Uniqueness Theorem for Abstract Hyperbolic Equations with Application to the Uniqueness of the Harmonic Coordinate System in General Relativity

An analog of the radiation condition is found for an abstract hyperbolic equation. When this condition holds, the uniqueness theorem for a&z&u) &(p8,u) + Au = f is valid. Here n, p, and A are some linear operators depending on t and Y. For example, Eq. 0 u = &(uoo(t, X) 8,~) &(u$~u) + q(t, x) u = f is of such a type. In what follows a, = a/a, , ai = a/axi , a, = a/a,. , x = (XI )...) xN), 1 < i, j < N, where repeating indexes are summed. The results obtained can be applied to the scattering theory of hyperbolic equations. They will be used in the proof of Fock’s conjecture of uniqueness (up to the Lorentz transformation) of the harmonic coordinate system. This conjecture has previously been shown to be valid only for gu” = -8uy, 1 < V, TV < 3, goV == 0, 1 < v < 3, go0 = 1, gUV being the fundamental tensor [l]. It follows from our results that the conjecture is valid for an arbitrary central-symmetric gravitation field which is Galilean at the infinity; in particular for the Schwarzschild tensor. If aoo , aij do not depend on t, the uniqueness theorem follows from the uniqueness theorems for elliptic equations. In this case the radiation condition is less restrictive than in our more general case [2, I]. In [2, II] a uniqueness theorem for some special type of Eq. 0 u = 0 with aoo depending on t was announced. In Section 1 the main result (Theorem I and a corollary) is given, in Section 2 its proof is presented, in Section 3 an application to partial differential equations and to general relativity is considered.