Dual spacetimes, Mach's Principle and topological defects

TL;DR

This paper introduces a duality transformation in spacetime geometry that interchanges electric parts of curvature, relates to topological defects, and modifies Schwarzschild solutions to align with Mach's Principle.

Contribution

It defines a novel duality transformation in spacetime curvature that connects topological defects with modified black hole solutions and Mach's Principle.

Findings

Dual spacetimes describe topological defects like monopoles and textures.

The duality transformation interchanges Ricci and Einstein curvatures.

Schwarzschild spacetime can be transformed into a dual spacetime with non-flat asymptotics.

Abstract

By resolving the Riemann curvature relative to a unit timelike vector into electric and magnetic parts, we define a duality transformation which interchanges active and passive electric parts. It implies interchange of roles of Ricci and Einstein curvatures. Further by modifying the vacuum/flat equation we construct spacetimes dual to the Schwarzschild solution and flat spacetime. The dual spacetimes describe the original spacetimes with global monopole charge and global texture. The duality so defined is thus intimately related to the topological defects and also renders the Schwarzschild field asymptotically non-flat which augurs well with Mach's Principle.