REVIEW

Testing MOND on Earth1

A.Yu. Ignatiev

Abstract: MOND is one of the most popular alternatives to dark matter (DM). While efforts to directly detect DM in laboratorieshave been steadily pursued over the years, the proposed Earth-based tests of MOND are still in their infancy. Some proposals thatrecently appeared in the literature are briefly reviewed, and it is argued that collaborative efforts of theorists and experimentersare needed to move forward in this exciting new area. Possible future directions are outlined.

PACS Nos.: 04.80.Cc, 45.20.D−.

Résumé : MOND est l’une des plus populaires options de remplacement de la matière noire (DM). Alors que les efforts pourdétecter directement la DM dans les laboratoires se poursuivent activement depuis des années, les tests de MOND proposés surTerre sont encore dans l’enfance. Nous passons ici en revue certaines propositions faites récemment dans la littérature et nousarguons que des efforts de collaboration entre théoriciens et expérimentateurs sont requis pour avancer dans cette excitantedirection. Nous soulignons quelques possibilités futures. [Traduit par la Rédaction]

1. IntroductionThe existing astrophysical evidence suggests that Newton’s sec-

ond law may need modification. This idea is part of the MONDparadigm [1]. There are two versions of MOND [1]: one modifies theuniversal gravitational law, and the other changes Newton’s sec-ond law. Here, we are only talking about the latter because theformer has been extensively reviewed before (see ref. 2 and refer-ences therein).

This whole idea was discussed and tested within an astro-physical context by looking into astronomy data. It was suggestedmore recently that one should look at laboratory experiments [3, 4]and how to test this idea in the ordinary experimental way that wetest other similar ideas.

The modification of dynamics takes effect only when the acceler-ation is of order a0 � 10−10 ms−2, much smaller than any accelerationthat we usually encounter, because under ordinary conditions everyphysical body is acted upon by many forces, which give the bodyan acceleration that is usually much larger than a0.

Because of that, observing these phenomena was consideredvery close to impossible. Recently it was found that some speciallyarranged conditions allow us to test the hypothesis in a normalterrestrial laboratory.

To do this, we have to provide conditions that would lead tocancellation of all large forces. All those large forces that act onour test body should cancel almost exactly leaving only a tinyresidue, which is, of course, not easy.

When we are on the Earth, each body is acted upon by inertialforces due to the Earth’s spinning around its axis, orbiting aroundthe Sun (centrifugal forces), etc. So because of Earth’s motions, wehave several inertial forces, and the main problem is makingthese forces cancel.

It appears that the answer depends crucially on the state ofmotion of the test body. Depending on how the body moves, theconditions of “entry” into the MOND regime could be very differ-ent, and the corresponding experiments could vary wildly in theirdifficulty. Initially, it was important to focus on the “proof-in-principle” that such experiments are at all possible, laying aside

more practical considerations of the required costs and efforts.Because of this, attention was drawn to the conceptually simplestcase: when the test body is at rest (relative to the Earth). We empha-size from the beginning that this is only one of the many possibilities [3].It may or may not be practical, but if it is not, there are plenty ofother options to contemplate. At least three of them have beendiscussed in the literature [3, 5]. The main objective of this paperis to draw attention to these alternatives, which are largely in thenascent stage and many details have yet to be filled in. (We arediscussing a challenging problem requiring collective efforts ofboth theorists and experimenters.)

Let us start with reviewing briefly the “rest” case to set the stageand for comparison with other scenarios.

The inertial forces can cancel in the static case, but not alwaysand not everywhere. Only at a certain time and certain place onthe Earth could the conditions satisfy the cancellation require-ment. These circumstances are very rare and special occasions onthe latitudes about 80° to the north or to the south — not placeswhere you would normally go for a holiday! But such places arenot new for physicists: in the 1990s they drilled a borehole inGreenland to see if there exist deviations from the gravitation law(the so-called “fifth force”).

However, we may prefer to perform the experiment in the com-fort of our laboratory at home. Then, we still have to exploit thesame idea of cancellation of forces, but in this case different forceswill be involved. This leads to the so-called cancellation betweenthe Coriolis and centrifugal (CCC) setup [3].

In this case, there is no restriction on the laboratory location,but the restriction on time (only twice a year) remains. Techni-cally, this stems from the fact that the centrifugal forces are due tothe Earth’s spin.

Finally, there are at least two approaches to doing the test at anytime and any place. One is to move a test body along a specialtrajectory with a prescribed speed and acceleration [3]. Anotherapproach [5] exploits the same basic idea of cancellation, but in adifferent experimental setup. The key point is to introduce extraman-made centrifugal forces into play (in addition to the usual

Received 31 March 2014. Accepted 13 May 2014.

