Gravitational complexity factor of anisotropic polytropes in coincident gauge $$f(\mathbb {Q})$$ gravity

Abstract This work presents a novel characterization of complexity for gravitationally bound astrophysical configurations arising from two factors: (i) inhomogeneity and (ii) anisotropy in the complex arrangement of stellar structures. For this purpose, we employ the non-metricity-motivated gravitational model with a linear choice of coincident gauge $$f(\mathbb {Q})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> gravity, given by $$f(\mathbb {Q}) = \beta _0 \mathbb {Q} + \beta _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>β</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>Q</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>β</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> , where $$\beta _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>β</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> and $$\beta _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>β</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> are model parameters. Our analysis begins by postulating that a fluid distribution exhibiting density uniformity and pressure isotropy is characterized by a minimal (or zero) gravitational complexity factor. The novelty of this study lies in its ability to study the effect of non-metricity on the intricate mechanism of dense-matter static stars while also considering the effectiveness of the complexity factor in determining fluctuations in the Tolman mass for both complex and non-complex compact structures within a non-metricity framework. The variation in the Tolman gravitational mass is caused by a suggested formulation of anisotropic pressure and density non-uniformity. It is observed that in Einstein’s gravitational model, a stellar system that features both can exhibit zero complexity ( $$Y_{TF}=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow> <mml:mi>TF</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ) if their contributions cancel out. On the other hand, the linear $$f(\mathbb {Q})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> model compels the gravitational configuration to maintain $$Y_{TF}\ne 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow> <mml:mi>TF</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> due to non-metricity contributions, even when the fluid exhibits density uniformity and pressure anisotropy. Furthermore, we discuss the construction of anisotropic self-gravitating polytropes by coupling the $$Y_{TF}=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Y</mml:mi> <mml:mrow> <mml:mi>TF</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> condition with a polytropic EoS. This underscores the importance of the zero-complexity criterion in modeling astrophysical compact systems.