On electrogravity duality and black hole with global monopole

TL;DR

The work investigates electrogravity duality by decomposing the Riemann curvature into electric and magnetic parts relative to a timelike vector and defining a dual transformation that interchanges $E_{ab}$ and $ ilde{E}_{ab}$. It shows that in static, spherically symmetric spacetimes the dual vacuum condition leads to Schwarzschild black holes with a global monopole, while in the axially symmetric (Kerr) case the dual vacuum equation yields Kerr as electrogravity self-dual, with asymptotic monopole behavior possible. By applying the Newman–Janis algorithm to Schwarzschild with a global monopole, the authors obtain Kerr with a global monopole, consistent with an asymptotic monopole structure and related stress-energy. These results illuminate how topological defects arise from dual vacuum conditions and broaden the space of exact solutions, linking global monopole physics to electrogravity duality with potential implications for rotating spacetimes.

Abstract

By resolving the Riemann curvature into electric and magnetic parts, Einstein's equation can accordingly be written in terms of electric (active and passive) and magnetic parts. The electrogravity duality is defined by the interchange of active and passive parts. It turns out that in static and stationary spacetimes, there is a subset of the equations (that identifies the effective vacuum equation) that is sufficient to yield the vacuum solution. In spherically symmetric spacetime, the electrograv dual of the effective equation solves to give the Schwarzschild black hole with a global monopole. Interestingly, this is not so for axial symmetry, where the Kerr vacuum solution turns out to be electrograv self-dual. However, in the asymptotic limit where the effect of rotation dies out, the situation reverts to the static case, admitting a global monopole. This is also what follows when we apply the Newman-Janis transformation to the static black hole with a global monopole.