⚛️ general relativity

Atmospheric Newtonian noise in torsion-balance measurements of the gravitational constant $G$

This paper establishes a GUM-consistent framework to quantify atmospheric Newtonian noise in torsion-balance measurements of the gravitational constant $G$ , deriving spatial transfer functions for benchmark geometries to demonstrate that while current atmospheric contributions are below reference uncertainty levels, they become significant as systematic errors approach the part-per-million regime.

Original authors: Jyotirmaya Mohanta, Yutaka Shikano

Published 2026-08-27

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CC BY 4.0

Original authors: Jyotirmaya Mohanta, Yutaka Shikano

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

Measuring the strength of gravity between two objects in a laboratory is one of the most difficult tasks in physics. Scientists have been trying to pin down the exact value of Newton's gravitational constant, a number that tells us how strongly matter pulls on other matter, for centuries. Despite their best efforts, the measurements still disagree with each other by a small but stubborn amount. This disagreement suggests that something is hiding in the background, a subtle influence that the experiments cannot quite filter out. While scientists have long known that vibrations from the ground or changes in air pressure can nudge sensitive instruments, the specific way that shifting air masses create tiny, unshieldable changes in gravity has remained difficult to quantify with the precision needed for the next generation of experiments.

To understand the problem, imagine a delicate balance scale that can detect the weight of a single grain of sand. If the air in the room shifts, the density of the air changes, and because air has mass, it exerts a tiny gravitational pull on the scale. This pull is not a force you can block with a wall or a shield; it is a fundamental property of gravity itself. In the world of gravitational wave detectors, this phenomenon is known as Newtonian noise, but for the specific experiments trying to measure the gravitational constant, the rules are different. These experiments use a torsion balance, a device where a bar hangs from a thin wire and twists when pulled by nearby heavy weights. The goal is to measure that twist to calculate the gravitational constant. The challenge is that the same air density fluctuations that create the background noise also create a torque, or twisting force, on the balance, potentially confusing the measurement.

A team of researchers has now developed a new way to calculate exactly how much this atmospheric noise matters. They did not build a new machine or run a new physical experiment. Instead, they built a mathematical framework, a set of rules that allows scientists to take the specific conditions of their laboratory—the shape of their balance, the way they average their data, and the local weather patterns—and translate them into a precise estimate of the error caused by the air. They treated the air as a fluctuating sheet of mass moving over the floor and calculated how those fluctuations would twist the balance. They tested their method using two extreme shapes for the balance: a simple bar with weights on the ends, which represents the worst-case scenario for noise, and a perfectly symmetrical cross shape, which represents the best-case scenario for blocking out the noise.

The researchers found that for the current generation of experiments, the noise from the atmosphere is small enough that it does not explain the disagreements between different measurements of the gravitational constant. The background noise from the air is currently well below the level of uncertainty that scientists are working with. However, the study shows that as experiments become more precise and aim for a level of accuracy one hundred times better than today, this atmospheric noise will become a major factor. It will no longer be a background whisper; it will become a shout that must be accounted for. The team showed that by using their new framework, future scientists can predict exactly how much their local air pressure will interfere with their results and can design their experiments to either cancel out this noise or subtract it from their data.

One of the key insights of the work is the distinction between steady, predictable noise and slow, drifting changes. The researchers separated the random, jittery fluctuations of the air, which can be averaged out over time, from the slow, wandering drift of the balance itself, which cannot be averaged away. They demonstrated that while the random noise can be managed with better data processing, the slow drift requires a different kind of solution. They also showed that simply making a balance perfectly symmetrical to block out the noise is not a perfect solution, because it would also block the signal the scientists are trying to measure. Instead, the best approach involves a careful balance, keeping just enough asymmetry to see the signal while suppressing the noise as much as possible.

The study also looked at how to use sensors to help. By placing multiple barometers around the experiment, scientists could map the air pressure changes in real time. The researchers simulated how well a ring of sensors could predict the noise on the balance. They found that the size of the ring matters; a ring that is too small acts like a single point and misses the big picture, while a ring that matches the size of the experiment's sensitive area can effectively predict and remove the noise. This suggests that future experiments will likely need to be equipped with arrays of sensors to monitor the environment, turning the air itself from a source of error into a source of data that can be corrected.

Ultimately, this work provides a roadmap for the next step in measuring gravity. It confirms that the current disagreements in the data are not caused by the atmosphere, but it warns that as scientists push for greater precision, the air will become a significant hurdle. The framework they created allows any research group to plug in their specific setup and determine exactly how much the atmosphere is affecting their results. This moves the field from guessing about environmental effects to calculating them with certainty. As experiments move toward the goal of measuring the gravitational constant with a precision of one part in a million, the ability to model and subtract this atmospheric noise will be essential. The researchers have shown that while the air is always there, pulling on the balance with invisible hands, its influence can now be measured, understood, and removed, clearing the path for a more accurate understanding of the universe's fundamental forces.

Technical Summary: Atmospheric Newtonian Noise in Torsion-Balance Measurements of $G$

Problem Statement
The measurement of Newton's gravitational constant ( $G$ ) remains limited by environmental and apparatus-dependent systematics, which are often treated less explicitly than instrumental noise. While instrumental (thermal) noise averages down as $T^{-1/2}$ , environmental Newtonian noise (NN)—specifically gravity gradients generated by atmospheric density fluctuations—exhibits long correlation times. These fluctuations produce unshieldable gravity gradients that couple directly to torsion-balance observables. As experiments push toward relative uncertainties at or below $10^{-6}$ , the metrological relevance of these persistent environmental contributions increases, necessitating a rigorous framework to evaluate their impact on the uncertainty budget.

