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

TL;DR

This paper provides a straightforward derivation of the Lagrangian for a barotropic perfect fluid with conserved particle number, applicable across various gravitational theories without extra fields or Lagrange multipliers.

Contribution

It introduces a simple proof for the fluid Lagrangian that requires no additional fields or multipliers, broadening its applicability in gravitational theories.

Findings

Derived the Lagrangian expression for barotropic perfect fluids

Proved the derivation requires only particle number conservation and no metric derivatives

Applicable to a wide range of gravitational theories

Abstract

In this paper we give a simple proof that when the particle number is conserved, the Lagrangian of a barotropic perfect fluid is $\mathcal{L}_m=-\rho [c^2 +\int P(\rho)/\rho^2 d\rho]$, where $\rho$ is the \textit{rest mass} density and $P(\rho)$ 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 ($\nabla_\sigma (\rho u^\sigma)=0$).