witten spinor field. DOI: 10.1103/PhysRevD.72.108502 PACS numbers: 04.50.+h, 04.20.Fy, 04.20.Gz Ever since Witten developed the spinor method to prove the positive energy theorem (PET) for gravity, the problem of comparing this method with the tensor methods has been a subject of continuing interest. Goldberg’s initial categori- cal negation of the possibility that connections exist be- tween these two methods [1] (‘‘For the first time spinors have an intrinsic role for which tetrads cannot be substi- tuted’’) was partially disproved by Dimakis and Mu ¨ller- Hoissen [2], and later by Frauendiener [3]. Dimakis and Mu ¨ller-Hoissen supposed that the spinor field could be ‘‘replaced’’ by some orthonormal frame field, so that the existence of a global solution to the Sen-Witten equation would imply the existence of globally defined orthonormal frames on the Cauchy surface. But, in general, the solution to the Sen-Witten equation will have zeros; from this Dimakis and Mu ¨ller-Hoissen concluded that each ortho- normal frame field, as well as Nester’s special orthonormal frame field (SOF, triad) on a spacelike hypersurface in an asymptotically Minkowskian manifold, can exist almost everywhere [2,4]. Frauendiener established that correspondence may exist between the spinor field A , which satisfies on a spacelike hypersurface the Sen-Witten equation (SWE) D A B B 0; and a triad, which satisfies on a certain gauge condition, and noted that this gauge is closely related to Nester’s. But this Frauendiener result is valid only under the additional assumption that the Sen-Witten spinor field has no zeros. Nester’s SOF consists of the variables that describe the physical degrees of freedom in general relativity. Analogously, the preferred lapse N A A and shift N a 2 p i A B , constructed by Ashtekar and Horowitz [5] from the Witten spinor, give an especially simple form of gravitational Hamiltonian. Nevertheless, degeneracy of Nester’s SOF or Ashtekar and Horowitz preferred time variables, which is due to the existence of zeros of the spinor field, and may occur on subsets of dimensions lower that 3 on the Cauchy hypersurface, puts the physical sense of these two constructions in doubt. Taking this degeneracy into account, Nester [6] had sup- posed that a SOF exists at least for geometries in a neigh- borhood of Euclidean space. Chee in his paper [7] states that the Nester gauge condition can be derived from Witten’s equation without any additional conditions for all geometries, even on nonmaximal hypersurfaces. Below we prove that this statement is not valid without additional assumptions and give a corrected proof of the PET for the case including momentum. Indeed, the correspondence between the spinor field, which satisfies the Sen-Witten equation, and a triad, which satisfies a certain gauge condition, is correctly defined by the Sommers transformation [8] 1 2 p 2 L L; 2 2 p 2i L L; 3 ~ L; (1) where a is a coframe basis, L A B , A A , and ~ L j L j 1 L ^ L if and only if the spinor field A vanishes nowhere on . This follows from the fact that the bilinear form 1 2 p n A _ A A _ A A A ; where n is the unit normal one-form to , is Hermitian positive definite, and does not vanish at a point on if the solution A does not have a zero at this point. But A is the solution of the SWE, which is of elliptic type; zeros of solutions to such equations not only may, but must exist, and these have a clear physical meaning: for example, zeros of solution to the equation for vibrations of a flat membrane are the node lines of standing waves. In Chee’s work, the possible existence of node mani- folds for the SWE is not excluded but it is ignored com- pletely —there even is no mention of the assumption 2 A A 0. As a result, the Sommers trans- * Electronic address: pelykh@lms.lviv.ua PHYSICAL REVIEW D 72, 108502 (2005) 1550-7998= 2005=72(10)=108502(3)$23.00 108502-1 2005 The American Physical Society