Metric variation of the energy-momentum tensor of a perfect fluid and its applications to cosmology and neutron stars

Abstract We show that the expressions for the matter Lagrangian $$L_m$$ L m and the metric variation $$\delta T_{\mu \nu }$$ δ T μ ν of a perfect fluid obtained in previous studies appear to be inconsistent with the standard energy-momentum tensor under general conditions. Consequently, a large number of studies in astrophysics and cosmology relying on these expressions may need to be re-examined. By performing a series of straightforward calculations directly on the standard energy-momentum tensor $$T_{\mu \nu } = (\epsilon + P) u_\mu u_\nu - P g_{\mu \nu }$$ T μ ν = ( ϵ + P ) u μ u ν - P g μ ν together with the particle number conservation condition, we derive an expression for $$\delta T_{\mu \nu }$$ δ T μ ν that is independent of the choice of $$L_m$$ L m . Applying this result to $$f(R,T)$$ f ( R , T ) gravity, we obtain the exact form of the tensor $$\Theta _{\mu \nu } = g^{\sigma \rho } \frac{\delta T_{\sigma \rho }}{\delta g^{\mu \nu }}$$ Θ μ ν = g σ ρ δ T σ ρ δ g μ ν , which remains an important yet long-standing controversial quantity. This expression is shown to hold also for radiation, regardless of whether particle number is conserved. A major result is that if the energy-momentum tensor $$T_{\mu \nu }$$ T μ ν of the Universe consists solely of standard components with EOS $$P = \omega \epsilon $$ P = ω ϵ where