Measuring Gravity on Small Length Scales


S. J. Smullin, A. A. Geraci, D. M. Weld, J. Chiaverini, A. Kapitulnik





Despite excellent confirmation of both Newtonian and Einsteinian gravitational theories at large length scales, the force of gravity at sub-millimeter length scales has not been fully explored.  Since no measurements have been made, it is not known if there is a deviation from inverse square scaling at small length scales. According to some recent theories, such a deviation could be evidence for either compact extra dimensions or gravitationally coupled scalar moduli (proposed particles as yet unseen).  The hypothesis of extra dimensions addresses a current theoretical challenge: the hierarchy problem, or why gravity is so much weaker than the other forces in nature.  Moreover, it opens up the wonderful possibility for discovering that the world is not, at a very basic level, the way we assume it to be.


The notion of extra spatial dimensions has captured the imagination of mathematical and artistic minds for generations.  As a recent example, superstring theories are consistent only with the existence of nine spatial dimensions.  Arkani-Hamed, Dimopoulos, and Dvali have taken a fairly phenomenological approach to the idea of extra dimensions.  Taking advantage of the unmeasured gravitational regime, in 1998 they proposed a framework in which to solve the hierarchy problem.  What if, they asked, there are extra spatial dimensions in which gravity – but not the other forces in nature – lives?  The gravitational field lines would reach into these extra dimensions.  However, since the extra dimensions are compactified, we would not normally observe effects of the extra dimensions at scales larger than the compactification radius.  Thus, when we measure gravity in only three spatial dimensions, we find it to be weak; we have not measured gravity in the entirety of the space that the field lines inhabit.  These extra dimensions could be compactified at a length scale just below the length scales at which we have successfully measured.  If that were the case, then one could propose a consistent unification theory, including gravity, at the electroweak scale (1 TeV in energy), instead of at the currently inaccessible Planck scale, (1019 GeV).  Even within the strict bounds imposed by past experiments, there are regimes in which we may still hope to measure such a deviation from 1/r2.  Indeed, it is the experimental accessibility of this theory that makes it so appealing.  In addition, the non-Newtonian force due to extra dimensions may be up to 100 times the strength of normal Newtonian gravity.  Experimental signatures of extra dimensions may be found in tabletop experiments like ours, as well as in high-energy experiments.  Our simple goal: to measure gravity at varying distances and see if the scaling deviates from 1/r2 on small length scales.


The 1/r2 scaling of Newton’s gravity is a geometrical factor – a consequence of having a divergence independent of radius over a three-dimensional sphere.  The existence of extra dimensions would alter this scaling relation.  For “n” extra dimensions, if we measured gravity inside the compactification radius, we would expect gravity to scale as 1/r (2+n).  Another approach is to examine how a massless graviton in extra dimensions would appear to an observer in three spatial dimensions.  The graviton would appear to be massive – summing the Kaluza-Klein modes, we would expect gravity to scale as (1/r2)*(1+a*e-r/l), where a is the strength of non-Newtonian force, and l is the length scale below which it is relevant.  This gives us a phase space of a vs. l that we must cover with our measurements; we must be able to measure small forces and small distances.  Some parts of this phase space have been ruled out by past experiments including direct gravitational measurements, Casimir force measurements, cosmological observations, and high-energy experiments.  Yet still, a section of phase space remains to be explored – this is our frontier.

Figure 1

Our exploration of the gravitational force resembles the original experiments of this type.  Cavendish measured the Newtonian force between masses on the scale of 0.1 to 1 meter using a torsion balance to measure the deflection of test masses toward large stationary masses.  Other groups have used similar methods to measure gravity at small distances. The basic premise of these experiments and of ours is the same:  the use of a precise, mechanical force measurement system to discern the force of gravity between a pair of known masses placed at known separation.  Toward this end, we have designed apparatus suitable to such measurement at small mass separations.





To measure gravity at such small distances, a highly sensitive force meter is necessary.  We use micro-machined single-crystal silicon cantilevers, similar to but more sensitive than cantilevers used for atomic force microscopy.  These devices work on the same principle as a scale (a certain deflection of a “spring” corresponds to a certain amount of force), though to a much finer degree.  We put a small test mass on the end of a cantilever, and below this mass, we oscillate a pattern of heavy and light masses.  The gravitational force between this oscillating drive mass pattern and the test mass excites the cantilever at its resonant frequency.  The amplitude of the motion is measured using laser interferometry; consequently, we can determine the gravitational force between the drive and test masses.  By varying the distance between masses, we can discover how the gravitational force scales with distance.


