# Is $F = G\dfrac{{m_1}{m_2}}{r^2}$ really true?

My book (Concepts of Physics by H.C. Verma) writes:

It has been reported (Phys. Rev. Lett. Jan 6, 1986) that the force between two masses may be better represented by $$F = \frac{G_{\infty} m_{1} m_{2}}{r^2} \left[1 + \left(1 + \frac{r}{\lambda} \right) \alpha e^{-\frac{r}{\lambda}}\right]$$ where $$\alpha \approx - 0.007$$ and $$\lambda \approx 200~\mathrm{m}$$.

What is this? Such a horrendous formula! So, what about Newton's? And what's the difference between $$G$$ & $$G_{\infty}$$?

• '86 would have been one of the very short-lived and unreproduced "fifth-force" claims. I won't blame the author for putting it in the book, but I think it is fair to suggest that more conservative language would have been better. Commented Nov 30, 2014 at 16:14
• Here is a link to the paper they cite. Commented Nov 30, 2014 at 16:16
• Yes, see Fifth force for more info, which cites the 1986 paper by Fischbach et al. As the article says, other experimenters weren't able to reproduce the result so it was presumably just due to experimental error. Commented Nov 30, 2014 at 16:19
• Note that Newton's gravitational law is still just an approximation; a more accurate treatment of gravitation is provided by Einstein's general relativity. However, for most purposes, Newton's $1/r^2$ gravitational law is more than good enough. Commented Nov 30, 2014 at 16:34
• This link is very relevant: tudtor.kfki.hu/eotvos1/onehund.html Commented Nov 30, 2014 at 16:41

Yes, Newton's formula is just fine. No, the formula in your book doesn't describe reality. At first this sounded like an exercise, where the next sentence is probably something like "calculate the effect this has..." These sorts of hypothetical questions are meant to show you how you could distinguish between competing physical theories.

Some more digging turns up the actual paper, though: Fischbach et al., Phys. Rev. Lett. 1986, 56, 3. Apparently another group kept getting the wrong value for $G$ -- which is, by the way, the most difficult physical constant to measure -- and so these authors proposed some extra force. This force has an extended range (the $200\ \mathrm{m}$ determines how quickly it falls off) compared to nuclear forces, but still vanishes exponentially for large distances.

The idea never really went anywhere, and it is just an example laboratory errors. The claim was countered by Keyser et al., Phys. Rev. Lett. 1986, 56, 2425, where they show how the extra force only appears when one cherry picks the data. All measurements have some error, and if you systematically only include the measurements that randomly happen to support your hypothesis, you can make your hypothesis seem true.

• Interesting, contradicts this review in 1990, tudtor.kfki.hu/eotvos1/onehund.html, "[no] group pinpointed an error in the Eötvös experiment which could be the source of their suggestive data" - but that is by some of the same authors, could be biased, I suppose. Commented Nov 30, 2014 at 16:44
• an impartial source: plato.stanford.edu/entries/physics-experiment/app4.html, "It is a fact of experimental life that experiments rarely work when they are initially turned on and that experimental results can be wrong, even if there is no apparent error. It is not necessary to know the exact source of an error in order to discount or to distrust a particular experimental result. Its disagreement with numerous other results can, I believe, be sufficient." No suggestion of cherry-picking data. Commented Nov 30, 2014 at 18:31
• a history of the affair in, "No Easy Answers: Science and the Pursuit of Knowledge", Allan Franklin. There's interesting material, including a letter by Feynman on the matter. There's no suggestion that Fischenbach cherry-picked data or acted unprofessionally. He comes across as careful and scrupulous. books.google.ee/… Commented Nov 30, 2014 at 18:48
• Cherry picking can be subconscious too. I only used the term because it's clearly what Keyser implied between the lines. In that paper they have a plot where they show the fit to data used by Fischbach, but they restore the Eötvös-Pekár-Fekete data discarded by Fischbach, showing there is no actual trend if the full dataset is used.
– user10851
Commented Nov 30, 2014 at 19:05
• On the matter of subconscious cherry-picking or otherwise diddling the data, it is always safer to blind the data if you can. Commented Nov 30, 2014 at 19:23

In principle not, because it is wrong when describing the behavior of the orbit of Mercury, but should be borne in mind that there is no absolute truth when describe the universe, they always talk about "good approximations" and Newton's law of gravitation rule! xD, (it can take you to the moon!).