Thermodynamic invariance of the energy-momentum tensor under matter-Lagrangian choices and its astrophysical implications in $f(R,T)$ gravity

The correct choice for the matter Lagrangian $({\mathcal{L}}_{\mathcal{M}})$ in the framework of $f(R,T)$ theory of gravity, has been a fundamental yet often overlooked ambiguity. It has been a long-standing issue, whether to choose ${\mathcal{L}}_{\mathcal{M}}=p$ or $-\rho$ as the proper definition of matter sector. In this work, we show that both choices lead to the same energy-momentum tensor, from thermodynamic point of view. However, for these two choices, the structure of the TOV equations are different. We construct and solve TOV equations using MIT bag model equation of state for ${\mathcal{L}}_{\mathcal{M}}=p$ and $-\rho$, and study the impact of the choices for matter Lagrangian on the maximum mass limit as well as M-R plot of compact stars. It is interesting to note that allowed range of gravity-matter coupling coefficient $({\alpha }_c)$ is also different for ${\mathcal{L}}_{\mathcal{M}}=p$ and ${\mathcal{L}}_{\mathcal{M}}=-\rho$, i.e., ${\alpha }_c$ can not be taken arbitrarily. Notably, through ${\mathcal{L}}_{\mathcal{M}}=p$, we achieve a maximum mass of $2.78M_{\odot }$, whereas for ${\mathcal{L}}_{\mathcal{M}}=-\rho$, we obtain a maximum mass of $2.41M_{\odot }$. So, despite the same energy-momentum tensor for different choices of ${\mathcal{L}}_{\mathcal{M}}$, the upper limit of maximum mass is significantly modified.