Asymptotically Newtonian Conditions for Poincare Gauge Theory

Abstract

A class of theories with gravity treated as a gauge theory for the Poincare group has been developed principally by Hehl l ) and Hayashi and Shirafuj?) and their respective associates. These theories are very much in accord with contemporary fundamental theoretical principles. We refer the reader to the works cited above for a detailed discussion of the history and motivation for this type of theory. For these theories space-time has a Riemann-Cartan geometry, i.e., a metric compatible connection with torsion. The field equations describe how the curvature and torsion are driven by the matter energy-momentum and spin density tensors. The gravitational Lagrangian is assumed to contain terms which are at the most quadratic in the curvature and torsion. With normal parity this leads to a ten parameter class of theories. The search for experimental and theoretical restrictions on these parameters continues.2H) Here we determine the parameter values that allow solutions with Newtonian behavior at large distance from the source. We refer to such solutions as 'asymptotically Newtonian'. 'Asymptotically Newtonian' is here understood physically: The gravitational field far from an isolated source should have spherically symmetric 1/r fall off. From a correspondence principle point of view one may be tempted to regard this as an absolute requirement for any tenable theory. However 'some caution is needed. Indeed the most popular cosmological models are not 'asymptotically Newtonian'. Moreover, Hehl's favorite choice of parameters does not admit any 'asymptotically Newtonian' solutions. He regards this fact as a virtue, making a distinction between the true microphysical law of gravity and its macroscopic average. Nevertheless, a theory that permits 'asymptotically Newtonian' solutions has at least an interesting and perhaps even very desirable property. It allows the Newtonian and postNewtonian limit to be calculated and compared directly with observations. It also allows the energy and momentum of an isolated gravitating system to be defined. As a consequence, the theoretical test recently advocated by some of us,S) based'upon the