Energy dominance conditions in the center of toroidal T^2- wormhole

Abstract

The conditions for the energy dominance of matter necessary for the existence of a toroidal-shaped T² neck are identified. The center of the wormhole, which is defined as a section of a wormhole with a minimum area and having the topology of a T²  2-torus is taken up. The positivity conditions of the 2nd derivatives of some components of the metric are used, which describe the change in the linear dimensions of the metric of the neck cross section at its center. The corresponding inequalities are obtained for the energy density, matter pressure and metric in the center of the neck. Some particular conditions of violation or non-violation of energy dominance conditions for constructing T² necks are examined. The energy conditions imposed on the matter forming the wormhole were analyzed and the need for these violations was considered. Inequalities were obtained that describe the energy conditions necessary for the existence of a toroidal T² neck, provided that the minima of all metric functions at the point  χ=0 are reached for all values of the angular coordinate  β simultaneously. These conditions are complex; therefore, in order to obtain more specific results clarifying the physical situation, the derivatives of the inequalities were assessed in some special cases. In one particular case, it was found that the throat of a toroidal  neck is able to exist only if there is a violation of a zero energy condition.