⚛️ general relativity

Revisiting the Entangled Relativity gravity framework

This paper revisits Entangled Relativity by embedding it in a broader class of scalar-tensor theories that admit General Relativity solutions under an "intrinsic decoupling" condition, demonstrating that the theory satisfies observational constraints on Eddington parameters for weak-field spherical stars while highlighting the need to address its non-strict compliance with conservation laws.

Original authors: Bertrand Chauvineau

Published 2026-09-21

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CC BY 4.0

Original authors: Bertrand Chauvineau

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Gravity is the invisible architecture of our universe, the force that keeps planets in orbit and stars burning. For over a century, the standard blueprint for this force has been Albert Einstein's General Relativity. In this view, gravity is not a pull but a curvature of space and time caused by mass and energy. This theory has passed every test thrown at it, from the bending of starlight to the ripples of gravitational waves. Yet, physicists have long wondered if Einstein's equations are the final word or merely the most successful approximation of something deeper. The question is not just academic; if the rules of gravity change even slightly under different conditions, our understanding of the cosmos, from the life cycles of stars to the expansion of the universe, would need rewriting.

A decade ago, a new idea called Entangled Relativity emerged, proposing a radical shift in how gravity and matter interact. Unlike Einstein's theory, where gravity exists as a background stage even in empty space, this new framework suggests that gravity and matter are inextricably linked, or "entangled." In this view, gravity cannot exist without matter to anchor it; if you were to remove all matter from the universe, the theory suggests gravity itself would vanish. This concept challenged the traditional notion of a vacuum, where empty space can still hold gravitational waves. However, the original theory was difficult to work with because it strictly forbade empty regions, making it hard to compare with real-world observations of stars and planets that have vast empty spaces around them.

A researcher named Bertrand Chauvineau has now revisited this framework, expanding it into a broader family of theories to see if they can survive the scrutiny of modern astronomy. He introduced a flexible parameter that allows the theory to be tuned, creating a spectrum of possibilities rather than a single rigid model. The goal was to see if these theories could mimic the success of Einstein's General Relativity in our solar system while offering a different underlying structure. The investigation focused on how these theories behave when applied to the most common form of matter in the universe: perfect fluids, which are substances like the gas and plasma inside stars that flow smoothly without friction.

The core of Chauvineau's work involved finding a way to describe the matter inside a star using a mathematical formula that fits the rules of this new gravity. In standard physics, the formula used to describe matter can be tweaked slightly without changing the physical outcome, much like adding a constant to a recipe that doesn't change the taste. However, in this entangled framework, the formula itself is part of the gravitational equation, so these tweaks matter immensely. Chauvineau discovered that for the theory to work smoothly and produce results close to what we observe, the matter formula had to follow a specific rule regarding how it depends on the geometry of space. He found that while no single formula perfectly satisfied this rule for all types of matter, he could construct a specific version that worked almost perfectly for the conditions found in stars, deviating only by a tiny, almost negligible amount.

When the researcher applied this nearly perfect formula to a model of a spherical star, like our Sun, the results were striking. The theory predicted how light would bend around the star and how the star's gravity would affect the passage of time. These predictions are measured by a specific number known as the Eddington parameter. In Einstein's theory, this number is exactly one. Chauvineau's calculations showed that for his new family of theories, this number is incredibly close to one, differing by a value so small it is about ten times smaller than the current limits of our most precise instruments. This suggests that these new theories are not just mathematical curiosities but viable alternatives to Einstein's gravity that could explain our solar system just as well as the current standard.

However, the story does not end with a simple victory. The paper highlights a significant complication: in this new framework, the conservation of energy and momentum, a fundamental law of physics that states energy cannot be created or destroyed, is not strictly obeyed. While the violation is minuscule—so small that it would take decades of planetary orbits to accumulate a shift of even a single meter—it means the usual way scientists test gravity theories might need to be rethought. The researcher points out that while the numbers look promising, a definitive conclusion requires a deeper look at how planets and light actually move when this slight violation of conservation is taken into account. The work suggests that these entangled theories are worthy of serious consideration as potential alternatives to General Relativity, but it stops short of declaring them the new standard, emphasizing that the path to confirmation requires careful navigation of these subtle deviations.

Technical Summary of "Revisiting the Entangled Relativity gravity framework"

Problem Statement
The paper addresses the theoretical framework of Entangled Relativity (ER), a gravity theory proposed by Minazzoli et al. where the gravitational action vanishes in the absence of matter, implying that spacetime geometry is intrinsically "entangled" with matter content. While the original ER theory ( $q=2$ ) is purely metric, it admits no vacuum solutions, creating a conflict with observational reality where vacuum regions exist. Although a scalar-tensor (ST) reformulation exists that allows for vacuum solutions, the relationship between the metric and ST formulations, and the viability of the theory against observational constraints (specifically the Eddington parameter $\gamma$ ), requires further clarification. Furthermore, the behavior of matter sources, particularly perfect fluids, within this non-standard framework needs rigorous definition to assess the theory's physical consistency and experimental viability.

