[Paper Review] On Rastall gravity formulation as a $f(R,\mathcal{L}_m)$ and a $f(R,T)$ theory

J. C. Fabris, Oliver F. Piattella+1 more|arXiv (Cornell University)|Nov 20, 2020

Cosmology and Gravitation Theories4 citations

TL;DR

This paper investigates whether Rastall gravity—where the stress-energy tensor's divergence is proportional to the Ricci scalar's gradient—can be derived from $f(R,\mathcal{L}_m)$ or $f(R,T)$ gravity actions. It shows that while Rastall's field equations resemble those of these extended gravity theories, the Rastall stress-energy tensor cannot be fully recovered from them except in special cases, highlighting the difficulty of embedding Rastall's covariant, invertible SET redefinition directly into standard action-based frameworks.

ABSTRACT

Rastall introduced a stress-energy tensor whose divergence is proportional to the gradient of the Ricci scalar. This proposal leads to a change in the form of the field equations of General Relativity, but it preserves the number of degrees of freedom. Rastall's field equations can be either interpreted as GR with a redefined SET, or it can imply different physical consequences inside the matter sector. We investigate limits under which the Rastall field equations can be directly derived from an action, in particular from two $f(R)$-gravity extensions: $f(R,\mathcal L_m)$ and $f(R,T)$. We show that there are similarities between these theories, but the Rastall SET cannot be fully recovered from them, apart from certain particular cases here discussed. It is remarkable that a simple, covariant and invertible redefinition of the SET, as the one proposed by Rastall, is hard to be directly implemented in the action.

Motivation & Objective

• To determine whether Rastall gravity's field equations can be derived from $f(R,\mathcal{L}_m)$ or $f(R,T)$ gravity actions.
• To examine the physical and mathematical compatibility between Rastall's modified stress-energy tensor conservation and standard action-based gravity theories.
• To clarify the role of the conserved tensor $T^{\text{(C)}}_{\mu\nu}$ in connecting Rastall's formulation to standard general relativity.
• To investigate the feasibility of deriving Rastall's non-standard SET directly from an action, particularly via field redefinitions.
• To assess whether $f(R,T)$ and $f(R,\mathcal{L}_m)$ theories can fully reproduce the dynamics of Rastall gravity under general conditions.

Proposed method

• Derives the Rastall field equations from a modified stress-energy tensor satisfying $\nabla^\mu T^{\text{(R)}}_{\mu\nu} \propto \nabla^\nu R$, with a free parameter $\gamma$.
• Introduces a conserved tensor $T^{\text{(C)}}_{\mu\nu}$ via a linear, invertible transformation of $T^{\text{(R)}}_{\mu\nu}$, enabling equivalence to standard GR with redefined matter content.
• Applies the standard GR definition $T^{\text{(S)}}_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta S_m}{\delta g^{\mu\nu}}$ to recover the conserved current from the Rastall framework.
• Uses field redefinitions (e.g., $\partial_\mu \phi \propto (\partial_\lambda \xi \partial^\lambda \xi)^{(1-k)/(2k)} \partial_\mu \xi$) to map Rastall's scalar field dynamics to a k-essence action.
• Analyzes $f(R,T)$ and $f(R,\mathcal{L}_m)$ theories by decomposing actions into gravitational and matter parts, evaluating whether the resulting $T_{\mu\nu}$ matches the Rastall form.
• Demonstrates that while $T^{\text{(C)}}_{\mu\nu}$ can be derived from a standard action, the original $T^{\text{(R)}}_{\mu\nu}$ cannot be directly obtained via metric variation of such actions.

Experimental results

Research questions

• RQ1 Can Rastall gravity's field equations be derived from an $f(R,\mathcal{L}_m)$ or $f(R,T)$ action?
• RQ2 What is the relationship between the conserved tensor $T^{\text{(C)}}_{\mu\nu}$ and the standard stress-energy tensor $T^{\text{(S)}}_{\mu\nu}$ in Rastall's framework?
• RQ3 Under what conditions can the Rastall stress-energy tensor be recovered from $f(R,T)$ or $f(R,\mathcal{L}_m)$ theories?
• RQ4 Is there a consistent action formulation that directly yields Rastall's non-conserved $T^{\text{(R)}}_{\mu\nu}$ via metric variation?
• RQ5 How do field redefinitions relate Rastall's scalar field dynamics to k-essence or standard scalar field actions?

Key findings

• Rastall's field equations can be mapped to standard GR with a redefined matter stress-energy tensor $T^{\text{(C)}}_{\mu\nu}$, which is conserved and equivalent to $T^{\text{(S)}}_{\mu\nu}$.
• The transformation between $T^{\text{(R)}}_{\mu\nu}$ and $T^{\text{(C)}}_{\mu\nu}$ is linear and invertible for $\gamma \neq 3/2$, allowing full reconstruction of the Rastall tensor from a standard action.
• The Rastall stress-energy tensor cannot be fully recovered from $f(R,T)$ or $f(R,\mathcal{L}_m)$ actions in general, except in specific cases such as when the action allows a clean split into gravitational and matter parts.
• For $f(R,T)$ theories, the $T_{\mu\nu}$ appearing inside the action is not necessarily conserved, and the conserved current does not generally match the $T_{\mu\nu}$ used in the action.
• A free scalar field in Rastall gravity can be mapped to a k-essence field via a field redefinition, and the corresponding action $S[\xi] = \int (\partial_\mu \xi \partial^\mu \xi)^{1/k} \sqrt{-g} \, d^4x$ yields the same conserved current as the Rastall formulation.
• The paper concludes that while Rastall gravity is dynamically equivalent to GR with a redefined matter content, its non-standard SET cannot be directly derived from standard action-based $f(R,\mathcal{L}_m)$ or $f(R,T)$ frameworks without additional field redefinitions or constraints.