Differential Forms and Wave Equations for General Relativity

Recently, Choquet-Bruhat and York and Abrahams, Anderson, ChoquetBruhat, and York (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 governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of aacy is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparling’s tetrad-dependent differential forms, and our wave equation governs the propagation of Sparling’s 2-form, which in the “time-gauge” is built linearly from the “extrinsic curvature 1-form.” The tensor-index version of our wave equation describes the propagation of (what is essentially) the Arnowitt-Deser-Misner gravitational momentum. Vienna, December 1996 Preprint TUW-96-09 (revised). Research supported by the “Fonds zur Förderung der wissenschaftlichen Forschung” in Austria (Lise Meitner Fellowship M-00182-PHY and FWF Project 10.221-PHY). email address: [email protected]