Spherically symmetric empty space and its dual in general relativity

TL;DR

This paper characterizes spherically symmetric empty space in general relativity using convergence densities and introduces a duality concept that relates vacuum solutions to dual spaces with different physical properties.

Contribution

It provides a novel effective characterization of vacuum solutions via convergence densities and introduces a duality framework that generates new dual spaces from known vacuum solutions.

Findings

Space around a static particle is specified by vanishing energy and null convergence density.

Dual vacuum solutions include Schwarzschild black hole with monopole charge or string dust.

The method applies to various dimensions and yields new dual spaces.

Abstract

In the spirit of the Newtonian theory, we characterize spherically symmetric empty space in general relativity in terms of energy density measured by a static observer and convergence density experienced by null and timelike congruences. It turns out that space surrounding a static particle is entirely specified by vanishing of energy and null convergence density. The electrograv-dual$^{1}$ to this condition would be vanishing of timelike and null convergence density which gives the dual-vacuum solution representing a Schwarzschild black hole with global monopole charge$^{2}$ or with cloud of string dust$^{3}$. Here the duality$^{1}$ is defined by interchange of active and passive electric parts of the Riemann curvature, which amounts to interchange of the Ricci and Einstein tensors. This effective characterization of stationary vacuum works for the Schwarzschild and NUT solutions. The most remarkable feature of the effective characterization of empty space is that it leads to new dual spaces and the method can also be applied to lower and higher dimensions.