Methodology
The author, Bertrand Chauvineau, employs the following methodological steps:

Generalization: The ER framework is extended to a one-parameter family of theories, denoted $q$ -ER, where the action is defined by $S_q = \int d^4x \sqrt{-g} R |L_m/R|^q$ . The original ER corresponds to $q=2$ .
Scalar-Tensor Reformulation: The metric equations are transformed into a scalar-tensor (ST) formulation ( $q$ -EST) involving a scalar field $\Phi$ derived from the ratio of the matter Lagrangian $L_m$ to the Ricci scalar $R$ .
Intrinsic Decoupling (ID) Analysis: The paper investigates the "Intrinsic Decoupling" condition, a specific relationship between the matter Lagrangian and the stress-energy tensor trace ( $L_m = \frac{q}{2(q-1)}T$ ). The author utilizes Euler's theorem on homogeneous functions to identify classes of matter Lagrangians that satisfy this condition off-shell.
Perfect Fluid Modeling: Recognizing that standard perfect fluid (PF) Lagrangians do not strictly satisfy the ID condition in $q$ -ER, the author proposes a modified Brown-like Lagrangian (including a divergence term with parameter $h$ ). The parameter $h$ is tuned to satisfy the ID condition up to the first Post-Newtonian (1PN) order.
Weak Field and Spherical Symmetry: The modified field equations are applied to a weak-field, spherically symmetric star. The external vacuum solution is identified as the Brans-Dicke (BD) Class I solution with $\omega=0$ . The internal structure is solved to determine the deviation of the Eddington parameter $\gamma_{Edd}$ from unity.
Conservation Analysis: The paper explicitly calculates the non-conservation of the stress-energy tensor ( $\nabla_b T^b_a \neq 0$ ) both inside and outside the star, estimating the magnitude of these violations for Solar System scales.

Key Contributions

Unification of Vacuum Solutions: The paper demonstrates that while $q$ -EST theories differ from one another and from standard Brans-Dicke theory in matter-filled regions, they all converge to the vacuum Brans-Dicke theory with $\omega=0$ in vacuum regions. This implies that the theory cannot be ruled out solely by the failure of $\omega=0$ BD in Solar System tests, as the external parameters depend on the internal matter structure.
Characterization of Matter Lagrangians: A set of matter Lagrangians satisfying the ID condition is completely characterized by their homogeneity with respect to the metric tensor. Specifically, Lagrangians that are $(\frac{1}{q} + 1)$ -homogeneous in the metric automatically satisfy ID.
1PN-Compliant Perfect Fluid Lagrangian: The author constructs a specific family of Brown-like perfect fluid Lagrangians (parameterized by $h$ ) that satisfy the ID condition up to 1PN order for any $q$ . This allows for the existence of stellar solutions that are close to General Relativity (GR) solutions.
Quantification of Non-Conservation: The paper provides explicit expressions for the violation of stress-energy conservation in $q$ -ER. It shows that while the stress tensor is not strictly conserved, the violation is suppressed by factors related to the pressure-to-density ratio and the compactness of the source.

Results

Eddington Parameter ( $\gamma_{Edd}$ ): For a spherical star modeled with the proposed 1PN-compliant Lagrangian, the deviation of the Eddington parameter from unity is derived as:
$1 - \gamma_{Edd} = -\frac{4}{9} \frac{q+1}{q} \Theta$
where $\Theta$ is a dimensionless parameter related to the ratio of total pressure to total mass-energy within the star.
Numerical Viability: Applying this to the Sun, the author utilizes the value of $\Theta \simeq 3.43 \times 10^{-6}$ taken from a previous study [17]. For the original ER theory ( $q=2$ ), this yields $|1 - \gamma_{Edd}| \approx 2.28 \times 10^{-6}$ . This value is approximately one order of magnitude smaller than current observational constraints, suggesting the theory is not immediately ruled out by Solar System data.
Stress Tensor Conservation: Inside a Sun-like star, the relative violation of stress-energy conservation is estimated to be of order $10^{-11}$ . Outside the star, for planetary bodies, the deviation is further suppressed by the ratio of the stellar radius to the orbital distance ( $R/r$ ), yielding deviations of order $10^{-13}$ .
Special Cases: The paper notes that for $q=-1$ , the ID condition can be satisfied exactly for perfect fluids, but the free particle Lagrangian fails to satisfy ID. Conversely, for $q=2$ , free particles satisfy ID exactly, but perfect fluids only satisfy it up to 1PN order.

Significance and Claims
The paper claims that the $q$ -ER theories represent "worthy alternatives" to General Relativity. The primary significance lies in the fact that these theories naturally admit vacuum solutions via their scalar-tensor reformulation while maintaining a distinct internal structure that allows the Eddington parameter $\gamma$ to deviate from the standard Brans-Dicke value of $0.5$ (which is experimentally excluded) and approach the GR value of $1$.

However, the author maintains a modest stance regarding definitive conclusions. The paper explicitly states that the claim of experimental viability is provisional. A "definitive clarification" requires a re-evaluation of the link between observational data and the Eddington parameters, specifically addressing how the non-conservation of the stress-energy tensor affects the motion of test bodies (planets modeled as perfect fluid balls) and the propagation of light. While light propagation remains along null geodesics, the dynamics of massive bodies in the Post-Newtonian limit require further study to confirm that the derived $\gamma$ values are directly comparable to current experimental bounds. The paper concludes that while the numerical results are promising, a precise study of the impact of stress tensor non-conservation on planetary motion is necessary before $q$ -ER can be firmly established as a viable alternative to GR.

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