Higher-Dimensional Vacuum Einstein Equations: Symmetry, New Solutions, and Ricci Solitons

TL;DR

This paper demonstrates a symmetry-based method to generate new solutions and Ricci solitons in higher-dimensional vacuum Einstein equations with specific symmetries, extending known four-dimensional results.

Contribution

It introduces a universal, algebraic approach to generate solution families in higher dimensions, expanding the symmetry analysis of vacuum Einstein equations.

Findings

One-parameter solution families can be generated algebraically from seed solutions.

The method extends four-dimensional symmetry results to higher dimensions.

First example of purely algebraic solution generation in higher-dimensional vacuum Einstein equations.

Abstract

We show that the system of vacuum Einstein equations (i.e., Ricci-flat metrics) with two hypersurface-orthogonal, commuting Killing vector fields in $d \ge 5$ dimensions is invariant under the action of a one-parameter Lie group, and the group action on any metric can be expressed in a closed, universal form. This enables the generation of a one-parameter family of solutions from any given ``seed" solution of the system without solving additional equations, as well as one-parameter families of local steady Ricci solitons. This extends the Lie point symmetry in four dimensions, found earlier for axisymmetric static vacuum systems, and provides the first example of solution generation in higher-dimensional vacuum Einstein equations that can be realized purely algebraically.