On attempts to avoid the contradiction between the first law of thermodynamics and energy-momentum tensor conservation in cosmology

Abstract For a general ideal fluid, the first law of thermodynamics (energy conservation in thermodynamics) and energy-momentum tensor conservation may be contradictory in the context of cosmology. In this work, it is found that the contradiction can be avoided by modifying the energy conservation equation of the ideal fluid in thermodynamics or redefining the equation of state of the ideal fluid in cosmology. We first study a special ideal fluid, i.e., ideal gas-like dust, whose thermodynamic properties are the same as the ones of the ideal gas. Employing these two schemes to the ideal gas-like dust respectively, we mathematically implement the consistency between energy conservation in thermodynamics and energy-momentum tensor conservation in cosmology. The entropy evolution of the ideal gas-like dust manifests that both schemes can be thermodynamically compliant under certain conditions. Finally, we extend the research to a general ideal fluid and obtain conditions for the ideal fluid to satisfy the second law of thermodynamics.