A.Yu. Ignatiev. Theoretical Physics Research Institute, Melbourne 3163, Australia.E-mail for correspondence: a.ignatiev@ritp.org.1This manuscript is part of a special issue whose topic is MOND: modified Newtonian dynamics.

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Can. J. Phys. 93: 1–3 (2015) dx.doi.org/10.1139/cjp-2014-0164 Published at www.nrcresearchpress.com/cjp on xx xxx 2014.

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terrestrial ones). To realize this, a spinning object, such as a ring,is needed. If the rotation of the ring is carefully controlled, thenthere is a chance to achieve the desired cancellation at any timeand any place.

All four setups proposed so far are technically challenging andit is hard to predict the ultimate winner, but the game is exciting:it is not every day that a 300-year-old law can be challenged!

2. Which reference frame?It is extremely important to clarify2 which reference frame

should be used for designing the MOND tests [3, 4]. The main pointis that the usual lab frame (i.e., frame fixed on the Earth) is not asuitable one. Although such a frame can be treated as inertial formany purposes, it is not inertial when we talk about MOND be-cause of the smallness of the key acceleration scale, a0. Comparedto that scale, the usually insignificant inertial forces becomehuge. So even if we manage to arrange for a test body that has avery small acceleration relative to the lab frame, and look forpossible deviations from Newtonian dynamics, this would be aninteresting experiment, but it would not be possible to confirm orrefute MOND using its outcome (see refs. 6–10).

What is needed is the Galactic reference frame rather thanthe laboratory one. Are alternative choices possible? For example,what about a frame with the origin at the center of mass of theLocal Group of galaxies? Fortunately, the acceleration due tothe neighbour galaxies is much smaller than a0. For example, theacceleration due to the Andromeda galaxy is less than 10−12 ms−2.Therefore, although such an alternative is possible, it would notaffect the results very much.

The preceding intuitive argument can be formalized as follows [4].Suppose we have two frames of reference: S (inertial) and S=

(noninertial). Let S= move with acceleration b with respect to S.The equation of motion in S reads

F � ma�� a

a0� (1)

where a is the test body acceleration relative to S.Assuming that a0 is invariant, in the S= frame we have the

following equation of motion:

F � ma ′��a ′

a0� � mb (2)

where a= = a − b represents the test body acceleration relative to S=.It is easily seen that (1) and (2) cannot be satisfied simultaneouslyfor all a and b. Indeed, if a = 0 then

m�� b

a0� � m (3)

for all b, which means that �(z) = 1 for all z. This contradictionproves that it is not possible to assume that the value of the criticalacceleration is independent of the reference frame. In other words,MOND should be formulated relative to the Galactic referenceframe, and not relative to the laboratory reference frame.

3. The static high-latitude equinox modified inertia(SHLEM) effect

To make the paper self-consistent, we briefly recall the essenceof what was dubbed “the SHLEM effect” in ref. 3. This effect refers

to the possibility of testing MOND using a static probe. The essenceof the effect is that the cancellation equations yield solutions thatare strictly localized, both in time and in space.

Time-wise, the solutions occur around the equinoxes, and space-wise, in the vicinity of 80° latitude.

The effect itself consists of a spontaneous tiny displacement ofthe test body at those special instants of time and at those specialpoints on Earth.

In addition to ground-based experiments, the approach pro-posed in ref. 3 allows one to discuss the possibility of using anEarth’s satellite for testing MOND. This would require a very high-altitude orbit Rorbit � REarth(g/as)1/2 � 40REarth and the inclination�23°27= (so that the orbit is in the ecliptic plane). For such asatellite, entering the MOND regime and, thus, the violation ofNewton’s second law would be expected to occur once per revo-lution around the Earth, that is, every 15 days. Further study isneeded to understand whether the anomalies in the satellite’smotion due to this effect would be observable or not. In particu-lar, lunar and planetary as well as nongravitational effects (see,e.g., refs. 11 and 12) should be taken into account.

4. Beyond SHLEMIt is important to realize that using the SHLEM setup is certainly

not a unique way to test MOND. Several alternatives were alreadydiscussed, along with SHLEM, in ref. 3. Another scenario was pro-posed in ref. 5. Even this list is unlikely to be exhaustive, and newopportunities could well be contemplated. However, the problemis challenging, and collaboration between experimentalists andtheorists would probably be needed to turn the general ideas intospecific proposals and to give the research a significant boost.

The necessary and sufficient condition for entering the MONDregime in the laboratory reads [3]

alab ≈ �a1(t) � � × �� × (r � r1)� � 2� × v � a2 (4)

where a1 is the acceleration of the Earth’s centre relative to theheliocentric reference frame; � is the Earth’s angular velocity; a2

is the Sun’s acceleration with respect to S0; r, v = r, and alab � r arethe position, velocity, and acceleration of the test body relative tothe lab frame; r1 is the position vector of the origin of the labframe relative to the terrestrial frame with origin at the Earth’scentre.

