Spherically symmetric empty space and its dual in general relativity

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.