Methodology
The authors develop a framework consistent with the Guide to the Expression of Uncertainty in Measurement (GUM) to propagate atmospheric Newtonian noise into the uncertainty budget of torsion-balance experiments. The approach involves:

Measurement Model: Treating $G$ estimation via the linear equation $\hat{G} = \hat{\tau}/C_G$ , where $\hat{\tau}$ is the estimated gravitational signal torque and $C_G$ is a geometrical response coefficient. The environmental contribution is isolated as a standard-uncertainty term $u_{env}(\hat{\tau})$ .
Estimator-Dependent Propagation: Unlike purely algebraic treatments, the paper explicitly evaluates the standard uncertainty of the torque estimator for correlated noise using the estimator transfer function. The variance is calculated via the power spectral density (PSD) of the noise and the Fourier transform of the estimator's weight function.
Spatial Coupling Analysis: The study derives closed-form spatial transfer functions for two benchmark geometries:
- Two-mass dumbbell: Serves as an upper-coupling reference (worst-case scenario).
- Perfectly symmetric cross: Serves as an idealized rejection limit. The analysis quantifies the trade-off between suppressing low-order environmental coupling (quadrupole cancellation) and preserving the signal response, introducing a residual quadrupole fraction parameter ( $\epsilon$ ) to model practical "engineered anisotropy."
Spatio-Temporal Modeling: The environmental input is modeled as surface-density fluctuations ( $\delta\Sigma$ $δ Σ$ ) on the plane $z=0$ $z = 0$ . The study separates stationary correlated inputs from non-stationary baseline drift. For the stationary component, it utilizes:
- An Ornstein–Uhlenbeck (OU) process as a minimal benchmark for finite correlation times.
- A data-driven atmospheric benchmark using global infrasound pressure noise envelopes (NLNM/NHNM) from the infrasound network, mapped to spatial wavenumbers via phase velocity assumptions (acoustic and slow advection).

Key Contributions
The paper makes five specific contributions to the metrology of $G$ :

GUM-Consistent Evaluation: An estimator-dependent evaluation of NN contributions to the relative standard uncertainty $u_r(G)$ , explicitly separating stationary inputs from non-stationary drift.
Analytic Transfer Functions: Derivation of analytic transfer functions from spatio-temporal surface-density fluctuations to torque, including baseline suppression factors.
Geometric Benchmarks: A closed-form extension from the dumbbell to a symmetric cross configuration, quantifying the limiting case of quadrupole cancellation and the associated signal–noise trade-off.
Metrological Separation: A clear distinction between stationary environmental inputs (amenable to PSD-based propagation) and non-stationary baseline wandering (requiring separate Type-B or drift-model treatment).
Atmospheric Benchmark: A realistic atmospheric pressure benchmark based on global pressure-noise envelopes, converted to torque uncertainty via the derived transfer functions.

Results

Transfer Functions: The baseline suppression factor for a symmetric cross scales as $(kl)^6$ for long wavelengths ( $kl \ll 1$ ), compared to $(kl)^2$ for a dumbbell, demonstrating significantly stronger suppression of long-wavelength gradients. However, perfect symmetry ( $\epsilon=0$ ) would also suppress the signal if it enters through the same quadrupole channel.
Atmospheric Impact: Using the pressure NLNM/NHNM envelopes, the resulting background torque uncertainty for a representative dumbbell ( $l=5$ cm, $z_0=1$ m) yields equivalent gradient uncertainties in the range of $10^{-16}$ to $10^{-14}$ s $^{-2}$ for integration times of $10^3$ – $10^5$ s.
Relative Uncertainty: When translated to relative uncertainty $u_r(G)$ using benchmark signal gradients ( $10^{-7}$ and $10^{-6}$ s $^{-2}$ ), the atmospheric background contribution is found to be well below the current CODATA relative standard uncertainty of $\sim 2.2 \times 10^{-5}$ .
Regression Potential: Simulations of regression subtraction using a ring array of barometers suggest that arrays with radii comparable to the coupled spatial scale perform better than compact arrays, though this is illustrated via a simplified dimensionless toy model.

Significance and Claims
The authors state modestly that their work does not claim atmospheric Newtonian noise dominates any specific historical $G$ experiment. Instead, the paper provides a practical benchmark framework for judging the relevance of environmental gravity gradients as systematic floors approach the part-per-million ( $10^{-6}$ ) regime.

The central claim is that while current atmospheric background NN is not large enough to explain the historical scatter in published $G$ values, it is approaching a regime where it must be explicitly bounded, monitored, and potentially subtracted in next-generation experiments. The framework allows for the incorporation of site-specific environmental models into future uncertainty budgets, distinguishing between stationary noise (which can be averaged or regressed) and non-stationary drift (which requires separate metrological treatment).

Technical Summary: Atmospheric Newtonian Noise in Torsion-Balance Measurements of GGG

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Technical Summary: Atmospheric Newtonian Noise in Torsion-Balance Measurements of $G$