The hidden truth of ideal relativistic fluids: The matter Lagrangian is its total energy density, not its pressure!

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Abstract
It has been established in the literature that the matter Lagrangian of an ideal fluid can be expressed either as its total energy density or as its pressure. In this work, we demonstrate that identifying the matter Lagrangian with the pressure leads to physical inconsistencies, which are resolved when the fluid is coupled to the gravitational field. In such a scenario, the matter Lagrangian necessarily assumes the value of the total energy density. We thus conclude that, for an ideal fluid, the only physically consistent choice for the matter Lagrangian is its total energy density.
Funding Information
  • Dirección General de Asuntos del Personal Académico, Universidad Nacional Autnoma de Mxico (IN110522)
  • Dirección General de Asuntos del Personal Académico, Universidad Nacional Autnoma de Mxico (IN118325)
  • Sistema Nacional de Investigadores (26344)
  • Secretaría de Ciencia, Humanidades, Tecnología e Innovación (1228643)