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Equations of motion of the mass centers in a scalar theory of gravity with a preferred frame

This paper derives the equations of motion for the mass centers of a system of weakly gravitating bodies within the second version of a scalar theory of gravity that interprets gravity as a pressure force, utilizing a post-Newtonian approximation framework to transition from local field equations to global body dynamics.

Original authors: Mayeul Arminjon

Published 2026-04-20

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CC BY 4.0

Original authors: Mayeul Arminjon

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

Imagine the universe not as a stage where gravity is a mysterious force pulling things together, but as a fluid, like a giant, invisible ocean. This is the core idea of the theory explored in Mayeul Arminjon's paper.

Here is a breakdown of what the paper does, using simple analogies:

1. The Big Idea: Gravity as Pressure

In most people's minds, gravity is like a magnet pulling a metal ball. But in this specific theory, gravity is more like wind pressure.

The Analogy: Imagine you are floating in a calm lake. If the water pressure around you changes, you get pushed. The author suggests that space itself is filled with a "micro-ether" (a tiny, invisible fluid). Matter (like planets) are just swirling vortices in this fluid. Gravity isn't a pull; it's a push caused by differences in pressure within this fluid.
The "Preferred Frame": Unlike Einstein's General Relativity, which says there is no "special" viewpoint, this theory says there is a master clock and a master ruler (a "preferred frame") that defines how the universe is moving. It's like having a fixed map of the ocean currents that everyone agrees on.

2. The Problem: Two Versions of the Theory

The author has been working on this theory for a while.

Version 1 (v1): The first draft had a flaw. It treated space differently depending on which way you were looking (like a stretched rubber sheet that is tighter in one direction than another). This broke a fundamental rule of physics called the "Weak Equivalence Principle" (which basically says a feather and a hammer should fall at the same rate in a vacuum).
Version 2 (v2): The author fixed the theory. In this new version, space is "isotropic," meaning it looks the same in every direction. The feather and the hammer fall together again. The theory is now mathematically sound.

3. The Mission: Calculating the Dance of the Planets

Now that the theory is fixed, the author wants to know: How do planets actually move?

The Challenge: In Newton's time, calculating the solar system was like solving a puzzle with a few pieces. In Einstein's time, it got much harder. In this new theory, it's even more complex. You can't just write down a simple formula to predict where Mars will be in 100 years.
The Solution: The author uses a method called "Post-Newtonian Approximation."
- The Analogy: Imagine trying to describe the path of a leaf floating down a river. First, you describe the leaf moving in a straight line (Newton). Then, you add the gentle push of the wind (1st correction). Then, you add the tiny ripples in the water (2nd correction).
- The author calculates these "corrections" up to the first level of complexity (1PN). This allows them to predict how the "centers of mass" (the average center point) of planets and stars move when they are close to each other.

4. The "Good Separation" Trick

To make the math work, the author assumes the planets are far enough apart that they don't crash into each other immediately.

The Analogy: Think of a crowded dance floor. If everyone is packed tight, you can't predict the next move. But if the dancers are spread out (well-separated), you can predict how they will spin and move based on who is closest to them. The author uses a mathematical "separation parameter" to ensure the planets are treated as distinct dancers rather than a messy pile.

5. The Result: A New Equation of Motion

The paper's main achievement is deriving a specific equation (Equation 53 or 55 in the text) that tells us exactly how a planet accelerates.

What's New? This equation looks very similar to the famous Einstein equations used today, but it has a few extra terms.
- The "Spin" Term: One of the new terms depends on how fast a planet is spinning. It's like a self-correcting mechanism: if a planet spins, it feels a tiny extra push or pull that doesn't exist in standard Einstein gravity.
- The "Structure" Terms: The equation suggests that the internal "squishiness" or energy of a planet matters slightly more here than in standard theory.

6. Why Does This Matter?

You might ask, "Why bother with a new theory if Einstein's works so well?"

The Reality Check: Einstein's theory is great, but it's not the only way to explain gravity. This paper provides a rival recipe for how the solar system works.
The Future: The author is preparing these equations to be put into computer software. Astronomers can run simulations using this theory and compare the results with real telescope data.
- If the predictions match the data perfectly, great!
- If there are tiny differences, we might discover that Einstein's theory needs a tweak, or that this "pressure-based" theory is actually the true description of our universe.

Summary

Mayeul Arminjon has taken a theory where gravity is a pressure force in a fluid-like universe, fixed its mathematical flaws, and calculated exactly how planets should move within it. He found that while the planets mostly follow the familiar paths we know, there are tiny, unique "wiggles" caused by their spin and internal structure that could potentially be detected by future, ultra-precise measurements.

1. Problem Statement

The paper addresses the derivation of the explicit equations of motion for the mass centers of a system of weakly gravitating bodies (e.g., a solar system) within a specific alternative theory of gravity.

The Theory: The theory interprets gravity as a pressure force within a "micro-ether" (a perfect fluid). It posits a preferred inertial frame (Lorentz-Poincaré ether) and utilizes a scalar gravitational field ( $\beta$ ) to define both the gravitational acceleration and the relationship between a flat background metric and a curved physical metric.
The Version: The paper focuses on Version 2 (v2) of this theory. Previous versions (v1) violated the Weak Equivalence Principle at the point-particle limit due to anisotropic spatial metrics. Version 2 corrects this by ensuring the spatial metric is isotropic.
The Gap: While the general form of the equations of motion for v2 had been established in previous work, the fully explicit form required for numerical implementation in celestial mechanics software had not yet been derived. The complexity of the theory, compared to Newtonian mechanics, necessitates numerical solutions rather than analytical ones.

