An Extended New General Relativity as a Reduction of \overlinePoincare Gauge Theory of Gravity: Generators of Internal and Coordinate Transformations

Abstract

An extended new general relativity (E. N. G. R.) is formulated as a reduction of Poincare gauge theory (P. G. T.) of gravity whose gauge group is the covering group of the Poincare group. Fundamental fields in the theory are a Higgs-type field <p', and the translational gauge potential A'p and a matter field ,pa. The components e'p of the duals of the vierbein fields are given by e'p =f7t<p'=<p',p+A'p. Internal translation and the field <p',. 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, e)) ® general coordinate transformations are given. For the specific gravitational Lagrangian LT = - (1/3K)t klm tklm +(1/3K)v'v. + a3a'a., the generators of internal Poincare 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 {<p',A'p, ,pa} with <p''''''e(O)'pY P + <p(0)' is employed as the set of il).dependent variables, (a) the generators of (internal translation ® SL(2, e)) give the energy-momentum and the total angular momentum for an isolated system, and (b) the generators of ~eneral affine coordinate transformations vanish and are trivial.