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Novel Setup for Detecting Short-Range Anisotropic Corrections to Gravity

In this letter, we argue that even though there are strong theoretical and empirical reasons to expect a violation of spatial isotropy at short distances, contemporary setups for probing gravitational interactions at short distances have not been configured to measure such spatial anisotropies. We propose a simple modification to the state-of-the-art torsion pendulum design and numerically demonstrate that it suppresses signals due to the large spatially-isotropic component of the gravitational force while maintaining a high sensitivity to short-range spatial anisotropies. We incorporate anisotropy using both Yukawa-type and power-law-type short-distance corrections to gravity. The proposed differential torsion pendulum is shown to be capable of making sensitive measurements of small gravitational anisotropies and the resulting anisotropic torques are largely independent of the details of the underlying short-distance modification to gravity. Thus, if there is an anisotropic modification to gravity, from any theory, in any form of the modified potential, the proposed setup provides a practical means of detecting it.

Even though general relativity, and its short-distance approximation in the form of Newton gravity, does not break the isotropy of space, there are strong reasons to suspect that at short distances, the general relativistic description will be modified and this could, in turn, break the isotropy at short distances. In fact, as general relativity is only an effective field theory approximation to some quantum theory of gravity, it is expected that the observed symmetries of spacetime might only be symmetries of this low-energy effective field theory, and there could exist high-energy sectors in the theory that break these symmetries. It has been proposed that Lorentz symmetry breaks due to such quantum gravitational effects [1,2]. Usually, the Lorentz symmetry is broken from SO(3, 1) → SO (3) [3,4], but even the breaking of Lorentz symmetry from SO(3, 1) → SO(2, 1) has been proposed [5,6]. Such symmetry breaking would produce anisotropies.
In any theory of quantum gravity, which could break isotropy, the leading order term has to be the Einstein-Hilbert action. The short-distance approximation to the Einstein-Hilbert action is Newtonian gravity, which preserves isotropy, so we expect any anisotropy to come from corrections to Newtonian gravity, via a Yukawa [7] or power-law [8] term, for example. As these corrections occur due to string theoretical brane world models [9], an anisotropic brane would produce anisotropic corrections. Such anisotropic branes have been throughly studied as they are the dual to the deformation of super-Yang-Mills theory by a positiondependent θ term [10][11][12][13]. Here, we note that most spin-independent corrections are a combination of the Yukawa-type and power-law-type corrections to Newtonian gravity [9,[14][15][16][17][18][19]. In brane world models, it is pos-sible for the effective Planck scale to be much larger than the actual Planck scale, and this can lead to corrections to the gravitational field at sub-millimeter and micrometer length scales [20,21]. If the actual cosmology is described by anisotropic branes [10][11][12][13], such effects can produce detectable anisotropic corrections to the gravitational field.
The anisotropies could also be motivated by the discreteness in the structure of geometry at the Planck scale [22,23], which can also be motivated by appealing to results from loop quantum gravity [24][25][26][27]. The continuous geometry of spacetime occurs as a low-energy approximation to this discrete structure. It is expected that the discrete structure will break the isotropy of space at sub-leading order. In fact, such sub-leading order corrections have been demonstrated to break isotropy in condensed matter systems due to the intrinsic discreteness of such systems [28][29][30][31]. If the underlying structure of spacetime is discrete, then such sub-leading corrections would break the isotropy of space [32,33]. It is well known that low-energy modifications to quantum mechanics arise from various theories of quantum gravity [34,35]. Moreover, such modifications have recently been proposed to break isotropy at a detectable scale [36].
In conjunction with the aforementioned theoretical reasons, there are strong observational indicators pointing towards an anisotropy in space. Recent cosmological data indicates that there are clear anisotropies in the Cosmic Microwave Background (CMB) [37,38]. In fact, it has already been suggested that these anisotropies could occur due to short-distance modifications of the gravitational field [39,40], which could have important consequences during inflation [41][42][43][44]. The CMB anisotropies [37,38] have also been studied using anisotropic brane world models [45,46]. In these models, the anisotropies occur at short distances, and thus should be detectable by sensitive tabletop experiments.
The most sensitive tests of the gravitational force at the sub-millimeter scale are made using precision torsion pendulums placed in close proximity to a rotating attractor [15,47]. These tests are, in part, designed to look for deviations from the Newtonian inverse-square law (ISL). For example, the "missing mass" torsion pendulum and attractor of the Eöt-Wash group each have N equallyspaced holes a radial distance R from the rotation axis. This geometry results in a periodic torque that completes N cycles with each rotation of the attractor [48].
Our objective was to design a modified pendulumattractor set that is insensitive to spatially-isotropic gravitational forces while maintaining a high sensitivity to a posited spatially-anisotropic component of the gravitational force. Our differential torsion pendulum design, shown in Fig. 1, consists of an attractor with N fins and a pendulum with N − 1 fins. Figure 2 shows a top view of pendulum (gold) and attractor (teal) fins. Semitransparent coloring has been used to show the overlap of pendulum and attractor fins. Figure 2(a) shows the traditional setup in which the pendulum and attractor have the same number of fins. In this configuration, which is sensitive to the isotropic component of the gravitational force, the net torque on the pendulum vanishes only when the pendulum and attractor are perfectly aligned (β = 0) or perfectly misaligned (β = π/N ). Figure 2(b), on the other hand, shows an attractor with N fins and a pendulum with N − 1 fins. In this case, to within a first order approximation, the net torque on the pendulum due to the isotropic component of the gravitational force vanishes for a large number of attractor orientations. As discussed below, the suppression of the isotropic component of the gravitational torque is enhanced as N is increased.
