# Why is the resolution or measurement uncertainty of $G$ so bad?

I sorta know how the Cavendish apparatus works, and I know that "Gravity is an extremely weak force" for subatomic particles in comparison with the E&M or nuclear forces that act on these particles. So I understand why such a measurement is extremely difficult and, in a single experiment why such a measurement would have a relatively high relative standard uncertainty, $$4.5 \times 10^{-5}$$, (or relatively few significant digits, ca. 5).

But if a quantity being measured is constant over time, I would normally understand that repeating an experiment many many times over a long period of time, and averaging the values, that should decrease the relative standard uncertainty with any measurement.

If I were measuring a DC voltage that had some noise added to it, with an analog-to-digital converter of relatively few bits resolution, I can add some known dither noise to the DC voltage and run the measurement for weeks, get millions or billions of measurements of that DC voltage, and average these measurements to get a value of increased resolution greatly exceeding the resolution of the A/D converter.

Why can't this U Washington experiment be set up and run continuously for weeks or months or years, to get a zillion slightly different values for $$G$$, and average those zillion values to get a good value of $$G$$ with 8 or 10 significant digits?

• "greatly exceeding the resolution of the A/D converter." If I understand correctly, you mean that if you are using a voltmeter that measures only volts, repeating measurement would allow you to measure nanovolts? Apr 23, 2019 at 7:36
• It is worth mentioning that one part per 2000 or so isn't all that shabby. Many fundamental physical constants are know with roughly the same or less accuracy: all 8 elements of the CKM and PMNS matrixes; the masses of the neutrinos (relative & absolute), Higgs, W and Z bosons, all 6 quarks & the tau lepton; the strong force coupling constant, the Hubble constant & the cosmological constant. The only experimentally determined fundamental constants measured to much better accuracy are the EM & weak force coupling constants, the electron & muon masses, the Higgs vev, Planck's constant, and c. Apr 23, 2019 at 7:55
• No experimentally determined fundamental constant is measured to better than 10 significant digits, although one or two non-fundamental constants (e.g. the proton mass) are measured to 11 significant digits. So 75% of other fundamental constants are know less accurately or at about the same accuracy and 25% are known more accurately. Apr 23, 2019 at 8:04
• @ohwilleke , i thought the Rydberg constant was 12 significant digits. relative standard uncertainty is $5.9 \times 10^{-12}$. but i don't expect that to be normative. Apr 23, 2019 at 19:28
• @robertbristow-johnson The Rydberg constant is a function of the speed of light, Planck's constant, the rest mass of the electron, and the charge to the electron, and so is not itself fundamental and is known to somewhat more precision than the fundamental constants from which it is derived because it is measured directly with very exact instrumentation. In that respect it is similar to the proton mass which is known with much more precision than the fundamental physical constants that give rise to the proton mass (mostly the strong force coupling constant and quark masses). Anyway, same idea. Apr 23, 2019 at 19:41

I would normally understand that repeating an experiment many many times over a long period of time, and averaging the values, that should decrease the relative standard uncertainty with any measurement.

First, your understanding is, well, wrong.

This happy situation, where averaging decreases the type A uncertainty component, does happen only when the random measurement fluctuations can be modelled as white noise, that is, when they can be modelled as uncorrelated random variables.

Unfortunately, most of the noise processes that occur in experimental apparatuses during long-term measurements cannot be modelled as white noise. White noise is dominant for short-term measurements but, in the long run, noises with long-range correlations appear (e.g. flicker noise, random walk and, more generally, power-law processes).

For noise with long-range dependence (or long memory) it is no longer true that the statistical uncertainty decreases with time. For instance, it can remain constant, as with flicker noise or, actually, it can even increase, as with random walk. A good starting point to understand this problem is [1].

Typical noise sources of the above kind are the electronic noise of the measurement equipment, the seismic noise coming from the environment and the mechanical noise of the experimental apparatus (e.g. this is a well-known problem in gravitational wave detectors).

The consequence is that in any real experiment there is somehow an optimum measuring time (which, sometimes, can be quite short) at which the type A uncertainty reaches a minimum value and above which it no longer decreases.

