A study of stable wormhole solution with non-commutative geometry in the framework of linear $$f(R,{\mathcal {L}}_m, T)$$ gravity

Abstract This research delves into the potential existence of traversable wormholes (WHs) within the framework of modified, curvature based gravity. The modification includes linear perturbations of the matter Lagrangian and the trace of the energy-momentum tensor with specific coupling strengths $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> and can thus be viewed as a special case of linear f ( R , T )-gravity with a variable matter coupling or as the simplest additively separable $$f(R,{\mathcal {L}}_m,T)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -model. A thorough examination of static WH solutions is undertaken using a constant redshift function; therefore, our work can be regarded as the first-order approximation of WH theories in $$f(R,{\mathcal {L}}_m,T)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . The analysis involves deriving WH shape functions based on non-commutative geometry, with a particular focus on Gaussian and Lorentzian matter distributions $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> . Constraints on the coupling parameters are developed so that the shape function satisfies both the flaring-out and asymptotic flatness conditions. Moreover, for positive coupling parameters, violating the null energy condition (NEC) at the WH throat $$r_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> demands the presence of exotic matter. For negative couplings, however, we find that exotic matter can be avoided by establishing the upper bound $$\beta +\alpha /2&lt;-\frac{1}{\rho r_0^2}-8\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>+</mml:mo> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:msubsup> <mml:mi>r</mml:mi> <mml:mn>0</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:mfrac> <mml:mo>-</mml:mo> <mml:mn>8</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> . Additionally, the effects of gravitational lensing are explored, revealing the repulsive force of our modified gravity for large negative couplings. Lastly, the stability of the derived WH solutions is verified using the Tolman–Oppenheimer–Volkoff (TOV) formalism.