Non-exotic wormholes in $f(R,L_m)$ gravity

In the present analysis, we examine the potential existence of generalized wormhole models within the framework of newly developed extended $f(R,L_m)$ gravity. We investigate both a linear model, $f(R,L_m)=\alpha R+\beta L_m$, and a non-linear model, $f(R,L_m)=\frac{R}{2}+ L^\alpha_m$, to analyze traversable wormholes. By employing the variational approach, we derive modified versions of the field equations under the influence of an anisotropic matter source. A power-law shape function is applied, resulting in a linear equation of state for both radial and lateral pressures. Furthermore, we explore solutions characterized by a variable equation of state parameter. It was observed that the violation of energy conditions is influenced by the parameters $\alpha$ and $\beta$. A wide range of non-exotic wormhole solutions was discovered, dependent on the specific parameters of the model. We demonstrate that wormholes with power-law shape functions yield solutions that comply with the energy conditions in both linear and non-linear forms of $f(R, L_m)$. It is shown that the non-exotic wormhole solutions obtained within this framework are not isotropic.