The general solution of (4) requires knowledge of a1(t) and a2(t),which can be obtained from astronomical data. Once they aregiven, the equation should be solved numerically with the re-quired accuracy. To get an idea of such a solution, let us makethese modelling assumptions:

1. acceleration a2(t) is neglected (in other words, we assume thatthe heliocentric frame is inertial);

2. acceleration a1(t) is taken as a harmonic oscillation with thefrequency �1 = 2�/(1a) (i.e., Earth’s orbit’s eccentricity and lu-nar effects are ignored); and

3. assume that the direction of � (taken as z-axis) is perpendicu-lar to the ecliptic.

Under these assumptions, the general solution of (4) becomes

x ≈ (x1 � x0)�2 � 2v0y� � R�1

2 � y1�3t � 3v0x�3t � x1�

4t2

y ≈ (y1 � y0)�2 � 2v0x� � 2x1�

3t � 3v0y�2t � v0x�2t

�3R�12�t � y1�

4t2

(5)

2Overlooking this point leads to a misguided perception that MOND has been already ruled out experimentally.

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2 Can. J. Phys. Vol. 93, 2015

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where x0, y0, v0x, and v0y are the initial position and velocity of thetest body; x1 and y1 are the coordinates of the origin of the labora-tory frame relative to the Earth’s centre; and R is the Earth–Sundistance.

Using integration once (or twice), (5) will yield the velocity (orthe trajectory) of the test body that will satisfy the conditions ofthe MOND regime at all times and not just at the special instantsaround the equinoxes. Also, the restrictions on the laboratorylocation disappear. An experiment based on (5) could be per-formed anywhere. Thus we have a strong argument that space–time unrestricted setups could be possible. However, a lot offurther work would be required to find out if an actual experi-ment could be designed along these lines.

Schematically, the setup could be as follows. Take a test body,and move it precisely along the “MOND trajectory” with preciselythe “MOND velocity”. Then the MOND prediction is that anoma-lous behaviour (e.g., unaccountable residues in the position orvelocity data) would be observed.

The next category of MOND tests could be termed “space–unrestricted, time–restricted” experiments. Like the above setup,they could be performed anywhere on Earth, but only at certainspecific times around the equinoxes. An example in this categorycould be the CCC setup [3] based on the cancellation between thecentrifugal and Coriolis inertial forces.

As one of the specific realizations of the CCC scenario, we mayconsider testing MOND in a time-of-flight laboratory experiment.Again, we would like to stress that the time-of-flight tests are justone of the many possible examples in the “space–unrestricted,time–restricted” category.

For this experiment to succeed, many factors should be takeninto account, some of which are in conflict with one another.Here, we restrict ourselves to listing all of these factors leavingfurther details for future work:

velocity magnitude;velocity direction;accuracy of velocity control;baseline length;accuracy of time measurement;projectile type;statistics;gravity compensation accuracy;gravity compensation mechanism;vacuum;cryogenics;screening of external fields; andrelative or absolute measurement.

5. Outside the MOND regimeThe scenarios mentioned earlier are very different and require

entirely separate setups, yet even they do not exhaust all possibleroutes to testing MOND in the laboratory. All these options, al-though differing in their approach, shared one thing in common:they attempt to create conditions for entry into the MOND re-

gime, so we could probably call all of them collectively the “insideMOND” experiments.

However, seeing how difficult it is to get inside, we could alsotry to test MOND using high-precision measurements without en-tering the MOND regime, but being, in some sense, close to it. Theseattempts could be termed “outside MOND” tests.

It remains to be seen whether a competitive setup can be de-signed along these lines.

6. Novel setupOwing to a recent breakthrough in precision accelerometry, a

completely new setup can be imagined. A new accelerometer builtat NIST has a resolution of 10�18 m/�Hz over a frequency range ofseveral kilohertz [13]. Because characteristic displacements of atest body due to the SHLEM effect are of the order of 10−14 m [4],using accelerometers could be a promising direction to pursue indesigning a sensitive SHLEM experiment.

7. Physics and astrophysicsMOND theory is so unusual for many physicists that the astro-

physical evidence alone, no matter how strong, would probablybe not sufficient to convince the sceptics. What is missing is evi-dence from the laboratory. Experimental proof of MOND will behard to deny or ignore.

In fact, a parallel with the dark matter case is entirely appropri-ate here. Searches for dark matter are conducted on Earth asvigorously as in the cosmos. MOND should be given the samechance.

The main thing is that searches for MOND effects, while chal-lenging, are not prohibitively difficult, though they do requirea lot of collaboration between experimenters and theorists. Itseems that only team efforts can bring further progress here.

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