Energy and Momentum in the Tetrad Theory of Gravitation

We study the energy and momentum of an isolated system in the tetrad theoryof gravitation, starting from the most general Lagrangian quadratic in torsion,which involves four unknown parameters. When applied to the static sphericallysymmetric case, the parallel vector fields take a diagonal form, and the fieldequation has an exact solution. We analyze the linearized field equation invacuum at distances far from the isolated system without assuming any symmetryproperty of the system. The linearized equation is a set of coupled equationsfor a symmetric and skew-symmetric tensor fields, but it is possible to solveit up to $O(1/r)$ for the stationary case. It is found that the generalsolution contains two constants, one being the gravitational mass of the sourceand the other a constant vector ${\grave B_\alpha}$. The total energy iscalculated from this solution and is found to be equal to the gravitationalmass of the source. We also calculate the spatial momentum and find that itsvalue coincides with the constant vector ${\grave B_\alpha}$. The linearizedfield equation in vacuum, which is valid at distances far from the source, doesnot give any information about whether the constant vector ${\grave B_\alpha}$is vanishing or not. For a weakly gravitating source for which the field isweak everywhere, we find that the constant vector ${\grave B_\alpha}$ vanishes.Comment: 17 pages, LaTeX, published in Prog. Theor. Phys. 98 No.6 (1997