Static axisymmetric Einstein equations in vacuum: Symmetry, new solutions, and Ricci solitons

Abstract

An explicit one-parameter Lie point symmetry of the four-dimensional vacuum Einstein equations with two commuting hypersurface-orthogonal Killing vector fields is presented. The parameter takes values over all of the real line and the action of the group can be effected algebraically on any solution of the system. This enables one to construct particular one-parameter extended families of axisymmetric static solutions and cylindrical gravitational wave solutions from old ones, in a simpler way than most solution-generation techniques, including the prescription given by Ernst for this system [17]. As examples, we obtain the families that generalize the Schwarzschild solution and the C-metric. These in effect superpose a Levi-Civita cylindrical solution on the seeds. Exploiting a correspondence between static solutions of Einstein's equations and Ricci solitons (self-similar solutions of the Ricci flow), this also enables us to construct new steady Ricci solitons.

Some remarks on Einstein–Randers metrics

“…Finally, it is worth to be mentioned that, by applying symmetry transformation T γ introduced in [1] on Schwarzschild metric (9.1), M. Akbar and M. MacCallum obtain one-parameter extended families of Ricci flat metrics as follows,…”

Ricci solitons on anisotropic fluid spacetime in f(R, T)-gravity

“…Ricci solitons therefore got rapid attention of the research community and numerous authors studied different aspects of Ricci solitons. Static and steady Ricci solitons are discussed in detail by Akbar and MacCallum [11]. Ricci solitons for 4-dimensional pseudo-Riemannian manifolds are discussed in [12,13].…”