# Equivalence Principle Tests

The Weak Equivalence Principle is the principle of nature which states that the gravity behaves as if the observer were on a surface which was accelerating upwards.

The Equivalence Principle assumes a constant 1g acceleration:

“ Another clarification needed is that the equivalence principle assumes a constant acceleration of 1g without considering the mechanics of generating 1g. ”

Johannes Fankhauser at the University of Oxford says:

“ Einstein’s equivalence principle (also called the weak equivalence principle) assumes that any experiment in a uniform gravitational field yields the same results as the analogous experiment performed in a frame removed from any source of gravitational field but moving in uniform accelerated motion with respect to an inertial frame [Norton, 1985].^{3} ”

Physicists also often assume a uniform gravitational field. Professor Vincent Buonomano at the State University of Campinas states:

“ We always assume a uniform gravitational intensity unless stated otherwise. ”

## Contents

## Searching for Extra Dimensions

From the introduction of *Searching for Extra Dimensions and New String-Inspired Forces in the Casimir Regime* (Archive), its authors Dr. Dennis E. Krause (bio)
and Physics Professor Ephraim Fischbach (bio) tell us:

“ When Isaac Newton formulated his law of universal gravity over 300 years ago, he provided the first mathematical description of one of the fundamental forces of nature. Yet, physicists have realized only relatively recently that tests of Newtonian gravity can still provide a unique window into new physics [1-7]. Within the past 20 years, experimentalists have put Newtonian gravity to the test for distance scales 10^{-3}-10^{15} m by searching for violations of the weak equivalence principle (WEP) and inverse square law (ISL). The fact that no such violations have been observed places stringent constraints on extensions of the Standard Model that would naturally lead to such effects [1]. ”

## Universality of Free Fall

**The Newtonian gravitational constant: recent measurements and related studies**

By George T Gillies

p.200

**5. Searches for variations in G**
5.1. Spatial dependence of G

“ Searches for a change in G with intermass spacing have constituted a compelling quest in laboratory gravitation, especially during the past 25 years. The motivations for carrying out this kind of study were originally empirical, with the results of various benchtop experiments being interpreted in terms of either a value for or limit on some distance-dependent form of the gravitational constant (i.e. a G(r) effect), or in terms of a breakdown in the inverse square law (i.e. a modification to it of the form 1/r^{2+δ}, where δ is the departure parameter). Then, in the 1980s, observations that seemingly revealed evidence for non-Newtonian gravity at larger distance scales (Stacey et al 1987) fuelled much additional interest in this line of work. The contemporaneous suggestion by Fischbach et al (1986) that there may be previously undiscovered, weak, long-range forces in nature provided further impetus for investigating the composition- and distance-dependence of gravity, since the presence of any such effect might reveal the existence of a new force. During this time, a theoretical framework for admitting non-Newtonian effects into discussions of the experimental results was emerging. It led to the practice of using the laboratory data to set limits on the size of the strength-range parameters in a Yukawa term added onto the Newtonian potential, and this has become a standard method for intercomparing the results of this class of experiments. **Even though convincing evidence in favour of such new weak forces was never found, the many resulting experiments, when viewed as tests of the universality of free-fall, did much to improve the experimental underpinnings of the weak equivalence principle (WEP) of general relativity. In fact, searches for departures from the inverse square behaviour of Newtonian gravity have now come to be interpreted as attempts to uncover violations of the WEP.** ”

The above says that searches for departures from the inverse square bahavior of Newtonian gravity would be a violation of the Equivalence Principle which says that gravity is indistinguishable from an experiment which takes place on an upwardly accelerating Earth or compartment.

p.202

“ Other recent experimental searches for a breakdown in Newtonian gravity at large distances include a second set of tower gravity measurements made by Romaides et al (1994). **Their data, taken at five points over a nearly 500m vertical rise, reconfirmed the exactness of the inverse square law.** A similar result over a vertical distance of approximately 320 m was obtained at a meteorological tower in China by Liu et al (1992). ”

It should be noted that 500 meters is 1640.42 feet, and about as high as the Shanghai World Financial Center, a skyscraper in China.

### Inverse Square Behavior

The 'inverse square law', the 'Universality of Free Fall,' and the 'Weak Equivalence Principle' refers to the 9.8 meters per second squared rate of acceleration. From *Foundations of Modern Cosmology* we see a description of the inverse square law and how Newton interpreted the phenomenon in coming up with his theory of gravity:

“ In MKS units, the acceleration due to gravity at the surface of the earth is 9.8 meters per second per second, or 9.8 m s^{-2}. That is, if an object falls from rest, and air resistance can be neglected, at the end of one second is will be traveling 9.8 meters per second; at the end of another second it will attain the speed of 19.6 meters per second; and so fourth, until it hits the ground or air resistance balances the force due to gravity.

