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

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

$$\omega = 0$$ ω = 0

,

$$1/3$$ 1 / 3

, or

$$-1$$ - 1

(baryonic/cold dark matter, radiation, and the cosmological constant), then

$$f(R,T)$$ f ( R , T )

gravity satisfies the conservation law

$$\nabla _\mu T^{\mu \nu } = 0$$ ∇ μ T μ ν = 0

for any function

$$f(R,T)$$ f ( R , T )

. This contrasts with previous studies, which found that the conservation law holds only for a restricted class of

$$f(R,T)$$ 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

$$f(R,T)$$ 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.