Volume 105A, number 7 PHYSICS LETTERS 29 October 1984 DETERMINATION OF ALL PETROV TYPE-N SPACE-TIMES ON WHICH THE CONFORMALLY INVARIANT SCALAR WAVE EQUATION SATISFIES HUYGENS' PRINCIPLE J. CARMINATI and R.G. McLENAGHAN Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada Received 10 August 1984 It is shown that the conformally invariant wave equation on a Petrov type-N space-time satisfies Huygens' principle if and only if the space-time is conformally related to a plane wave space-time. Hadamard [ 1 ] posed the general problem, as yet unsolved, of determining up to equivalence all the second- order linear hyperbolic partial differential equations in n independent variables that satisfy Huygens' principle in the strict sense. We recall that such an equation may be written in coordinate invariant form as gabu;a b + Aau,a + Cu = 0, (1) where gab are the contravariant components of the metric tensor of an n-dimensional lorentzian space L n of sig- nature 2 - n, and ";" denotes the covariant derivative with respect to the Levi-Civita connection. The coeffi- cients gab, A a, and C and L n are assumed to be of class C °*. Cauchy's problem for eq. (1) is the problem of determining a solution which assumes given values of u and its normal derivative on a given space-like (n - 1)-dimensional submanifold S. These given values are called Cauchy data. The local existence and uniqueness of the solution of Cauchy's problem for (1) has been proved by Hada- mard. A modern treatment using distributions has been given by Friedlander [2]. The considerations of this paper will be purely local. Of particular importance in Cauchy's problem is the domain of dependence of the solution. In this regard, Hadamard has shown that for any Xo, U(Xo) depends only on the data in the interior of the intersection of the past null conoid C-(xo) with S and on the intersection itself. Eq. (1) satisfies Huygens' principle (in the strict sense) iff for every Cauchy problem and every point x 0 E L n the solution depends only on the data in an arbi- trarily small neighbourhood of S n C-(xo). Such an equation is called a Huygens' differential equation. The most familiar of these equations are the ordinary wave equations that may be obtained from (1) by setting gab = diag(1, -1 ..... -1), A a = C = 0, and n = 2m, m = 2, 3 ..... Hadamard showed that in order that (1) be a Huygens' differen- tial equation it is necessary that n/> 4 be even. He also obtained a complicated necessary and sufficient condition for (1) to be a Huygens' differential equation and wondered if every such equation was equivalent to the ordinary wave equation with the appropriate number of independent variables. This is often called "Hadamard's conjecture" in the literature. We recall that two equations of the form (1) are equivalent iff one may be transformed to the other by any of the following trivial transformations that preserve the Huygens' character of the equation: (a) a transformation of coordinates; (b) multiplication of both sides of (1) by a non-vanishing factor e-2¢(x), which transformation induces a con- formal transformation of the metric: gab = e2~gab ; (2) 0.375-9601 / 84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 351