Nonglobal proof of the thin–sandwich conjecture

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Abstract
A gravitational thin–sandwich conjecture was first proposed by Wheeler and coworkers during the period 1962–4. The present paper contains a proof of the nonglobal form of this gravitational thin–sandwich conjecture. The proof (a) applies for arbitrary choices of the spatial metric and its time derivative; and (b) demonstrates the existence on a spacelike three-surface of solutions which satisfy conditions of continuity known to be sufficient to obtain existence and uniqueness of solutions to Einstein’s equations off the three-surface and existence and uniqueness of geodesics. Riquier’s existence theorem plays an important role in the proof. The relationship of the present results to previous work is discussed. Some global questions associated with the thin–sandwich conjecture are clarified. Some aspects of the relationship of the thin–sandwich conjecture to the problem of the quantization of the gravitational field are noted. Both the vacuum case and the case of a nonviscous fluid are included. The discussion allows for an arbitrary equation of state p = p( ρ).

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