The Role of Elliptic Operators in the Initial-Value Problem for General Relativity

Abstract. The Arnowitt–Deser–Misner (ADM) equations are deeply in-tertwined with discrete spectral resolutions of an elliptic operator of Laplacetype associated with the spacelike hypersurfaces which foliate the space-timemanifold, and the non-linearities of the four-dimensional hyperbolic theoryare mapped into the potential term occurring in this operator. The ADMequations are here re-expressed as a coupled first-order system for the in-duced metric and the trace-free part of the extrinsic-curvature tensor, andtheir formulation in terms of integral equations is studied.The canonical formulation of general relativity relies on the assumptionthat space-time (M,g) is topologically Σ×Rand can be foliated by a familyof spacelike hypersurfaces Σ t , all diffeomorphic to the three-manifold Σ. Thespace-time metric g is then locally cast in the formg = −(N 2 −N i N i )dt ⊗dt+N i (dx i ⊗dt+dt ⊗dx i )+h ij dx i ⊗dx j , (1)where N is the lapse function and N i are components of the shift vectorof the foliation [1–3]. The induced metric h