Abstract
The story of the Fifth Force begins with a seeming digression because it involves not a modification of gravitational theory, but rather an experimental test of and confirmation of that theory. In 1975 Colella, Overhauser, and Werner measured the quantum mechanical phase difference between two neutron beams caused by a gravitational field. Although these experiments showed the effects of gravity at the quantum level, they did not, in fact, distinguish between General Relativity and its competitors, as Fischbach pointed out (1980; Fischbach and Freeman 1979). This was because these experiments were conducted at low speeds, and in the nonrelativistic limit all existing gravitational theories, such as General Relativity and the Brans–Dicke theory, reduce to Newtonian gravitation. Fischbach also discussed how one might test general relativity at the quantum level by considering gravitational effects in hydrogen.
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Notes
- 1.
There are two different K0 mesons, the short-lived \(\mathrm{K}_{\mathrm{S}}^{0}\), and the longer lived \(\mathrm{K}_{\mathrm{L}}^{0}\). CP symmetry allows the \(\mathrm{K}_{\mathrm{S}}^{0}\), but not the \(\mathrm{K}_{\mathrm{L}}^{0}\), to decay into two pions. For a detailed discussion of CP symmetry and the discovery of its violation see Franklin (1986, Chap. 3).
- 2.
The scale of the gravitational effect was given by \(g\hslash /c\), where g is the local acceleration due to gravity, \(\hslash\) is Planck’s constant/2π, and c is the speed of light. At the surface of the Earth this quantity is 2. 2 × 10−23 eV.
- 3.
Here ε, which is a complex number, is a measure of the decay rate of \(\mathrm{K}_{\mathrm{L}}^{0}\) mesons into two pions.
- 4.
The phenomenon of regeneration was one of the unusual properties of the K0 mesons. If one produced these mesons in an accelerator, one obtained a beam that was 50 % \(\mathrm{K}_{\mathrm{S}}^{0}\) and 50 % \(\mathrm{K}_{\mathrm{L}}^{0}\). If one allowed all of the \(\mathrm{K}_{\mathrm{S}}^{0}\) mesons to decay and allowed the remaining \(\mathrm{K}_{\mathrm{L}}^{0}\) mesons to interact with matter, one found that the beam once again contained \(\mathrm{K}_{\mathrm{S}}^{0}\) mesons. They were regenerated. See Franklin (1986, Chap. 3) for details.
- 5.
A stronger statement had appeared earlier (Roehrig et al. 1977, p. 1118): “The results are clearly consistent with constant phase […].”
- 6.
The graph actually shows the energy dependence of ϕ 21, assuming \(\phi _{+-}\) was constant. If ϕ 21 is considered to be a constant, the graph shows the energy dependence of \(\phi _{+-}\).
- 7.
An earlier paper (Fischbach et al. 1982) had examined the same question, although in less detail, and reached the same conclusion.
- 8.
These results were not greeted with enthusiasm or regarded as reliable by everyone within the physics community. Commenting on the need for new interactions to explain the effects, an anonymous referee remarked: “This latter statement also applies to spoon bending.” (A copy of the referee’s report was given to me by Fischbach.) The paper was, however, published.
- 9.
This possible energy dependence played an important role in the genesis of the Fifth Force hypothesis, as discussed in detail below. It may very well have been a statistical fluctuation.
- 10.
- 11.
I will be discussing here modifications of the Newtonian inverse square law, and not the well-established relativistic post-Newtonian corrections, which are of order GM∕c 2 r.
- 12.
- 13.
The history of gravitational theory is not a string of unbroken successes. Newton himself could not explain the motion of the moon in the Principia and his later work on the problem, in 1694–1695, also ended in failure (Westfall 1980, pp. 442–443, 540–548). The law was also questioned during the nineteenth century when irregularities were observed in the motion of Uranus. The suggestion of a new planet by Adams and LeVerrier and the subsequent discovery of the planet Neptune turned the problem into a triumph. During the nineteenth century it was also found that the observed advance of the perihelion of Mercury did not match the predictions of Newtonian theory. This remained an anomaly for 59 years until the advent of Einstein’s General Theory of Relativity, the successor to Newtonian gravitation.
- 14.
Some readers might worry that a variable constant is an oxymoron, but it does seem to be a useful shorthand.
- 15.
- 16.
The influence of Long’s work is apparent in the first sentence of the abstract (Mikkelsen and Newman 1977, p. 919): “D.R. Long and others have speculated that the gravitational force between point masses in the Newtonian regime might not be exactly proportional to 1∕r 2.”
- 17.
Fischbach (private communication) attributes this to a conversation with Wick Haxton.
- 18.
Fischbach’s first calculation was for a δ-function force.
- 19.
- 20.
Because the energy dependence of the K0–\(\mathrm{\overline{K}}^{0}\) parameters might have indicated a violation of Lorentz invariance, Fischbach et al. (1985) had looked at the consequences of such a violation for the Eötvös experiment.
- 21.
