## Weighing the Earth

The Cavendish experiment to weigh the Earth, conducted in 1798, is one of those iconic experiments that everyone should know. It was the definitive test of Newtonian gravity – it proved beyond reasonable doubt that not only are the apples attracted to the Earth, but they are also attracted to each other. I remember this experiment being done in physics lessons at school, and the sense of awe I felt that you could actually measure the gravitational pull of one stainless steel ball on another. I do remember that we were a little disappointed at the level of accuracy we could obtain, but I have since learnt that this is a feature of the Cavendish experiment that applies even to the best experts. We may have struggled to get 2 significant figures (which is what Cavendish got), but we certainly got 1, and nobody convincingly got more than 4 significant figures before the 21st century.

There are almost no other methods of directly measuring active gravitational mass, although some variations are now used, and accuracy is improving as a result. But as accuracy improves, the inconsistency between different experiments becomes more noticeable. It is already at the level of serious concern that there must be some errors that are not understood in the experiments. It is not yet at the level of actual disproof of the theory of gravity. But it may not be far off that.

So what can theory do to help us understand this situation? The real problem with gravity is that matter consists of electrons, protons and neutrons, and the proton and neutron masses differ by about 0.14%, and the electron mass is less than half of that difference. So until you get at least 4-figure accuracy, you cannot distinguish at all between the contributions to gravity from the electrons, protons and neutrons, and simply have to treat all matter particles the same. Until you get to 6-figure accuracy, you’re not going to get any significant evidence as to whether gravity treats electrons, protons and neutrons in the same way or not.

But one thing we do know is that that there is nothing in quantum physics that treats electrons, protons and neutrons the same. So it is certainly not safe to assume that quantum gravity treats them the same. That means that atoms that have relatively few electrons (like lead, gold, platinum) may not behave in quite the same way as atoms with relatively many electrons (like iron, copper). Experiments are being done to try to detect differences between copper and gold – essentially testing whether the inertial mass ratio of copper to gold atoms is the same as the gravitational mass ratio.

My models make predictions for the magnitude of the differences that will be detected in such experiments. They are of the same order of magnitude as the reported anomalies. So why are the experimenters not directly testing my predictions, rather than looking for differences without having a theory to test? I suspect the reason is sociological – experimenters have given up on the theorists, who have produced so many crap predictions over so many years that the experimenters have stopped listening. I don’t blame them.

### 18 Responses to “Weighing the Earth”

1. James Arathoon Says:

There is the recent work of Norbert Klein at Imperial College to account for errors in measuring big G using MOND

https://iopscience.iop.org/article/10.1088/1361-6382/ab6cab/pdf

There was a lot of experimental work done a few years ago that showed that the statistical variation in experimental results was significantly greater than the best estimates of the size experimental errors. I will post the link to that work when I can find it again – a reference to this work may be in Norbert’s paper.

There is little mainstream interest because Einstein’s General Relativity can’t account for the failure of the strong equivalence principle these measurements would seem to imply.

• Robert A. Wilson Says:

Thanks for that reference. A while back I found some papers discussing discrepancies in big G measurements, but I’ve never before seen a suggestion that MOND may account for them. What I am really interested in, though, is a common explanation for both MOND and the big G anomalies. Essentially what I am writing is a series of papers about why the strong equivalence principle is false, giving arguments from statistics, pure mathematics, history, philosophy, experimental physics and wherever else I can find them.

The physical mechanism of both MOND and big G anomalies must, in my opinion, be neutrino oscillations. This is the only known unknown that could plausibly be involved. I am not interested in working on the unknown unknowns that most other theorists are working on. If neutrinos oscillate into or out of the electron flavour state, then the effective gravitational mass of the electron will fluctuate relative to its inertial mass. That is why I suggest doing the Cavendish experiment with different materials, in which the proportion of electrons varies. One can then do a back-of-an-envelope calculation to see what magnitude of effect one might expect – and it turns out to be comparable to the observed anomalies.

My latest mathematical model has two definitions of mass, one using (a modification of) the Dirac equation and one using (a modification of) general relativity, and they are different. The model also contains conformal symmetries that change one type of mass and not the other. In other words, in this model, which is not very different from a load of other E_8 models, the strong equivalence principle is false. Moreover, it should be possible to calculate explicitly and quantitatively some deviations from the SEP. It’ll take me a while to write the paper, though. I may not get it done in time for Christmas!

A further characteristic of this model is that the Riemann curvature tensor is changed – the Einstein tensor stays the same, so the field equations do not need to be modified, but the Weyl tensor is replaced by something quite different. Hence the model has no spin 2 graviton, but appears to have neutrinos instead. In particular, gravitational waves do not consist of spin 2 gravitons, as GR supposes, but of neutrino oscillations. Gravitational waves, therefore, have already been detected by experiment, half a century before LIGO.

• James Arathoon Says:

Also there was a special issue of the Royal Society Philosophical Transactions on the experimental determination of Big G in 2014

https://royalsocietypublishing.org/toc/rsta/2014/372/2026

• Robert A. Wilson Says:

Thanks for the additional reference. It provides an excellent summary of the problem as it was in 2014, without being too technical. From reading a few of these papers, I realise I should emphasise that when I am talking about changing the materials used in the experiment, I am only talking about the material of the source mass, not the test masses. Experiment confirms the universality of free-fall to high accuracy, so it is pointless to vary the material of the test masses. But it is not pointless to vary the material of the source mass.

