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 (δT_{μν}) 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_{μν}) together with the particle number conservation condition, we derive an expression for (δT_{μν}) 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 (Θ_{μν} = g^{σρ} \frac{δT_{σρ}}{δg^{μν}}), 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_{μν}) 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 (\nabla_μT^{μν} = 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.

Source: arXiv:2610.05763v1 - http://arxiv.org/abs/2610.05763v1 PDF: https://arxiv.org/pdf/2610.05763v1 Original Link: http://arxiv.org/abs/2610.05763v1