Abstract
It is demonstrated that J. Nester's tensor method for proving the theorem on the positive definiteness of gravitational energy in an asymptotically Minkowsky space is equivalent to E. Witten's spinor method, in which the Saint–Witten equation must be completed with a linear term that includes the gradient of a certain scalar potential. A new proof of the theorem on the positive definiteness of energy is proposed.
Similar content being viewed by others
References
Y. Choquet-Bruhat, “Mathematical problems in general relativity theory, ” Usp. Mat. Nauk, 40, No. 6, 3–39 (1985); English transl. in Russian Math. Surveys.
A. Ashtekar and G. T. Horowitz, “Phase space of general relativity revisited: A canonical choice of time and simplification of the Hamiltonian, ” J. Math. Phys., 25, No. 5, 1473–1480 (1984).
Y. Choquet-Bruhat and D. Christodoulou, “Elliptic systems in \({\mathcal{H}}\) sδ spaces on manifolds which are Euclidean at infinity, ” Acta Mathematika, 146, No. 2, 124–150 (1981).
A. Dimakis and F. Muller-Hoissen, “On a gauge condition for orthonormal three-frames, ” Phys. Lett. A., 112, No. 2–3, 73–74 (1989).
A. Dimakis and F. Muller-Hoissen, “Spinor fields and the positive energy theorem, ” Class. Quantum Grav., 7, No. 3, 283–295 (1990).
J. Frauendiener, “Triads and the Witten equations, ” Class. Quantum Grav., 8, No. 10, 1881–1887 (1991).
J. Kijowski, in: Marsel Grossman Meeting on General Relativity, Elsevier (1986), pp. 1681–1687.
J. Nester, “A new general energy expression with a simple positivity proof, ” Phys. Lett., 83A, No. 6, 241–244 (1981).
J. Nester, “Gauge conditions for orthogonal three-frames, ” J. Math. Phys., 30, No. 3, 624–626 (1989).
J. Nester, “A positive gravitational energy proof, ” Phys. Lett., 139A, No. 3–4, 112–114 (1989).
V. Pelykh, “Squared Weyl and Dirac fields with Sommers-Sen connection, associated with V 3 4 distribution, ” Condens. Matt. Phys., 1, No. 3, 83–89 (1998).
O. Reula, “Existence theorem for solutions of Witten's equation and nonnegativity of total mass, ” J. Math. Phys., 22, No. 8, 810–814 (1981).
P. Sommers, “Space spinors, ” J. Math. Phys., 21, No. 10, 2567–2571 (1980).
E. Witten, “A new proof of the positive energy theorem, ” Comm. Math. Phys., 80, No. 3, 381–402 (1981).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pelykh, V.O. Conditions of Equivalence of Spinor and Tensor Methods in the Problem of Positive Definiteness of the Gravitational Problem. Journal of Mathematical Sciences 107, 3592–3596 (2001). https://doi.org/10.1023/A:1011994223951
Issue Date:
DOI: https://doi.org/10.1023/A:1011994223951