# Wormholes with $\rho(R,R^{\prime})$ matter in $f(\textit{R}, \textit{T})$ gravity

## TL;DR

This paper derives exact static wormhole solutions in $f(R,T)$ gravity with specific matter-Ricci scalar relations, demonstrating conditions for traversability and expanding the understanding of wormhole matter content models.

## Contribution

It introduces new exact wormhole solutions in $f(R,T)$ gravity with specific matter-Ricci scalar couplings, exploring their traversability and properties.

## Key findings

- Exact wormhole solutions for specific matter-Ricci scalar relations.
- Traversable wormhole models with constant shape functions.
- Existence of static traversable wormholes in extended $f(R,T)$ gravity.

## Abstract

Models of static wormholes are investigated in the framework of $f(\textit{R}, \textit{T})$ gravity ($\textit{R}$ being the curvature scalar, and $\textit{T}$ 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 $R(r) = 2b^{\prime}(r)/r^{2}$. Exact wormhole solutions are obtained for three particular cases: $\rho(r) = \alpha R(r) + \beta R^{\prime}(r)$, $\rho(r) = \alpha R^{2}(r) + \beta R^{\prime}(r)$, and $\rho(r) = \alpha R(r) + \beta R^{2}(r)$, when $f(\textit{R}, \textit{T}) = R + 2 \lambda T$. For the two first ones, traversable wormhole models are obtained. When the matter energy density is of the 3rd type, only solutions with constant shape may correspond to traversable wormholes. Exact wormhole solutions with same properties can be constructed for $\rho = \alpha R(r) + \beta R^{-2}(r)$, $\rho = \alpha R(r) + \beta r R^{2}(r)$, $\rho = \alpha R(r) + \beta r^{-1} R^{2}(r)$, $\rho = \alpha R(r) + \beta r ^{2} R^{2}(r)$, $\rho = \alpha R(r) + \beta r^{3} R^{2}(r)$ and $\rho = \alpha r^m R(r) \log (\beta R(r))$, as well. On the other hand, for $f(\textit{R}, \textit{T}) = R + \gamma R^{2} + 2 \lambda T$ gravity, two wormhole models are constructed, assuming that the energy density of the wormhole matter is $\rho(r) = \alpha R(r) + \beta R^{2}(r)$ and $\rho(r) = \alpha R(r) + \beta r^{3} R^{2}(r)$, respectively. In this case, the functional form of the shape function is taken to be $b(r) = \sqrt{\hat{r}_{0} r}$ and possible existence of appropriate static traversable wormhole configurations is proven. These results can be viewed as an initial step towards using specific properties of the new exact wormhole solutions, in order to propose new functional forms for describing the matter content of the wormhole.