Measuring G: The ultimate metrology challenge?

Four fundamental forces of nature—gravity, electromagnetism, strong nuclear force, and weak nuclear force—govern all known physical interactions in the universe. Of these four, gravity is the one with which we are all personally familiar, as we deal with it in our daily routine. Knowing these forces, along with the other defining constants of the International System of Units (SI), form the foundation of much of modern science and engineering (Figure 1).

Figure 1 This wallet card displays the fundamental constants and other physical values that will define a revised international system of units. Source: NIST

A good semi-technical, highly readable overview of the development of metrology, the people who made it happen, and its role in civilization and the industrial and technology revolution is the book “Beyond Measure: The Hidden History of Measurement from Cubits to Quantum Constants” by James Vincent (Figure 2).

Figure 2 This enjoyable book provides great insight into the hard-fought efforts of metrologies over the centuries, even if they were not called that. Source: W. W. Norton

The gravitational constant, informally dubbed “big G”, determines the strength of the attraction between two masses anywhere in the universe. It’s approximately 6.67 x 10^-11 meters³/kilogram-second². It is, of course, associated with Isaac Newton’s brilliant insight and law of universal gravitation, published in 1687, which states that every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them (Figure 3).

Figure 3 We can thank Isaac Newton for this simple equation that quantifies all-pervasive yet mysterious gravity. Source: Wikipedia

This “big G” is distinct from “little g”, which describes the acceleration that an object experiences due to the gravitational pull of a large mass, such as Earth, and it varies from location to location. For instance, the value of little g is approximately 9.8 meters/second² at Earth’s surface but only 1.62 meters/second² on the Moon.

The first implicit measurement is attributed to Henry Cavendish in a 1798 experiment with an accuracy of about 1%, which is impressive considering the year and available tools and technology. Yet, while other fundamental physical constants are known to 6 or more digits of confidence, measurement of this oldest-known force to comparable precision has eluded physicists, and it’s known with confidence only to about 4 digits.

While a better value for G wouldn’t affect the lives of most people or projects, there are some cases for which it would be needed, and it’s also a part of the broader science “quest”.

Why is G so hard to measure? There are three main reasons:

• Gravity is the weakest of the four fundamental forces of physics; for comparison, it’s approximately 10³⁸ times weaker than the strongest force.
• The masses used in the experiment must fit inside a relatively small, constrained space of the experimental lab, and small masses generate small gravitational forces.
• Since gravitational force is inherent by every object, it’s extremely challenging to make sure the force you measure in the laboratory really comes from the intended mass.

What was the next step?

Trying to determine G to higher accuracy has been an ongoing project for many institutions and researchers. I recently came across a news report from National Institute of Standards and Technology (NIST) site summarizing the 10-year quest led by physicist Stephan Schlamminger to improve that measurement (“NIST Weighs In on the Mystery of the Gravitational Constant”).

His team’s strategy was to painstakingly replicate a precision experiment conducted by the International Bureau of Weights and Measures (BIPM) in Sèvres, France, in 2007, which provides the value of G in use now. To do this, the team not only improved the precision of the physical parts of the experimental setup but also dived deeper to more fully identify sources of error, and then either reduce them or work out ways to have them self-cancel.

The basic arrangement is very simple and begins with a torsion balance similar to the one used by Cavendish (Figure 4). He placed two lead balls on opposite ends of a wooden beam horizontally suspended at its center by a thin wire. Nearby, he positioned two much heavier masses, suspended separately.

Figure 4 The traditional Cavendish experiment for measuring the strength of gravity was a torsion balance with an “optical” readout. Source: NASA

The gravitational attraction between the smaller and heavier masses caused the wooden beam to rotate, twisting the wire until the torque it exerted by counterbalancing the gravitational force. The motion of the wooden beam, measured with a mirror and light pointer, indicated the value of big G.

The Schlamminger team upgraded to eight cylindrical metal masses. Four of the cylinders sat on a rotating carousel, resembling four candlesticks in an old-fashioned chandelier. The other four smaller masses were placed inside the carousel, on a disk suspended by a ribbon of copper-beryllium about the thickness of a human hair.

They then added a modern-day “twist” not available to Cavendish: applying a voltage to electrodes placed alongside each of the inner masses (Figure 5). These voltages created an electrostatic torque that twisted the wire in a direction opposite to the gravitationally induced torque. By carefully setting a voltage that exactly counterbalanced and nulled the gravitational torque, the researchers prevented the torsion balance from rotating. The magnitude of the voltage provided another estimate of big G.

Figure 5 The latest version of the torsion balance is loosely based on the Cavendish design, but adds advanced features, including electrostatic torque nullification. Source: NASA

Of course, the actual unit is larger and much more sophisticated (Figure 6).

Figure 6 The NIST version of the Cavendish torsion balance bears little resemblance in actual implementation. Source: NASA

What’s the result?

The Committee on Data of the International Science Council, or CODATA, issues recommended values of fundamental physical constants. Its recommended numerical value for big G is a four-digit number with a measurement uncertainty of 22 points per million. To put this in perspective, a watch that runs 22 ppm late would measure the year 12 minutes too long.

So, to cut to the chase: How did the team do? Bluntly, not as well as they would like. In fact, their answer differed significantly from the BIPM number—which would be OK if it was “more correct” —but it had greater uncertainty as well. It was 0.0235% lower than the result that the researchers had attempted to replicate and is at odds with the CODATA figure.

I won’t try to summarize the project, as it has so many nuances and details. Fortunately, the published Metrologia paper titled “Redetermination of the gravitational constant with the BIPM torsion balance at NIST” is not a dry, academic-style presentation of the project. Instead, it’s a fascinating, highly readable 30-page recounting of a story that begins with an overview of the history of G measurement, then goes on to review the project step by step, covering the rationales for each step; the improbable, possible, and likely sources of error; the dilemmas they addressed; the qualitative issues as well as the quantitative analysis; and much more.

You could almost say it could be the basis for a scripted TV show or even a movie, perhaps not as dramatic as the 2023 Oppenheimer but still a “grabber.” As a very nice consideration for the readers, the paper begins with a full list of acronyms and abbreviations, which I wish all papers would do. It even includes a group photo of the 40+ team participants.

There’s one other interesting aspect of the project that I have very rarely seen. Schlamminger worried that he might unconsciously skew his measurement so that it agreed with the value of G that researchers found in the French experiment. To satisfy his own meticulous standards, he asked a colleague to scramble the data.

To accomplish this, colleague at NIST’s Mass and Force Group multiplied each Source Mass value by an unknown factor (1 + r) with r ∈ [−1 × 10⁻³, +1 × 10⁻³], stored in a secure envelope to be hidden from Schlamminger until the work was complete. This random offset number for the masses served to “blind” Schlamminger to the actual measurement he was taking.

By employing that strategy, Schlamminger would not know the actual value of big G that his team had measured. The envelope with the secret number was unsealed on a conference stage at the July 2024 Conference on Precision Electromagnetic Measurements (CPEM), and Schlamminger and his team finally found out the somewhat disappointing real results of their work.

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