Narrow Results

Models of static wormholes within the $f(R,T)$ extended theory of gravity are investigated, in particular the family $f(R,T)=R+\lambda T$, with $T=\rho +P_r+2P_l$ being the trace of the energy–momentum tensor. Models corresponding to different relations for the pressure components (radial and lateral), and several equations-of-state (EoS), reflecting different matter content, are worked out explicitly. The solutions obtained for the shape functions of the generated wormholes obey the necessary metric conditions, as manifested in other studies in the literature. The respective energy conditions reveal the physical nature of the wormhole models thus constructed. It is found, in particular, that for each of those considered, the parameter space can be divided into different regions, in which the exact wormhole solutions fulfill the null energy conditions (NEC) and the weak energy conditions (WEC), respectively, in terms of the lateral pressure. Moreover, the dominant energy condition (DEC) in terms of both pressures is also valid, while $\rho +P_r+2P_l=0$. A similar solution for the theory $P_r={\omega }_1\rho +{\omega }_2{\rho }^2$ is found numerically, where ${\omega }_1$ and ${\omega }_2$ are either constant or functions of $r$, leading to the result that the NEC in terms of the radial pressure is also valid. For nonconstant ${\omega }_i$ models, attention is focused on the behavior ${\omega }_i\propto r^m$. To finish, the question is addressed, how $f(R)=R+\alpha R^2$ will affect the wormhole solutions corresponding to fluids of the form $P_r={\omega }_1\rho +{\omega }_2{\rho }^2$, in the three cases such as NEC, WEC and DEC. Issues concerning the nonconservation of the matter energy–momentum tensor, the stability of the solutions obtained, and the observational possibilities for testing these models are discussed in Sec. 6.