Differential forms and wave equations for general relativity

In recent papers, Choquet-Bruhat and York and Abrahams, Anderson, Choquet-B ruhat, and York (we refer to both works jointly as AACY) have cast the 3 1 evolution equations of general relativity in gauge-covariant and causal " first-order symmetric hyperbolic form," thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity . A key ingredient in their construction is a certain wave equation which g overns the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equ ation in a system, describes vacuum general relativity. Whereas the approac h of AACY is based on tenser-index methods, the present formulation is writ ten solely in the language of differential forms. Our approach starts with Sparling's tetrad-dependent differential forms, and our wave equation gover ns the propagation of Sparling's two-form, which in the "time-gauge" is bui lt linearly from the "extrinsic curvature one-form." The tensor-index versi on of our wave equation describes the propagation of (what is essentially) the Arnowitt-Deser-Misner gravitational momentum.