MOND on Earth

Is Violation of Newton's Second Law Possible?

A.Y. Ignatiev

PRL 98, 101101 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e101101

Ignatiev analyzes the classical equations of motion for a particle in a noninertial frame to determine the feasibility of testing theories of Modified Newtonian Dynamics on earth.

MOND is the idea that Newton's Second Law, F=ma, is not valid for very small accelerations. Below some characteristic acceleration a0, the force vanishes faster than a --- something like F=ka^p, where p>1. The theory can explain all current data on galactic rotation, and Bekenstein has generalized MOND into a covariant theory. Ignatiev now proposes that we could test MOND with a ground-based experiment.

He starts by looking at the equation of motion for a particle on earth with respect to the center of mass of the galaxy. The acceleration of the particle in this noninertial frame depends on five parameters, and Ignatiev has already neglected several terms that are small compared to a0: Coriolis acceleration of the sun, variations in the length of the day on Earth, precession and nutation of the earth's rotation axis, polar motion, and something called Chandler's wobble.

Ignatiev uses a pretty clever continuity argument to show that there are at least two times during the year where the acceleration vanishes. The acceleration during the summer solstice is greater than zero, and that during the winter solstice is less than zero; therefore, the acceleration must vanish at least twice a year (roughly at the time of the spring and autumn equinoxes).

For a brief window of time --- on the order of a second --- the acceleration parallel to the Earth's angular velocity vanishes. This can be used to find two specific locations on the Earth's surface where the perpendicular acceleration vanishes as well.

This leads the author to predict that there is a window of opportunity twice a year of about 1 second at two antipodal regions on the earth where the acceleration relative to the center of the galaxy is smaller than a0. At this instant, it should be possible to observe any effects of MOND that are believed to occur in astrophysical processes.

Later, the author goes on to analyze the case where the apparatus is moving with constant velocity relative to the lab frame. This relaxes the constraint on geographic location, because the velocity can be tuned to effect the same cancellation of terms that resulted from location for a stationary apparatus.

The effect one should look for in one of these experiments is a spontaneous displacement of a test body at the instant the total acceleration changes sign.

Ignatiev predicts a displacement amplitude on the order of 10^-17 m in a time interval of about 0.5 ms. This sounds undetectable, but he points out that LIGO is supposed to measure displacements an order of magnitude smaller than this. He also mentions the possibility of measuring the effects with a torsional balance of the variety used to look for deviations from the 1/R^2 dependence of the gravitational force at small distances.

Ignatiev's point seems to be that although MOND was created to explain phenomena on astrophysical scales, it makes predictions for terrestrial objects that can be tested with existing technology.

It seems to me that the ability to gather data only twice a year for less than a second would put serious limits on statistics. But I guess the guys at Fermilab are claiming to have observed the Higgs boson based on three or four events. An entire second worth of data might convince these guys!

I can't imagine anyone getting the funding to build another LIGO experiment in Greenland to test this prediction. If you take money out of the equation, you can build your detector in space, free from the earth's rotation. If you put it at one of the Lagrange points, you could also eliminate most gravitational effects. Might this allow for longer observation windows? Or are the gravitational and rotational effects necessary to cancel one another?

The idea that MOND has a preferred reference point --- the center of the galaxy --- that gives rise to special points on the Earth's surface gives it a mystical quality. I could see people building something like Stonehenge around these points, gathering twice a year to watch the needle on a detector move, indicating the exact cancellation of all accelerations.

Of course, there is no universal reference frame. The center of the galaxy is only preferred in this case because it's the biggest gravitational mass around. In Andromeda, it would be the center of Andromeda: i.e., there is a locally preferred reference frame, but not a universally preferred one.