Self-dual Lorentzian Wormholes and Energy in Teleparallel Theory of Gravity

Two spherically symmetric, static Lorentzian wormholes are obtained in tetrad theory of gravitation as a solution of the equation $\rho ={\rho }_t=0$, where $\rho =T_{ij}{u}^i{u}^j,{\rho }_t=(T_{ij}-1\mathrm{/}2T{g}_{ij})u^i{u}^j$ and $u^i{u}_i=-1$. This equation characterizes a class of spacetime which are “self-dual” (in the sense of electrogravity duality). The obtained solutions are characterized by two-parameters $k_1,k_2$ and have a common property that they reproduce the same metric spacetime. This metric is the static Lorentzian wormhole and it includes the Schwarzschild black hole. Calculating the energy content of these tetrad fields using the {\it superpotential method given by M{\o}ller in the context of teleparallel spacetime} we find that $E=m$ or $2m$ which does not depend on the two parameters $k_1$ and $k_2$ characterize the wormhole.