Wormholes with ρ(R, R′) matter in f(R, T) gravity

Abstract

Models of static wormholes are investigated in the framework of $f(R,T)$ gravity (where $R$ is the curvature scalar and $T$ is the trace of the energy-momentum tensor). An attempt to link the energy density of the matter component to the Ricci scalar is made, which for the Morris and Thorne wormhole metric with constant redshift function yields $R(r)=2{b}^{\ensuremath{'}}(r)/{r}^{2}$. Exact wormhole solutions are obtained for three particular cases when $f(R,T)=R+2\ensuremath{\lambda}T$: $\ensuremath{\rho}(r)=\ensuremath{\alpha}R(r)+\ensuremath{\beta}{R}^{\ensuremath{'}}(r)$, $\ensuremath{\rho}(r)=\ensuremath{\alpha}{R}^{2}(r)+\ensuremath{\beta}{R}^{\ensuremath{'}}(r)$, and $\ensuremath{\rho}(r)=\phantom{\rule{0ex}{0ex}}\ensuremath{\alpha}R(r)+\ensuremath{\beta}{R}^{2}(r)$. Additionally, traversable wormhole models are obtained for the two first cases. However, when the wormhole matter energy density is of the third type, only solutions with constant shape correspond to traversable wormholes. Exact wormhole solutions possessing the same properties can be constructed when $\ensuremath{\rho}=\ensuremath{\alpha}R(r)+\ensuremath{\beta}{R}^{\ensuremath{-}2}(r)$, $\ensuremath{\rho}=\ensuremath{\alpha}R(r)+\ensuremath{\beta}r{R}^{2}(r)$, $\ensuremath{\rho}=\ensuremath{\alpha}R(r)+\ensuremath{\beta}{r}^{\ensuremath{-}1}{R}^{2}(r)$, $\ensuremath{\rho}=\ensuremath{\alpha}R(r)+\phantom{\rule{0ex}{0ex}}\ensuremath{\beta}{r}^{2}{R}^{2}(r)$, $\ensuremath{\rho}=\ensuremath{\alpha}R(r)+\ensuremath{\beta}{r}^{3}{R}^{2}(r)$, and $\ensuremath{\rho}=\ensuremath{\alpha}{r}^{m}R(r)\mathrm{log}(\ensuremath{\beta}R(r))$, as well. On the other hand, for $f(R,T)=R+\ensuremath{\gamma}{R}^{2}+2\ensuremath{\lambda}T$ gravity, two wormhole models are constructed, assuming that the energy density of the wormhole matter is $\ensuremath{\rho}(r)=\ensuremath{\alpha}R(r)+\ensuremath{\beta}{R}^{2}(r)$ and $\ensuremath{\rho}(r)=\ensuremath{\alpha}R(r)+\ensuremath{\beta}{r}^{3}{R}^{2}(r)$, respectively. In this case, the functional form of the shape function is taken to be $b(r)=\sqrt{{\stackrel{^}{r}}_{0}r}$ (where ${\stackrel{^}{r}}_{0}$ is a constant) and the possible existence of appropriate static traversable wormhole configurations is proven. The explicit forms of the pressures ${P}_{r}$ and ${P}_{l}$ leading to this result are found in both cases. As a general feature, the parameter space can be divided into several regions according to which of the energy conditions are valid. These results can be viewed as an initial step towards using specific properties of the new exact wormhole solutions to propose new functional forms for describing the matter content of wormholes.