Fischbach keeps detailed chronological notes of papers read and calculations done. He reports that he has notes on Fujii’s work at this time, but does not recall it having any influence on his work.
- 22.
Eötvös was originally interested in measuring gravity gradients so the weights were suspended at different heights. This introduced a source of error into his tests of the equivalence principle, the equality of gravitational and inertial mass.
- 23.
According to one source, the torsion balance was suggested by Juan Hernandez Torsión Herrera (Lindsay and Ketchum 1962):
Of Juan Hernandez Torsión Herrera very little is known. He was born of noble parents in Andalusia about 1454. He traveled widely and on one of his journeys in Granada with his cousin Juan Fernandez Herrera Torsión both were captured by Moorish bandits. Herrera Torsión died in captivity but Torsión Herrera managed to escape after a series of magnificent exploits of which he spoke quite freely in his later years. During these years he was affectionately known as the ‘Great Juan’ or as the ‘Juan Who Got Away.’
Although not a scientist in his own right, Torsión Herrera passed on to a Jesuit physicist the conception of his famous Torsión balance. The idea apparently came to him when he observed certain deformations in the machinery involved when another cousin, Juan Herrera Fernandez Torsión was being broken on the rack.
There are some reasons for doubting the veracity of this story.
- 24.
Fischbach noted that in the limit of infinite range their suggested force agreed with that proposed earlier by Lee and Yang (1955) on the basis of gauge invariance.
- 25.
In a later paper (Aronson et al. 1986) the group suggested other experiments, particularly on K meson decay, that might show the existence of such a hyperphoton.
- 26.
An interesting sidelight to this reanalysis is reported in a footnote to the Fischbach paper. Instead of reporting the observed values of \(\Delta k\) for the different substances directly, Eötvös and his colleagues presented their results relative to platinum as a standard (Fischbach et al. 1986, p. 6): “The effect of this combining say \(\Delta k(\mathrm{H}_{2}\mbox{ O}\textendash \mbox{Cu})\) and \(\Delta k(\mbox{ C}\textendash \mbox{Pt})\) to infer \(k(\mathrm{H}_{2}\mbox{ O}\textendash \mbox{Pt})\) is to reduce the magnitude of the observed nonzero effect [for water and platinum] from \(5\sigma\) to \(2\sigma\).” \(\Delta k(\mathrm{H}_{2}\mbox{ O}\textendash \mbox{Cu}) = (-10 \pm 2) \times 10^{-9}\) and \(\Delta k(\mbox{ Cu}\textendash \mbox{Pt}) = (+4 \pm 2) \times 10^{-9}\), respectively. Adding them to obtain \(\Delta k(\mathrm{H}_{2}\mbox{ O}\textendash \mbox{Pt})\) gives \((-6 \pm 3) \times 10^{-9}\).
Figure 1.9 shows both the final summary reported by Eötvös as well as Fischbach’s reanalysis, along with best-fit straight lines for both sets of data separately (this is my own analysis). Although several of the experimental uncertainties have increased, due to the calculation process, the lines have similar slopes. The major difference is in the uncertainty of the slopes. If one looks at the 95 % confidence level, as shown separately for the Fischbach and Eötvös data, respectively, in Figs. 1.10 and 1.11, one finds that at this level the published, tabulated Eötvös data is, in fact, consistent with no effect, or a horizontal straight line. This is certainly not true for the Fischbach reanalysis.
A skeptic might remark that the effect is seen only when the data are plotted as a function of \(\Delta (B/\mu )\), a theoretically suggested parameter. As De Rujula remarked (1986, p. 761): “In that case, Eötvös and collaborators would have carried their secret to their graves: how to gather ponderous evidence for something like baryon number decades before the neutron was discovered.” It is true that theory may suggest where one might look for an effect, but it cannot guarantee that the effect will be seen. Although one may be somewhat surprised, along with De Rujula, that data taken for one purpose takes on new significance in the light of later experimental and theoretical work, it is not unheard of.
There is a possibility that Eötvös and his collaborators might actually have seen something of this effect, but discounted it. They report (Eötvös et al. 1922, p. 164): “The probability of a value different from zero for the quantity \(x[\Delta k]\) even in these cases is vanishingly little, as a review of the according observational data shows quite long sequences with uniform departure from the average [emphasis added], the influence of which on the average could only be annulled by much longer series of observations.” The original summary, given in Table 1.1, gives an average value for \(x = (-0.002 \pm 0.001) \times 10^{-6}\), which seems to justify Eötvös’ original conclusion (Eötvös et al. 1922, p. 164): “We believe we have the right to state that x relating to the Earth’s attraction does not reach the value of 0. 005 × 10−6 for any of these bodies.”
Table 1.1 The value of \(x - x_{\mathrm{Pt}}\) (\(\Delta k\) relative to platinum) for various substances (From Eötvös et al. 1922, p. 164)
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Franklin, A., Fischbach, E. (2016). The Rise … . In: The Rise and Fall of the Fifth Force. Springer, Cham. https://doi.org/10.1007/978-3-319-28412-5_1
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