• Robert A. Wilson Says:

For full disclosure, I should say that one of the editors of this special issue has in fact spent a significant amount of time listening to my theories. He isn’t convinced, of course, but neither is he dismissive of the ideas I have put forward. What he wants from me, and what I am trying hard to provide, is some more mathematical detail, and specific physical predictions, solidly justified by the mathematics.

2. Mark Thomas Says:

I am just throwing this out there (it may be out there). Lead-208 is a stable isotope and is neutron rich. It is a ”doubly magic number’ isotope which kind of means that its orbital shells are complete and that the neutron number is even. It is known that the strong force is greater between a proton and a neutron (probably not much) than between a proton and a proton. If you use the dense 208 as a source mass (and it is an element) the test masses used could be elements with lower binding energies say copper or gold, what have you. If you can get different Big G numbers with elements having different binding energies then that may show a relation between QCD and gravity. Using masses that are molecular my induce discrepancies that are harder to interpret.

• Robert A. Wilson Says:

Yes, this is the kind of thing I had in mind. Both the binding energies of the nucleus and of the electrons complicate the picture, so that there is a lot more to the problem than just counting neutrons and protons. However, if you introduce too many variables at once, then it becomes too easy to fit the data by some arbitrary kludge. Ideally, one would have a single parameter to estimate/adjust. Or, perhaps, one can look at a number of parameters, but only vary one at a time.

I haven’t really looked at the strong force, as it just seems too difficult to get accurate experimental data. But a mixing of gravity with electromagnetism is fairly clear in the muon g-2 data, quite apart from the electron/proton mass ratio, and a mixing of gravity with the weak force in the W mass anomaly, as well as the various kaon decay mode anomalies.

3. Stefan Freundt Says:

Hi Robert, I am missing a link to your work/theories…

• Robert A. Wilson Says:

Ah, sorry. There are lots of links scattered through this blog, but I’m afraid they are not very well organised. I suppose I ought to do something about that. There are several papers on the arXiv, of which the most relevant and interesting (to me) are https://arxiv.org/abs/2202.08623/ and https://arxiv.org/abs/2205.05443/. I am no longer quite sure exactly what is in which paper, and most of them have not received constructive referee’s reports, so are in a fairly rough-and-ready state, and doubtless full of mistakes.

• Stefan Freundt Says:

“is irredeemably a local concept, that cannot be meaningfully extended outside the Solar System.”
How do you want to explain the H-spectra of the stars? These all get along with exactly one electron mass.

• Robert A. Wilson Says:

No, you misunderstand me. They get along perfectly well with one electron INERTIAL mass. They do NOT get along perfectly well with one electron GRAVITATIONAL mass.

• Stefan Freundt Says:

Okay. And what is the reason to distinguish between the two?

• Robert A. Wilson Says:

The reason to distinguish between the two is simply the fact that there is no good reason to assume that the two are the same. Moreover, experiment has been able to equate inertial mass with passive gravitational mass, to an accuracy of about 12 significant figures, if I understand correctly, but despite enormous efforts has not been able to equate passive and active gravitational mass to more than about 4 significant figures in local experiments carried out on the Earth. When measurements of inertial mass of stable particles are routinely quoted to 10 significant figures, this discrepancy is hugely significant.

• Stefan Freundt Says:

You want the gravitational electron mass to be variable in the range from 0 to m(neutron)-m(proton)?
Have I understood you correctly therein?

• Robert A. Wilson Says:

I am not sure I would go that far. What I think the experimental evidence does show is that the gravitational mass of the electron transforms differently from the inertial mass under non-inertial transformations. It does not tell us *how* these two masses transform. In particular it does not tell us how much of the range from 0 up to m(n) – m(p) is accessible in practice. Indeed, it may not necessarily be restricted to this range – there is no theory that I am aware of that says it must be so, if the strong equivalence principle does not hold.

Therefore I am not predicting any particular form for these variations in gravitational mass relative to inertial mass, but merely trying to draw attention to the “fact” (as I see it) that such variations do seem to occur.

• Robert A. Wilson Says:

Or perhaps, to put it another way, the electron has THREE masses, not one. This is a well-established and ill-understood fact – that there are three generations of electron, differing ONLY in mass. Quantum transitions between the three generations are (I am convinced) an essential part of quantum gravity, and if you ignore this fact, as most people do, you will never find a consistent model of quantum gravity.

4. zeynel Says:

To say that Cavendish experiment proved “beyond reasonable doubt” the existence of the supernatural force defined by Newton is not correct. Cavendish assumed the Newtonian force. The experiment was not designed to prove or disprove the Newtonian force. Cavendish made just 17 experiments in a year and a half. There is no mention of him calibrating the instrument during this time before each experiment. In the first experiment the wire he used was not stiff enough and Cavendish observed that “as the attraction of the weights drew the balls so much aside, as to make them touch the sides of the case…” This is circumstantial evidence, hearsay, not proving “beyond reasonable doubt” the Newtonian attraction. Furthermore, the arm of the pendulum is in continous motion, Cavendish only measures the motion of the imaginary middle point. There are many problems with the Cavendish experiment. And the toy pendulum that comes factory set to oscillate to give the correct value of G used in physics labs is no evidence of the Newtonian miracle.

• Robert A. Wilson Says:

Well, perhaps I over-stated the case somewhat. But I am using “proof” in the usual physical sense – where it always means “circumstantial evidence”. There is never any proof in the mathematical sense of any physical “fact”. When physicists tell me they have “proved” that general relativity holds, or that neutrinos have non-zero mass, or that dark matter exists, they always mean that they have found some circumstantial evidence. It is never a real proof.