Static Axisymmetric Einstein Equations in Vacuum: Symmetry, New Solutions and Ricci Solitons

TL;DR

This paper identifies a new symmetry in vacuum Einstein equations with two Killing vectors, allowing for simpler generation of extended axisymmetric solutions and the construction of new Ricci solitons, including generalizations of Schwarzschild and C-metric.

Contribution

It introduces an explicit one-parameter Lie symmetry for vacuum Einstein equations with two Killing vectors, facilitating solution generation and Ricci soliton construction.

Findings

Derived new families of axisymmetric static solutions including generalizations of Schwarzschild and C-metric.

Developed a method to generate solutions using the symmetry, simplifying previous techniques.

Constructed new steady Ricci solitons from static Einstein solutions.

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. 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.