$R=0\mathrm{}$ spacetimes and self-dual Lorentzian wormholes

A two-parameter family of spherically symmetric, static Lorentzian wormholes is obtained as the general solution of the equation $\rho ={\rho }_t=0,$ where $\rho {=T}_{\mathrm{ij}}{u}^i{u}^j,$ ${\rho }_t{=(T}_{\mathrm{ij}}-\frac{1}{2}{\mathrm{Tg}}_{\mathrm{ij}}{)u}^i{u}^j,$ and $u^i{u}_i=-1.$ This equation characterizes a class of spacetimes which are “self-dual” (in the sense of electrogravity duality). The class includes the Schwarzschild black hole, a family of naked singularities, and a disjoint family of Lorentzian wormholes, all of which have a vanishing scalar curvature $\mathrm{}(R=0\mathrm{}).$ The properties of these spacetimes are discussed. Using isotropic coordinates we delineate clearly the domains of parameter space for which wormholes, nakedly singular spacetimes and the Schwarzschild black hole can be obtained. A model for the required “exotic” stress-energy is discussed, and the notion of traversability for the wormholes is also examined.