Wormhole geometries in modified gravity

TL;DR

This paper explores traversable wormhole solutions in a modified gravity theory, showing they can satisfy all energy conditions without fine-tuning, and develops methods for matching these solutions to external spacetimes.

Contribution

It introduces a recursive algorithm for deriving wormhole solutions in energy-momentum squared gravity that satisfy all energy conditions and establishes junction conditions for smooth matching.

Findings

Existence of wormhole solutions satisfying all energy conditions.

Development of an analytical recursive algorithm for solutions.

Derivation of junction conditions for smooth spacetime matching.

Abstract

In this thesis, we investigate traversable wormhole spacetimes within the context of a covariant generalization of Einstein's General Relativity, namely the energy-momentum squared gravity, denoted as $f\left(R,T_{ab}T^{ab}\right)$. Here, $R$ represents the Ricci scalar and $T_{ab}$ is the energy-momentum tensor. Specifically considering the linear form $f\left(R,T_{ab}T^{ab}\right)=R+\gamma T_{ab}T^{ab}$, we demonstrate the existence of numerous wormhole solutions, wherein the matter fields satisfy all energy conditions, these being the null, weak, strong, and dominant energy conditions. Remarkably, these solutions do not require fine-tuning of the free parameters inherent to the model. Given the complicated nature of the field equations, we develop an analytical recursive algorithm to derive these solutions. One limitation is that these solutions lack natural localization, necessitating then a matching with an external vacuum spacetime. To address this, we derive the junction conditions for the theory, establishing that any matching between two spacetimes must be smooth, i.e. without the presence of thin-shells at the boundary. Ultimately, applying these junction conditions allows us to match the interior wormhole spacetime with an exterior vacuum described by the Schwarzschild solution. Therefore, we obtain traversable, localized, static, and spherically symmetric wormhole solutions that satisfy all energy conditions across the entire spacetime range. Additionally, we demonstrate that the approach used in this study can be easily generalized to more complicated dependencies of the action on $T_{ab}T^{ab}$, provided there are no crossed terms between $R$ and $T_{ab}T^{ab}$.