Spherically symmetric empty space and its dual in general relativity

In the spirit of the Newtonian theory, me characterize spherically symmetri c empty space in general relativity (GR) in terms of energy density measure d by a static observer and convergence density experienced by null and time -like congruences, It turns out that the space surrounding a static particl e is entirely specified by the vanishing of energy and null convergence den sity. The electrogray-dual(1) to this condition would be the vanishing of t ime-like and null convergence density which gives the dual-vacuum solution representing a Schwarzschild black hole with global monopole charge(2) or w ith a cloud of string dust(3). Here the duality is defined by interchange o f active and passive electric parts of the Riemann curvature, which amounts to interchange of the Ricci and Einstein tensors. This effective character ization of stationary vacuum works for the Schwarzschild and NUT solutions, The most remarkable feature of the effective characterization of empty spa ce is that it leads to new dual spaces and the method can also be applied t o lower and higher dimensions.