In the absence of Yukawa or power-law corrections, the force on pendulum fin i due to attractor fin j is given by F ij = Gm p m arij /r 2 ij , where m p and m a are the masses of the pendulum and attractor fins, respectively, r ij = r a,j − r p,i is a vector that points from the center of mass of pendulum fin i to the center of mass of attractor fin j, andr ij is a unit vector in the direction of r ij . Expressions for the centre-of-mass positions of the pendulum and attractor fins are given in Equation (4) of the Appendix. Finally, the net torque on the pendulum about the vertical axis is given by the z-component of For verification purposes, the net torque on a pendulum with N = 10 fins suspended a distance d = 50 µm above an attractor with N = 10 fins was calculated as a function of φ assuming a spatially-isotropic gravitational force. For this calculation, and all subsequent calculations, the center-of-mass positions of the pendulum and attractor fins were assumed to be 25 mm from the pendulum's rotation axis. Additionally, the fins were given a thickness t = 1 mm and assumed to be made of copper. Figure 3 shows the results of the calculation. The calculated torque has the expected periodicity of 2π/N and is in good agreement with the experimental results presented in Fig. 2 of Ref. [48].
To investigate the effects that a short-range, spatiallyanisotropic component of the gravitational force would have on our torsion pendulum, we considered both Yukawa-like and power-law-like corrections to Newtonian gravity. The gravitational anisotropy was applied only to the correction terms added to conventional Newtonian gravity which allowed us to gauge the universality of such anisotropies when applied to different gravitational models. As the Einstein-Hilbert action preserves isotropy, only the correction terms can break the isotropy of space.
The anisotropic Yukawa-corrected force of attractor fin j on pendulum fin i is expressed as Here, α is the strength correction parameter and λ is the scale correction parameter. As described in the Appendix, the gravitational anisotropy is introduced by defining an anisotropic unit vectorr ij (ε) that is characterized by an anisotropy parameter ε 1. In a similar way, an anisotropic power-law corrected force [8] can be expressed as In this case, the correction term is parameterized by a single constant k. Figure 7 of the Appendix shows the numericallycalculated torque on a pendulum with N − 1 fins suspended above an attractor with N fins as a function of the attractor's angular position φ. For each value of N considered, the torque about the rotation axis of the pendulum was calculated assuming an isotropic gravitational force (ε = 0) and a gravitational force with a short-range anisotropy (ε = 10 −5 ) included. The calculations assumed that the pendulum-attractor separation distance was d = 50 µm. For the Yukawa correction, the parameters α = 0.1 and λ = 1 mm were used and, for the power-law correction, we used k = 2 µm.
We note that, for the Yukawa correction, the selected combination of α and λ has already been excluded by the current state-of-the-art experimental searches for deviations away from the Newtonian ISL [15,47]. However, the combination of parameters (α, λ, and ε or k and ε) merely set the scale of the anisotropic contribution to the net torque therefore, besides an overall scaling factor, the same qualitative features observed in the calculated data are expected for any combination of parameters.
The N -dependencies of the numerically-calculated torques for the Yukawa and power-law corrections were qualitatively similar ( Figure 7   The solid lines were obtained assuming perfectly isotropic gravity and the dotted lines assume a small short-range gravitational anisotropy. For small N , the Fourier transform of the torque data is dominated by a peak at N (N − 1) /2π in angular reciprocal space due to the isotropic component of gravity. As N is increased the isotropic peak is suppressed and the anisotropic peak at N/π begins to dominate. are suppressed. For N ≥ 35, the isotropic peaks fall below the noise floor of the numerical calculations. A plot of the isotropic and anisotropic peak heights as a function of N is shown in Fig. 5. For both the Yukawa and power-law cases, increasing N from 15 to 30 suppresses the fundamental isotropic peak by nearly ten orders of magnitude. The black squares in Fig. 5, represent the heights of the isotropic peaks for both the Yukawa and power-law cases. Because the corrections to Newtonian gravity are small, the isotropic component of the torque oscillations are largely independent of the details of the correction term.
Over the same range of N values, the anisotropic peak heights remain virtually unchanged. In Fig. 5, the red circles and blue triangles represent the heights of the anisotropic peaks for Yukawa-and power-lawcorrections, respectively. The two datasets follow similar trends as N is increased. The difference in the heights, a factor between 30 and 40 depending on the value of N , is due to the choice of parameters for the correction term (α = 0.1 and λ = 1 mm for the Yukawa correction and k = 2 µm for the power-law correction). In both cases, increasing N from 20 to 100 decreases heights of the anisotropic peaks by only a factor of 50. As a result, one is able to select N such that the net torque is completely dominated by the anisotropic contribution.
