Other
Scientific paper
Jun 1969
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1969phrv..182.1361h&link_type=abstract
Physical Review, vol. 182, Issue 5, pp. 1361-1368
Other
7
Scientific paper
It is shown that a necessary condition that normal-hyperbolic solutions of the Einstein vacuum field equations for the metric tensor defined by the quadratic differential form ds2=fdu2-2mdudv-ldv2- e2γ(dx2+dz2) (where f, l, m, and γ are functions of x and z, and fl+m2=x2) be of type III or N is that x-1f, x-1l, and x-1m be functions of a single function μ it is further shown that no such nonflat solutions exist. Solutions having this functional dependence are found to belong to one of three classes: the Weyl class and two classes which may be obtained from it. One of these classes is characterized by Sachs-Penrose type-I stationary solutions having one real and two distinct complex-conjugate eigenvalues. The other class is characterized by Sachs-Penrose type-II stationary solutions admitting a single shear-, twist-, and expansion-free doubly degenerate geodesic ray which is also a null, hypersurface-orthogonal Killing vector. Further invariant properties of these classes are discussed, as well as the special case where μ depends only upon x.
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