Lagrangian Extensions of Newtonian Gravity constrained by Solar System tests

This paper proposes a Lagrangian extension to Newtonian gravity involving a second dynamical scalar field, derives its post-Newtonian potential and N-body equations, and constrains the model's free parameter using observational data from the Nordtvedt effect and Mercury's perihelion shift.

The Gist

Imagine gravity not as a rigid, unchangeable law, but as a flexible fabric that can be tweaked. For centuries, Isaac Newton's version of gravity was the gold standard, perfectly explaining how apples fall and planets orbit. However, we know from Einstein that Newton's rules aren't the whole story; they miss some subtle, "relativistic" effects, like the way Mercury's orbit slowly wobbles over time.

This paper asks a fascinating question: Can we upgrade Newton's simple gravity to include these fancy relativistic effects without throwing away the simplicity of Newton's math?

Here is the breakdown of their journey, using some everyday analogies:

1. The "Two-Engine" Upgrade

Newton's original theory is like a car with a single, reliable engine. It works great for most driving. The authors wanted to add a "second engine" to this car to make it run smoother on bumpy roads (strong gravity), but they wanted to keep the dashboard simple.

They introduced a new, invisible field (a scalar field) alongside the usual gravity field. Think of the usual gravity field as the road itself, and this new field as a "wind" blowing over the road.

• The Goal: To see if this "wind" could explain the weird behaviors of planets that Newton couldn't, while still looking like Newton's gravity when you aren't looking too closely.

2. The "Weak-Field" Test Drive

The authors didn't try to simulate a black hole (where gravity is insane). Instead, they looked at our Solar System, where gravity is relatively "weak." They treated the new "wind" field as a gentle breeze that gets stronger or weaker depending on how much matter is around.

By doing some heavy math (which they call a "weak-field approximation"), they derived a new formula for gravity. This new formula has a few extra terms that act like a correction factor.

• The Result: In this new theory, the "weight" of an object (how hard gravity pulls it) isn't exactly the same as its "mass" (how much stuff is inside it). It's as if a heavy rock and a light rock, if they have different internal structures, might fall at slightly different speeds in a specific gravitational wind.

3. The "Nordtvedt Effect" (The Moon's Wobble)

One of the first tests they ran was on the Earth and the Moon.

• The Analogy: Imagine the Earth and the Moon are two dancers holding hands, spinning around the Sun. If the "wind" (the new gravity field) pushes on the Earth differently than it pushes on the Moon because they have different internal "heaviness," their dance would get out of sync.
• The Constraint: Scientists have measured the Moon's orbit with lasers for decades. They found that the Earth and Moon fall toward the Sun at exactly the same rate, to an incredibly precise degree.
• The Paper's Finding: For the authors' theory to match this reality, the "wind" must be incredibly weak. If it were any stronger, the Moon's orbit would wobble in a way we would have already seen. This puts a very strict limit on how strong their new theory can be.

4. The "Mercury Problem" (The Wobbly Orbit)

The second test was Mercury, the planet closest to the Sun.

• The Analogy: Mercury's orbit is like an oval track that slowly rotates, so the point where Mercury is closest to the Sun (the perihelion) moves forward a tiny bit every century. Newton's math predicted almost all of this movement, but there was a tiny "missing piece" of about 43 seconds of arc per century. Einstein's General Relativity filled that gap perfectly.
• The Paper's Finding: The authors tried to use their new "two-engine" gravity to fill that same gap. They calculated that to match Mercury's wobble, the "wind" parameter (called $\kappa$) needs to be a specific, non-zero number.

5. The Big Contradiction

Here is where the plot twist happens. The paper concludes with a bit of a "catch-22":

• To satisfy the Moon test (where Earth and Moon must fall together), the new gravity effect must be tiny (almost zero).
• To satisfy the Mercury test (where the orbit must wobble), the new gravity effect must be much larger.

The Verdict: You cannot have a theory that satisfies both tests at the same time. The specific version of "upgraded Newtonian gravity" they built cannot explain Mercury's wobble without breaking the rules of the Moon's dance.

Why Do This If It Doesn't Work?

You might ask, "If it fails, why write the paper?"
The authors explain that this isn't about replacing Einstein. Instead, it's like a training simulator.

• They wanted to see if a simpler, non-relativistic version of gravity could mimic the complex rules of Einstein's theory.
• Even though this specific model failed the Solar System tests, the exercise helps scientists understand how complex theories work and where the boundaries lie between simple Newtonian physics and complex Relativistic physics.
• It serves as a "map" showing us which simple modifications to gravity are possible and which ones are impossible, helping us better understand the universe's rules.

In short: They tried to build a "Newtonian 2.0" with a secret extra ingredient. They found that while the ingredient could explain Mercury's wobble, it made the Moon dance out of step. Therefore, this specific recipe doesn't work for our Solar System, but the cooking process taught them a lot about the nature of gravity.

