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

Abstract

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 $$f\left( R,{\mathcal {T}}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mfenced> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> gravity, where R is the Ricci scalar, $${\mathcal {T}}=T_{ab}T^{ab}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>ab</mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>ab</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , and $$T_{ab}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>ab</mml:mi> </mml:mrow> </mml:msub> </mml:math> is the energy–momentum tensor. Considering a linear $$f\left( R,{\mathcal {T}}\right) =R+\gamma {\mathcal {T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mfenced> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> <mml:mi>γ</mml:mi> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> 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 $${\mathcal {T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> , as long as crossed terms between R and $${\mathcal {T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> are absent.