Non-exotic traversable wormholes in $f\left(R,T_{ab}T^{ab}\right)$ gravity

TL;DR

This paper demonstrates that in a generalized gravity theory called $f(R, T_{ab}T^{ab})$ gravity, it is possible to find traversable wormhole solutions that satisfy all classical energy conditions without exotic matter, using an analytical recursive approach.

Contribution

The work introduces a method to find energy-condition-satisfying wormholes in $f(R, T_{ab}T^{ab})$ gravity and derives junction conditions for matching interior solutions to exterior vacuum spacetimes.

Findings

Existence of wormhole solutions satisfying all energy conditions.

Analytical recursive algorithm for solving complex field equations.

Successful matching of wormhole interiors to Schwarzschild exterior.

Abstract

In this work we analyze traversable wormhole spacetimes in the framework of a covariant generalization of Einstein's General Relativity known as energy-momentum squared gravity, or $f\left(R,\mathcal T\right)$ gravity, where $R$ is the Ricci scalar, $\mathcal T=T_{ab}T^{ab}$, and $T_{ab}$ is the energy-momentum tensor. Considering a linear $f\left(R,\mathcal T\right)=R+\gamma \mathcal T$ form, we show that a wide variety of wormhole solutions for which the matter fields satisfy all the energy conditions, namely the null, weak, strong and dominant energy conditions, exist in this framework, without the necessity for a fine-tuning of the free parameters that describe the model. Due to the complexity of the field equations these solutions are obtained through an analytical recursive algorithm. A drawback of the solutions obtained is that they are not naturally localized, and thus a matching with an external vacuum is required. For that purpose, we derive the junction conditions for the theory, and we prove that a matching between two spacetimes must always be smooth, i.e., no thin-shells are allowed at the boundary. Finally, we use these junction conditions to match the interior wormhole spacetime to an exterior vacuum described by the Schwarzschild solution, thus obtaining traversable localized static and spherically symmetric wormhole solutions satisfying all energy conditions for the whole spacetime range. We also prove that the methods outlined in this work can be straightforwardly generalized to more complicated dependencies of the action in $\mathcal T$, as long as crossed terms between $R$ and $\mathcal T$ are absent.