A photograph of micro-machined cantilevers (similar to those used in our gravity experiments) is shown below.


Figure 2:  Micro-machined cantilevers


Our cantilevers are typically 50 mm wide, 250 mm long, and 0.3 mm thick.  They have a quality factor, Q, of about 80,000 at a temperature of 4 K (room temperature is approximately 300 K).  The force sensitivity of these cantilevers is better for long, thin, skinny cantilevers with high Q, measured at low temperatures over long periods of time.  This experiment is performed under vacuum in a liquid helium cryostat. 


The test mass is a 1 mg gold rectangular prism, a precisely formed “piece of dust.”  This test mass is placed on the end of a cantilever.  The test mass/cantilever system is driven by a pattern of gold/silicon/gold/silicon.  This pattern is made by etching trenches in silicon and filling these trenches with gold.  The drive mass is placed on the end of a piezoelectric bimorph that oscillates below the cantilever and test mass.  The frequency of the bimorph’s oscillation is such that the test mass feels a time-varying force at the resonant frequency of the cantilever.  The Newtonian gravitational force between the test and drive masses is on the order of attoNewtons (10-18 N) – we expect to have a sensitivity of 1 – 10 times this.  The resulting motion of the cantilever is on the order of 1 Angstrom, easily measured with our fiber interferometer.  A schematic of this setup (not to scale) and a photo are shown below.


Figure 3:  Experimental apparatus

(a)                                                                                                                   (b)



In (b) the cantilever with the test mass is seen at an angle from above the gold-covered cover wafer.  The shield wafer is not in place in this image and thus the drive mass pattern is visible below, through the trench in the cover and cantilever wafers.


Since the force we are measuring is so small, we must be careful that there are no spurious forces causing the test mass to move.  The cover wafer and the shield wafer are both metallized, providing a Faraday cage to shield the mass from electromagnetic fields.  The Casimir force, an attraction between two parallel conducting surfaces due to a quantum effect, is stronger than the Newtonian gravitational force at this scale.  The stiff 3 mm thick nitride film that comprises the shield wafer prevents the cantilever from being affected by the oscillating Casimir force between the shield and the drive mass. 


A further challenge is mechanically isolating the cantilever from the oscillations of the bimorph.  We want to make sure the motion we are measuring is due to the gravitational force between the masses, not the direct mechanical shaking of the cantilever by the bimorph.  The drive mass pattern of gold and silicon has multiple “periods”.  It is shaken at 1/3 of the resonant frequency of the cantilever so that the test mass feels a gravitational oscillation at the cantilever resonant frequency while the bimorph is (ideally) moving only at the drive frequency.  An animation (for Internet Explorer) of the bimorph/drive mass and cantilever/test mass shows this effect.  However, the bimorph is not ideal – the non-linearity of the piezo motion means that even with a pure frequency AC voltage at 1/3 of the cantilever resonant frequency, the piezo has a non-zero Fourier component at the resonant frequency.  In other words, the bimorph still oscillates a bit at multiples (including the 3rd) of the frequency at which it is being driven.  This component is small and yet it still excites the cantilever enough to swamp the gravitational effect we expect to see.  To alleviate this problem, the cantilever and bimorph stages must be isolated mechanically.  The usual mechanical coupling is attenuated by eight orders of magnitude with two vibration isolation stages (spring and mass systems at 2.2 Hz) between cantilever and bimorph.  A schematic and a photo of the probe (on a testing jig) are shown below. 


Figure 4:  Vibration isolation


Figure 5: The Probe



The probe, about 14 inches in height and 5 inches in diameter.  The optical fiber is not visible due to lighting.  The bimorph and wafer are obscured by brass pieces.  The outer frame holds the bimorph in its positioner.  The second mass of the vibration isolation system holds the “wafer sandwich.”



For more information:

Here is our recent preprint.

Schedule for Rencontres de Moriond (Mar 03). See Friday morning for talk on this experiment.

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Last updated by SJS on 12-Apr-03.