In addition to this, in such kind of experiments, the ultimate measurement uncertainty does not depend on the type A uncertainty component but on the type B uncertainty component, that is, the uncertainty component which depends on the non-idealities of the experimental apparatus that lead to systematic errors. Examples are the inaccuracies in the determination of geometrical dimensions, the inhomogeneity of the test masses, etc.

For what concerns your idea, expressed in a comment, of averaging across different experiments, this is actually done. The recommended values of the fundamental constants given by CODATA that you can find in tables are actually averages across the state-of-the-art experiments. It should be noted, however, that some experiments can be affected by systematic errors with the same direction, and averaging would not reduce the effect. This probably actually happened in some of the past $$G$$ experiments (before the 1990s), because it was later found that the inelasticity of the fibre used in dynamic (time-of-swing) experiments could produce a systematic error [2].

[1] J. Beran, Statistics for Long-Memory Processes, Chapman and Hall, 1994.

[2] K. Kuroda, "Does the time-of-swing method give a correct value of the Newtonian gravitational constant?", Phys. Rev. Lett., 75, 2796, 1995.

• the "random measurement fluctuations" don't have to be white noise (which actually has infinite power and does not truly exist in reality), but can be colored noise or bandlimited, as long as they don't have a constant DC value (which is what i would call a "systemic error"). i'm very familiar with various colors of noise and what you can do with it. we actually add noise with zero-mean to some DC voltages to get a better measurement of that voltage. Apr 23, 2019 at 6:21
• the issue really is whether the error is systemic and manifests as a DC bias. then, i realize that you can average until the cows come home and you will not reduce that error. but it seems to me that systemic errors would be coming from errors of dimension of the machined parts. Apr 23, 2019 at 6:23
• @robertbristow-johnson as I said, for coloured noise like flicker ("pink" noise) and random walk, it is no longer true that the uncertainty decreases by averaging. Apr 23, 2019 at 6:23
• if the DC component is zero, you can used pink noise for averaging out other errors. i have done so myself. and remember, there is no such thing as "white noise". it's all bandlimited in reality. even Johnson noise is not really white. Apr 23, 2019 at 6:24
• @robertbristow-johnson No, you can't, but you may be so convinced ;-) Apr 23, 2019 at 6:26

The apparatus may (does) have systematic biases (errors). You can't get rid of those by doing the same experiment over and over again.

In analyzing their experiment, they will have determined what the maximum value of these systematic errors is. For example, because of the tolerances on machining the parts of the apparatus, or their limited ability to control the temperature of the equipment.

Presumably the designers of the experiment have already chosen to run the experiment enough times that the residual random error is much lower than the limits they have achieved on their systematic errors.

• what would those be, other than the precision of the dimensions of the components? Why can't these positions of the spheres and whatnot be measured to whatever obscene degree of accuracy you dudes can do nowadays? Apr 23, 2019 at 5:36
• I remember doing an undergraduate version, and one major inconvenience with measuring $G$ was that, in a way, everything affected the measurements. Just standing next to the apparatus was enough to affect the results. So one possible source of error is mere presence of objects inside and outside the room. I'd imagine you'd need a lot of space to make sure everything is far away from the apparatus. Apr 23, 2019 at 5:49
• @SpiralRain, in this experiment, they rotate (with different angular velocities) both a test pendulum and the masses it interacts with, to avoid that problem. Apr 23, 2019 at 5:53
• well, i would expect nearby objects to affect it, if they are moving. if they are not moving, i don't see how that would be anything other than a DC bias on the twisting pendulum. and if they are moving, then averaging should come into play. if fact, i would devise some additive dither, in the form of some moving objects to bias the result in different ways between different runs of the experiment, to try to correct a systemic bias. that's how we do it in electrical engineering in trying to determine a difficult voltage that has very low bandwidth. we add dither. Apr 23, 2019 at 5:56
• @robertbristow-johnson, the final reference paper at your link (Gundlach and Merkowitz 2000) goes through the uncertainties in detail. I don't really have the background to understand them all, certainly not on a quick read. Apr 23, 2019 at 5:57