Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy Sim(n-2)

TL;DR

This paper classifies and constructs new time-dependent solutions with SIM(n-2) holonomy in Lorentzian geometry, reducing the problem to solving linear equations on Einstein manifolds, leading to novel black hole and monopole solutions.

Contribution

It provides a general method to find Einstein metrics with SIM(n-2) holonomy using linear equations, extending known solutions to higher dimensions and cosmological backgrounds.

Findings

Explicit construction of Einstein metrics with SIM(n-2) holonomy.

Reduction to solving linear Laplace and Poisson equations.

New time-dependent Kaluza-Klein black holes and monopoles.

Abstract

The classifications of holonomy groups in Lorentzian and in Euclidean signature are quite different. A group of interest in Lorentzian signature in n dimensions is the maximal proper subgroup of the Lorentz group, SIM(n-2). Ricci-flat metrics with SIM(2) holonomy were constructed by Kerr and Goldberg, and a single four-dimensional example with a non-zero cosmological constant was exhibited by Ghanam and Thompson. Here we reduce the problem of finding the general $n$-dimensional Einstein metric of SIM(n-2) holonomy, with and without a cosmological constant, to solving a set linear generalised Laplace and Poisson equations on an (n-2)-dimensional Einstein base manifold. Explicit examples may be constructed in terms of generalised harmonic functions. A dimensional reduction of these multi-centre solutions gives new time-dependent Kaluza-Klein black holes and monopoles, including time-dependent black holes in a cosmological background whose spatial sections have non-vanishing curvature.