It is important to emphasize that, despite having very different dependencies on r ij , the Yukawa and power-law corrections produce torques that evolve in nearly identical manners as the value of N is changed. This is a crucial observation because, in most cases, modifications to general relativity are expected to introduce corrections to Newtonian gravity that are linear combinations of the Yukawa-and power-law-corrections that we have considered. Therefore, if short-range gravitational anisotropies do exist in nature, regardless of the underlying theory, the proposed apparatus will be a sensitive probe of the resulting anisotropic forces.
Motivated by theoretical and empirical arguments for short-distance anisotropy, and the absence of a suitable experiment to detect such anisotropy, we have proposed novel apparatus based on a simple, but novel, modification to the torsion-pendulum experiments. The proposed apparatus has the capability to probe anisotropies at micrometer length scales. We have incorporated anisotropy using both Yukawa-type and power-law-type short-distance corrections to gravity, and observed that this setup is largely independent of the details of the underlying short-distance modification to gravity. Thus, any form of short-distance anisotropic correction can be detected using the proposed apparatus. An important aspect of this proposal is that it only requires a small modification to proven experimental methods that are already used to search for short-range deviations from the ISL. Thus, the differential torsion pendulum describe in this letter can be constructed relatively easily and used to either verify the existence of short-distance anisotropy, or at least set a strong bound on the scale at which such anisotropies can occur. It would also be interesting to use the insights presented in this letter to develop experiments to search for anisotropies in other theories, such as those that may arise from short-range corrections to quantum mechanics [36].

APPENDIX
This Appendix provides additional details about the numerical calculations used to characterize the proposed differential torsion pendulum. First, we present the geometry of pendulum and attractor fins and relate their positions to the calculated torque. Second, we discuss how anisotropy was incorporated into the short-range corrections to Newtonian gravity. Finally, we show the dependence of the numerically-calculated torque on the angular position of the attractor for different values of N . We present results for both Yukawa-type and power-lawtype corrections and we consider both purely isotropic gravity and gravity with a small anisotropic component included. Figure 6 reproduces Fig. 2(a) of the main manuscript. It shows a top-view of a pendulum with N = 4 fins (gold) and an attractor, also with N = 4 fins (teal). The centreof-mass positions of pendulum fin i and attractor fin j are given by where it has been assumed that the centre of mass of each pendulum fin lies in the xy-plane (z = 0) and the centre of mass positions of the attractor fins are a distance z below the xy-plane. If the pendulum and attractor The angles θ and φ give the orientations of the pendulum and attractor, respectively, relative to a horizontal x-axis. The center of mass positions of the pendulum and attractor fins are a distance r 1 and r 2 from the rotation axis, respectively. As discussed in the main manuscript, the force on of attractor fin j on pendulum fin i due to Newtonian gravity is given by F ij = Gm p m arij /r 2 ij , where m p and m a are the masses of the pendulum and attractor fins, respectively, r ij = r a,j − r p,i , andr ij is a unit vector in the direction of r ij .
The isotropic form of the Yukawa-corrected gravitational potential is given by [7] Here, α is the strength correction parameter and λ is the scale correction parameter. We consider anisotropic corrections only to the strength parameter α. This is because any correction (for example, from some theory of quantum gravity) would be a higher-order correction to the Einstein-Hilbert action. As the Einstein-Hilbert action preserves isotropy, we expect only the correction terms to break the isotropy of space. The isotropic form of the Yukawa-corrected force is thus given by To make the Yukawa term anisotropic, we construct a normalized anisotropic unit vector given bŷ The angles θ ij and φ ij specify the direction ofr ij in spherical coordinates and the parameter ε determines the strength of the anisotropy. The anisotropic Yukawacorrected force can then be expressed as In a similar way, an anisotropic power-law corrected force can be expressed as Equations (8) and (9) reproduce Equations (1) and (2) of the main manuscript. Figure 4 of the main manuscript shows the Fourier transform of the calculated torque on the proposed differential pendulum for various values of N . In Figure 7, we show the raw torque versus φ results for both Yukawaand power-law corrections at several values of N . As stated in the main manuscript, these calculations were done assuming α = 0.1 and λ = 1 mm for the Yukawa correction and k = 2 µm for the power-law correction. In all cases, a pendulum-attractor separation distance of d = 50 µm was used. Furthermore, we assumed all fins had a thickness of t = 1 mm and were made of copper. The center-of-mass positions of the fins were assumed to be 25 mm from the rotation axis.  Figure 7), the torque is dominated by the isotropic contribution such that the isotropic and anisotropic curves overlap one another and are nearly indistinguishable. Parts (b) and (e) of the figure show that a modest increase in N suppresses the isotropic contribution to the torque by approximately four orders of magnitude. This suppression allows the contribution due to the anisotropic component of the gravitational force to be clearly observed as a superposition of the of the isotropic and anisotropic torque oscillations. Another modest increase to N (parts (c) and (f) of Figure 7) suppresses the isotropic contribution by another two orders of magnitude such that the anisotropic contributions completely dominate.