Probing the Strong Equivalence Principle through the External Field Effect. How Do Two Masses Fall?

Technical Summary: Probing the Strong Equivalence Principle through the External Field Effect

Problem Statement
Despite extensive astrophysical evidence for dark matter, the lack of a confirmed particle candidate sustains interest in Modified Newtonian Dynamics (MOND) as an alternative. MOND posits that gravitational laws change at low accelerations ($a \lesssim a_0 \approx 1.2 \times 10^{-10} \, \text{m/s}^2$). A critical, non-canonical prediction of MOND is the External Field Effect (EFE), which violates the Strong Equivalence Principle (SEP). Unlike Newtonian mechanics or General Relativity, where a uniform external field does not influence the internal dynamics of a system (due to the superposition principle or the equivalence principle), MOND predicts that the internal gravitational attraction between masses depends on the orientation of the system relative to the external field. This paper investigates the feasibility of detecting this anisotropy in a terrestrial or space-based laboratory setting using two spherical masses.

Methodology
The authors propose a precision experiment comparing the mutual gravitational attraction of two identical spherical masses arranged such that their symmetry axis is either parallel or perpendicular to a strong external gravitational field (e.g., Earth's gravity). The study proceeds through several analytical stages:

1. Baseline Dynamics (No External Field): The authors first establish the conditions for observing mutual attraction in the absence of an external field. They derive constraints on bead size, surface-to-surface separation, and material properties (specifically Platinum) to ensure the mutual acceleration remains within the MOND regime ($< a_0$) while minimizing non-gravitational interference from van der Waals forces, Casimir forces, and air drag.
2. Uniform External Field Analysis: The paper derives solutions for the internal gravitational field of a point mass embedded in a uniform external field. Two distinct solution classes are considered:
   • Vanishing Curl: Solutions where the field is the gradient of a scalar potential (analogous to the Blanchet–Novak solution).
   • Non-Vanishing Curl: Solutions where the field possesses a curl, a possibility allowed by the general MOND field equation but often excluded in specific relativistic formulations.
3. Trajectory Calculation: Using these field solutions, the authors calculate the spacetime trajectories of the masses for both parallel and perpendicular configurations. They derive the effective gravitational constants ($G_{\parallel}$ and $G_{\perp}$) for each case, which differ based on the interpolating function $\mu(x)$ and the parameter $\lambda_e$.
4. Radial External Field Extension: The analysis is generalized to a radial external field (Earth's actual field). The authors examine the effects of tidal forces and the non-uniformity of the external field, specifically looking at the perpendicular configuration where the symmetry axis is orthogonal to the radial field.
5. Sensitivity Assessment: The paper quantifies the spatial and temporal resolution required to distinguish between Newtonian predictions, the two MOND solution types, and different interpolating functions (characterized by parameter $\alpha$).

Key Contributions and Results

• Experimental Feasibility Constraints: The study identifies that for Platinum spheres, a radius of approximately $215 \, \mu\text{m}$ with a final surface-to-surface separation of $274 \, \mu\text{m}$ is a viable benchmark configuration. This setup minimizes van der Waals forces while keeping accelerations within the MOND regime. Larger masses (e.g., $2 \, \text{cm}$ radius) are also discussed as alternatives that could operate without high-vacuum conditions.
• Anisotropy in Mutual Attraction:
  • In a uniform external field, the mutual attraction is orientation-dependent. For the "non-vanishing curl" solution, the effective gravitational constant in the perpendicular configuration ($G_{\perp}$) differs from the parallel configuration ($G_{\parallel}$) by a factor related to $\lambda_e$.
  • In a radial external field, the perpendicular configuration exhibits a unique feature: the mutual attraction is not strictly aligned with the symmetry axis. The internal dynamics acquire a small transverse component, causing the masses to move toward each other along an arc, even if the external field is perfectly balanced by non-gravitational forces. This represents a violation of Newton's third law in the context of the internal forces.
• Resolution Requirements:
  • Temporal Precision: Distinguishing the "simple" interpolating function ($\alpha=1$) from Newtonian gravity requires temporal precision better than $2 \, \text{ns}$ for a 30-minute evolution. However, testing smaller values of $\alpha$ (e.g., $\alpha=0.1$) requires only $\delta t < 17 \, \text{s}$.
  • Spatial Precision: The spatial sensitivity requirements are more stringent. For $\alpha=0.1$, a spatial resolution of $\delta x < 2 \, \mu\text{m}$ is required. For the "simple" function ($\alpha=1$), the requirement tightens to $\delta x < 3 \times 10^{-16} \, \text{m}$, and for the standard function ($\alpha=2$), it reaches $\delta x < 3 \times 10^{-27} \, \text{m}$.
  • The authors note that while current technology struggles with the "simple" and "standard" functions, optomechanical techniques are approaching the regime relevant for smaller $\alpha$ values.

Significance and Claims
The paper claims that the proposed experiment serves as a direct test of the Strong Equivalence Principle. Even if the sensitivity is insufficient to confirm specific MOND interpolating functions, the setup provides an independent verification of whether the internal dynamics of a system are influenced by an external gravitational field.

The authors emphasize that the detection of the transverse component in the radial field configuration (the "arc" trajectory) would be a distinct signature of the non-vanishing curl solution, offering a way to differentiate between various theoretical extensions of MOND. The study concludes that while detecting the "simple" or "standard" MOND predictions is currently beyond experimental reach due to extreme spatial resolution requirements, experiments with lower resolutions (targeting smaller $\alpha$ values or simply testing the SEP) remain scientifically valuable and potentially achievable with levitated particles or space-based environments.