W. E. Couch

Tail of a Gravitational Wave

A first‐order quadrupole sandwich wave of gravitational radiation exploding from a first‐order Schwarzschild mass is examined to second order. If the second‐order field preceding the sandwich wave vanishes, it is shown that the region of space‐time following the sandwich wave contains a second‐order, imploding quadrupole wave. The rest of the second‐order field in the space‐time region following the sandwich wave is also given, and it is seen to consist of monopole, quadrupole, and 16‐pole nonradiative motions.

Perfect fluid perturbations of cosmological spacetimes in Stewart's variables

A formalism suggested by Stewart for the study of perturbations of cosmological spacetimes is applied to the case of isentropic perfect fluid perturbations. Autonomous systems of differential equations for distinct types of perturbations, including gravitational waves, vorticity perturbations, and waves of density perturbations, arise in a natural way.

Conformal invariance under spatial inversion of extreme Reissner-Nordström black holes

The extreme Reissner-Nordström geometry is shown to be conformally invariant under a spatial inversion. Generalization of this result to other geometries is briefly discussed.

A class of wave equations with progressive wave solutions of finite order

Abstract We obtain a family of wave equations with progressive wave solutions of finite order that combines and generalizes a subfamily involving the nonreflecting Poschl-Teller potentials and several subfamilies reported by Chang and Janis. The elementary equivalence of some of the known families is derived.

Algebraically special perturbations of the Schwarzschild metric

Algebraically special perturbations of the Schwarzschild metric are found and expressed in a simple form. They become singular on the event horizon.

Asymptotic Behavior of Vacuum Space‐Times

Newman and Penrose have given conditions on the asymptotic form of the Weyl tensor in empty space‐time that are sufficient to insure that the space‐time is asymptotically flat at null infinity and has the peeling property. We give considerably weaker conditions and show them to be sufficient for asymptotic flatness. Under the weaker conditions the asymptotic behavior of the Weyl tensor is more general than the case where the peeling property holds. The asymptotic dependence on a suitably defined radial coordinate is given for the basis null tetrad, the spin coefficients, and the tetrad components of the Weyl tensor.

Note on Kantowski-Sachs spacetimes

It is shown that a portion of de Sitter space can be expressed in the formds2=dt2−A2(t)dr2−B2(t)(dθ2+sin2θdφ2). It follows that it is a Kantowski-Sachs spacetime, according to the usual definition. This disproves the statement sometimes seen in the literature that all Kantowski-Sachs spacetimes are anisotropic.

Wake‐free waves in one and three dimensions

A recent paper by Gottlieb [J. Math. Phys. 29, 2434 (1988)] provides examples of acoustic wave equations, in various dimensions, that have nontrivial families of solutions that are progressing waves of order 1, and relates this to whether or not these equations satisfy Huygens’ principle. A statement made in that paper related to Huygens’ principle in one space dimension is clarified, and it is shown in this connection that, in general, the relationship between the possession of progressing wave solutions and the satisfaction of Huygens’ principle is more complex than the situation described by Gottlieb. In addition, the attractive properties of progressing waves of order 1 are retained by progressing waves of any finite order, and we use this to generalize in several ways Gottlieb’s results on ‘‘wake‐free’’ solutions of the acoustic equation in three dimensions.

Spherically symmetric space-times transparent to scalar multipole waves

Using nonscattering potentials of Chang and Janis, a large class of spherically symmetric space-times is constructed on which all multipole solutions to the minimally coupled scalar wave equation are expressible in terms of characteristic data functions in essentially as simple a fashion as for flat space-time. The space-times are transparent to multipole waves in the same sense that flat space-time is. Both conformally flat and not conformally flat space-times are obtained. Some examples are discussed which show that the variety of transparent space-times is large even within the class of Robertson-Walker spaces.

Solutions to wave equations on black hole geometries. II

Methods from the author’s previous work and the classes of solutions which they produce are extended to the wave equations which govern the massive scalar field and the massless spin‐ 1/2 field on the Kerr–Newman geometry and the massless fields of spin 1 and 2 on the Kerr geometry. The solutions found are exact and expressed in simple closed forms in terms of elementary functions, but they only exist when appropriate constraints hold on some of the black hole parameters and on the frequency of the field in some cases. The behavior on the horizon, at null infinity, and with respect to the angular variables is analyzed for some example solutions. For the examples studied, it is found that the ones having radial behavior of a normal mode are not free of angular singularities. An exact relation is established between the scalar wave equation on the extreme Kerr–Newman geometry and the Whittaker–Hill equation.