Paper detail

New derivation of the Lagrangian of a perfect fluid with a barotropic equation of state

In this paper we give a simple proof that when the particle number is conserved, the Lagrangian of a barotropic perfect fluid is $\mathcal{L}_m=-ρ[c^2 +\int P(ρ)/ρ^2 dρ]$, where $ρ$ is the \textit{rest mass} density and $P(ρ)$ is the pressure. To prove this result nor additional fields neither Lagrange multipliers are needed. Besides, the result is applicable to a wide range of theories of gravitation. The only assumptions used in the derivation are: 1) the matter part of the Lagrangian does not depend on the derivatives of the metric, and 2) the particle number of the fluid is conserved ($\nabla_σ(ρu^σ)=0$).