Matter Lagrangian of particles and fluids

Abstract: We consider a model where particles are described as localized concentrations of energy, with fixed rest mass and structure, which are not significantly affected by their self-induced gravitational field. We show that the volume average of the on-shell matter Lagrangian ${\mathcal L_m}$ describing such particles, in the proper frame, is equal to the volume average of the trace $T$ of the energy-momentum tensor in the same frame, independently of the particle's structure and constitution. Since both ${\mathcal L_m}$ and $T$ are scalars, and thus independent of the reference frame, this result is also applicable to collections of moving particles and, in particular, to those which can be described by a perfect fluid. Our results are expected to be particularly relevant in the case of modified theories of gravity with nonminimal coupling to matter where the matter Lagrangian appears explicitly in the equations of motion of the gravitational and matter fields, such as $f(R,{\mathcal L_m})$ and $f(R,T)$ gravity. In particular, they indicate that, in this context, $f(R,{\mathcal L_m})$ theories may be regarded as a subclass of $f(R,T)$ gravity.