Some new results on the validity of Huygens’ principle for the scalar wave equation on a curved space-time. (English) Zbl 0557.53046

Carminati, J.; McLenaghan, R. G.

Gravitation, geometry and relativistic physics, Proc., Aussois/France 1984, Lect. Notes Phys. 212, 138-142 (1984).

[For the entire collection see Zbl 0546.00038.]
A space-time on which the conformally invariant scalar wave equation satisfies Huygens’ principle is called a Huygens’ space-time. The only known examples of Huygens’ space-times are conformally flat or conformally related to the plane wave space-time whose metric is \[ ds^ 2=2dv(du+(Dz^ 2+\bar D\bar z^ 2+ez\bar z)dv)-2dzd\bar z, \] (1) where \(D\neq 0\), and e are functions of v only. The authors have obtained the following result: For every Huygens’ space-time of Petrov type N there exist a coordinate system \((u,v,z,\bar z)\) and a function \(\Phi\) such that the metric has the form \[ ds^ 2=e^{2\Phi}\{2dv[du+(a(z+\bar z)u+m)dv]-2(dz+az^ 2dv) (d\bar z+a\bar z^ 2dv)\} \] where a is a function only of v, and the function m has the form \(m(v,z,\bar z)=\bar zG(v,z)+z\bar G(v,z)+H(v,z)+\bar H(v,\bar z),\) where the functions G and H are given by either \(G(v,z)=e(v)z+f(v)z+f(v),\quad H(v,z)=g(v)z^ 2+h(v)z,\) and e,f,g, and h are arbitrary functions or G and H satisfy the differential equations \[ (\partial^ 2G(v,z)/\partial z^ 2)=f(v) (d(v)z+e(v))^{1/d(v)}, \]
\[ \partial^ 2H(v,z)/\partial z^ 2=\frac{f(v)}{1+d(v)}(d(v)z+e(v))^{1/d(v)}g(v) z+h(v) (1+d(v))- e(v)g(v), \] where the d,e,f,g, and h satisfy certain additional algebraic constraints. The metric given in the above theorem contains, as special cases, the metric (1) and the metrics of other space-times which are not Huygens’ space-times.

Reviewer: I.Gottlieb

MSC:

53B50	Applications of local differential geometry to the sciences
58J60	Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53B30	Local differential geometry of Lorentz metrics, indefinite metrics

Keywords:

conformally invariant scalar wave equation; Huygens’ principle; Huygens’ space-times; conformally flat; plane wave space-time

Citations:

Zbl 0546.00038