2. Methodology

The author employs an asymptotic Post-Newtonian (PN) approximation scheme tailored to this specific scalar theory. The methodology involves several distinct steps:

Asymptotic Expansion: The theory assumes a family of gravitational systems indexed by a small field-strength parameter $\lambda \propto 1/c^2$ . Fields (pressure, velocity, scalar field) are expanded in powers of $1/c^2$ .
Local Field Equations: The local equations of motion for a continuous medium (perfect fluid) are derived up to the first Post-Newtonian order (1PN). This involves expanding the energy-momentum tensor conservation law and the scalar field wave equation.
Good Separation Framework: To handle a system of $N$ distinct bodies, an asymptotic framework for "good separation" is used. This assumes bodies are well-separated ( $\eta \ll 1$ , where $\eta$ is the ratio of body radius to inter-body distance) and undergo rigid rotation at the Newtonian order.
Integration over Body Domains: The local 1PN field equations are integrated over the volume of each body ( $D_a$ ) to derive the global equations of motion for the mass centers.
Integral Evaluation: The derivation requires the explicit calculation of three complex integrals ( $I_a, J_a, K_a$ ) representing momentum, active mass density effects, and pressure/velocity coupling terms. These are evaluated in the Appendix under the assumption of spherical symmetry for the Newtonian density distribution.

3. Key Contributions

The primary contribution is the derivation of the explicit 1PN equations of motion for the mass centers in the v2 scalar theory.

Explicit Formulation: The paper provides the complete mathematical expression for the acceleration $\ddot{\mathbf{a}}^{(1)}$ of the $a$ -th body, including all 1PN correction terms.
Structure Parameters: The derivation explicitly identifies how the internal structure of the bodies influences their motion through specific parameters:
- $\varepsilon_a$ : Related to the internal kinetic energy.
- $T_a$ : Related to the internal rotational energy.
- $\xi_a$ : A parameter related to the coupling between rotation and the scalar field (spin term).
Redefinition of Mass: The paper demonstrates that certain structure-dependent terms (involving $\varepsilon_a$ and $T_a$ ) can be absorbed into a redefinition of the Newtonian mass ( $M'_a$ ), effectively "hiding" them from the observable dynamics, similar to the Strong Equivalence Principle behavior in General Relativity (GR).
Spin-Dependent Acceleration: A unique term involving $\xi_a$ is derived, representing a "self-acceleration" dependent on the body's spin ( $\omega_a$ ) and velocity. This term persists even for a single isolated body.

4. Key Results

The final equation of motion for the mass center $\mathbf{a}^{(1)}$ of body $a$ is given (Eq. 53/55) as:

$\ddot{\mathbf{a}}^{(1)} = \sum_{b \neq a} \frac{-GM_b \mathbf{N}_{ab}}{R_{ab}^2} \left[ 1 + \delta_b + \frac{1}{c^2} \left( v_a^2 + v_b^2 - \frac{3}{2}(\mathbf{N}_{ab} \cdot \mathbf{v}_b)^2 - 4U^{(a)}(a) - U^{(b)}(b) \right) \right] + \dots$

Notable features of the result:

Comparison with GR: The equation resembles the Lorentz-Droste-Einstein-Infeld-Hoffmann (LDEIH) equations of General Relativity but contains specific differences in the coefficients of the velocity and potential terms.
The $\xi_a$ Term: The equation includes a term $3 \frac{\xi_a}{c^2 M_a} (\boldsymbol{\omega}_a \times \dot{\mathbf{a}})$ . This is a distinct signature of the scalar theory's coupling to rotation, which does not appear in the standard LDEIH equations of GR in the same form.
Mass Redefinition: The parameter $\delta_b$ (Eq. 54) contains internal structure terms. The author shows that by redefining the mass $M_a \to M_a(1+\delta_a)$ , these terms vanish from the main acceleration equation, leaving only the spin term $\xi_a$ as a direct structural effect.
Numerical Viability: The derived equations are structured to be implemented in numerical integrators for celestial mechanics, allowing for the testing of this scalar theory against observational data (e.g., planetary ephemerides).

5. Significance

Theoretical Consistency: The work validates that the v2 scalar theory is mathematically consistent at the 1PN level and capable of describing multi-body systems without violating the Weak Equivalence Principle.
Observational Testing: By providing explicit equations, the paper enables the comparison of this alternative theory with high-precision solar system data. The author notes that while the equations differ from GR, parameter adjustments might allow for a fit to current data, particularly if "direct" data (uncorrected by standard GR-based reductions) are used.
Distinctive Signatures: The presence of the spin-dependent self-acceleration term ( $\xi_a$ ) offers a potential observational signature to distinguish this scalar theory from General Relativity. If $\xi_a$ is non-zero, it implies that the internal rotation of a body affects its trajectory in a way not predicted by standard GR.
Methodological Framework: The paper reinforces the utility of the specific asymptotic PN scheme used here, which expands all fields (including matter) rather than just the gravitational field, providing a rigorous alternative to standard PN approximations used in GR.

In summary, this paper bridges the gap between the theoretical formulation of a preferred-frame scalar gravity theory and its practical application in celestial mechanics, providing the necessary tools to test the theory against the solar system.