Once Newton had determined that gravity followed an inverse square force law, he was able to prove that Kepler's first and second laws followed necessarily. ”

## Encyclopedia Britannica

Encyclopedia Britannica has provided an article on the experimental study of gravity. Aside from the gravimeter devices which have been shown to be seismometers (See: Gravimetry) and the Cavendish Experiment, which is an inconsistent short range experiment, Encyclopedia Britannica agrees that there are no variations in gravity:

https://www.britannica.com/science/gravity-physics/Experimental-study-of-gravitation (Archive)

“ Early in the 1970s an experiment by the American physicist Daniel R. Long seemed to show a deviation from the inverse square law at a range of about 0.1 metre. Long compared the maximum attractions of two rings upon a test mass hung from the arm of a torsion balance. The maximum attraction of a ring occurs at a particular point on the axis and is determined by the mass and dimensions of the ring. If the ring is moved until the force on the test mass is greatest, the distance between the test mass and the ring is not needed. Two later experiments over the same range showed **no deviation** from the inverse square law. In one, conducted by the American physicist Riley Newman and his colleagues, a test mass hung on a torsion balance was moved around in a long hollow cylinder. The cylinder approximates a complete gravitational enclosure and, allowing for a small correction because it is open at the ends, the force on the test mass should not depend on its location within the cylinder. **No deviation from the inverse square law was found.** In the other experiment, performed in Cambridge, Eng., by Y.T. Chen and associates, the attractions of two solid cylinders of different mass were balanced against a third cylinder so that only the separations of the cylinders had to be known; it was not necessary to know the distances of any from a test mass. **Again no deviation of more than one part in 10 ^{4} from the inverse square law was found.** Other, somewhat less-sensitive experiments at ranges up to one metre or so also have failed to establish any greater deviation.

The geophysical tests go back to a method for the determination of the constant of gravitation that had been used in the 19th century, especially by the British astronomer Sir George Airy. Suppose the value of gravity g is measured at the top and bottom of a horizontal slab of rock of thickness t and density d. The values for the top and bottom will be different for two reasons. First, the top of the slab is t farther from the centre of Earth, and so the measured value of gravity will be less by 2(t/R)g, where R is the radius of Earth. Second, the slab itself attracts objects above and below it toward its centre; the difference between the downward and upward attractions of the slab is 4πGtd. Thus, a value of G may be estimated. Frank D. Stacey and his colleagues in Australia made such measurements at the top and bottom of deep mine shafts and claimed that there may be a real difference between their value of G and the best value from laboratory experiments. The difficulties lie in obtaining reliable samples of the density and in taking account of varying densities at greater depths. Similar uncertainties appear to have afflicted measurements in a deep bore hole in the Greenland ice sheet.

**New measurements have failed to detect any deviation from the inverse square law.** The most thorough investigation was carried out from a high tower in Colorado. Measurements were made with a gravimeter at different heights and coupled with an extensive survey of gravity around the base of the tower. Any variations of gravity over the surface that would give rise to variations up the height of the tower were estimated with great care. Allowance was also made for deflections of the tower and for the accelerations of its motions. **The final result was that no deviation from the inverse square law could be found.**

...**Thus far, all of the most reliable experiments and observations reveal no deviation from the inverse square law.** ”

### Torsion Balances

On the topic of the Torsion Balance tests discussed above which attempted to measure the gravitation of the sun, Encyclopedia Britannica says:

“ Experiments with ordinary pendulums test the principle of equivalence to no better than about one part in 10^{5}. Eötvös obtained much better discrimination with a torsion balance. His tests depended on comparing gravitational forces with inertial forces for masses of different composition. Eötvös set up a torsion balance to compare, for each of two masses, the gravitational attraction of Earth with the inertial forces due to the rotation of Earth about its polar axis. His arrangement of the masses was not optimal, and he did not have the sensitive electronic means of control and reading that are now available. Nonetheless, **Eötvös found that the weak equivalence principle (see above Gravitational fields and the theory of general relativity) was satisfied to within one part in 10 ^{9}** for a number of very different chemicals, some of which were quite exotic. His results were later confirmed by the Hungarian physicist János Renner. Renner’s work has been analyzed recently in great detail because of the suggestion that it could provide evidence for a new force. It seems that the uncertainties of the experiments hardly allow such analyses.

Eötvös also suggested that the attraction of the Sun upon test masses could be compared with the inertial forces of Earth’s orbital motion about the Sun. He performed some experiments, verifying equivalence with an accuracy similar to that which he had obtained with his terrestrial experiments. The solar scheme has substantial experimental advantages, and the American physicist Robert H. Dicke and his colleagues, in a careful series of observations in the 1960s (employing up-to-date methods of servo control and observation), found that **the weak equivalence principle held to about one part in 10 ^{11} for the attraction of the Sun on gold and aluminum. A later experiment by the Russian researcher Vladimir Braginski, with very different experimental arrangements, gave a limit of about one part in 10^{12} for platinum and aluminum.** ”

### Galileo Type

“ Galileo’s supposed experiment of dropping objects from the Leaning Tower of Pisa has been reproduced in the laboratory with apparatuses used to determine the absolute value of gravity by timing a falling body. Two objects, one of uranium, the other of copper, were timed as they fell. **No difference was detected.** ”

### Conclusion

The Britannica article concludes:

“ By the start of the 21st century, all observations and experiments on gravitation had detected that there are no deviations from the deductions of general relativity, that **the weak principle of equivalence is valid, and that the inverse square law holds over distances from a few centimetres to thousands of kilometres.** ”