Non-exotic traversable wormhole solutions in linear $f\left(R,T\right)$ gravity

In this work we analyze traversable wormhole solutions in the linear form of $f\left(R,T\right)=R+\lambda T$ gravity satisfying the Null, Weak, Strong, and Dominant Energy Conditions (NEC, WEC, SEC, and DEC respectively) for the entire spacetime. These solutions are obtained via a fully analytical parameter space analysis of the free parameters of the wormhole model, namely the exponents controlling the degree of the redshift and shape functions, the radius of the wormhole throat $r_0$, the value of the redshift function at the throat $\zeta_0$, and the coupling parameter $\lambda$. Bounds on these free parameters for which the energy conditions are satisfied for the entire spacetime are deduced and two explicit solutions are provided. Even if some of these bounds are violated, leading to the violation of the NEC at some critical radius $r_c>r_0$, it is still possible to find physically relevant wormhole solutions via a matching with an exterior vacuum spacetime in the region where the energy conditions are still satisfied. For this purpose, we deduce the set of junction conditions for the form of $f\left(R,T\right)$ considered and provide an explicit example. These results seem to indicate that a wide variety of non-exotic wormhole solutions are attainable in the $f\left(R,T\right)$ theory without the requirement of fine-tuning.