An Extended New General Relativity as a Reduction of $\overline{\mbox{Poincar\'{e}}}$ Gauge Theory of Gravity

Abstract

An extended new general relativity (E. N. G. R.) is formulated as a reduction of Poincaré gauge theory (P, G. T.) of gravity whose gauge group is the covering group of the Poincaré group. Fundamental fields in the theory are a Higgs-type field ψ k , and the translational gauge potential Akµ and a matter field φ a . The components ekµ of the duals of the vierbein fields are given by ekµ = ∇ *µ ψ k = ψ k,µ + Akµ . Internal translation and the field ψ k , which the conventional new general relativity lacks, play fundamental roles in E. N. G. R. as well as in P, G. T. The restrictions on the Lagrangian density, identities and differential conservation laws following from the ( local internal translation ⊗ global SL (2, C ) ) ⊗ general coordinate transformations are given. For the specific gravitational Lagrangian LT = -(1/3κ) tklmtklm + (1/3κ) vkvk + a3akak , the generators of internal \overlinePoincaré and general affine coordinate transformations for an isolated system are examined. The results are quite parallel to those in P, G. T., but the former is not trivial reductions of the latter. The following is worth noticing also in E. N. G. R.: For the case in which {ψ k , Akµ , φ a } with ψ k ≃ e (0) kµyµ + ψ (0) k is employed as the set of independent variables, (a) the generators of ( internal translation ⊗SL (2, C )) give the energy-momentum and the total angular momentum for an isolated system, and (b) the generators of general affine coordinate transformations vanish and are trivial.