Wormhole solution and Energy in Teleparallel Theory of Gravity

Abstract

An exact solution is obtained in the tetrad theory of gravitation. This solution is characterized by two-parameters $k_1, k_2$ of spherically symmetric static Lorentzian wormhole which is obtained as a solution of the equation $\rho=\rho_t=0$ with $\rho=T_{i,j}u^iu^j$, $\rho_t=(T_{ij}-\displaystyle{1 \over 2}Tg_{ij}) u^iu^j$ where $u^iu_i=-1$. From this solution which contains an arbitrary function we can generates the other two solutions obtained before. The associated metric of this spacetime is a static Lorentzian wormhole and it includes the Schwarzschild black hole, a family of naked singularity and a disjoint family of Lorentzian wormholes. Calculate the energy content of this tetrad field using the gravitational energy-momentum given by M{\o}ller in teleparallel spacetime we find that the resulting form depends on the arbitrary function and does not depend on the two parameters $k_1$ and $k_2$ characterize the wormhole. Using the regularized expression of the gravitational energy-momentum we get the value of energy does not depend on the arbitrary function.