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

We show that the expressions for the matter Lagrangian Lₘ L m and the metric variation δ Tμ ν 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 Tμ ν = (ε + P) u_μ u_ν - P gμ ν 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 δ Tμ ν δ T μ ν that is independent of the choice of Lₘ L m . Applying this result to $$f(R,T)$$ f ( R , T ) gravity, we obtain the exact form of the tensor Θ μ ν = gσ ρ δ Tσ ρδ gμ ν 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 Tμ ν T μ ν of the Universe consists solely of standard components with EOS P = ω ε P = ω ϵ where