Mass Anisotropy Searching for the Effects in a Foucault Pendulum Benjamin C. Auerbach, Seth A. Major Department of Physics, Hamilton College † email: {bauerbac, smajor}@hamilton.edu Introduction In his Principia, Newton wrote about his bucket ex- periment [1]. He filled a bucket with water and tied a piece of string around the handle. He then spun the bucket slightly and showed that the water in the bucket did not move much. He then twisted the string and allowed the bucket to spin as the string uncurled, and noticed that the water assumed a parabolic shape once the motion of the bucket had been communicated to the water in the bucket. His conclusion was that so long as the water was not mov- ing relative to absolute space then there was no effect on the water. Newton defines, “Absolute space is in its own nature without relation to anything external, remains always sim- ilar and immovable.” Once the water began to follow the bucket and hence have motion in absolute space then the water would assume the parabolic shape. In the 1800’s Ernest Mach approached Newton’s ex- periment differently. If we imagine the mass distribution spinning about the bucket we can see no physical difference between this scenario and Newton’s bucket experiment. So perhaps the shape of the water is caused by something other than absolute motion. The mass of the universe may be able to affect the inertia of the water. Using Cocconi and Salpeter’s Work Cocconi and Salpeter gave a formulation of Mach’s ideas [2]. They began with the assumptions: • Inertial mass is proportional to the mass distribution of the universe around the object. Hence any non- uniformities in the mass distribution in the universe could lead to anisotropic mass. • Mass anisotropy, Δ M ΔM ∝ M r ν M is the mass of a massive object impacting the anisotropy, and r is the vector distance away. • Using the above formula we see that ν lies between 0 and 0.4. It is ensured that objects further away from the Earth than the Sun have a larger effect on the Anisotropy, than the mass of Earth and the Sun. anisotropy v nu 1E-24 1E-22 1E-20 1E-18 1E-16 1E-14 1E-12 1E-10 1E-08 1E-06 0.0001 0.01 1 100 0 0.2 0.4 0.6 0.8 1 nu anisotropy Anistropy due to sun Anisotropy due to GC Anisotropy due to Earth Anisotropy Due to Nearby Massive Objects as a Function of ν This anisotropy is then directly related to the mass at the Galactic Center. • This contribution depends on the angle θ between r and the acceleration of the test body. • The vector pointing towards the Galactic Center de- fines a preferred direction. The most attractive depen- dence, which maintains energy conservation, has a maxi- mum value cos(θ )= 1. The angular dependence between acceleration and the preferred direction transforms New- ton’s Law into: F i = M ij a j Cocconi and Salpeter tell us what they believe the entries for M ij should be. Using basic Linear Algebra we can find the entries for the surface of the Earth rotating relative to the Galactic Center, the preferred direction. A Foucault Pendulum The Foucault Pendulum A Foucault Pendulum is similar to a typical pendulum however the top of the string is free to rotate. What we see is that the pendulum bob goes to relative maxima and minima, and while so doing it also rotates relative to the rotation of the Earth. In this figure the string’s length has been adjusted to illustrate the path of the Foucault pendulum as seen from above. A Top View of the Path of a Foucault Pendulum Coordinate System We will be using Cartesian coordinates for the following calculations. We choose our coordinates such that the ˆ q 3 axis points away from the surface of he Earth, the ˆ q 2 axis is perpendicular to the ˆ q 3 axis and points toward the North Pole, and the ˆ q 1 axis is mutually orthogonal. Review of Classical Mechanics We first write the Lagrangian for the Original Foucault Pendulum. Because we are on the surface of a rotating planet we are in a non-inertial reference frame, thus there will be a Coriolis Effect which is proportional to  ω × v [3]. We know that: L = m 2 (˙ q 1 2 +˙ q 2 2 + 2(ε ijk ω j q k )˙ q i ) - mgq 3 By using a small angle approximation we may write q 3 as q 2 1 +q 2 2 2l . We now use the Lagrangian, L, to find the Hamilto- nian. By definition we know that p l = ∂L ∂ ˙ x l . Which implies that p 1 = m ˙ q 1 - mωq 2 sin(λ) and p 2 = m ˙ q 2 + mωq 1 sin(λ). Thus the Hamiltonian is: H = p 2 1 + p 2 1 2m + p 1 ε 123 q 2 ω 3 + p 2 ε 213 q 1 ω 3 + mg q 2 1 + q 2 2 2l Which result in the equations of motion: ¨ q 1 = -2ε 123 ω 3 ˙ q 2 + gq 1 l ¨ q 2 =2ε 213 ω 3 ˙ q 1 + gq 2 l with solutions q 1 = cos(sin(λ)ωt) cos(  g l t) q 2 = sin(sin(λ)ωt) cos(  g l t). Our Coordinate System Foucault Pendulum with Mass Anisotropy Suppose we were to consider anisotropic mass in the con- text of a Foucault Pendulum. In a non-inertial reference frame M ij is a function of time, M ij = M ij (t). H = p j p n M -1 nj 2 - M ij ǫ jkl ω k q l p n M -1 ni + M 33 g 3 q 2 1 + q 2 2 2l Which yield the following equations of motion: M bj ¨ q j + ˙ M bj ˙ q j +2M bj ǫ jkl ω k ˙ q l + M 33 g 3 q b l = - ˙ M bj ˙ q j - ˙ M bj ǫ jkl ω k q l These equations of motion are the same as the Isotropic Mass Foucault Pendulum but there are two additional terms the -2 ˙ M bj ˙ q j - ˙ M bj ǫ jkl ω k q l . These two terms serve to drive the Anisotropic Foucault Pendulum and have an interesting result. Results The path of the anisotropic mass and isotropic mass Foucault pendulum look nearly identical. However, if we compare the angular frequencies we can see a difference: The Difference in Precession Frequencies between Anisotropic and Isotropic mass Foucault Pendulum This graph shows the Anisotropic Foucualt Pendulum Fre- quency less the Isotropic Foucualt Pendulum Frequency. We notice there is a very small change in frequency but look at the relative maxima and minima. If we take the time dis- tance between two maxima or minima we notice there is a difference of about 44,000 seconds, or a half sidereal day. Twice a day the mass of the pendulum is increased and its precession slows down, and another two times a day the mass is decreased and the pendulum frequency is increased. Conclusion and extensions In the anisotropic case we observe that the pendulum is being driven relative to its orientation toward the Galac- tic Center. The anisotropic mass is less than the isotropic mass when ˆ q 1 is perpendicular to the Galactic Center, and the anisotropic is greater than the isotropic mass when ˆ q 2 is parallel or anti-parallel to the Galactic Center. The driving of the angular frequency is proportional to the latitude of the Earth. The driving is greatest at the North/South Pole and the least at the Equator. The effects of mass anisotropy can help in the search of finding discrete space time. Acknowledgements We are grateful to Hamilton College, and to the Hamilton College Physics Department for help and advice. References [1] Newton, I. Principia Mathematics . [2] Cocconi, G., Salpeter, E. Nuovo Cimento X (1958) 3608. [3] Hand, L., Finch, J Analytical Mechanics Cambridge, 1998.