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Let A be a C1- Lorentz metric on ^ and let B, C, and F be locally bounded. Let <= Jt be a three-surface which is spacelike and acausal with respect to A. Then if "V is a set in A), the solution on "V of the linear equation (7.20) is uniquely determined by its values and the values of its first derivatives on 0 A).

By proposition 6.6.7, , A) is of the form JlfxR1. liqe-f, then by proposition 6.6.6, J~(q) 0 J+(3t) is compact and so may be taken for <t+. □

Thus a physical field obeying a linear equation of the form (7.20) will satisfy the causality postulate (a) of §3.2 provided the null cone of A coincides with or lies within the null cone of the space-time metric g.

In order to prove the existence of solutions of the equations (7.20) we shall need inequalities for higher order derivatives of K. We shall now take the background metric § to be at least Cs+<2 where a is a nonnegative integer and we shall take to be such that 0) n has a smooth boundary and such that there is a diffeomorphism

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