Technical Summary

Technical Summary: Lagrangian Extensions of Newtonian Gravity Constrained by Solar System Tests

Problem Statement
While General Relativity (GR) remains the dominant framework for gravitational interactions, alternative theories are often motivated by cosmological evidence of a dark sector (dark matter and dark energy). However, many relativistic theories possess classical limits that differ from Newtonian gravity, necessitating "screening mechanisms" to recover locally tested GR results. Conversely, Newtonian gravity offers significant mathematical and numerical simplicity for analyzing weak-field astrophysical systems. The central problem addressed is whether a non-relativistic extension of Newtonian gravity can be formulated that incorporates dynamical degrees of freedom (specifically a varying gravitational coupling) and reproduces key relativistic effects—such as the Nordtvedt effect and perihelion precession—without relying on ad hoc assumptions for the evolution of extra fields.

Methodology
The authors propose a generalized Lagrangian formulation for Newtonian gravity involving two scalar fields, $\psi$ (analogous to the Newtonian potential) and $\sigma$ (a new dimensionless dynamical field). The Lagrangian is defined as:
$$\mathcal{L} = \frac{k(\sigma)}{8\pi G_0}|\nabla\psi|^2 - \frac{\omega}{8\pi G_0}\left[f(\psi, \sigma)\dot{\sigma}^2 - g(\psi, \sigma)|\nabla\sigma|^2\right] + \rho h(\sigma)\psi$$
where $k, f, g, h$ are coupling functions and $\omega$ is a dimensionless constant.

The study proceeds through the following steps:

1. Derivation of Field Equations: The Euler-Lagrange equations are applied to derive the complete field equations for $\psi$ and $\sigma$.
2. Weak-Field Approximation: The scalar fields are expanded in a perturbation series ($\psi = \psi_0 + \psi_1 + \psi_2...$, $\sigma = \sigma_0 + \sigma_1 + \sigma_2...$) around a constant background. The authors solve the equations order-by-order to derive an effective post-Newtonian gravitational potential ($U_{eff}$).
3. Equations of Motion: The acceleration of massive bodies in an $N$-body system is derived by integrating the gradient of the effective potential. This leads to a distinction between inertial mass ($m$) and gravitational mass ($M$).
4. Orbital Dynamics: The method of osculating orbits is employed to calculate the secular variation of orbital elements (specifically the pericenter argument $\omega$) for a two-body system.
5. Observational Constraints: The derived parameters are constrained using two specific Solar System datasets:
   • The Nordtvedt effect (via Lunar Laser Ranging data) to test the Strong Equivalence Principle (SEP).
   • The anomalous perihelion shift of Mercury (via MESSENGER spacecraft data).

Key Contributions

• Generalized Framework: The paper establishes a general Lagrangian framework for Newtonian gravity with a dynamical scalar field, moving beyond previous specific models (e.g., Refs. [19–21]) to a broader class of theories defined by arbitrary coupling functions.
• Effective Potential: The derivation of an effective potential containing terms proportional to $U^2$ and $\Phi_2$ (where $\Phi_2$ is the integral of $\rho U$), which are characteristic of the Post-Newtonian (PPN) expansion in GR.
• Mass Distinction: The formalism explicitly demonstrates that in this extended theory, the gravitational mass ($M$) differs from the inertial mass ($m$) due to the body's gravitational self-energy ($\Omega$), defined as $M = m(1 + 4\kappa \Omega/m)$.
• Parameter $\kappa$: The authors consolidate the complex coupling functions into a single effective parameter, $\kappa$, which governs the magnitude of post-Newtonian corrections.

Results
The application of Solar System constraints yields conflicting requirements for the parameter $\kappa$:

1. Nordtvedt Effect (SEP Violation): Using Lunar Laser Ranging data, which constrains the differential acceleration of the Earth and Moon toward the Sun, the paper derives an upper limit of $|\kappa| \lesssim 10^{-22} \, (\text{m/s})^{-2}$.
2. Mercury's Perihelion Shift: To account for the observed anomalous precession of Mercury ($42.9799 \pm 0.0009$ arcseconds/century), the model requires $\kappa \approx 3.3 \times 10^{-17} \, (\text{m/s})^{-2}$.

The paper concludes that it is impossible to formulate a single coherent theory within this specific class of extensions that simultaneously satisfies both the Nordtvedt constraint and the Mercury perihelion shift observation. The required values for $\kappa$ differ by five orders of magnitude and have opposite signs in specific model configurations (e.g., Einstein's formulation implies $\kappa=0$, while Nordström's implies $\kappa=-1$).

Significance and Claims
The authors position this work not as a viable replacement for General Relativity, but as a "fundamental study" aimed at mapping the capabilities and limitations of Newtonian-like extensions.

• Bridging the Gap: The research elucidates the contexts in which simpler, non-relativistic models can reproduce known metric theory effects and where they fail.
• Utility in Weak Fields: Despite the failure to satisfy all Solar System constraints simultaneously, the authors argue that such extensions remain valuable as effective frameworks for exploring weak-field systems where full relativistic treatment is computationally expensive or unnecessary, provided the specific constraints of the system are understood.
• Model Specificity: The paper highlights that specific choices of coupling functions (e.g., those decoupling $\sigma$ from matter) can eliminate post-Newtonian corrections entirely, demonstrating that the presence of a scalar field does not automatically guarantee relativistic-like deviations in the Newtonian limit.

The study serves as a cautionary analysis, showing that while Newtonian extensions can mimic relativistic effects, they face severe observational hurdles when attempting to replicate the full spectrum of Solar System tests simultaneously.