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 $L_m$ and the metric variation $\delta T_{\mu \nu }$ 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 $T_{\mu \nu }$ together with the particle number conservation condition, we derive an expression for $\delta T_{\mu \nu }$ that is independent of the choice of $L_m$. Applying this result to $f(R,T)$ gravity, we obtain the exact form of the tensor ${\mathrm{\Theta }}_{\mu \nu }=g^{\sigma \rho }\frac{\delta T_{\sigma \rho }}{\delta g^{\mu \nu }}$, 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 $T_{\mu \nu }$ of the Universe consists solely of standard components (baryonic/cold dark matter, radiation, and the cosmological constant), then $f(R,T)$ gravity satisfies the conservation law ${\mathrm{\nabla }}_{\mu }{T}^{\mu \nu }=0$ for any function $f(R,T)$. This contrasts with previous studies, which found that the conservation law holds only for a restricted class of $f(R,T)$ functions. Applying the same formalism to stellar interiors, we derive a class of functions that preserve the conservation law. We construct a specific $f(R,T)$ model that is consistent at both cosmological scales and in high-density objects such as neutron stars. Remarkably, the same parameter set in this model simultaneously alleviates the Hubble tension and reproduces the observed mass-radius (M–R) relation of neutron stars.