Ambiguity in matter sector for modified gravity involving δ ² Lₘ/δ gμ νδ gα β and its implications to astrophysics and cosmology

Matter density ( ρ ρ ) and radial pressure ( p ) are often used as the matter Lagrangian density ( Lₘ 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 Lₘ L m relative to the metric tensor. So, picking either p or -ρ - ρ for Lₘ L m gives different effective EFEs. This confusion appears because one usually treats the four-velocity ( u_μ u μ ) and the metric tensor ( gμ ν g μ ν ) 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 Lₘ L m . Finally, we test this approach on neutron and quark stars (ultraviolet region) and on cosmological situations with matter-dominated or dust ( $$p=0$$ p = 0 ), radiation-dominated ( p=ρ /3 p = ρ / 3 ), and vacuum energy ( p=-ρ p = - ρ ) equations of state (infrared region), showing how it clarifies the ambiguity in picking Lₘ L m for gravity models.