Gravity with a Non-Dynamical Metric in a Yang-Mills Framework
arXiv:1205.2690
Abstract
General relativity assigns the spacetime metric two logically distinct roles: it defines physical measurements and it carries the propagating degrees of freedom of gravity. We investigate whether these roles must be assigned to the same field. We formulate a classical Yang-Mills framework in which the metric retains its operational role in defining distances, time intervals, causal structure, index contractions, and spacetime volume, but carries no kinetic term and no independent propagating degrees of freedom. The metric is nevertheless varied as an auxiliary field, producing an algebraic consistency condition rather than a propagation equation. The gravity dynamics are instead assigned to a local GL(4,R) Yang--Mills connection with sixteen gauge components. An explicit dictionary relates the Yang-Mills gauge potential to a connection-like geometric variable, under which the field strength is rewritten in curvature form. The distinctive feature of the construction is therefore not the use of GL(4,R) or a Yang-Mills-type gravitational action by themselves, but the variational hierarchy in which a physical measuring metric is non-propagating and algebraically constrained while the connection carries the dynamics. We clarify the relation of this framework to Poincaré gauge theory, metric-affine and Palatini formulations, pure-connection gravity, teleparallel approaches, metric quadratic gravity, and the historical Stephenson-Kilmister-Yang/Fairchild sector. In the restricted torsionless, metric-compatible sector, Ricci-flat geometries including the Schwarzschild metric remain admissible exact solutions, while additional branches are retained only as formal sectors requiring separate phenomenological assessment.
Substantially revised version. Changed the title, focus on non-dynamic metric and dynamic term comes from YM gauge bosons. Clarified the relation to SKY/Fairchild-type equations and to contemporary gauge/affine gravity frameworks; added discussion of the non-dynamical metric interpretation and generalized balance-law structure