AIP Digital Archive
We address the question of the existence and construction of nontrivial, regular solutions of the Einstein–conformally invariant massless scalar field equations, i.e., solutions (g,Φ) satisfying (1−αΦ2)Rμν=α(4∇μΦ∇νΦ−2Φ∇μ∇νΦ−gμν∇σΦ∇σΦ), ∇μ∇μΦ=0, and additionally geometry and the scalar field are regular across the degeneracy region defined as the zeros of (1−αΦ2). Under the assumptions (1) the solution (g,Φ) is minimally of class C3 and admits a hypersurface orthogonal, timelike Killing vector field ξ, and (2) relative to the three spacelike hypersurfaces perpendicular to the Killing field, the degeneracy region constitute regular two-surfaces, and the induced positive definite three metric possesses a degenerate Ricci, we show that the conformal system admits nontrivial, regular across the degeneracy region solutions and we demonstrate that any such solution necessarily admits an additional local G(3) group of isometries possessing two-dimensional orbits of constant Gaussian curvature coinciding with the Φ=cons- equipotential two surfaces. Those solutions exhibit similar properties as the Levi–Civita–Ehlers–Kundt class of static solutions of Einstein's vacuum equations. We investigate this coincidence and in particular we probe the origin of the additional local G(3) group of isometries exhibited by both classes of solutions. From the partial differential equations point of view, both systems, i.e., conformal system as well as the vacuum system, degenerate or become singular, the conformal system along solutions subject to αΦ2=1 within the static region, the vacuum along solutions subject to V=(−ξ⋅ξ)1/2→0+. We demonstrate that as a consequence of the singular nature of the dynamical equations, among all solutions possessing degenerate Ricci in the open vicinity of αΦ2=1, respectively, V→0+, the only regular across degeneracy region solutions are those characterized by a vanishing York–Cotton tensor and, furthermore, such solutions necessarily admit an additional local G(3) group of isometries possessing two-dimensional orbits of constant Gaussian curvature. © 2002 American Institute of Physics.
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