[Paper Review] Reexamining $f(R,T)$ gravity

Sarah B. Fisher, Eric D. Carlson|arXiv (Cornell University)|Aug 14, 2019

Geophysics and Gravity Measurements4 citations

TL;DR

This paper argues that in $f(R,T)$ gravity with separable $f(R,T) = f_1(R) + f_2(T)$, the $f_2(T)$ term should be absorbed into the matter Lagrangian $\mathcal{L}_m$ rather than treated as a gravitational modification, rendering it physically meaningless. The authors demonstrate this via explicit field redefinition for free fields and perfect fluids, showing that all apparent effects of $f_2(T)$ are artifacts of Lagrangian decomposition, not new physics.

ABSTRACT

We study $f(R,T)$ gravity, in which the curvature $R$ appearing in the gravitational Lagrangian is replaced by an arbitrary function of the curvature and the trace $T$ of the stress-energy tensor. We focus primarily on situations where $f$ is separable, so that $f(R,T) = f_1(R) + f_2(T)$. We argue that the term $f_2(T)$ should be included in the matter Lagrangian ${\cal L}_m$, and therefore has no physical significance. We demonstrate explicitly how this can be done for the cases of free fields and for perfect fluids. We argue that all uses of $f_2(T)$ for cosmological modeling and all attempts to place limits on parameters describing $f_2(T)$ are misguided.

Motivation & Objective

• To challenge the physical interpretation of $f_2(T)$ terms in $f(R,T)$ gravity, which are often treated as new gravitational physics.
• To show that when $f(R,T) = f_1(R) + f_2(T)$, the $f_2(T)$ contribution can be fully reinterpreted as part of the matter Lagrangian $\mathcal{L}_m$.
• To demonstrate that cosmological and astrophysical limits on $f_2(T)$, such as those on $\chi$ in $f_2(T) = -2\chi T$, are based on a fundamental misinterpretation of the Lagrangian structure.
• To clarify that only non-separable, curvature-dependent terms in $f(R,T)$ can yield genuine new physical effects, not $f_2(T)$ alone.

Proposed method

• Use of variational calculus to derive modified Einstein equations in $f(R,T)$ gravity, identifying the role of $f_2(T)$ in the stress-energy tensor.
• Explicit field redefinition of the metric and matter fields to absorb $f_2(T)$ into $\mathcal{L}_m$, showing equivalence of equations of motion.
• Application of the method to a free scalar field, deriving a new effective Lagrangian $\mathcal{L}'$ that includes $f_2(T)$ in $\mathcal{L}_m$.
• Extension to a generalized perfect fluid, deriving a rescaled energy density and pressure that preserve the equation of state.
• Use of on-shell conditions to verify that the rescaled stress-energy tensor remains conserved and satisfies the correct thermodynamic relation $\rho' + p' = n' \partial \rho' / \partial n'$.
• Comparison of the original and redefined Lagrangians to show identical dynamics, proving $f_2(T)$ is unphysical when separable.

Experimental results

Research questions

• RQ1 Can the $f_2(T)$ term in $f(R,T) = f_1(R) + f_2(T)$ be physically separated from gravity, or is it merely a redefinition of matter?
• RQ2 What happens to the stress-energy tensor and conservation laws when $f_2(T)$ is absorbed into $\mathcal{L}_m$?
• RQ3 Why are observational limits on $f_2(T)$, such as those on $\chi$ in $f_2(T) = -2\chi T$, physically unjustified?
• RQ4 How does the equation of state of a perfect fluid transform under the redefinition that absorbs $f_2(T)$ into $\mathcal{L}_m$?
• RQ5 Under what conditions do cross-terms in $f(R,T)$, such as $f(R,T)$ with $R$ and $T$ mixed, produce genuine new physics?

Key findings

• The $f_2(T)$ term in $f(R,T) = f_1(R) + f_2(T)$ can be fully absorbed into the matter Lagrangian $\mathcal{L}_m$, making it physically indistinguishable from a redefined matter sector.
• For a free scalar field, the effective Lagrangian $\mathcal{L}'$ is equivalent to the original $\mathcal{L}$, with $f_2(T)$ reinterpreted as part of $\mathcal{L}_m$, and equations of motion remain identical.
• For a perfect fluid, the rescaled energy density $\rho'$ and pressure $p'$ satisfy the correct thermodynamic relation $\rho' + p' = n' \partial \rho' / \partial n'$, confirming consistency.
• The stress-energy tensor remains conserved in the redefined theory, showing that $f_2(T)$ does not introduce non-conservation or new dynamics.
• Observational limits on $f_2(T)$, such as those on $\chi$ in $f_2(T) = -2\chi T$, are invalid because they assume $f_2(T)$ is a gravitational correction rather than a matter redefinition.
• Only non-separable terms in $f(R,T)$—those mixing $R$ and $T$—can yield genuine new physical effects; $f_2(T)$ alone is unphysical.