Ambiguity in matter sector for modified gravity involving $\delta^2 \mathcal{L}_{m}/\delta g^{\mu\nu}\delta g^{\alpha\beta}$ and its implications to astrophysics and cosmology

Abstract

Matter density ($\rho$) and radial pressure ($p$) are often used as the matter Lagrangian density ($\mathcal{L}_{m}$) because both are thermodynamically consistent and produce the same Einstein field equation (EFE) in general relativity (GR). New gravity models with explicit links between matter and geometry instead involve second-order derivatives of $\mathcal{L}_{m}$ relative to the metric tensor. So, picking either $p$ or $-\rho$ for $\mathcal{L}_{m}$ gives different effective EFEs. This confusion appears because one usually treats the four-velocity ($u_\mu$) and the metric tensor ($g_{\mu \nu}$) as independent. Here, we revisit the basics and offer a consistent framework by relaxing that assumption, thereby making the modified gravity theory independent of the choice of $\mathcal{L}_{m}$. Finally, we test this approach on neutron and quark stars (ultraviolet region) and on cosmological situations with radiation-dominated ($p=\rho/3$) equations of state (infrared region), showing how it clarifies the ambiguity in picking $\mathcal{L}_{m}$ for gravity models.