Laboratory Test of Newton’s Second Law for Small Accelerations

We have tested the proportionality of force and acceleration in Newton's second law, F = ma, in the limit of small forces and accelerations. Our tests reach well below the acceleration scales relevant to understanding several current astrophysical puzzles such as the flatness of galactic rotation curves, the Pioneer Anomaly, and the Hubble acceleration. We find good agreement with Newton's second law at accelerations as small as 5× 10 14 m/s2. F = m~ a is perhaps the most famous and most often used equation of physics. Together with its relativistic and quantum mechanical variants, this law is implicitly tested in many applications and experiments, and its validity is simply assumed at all acceleration scales. Any deviation from ~ F = m~ a would have profound consequences as it would imply a violation of crucial conservation laws such as energy and momentum in their conventional defini- tion. At very small accelerations a deviation from New- ton's second law could remain hidden in most laboratory scale experiments, but might appear in astrophysical and cosmological observations. One observed fact is the flatness of galactic rotation curves. The tangential velocity of stars measured as a function of distance from the galactic center rises first and flattens for larger distances. Newton's second law together with the gravitational effect of known matter predicts a decrease in the velocities for larger distances, and dark matter has been introduced to resolve this discrepancy (1). Alternatively, Milgrom discovered that Newton's second law can be modified with a single ad- ditional parameter a0 to describe the measured galac- tic rotation curves extremely well without invoking dark matter (2, 3). While Milgrom's full formalism MOND (Modified Newtonian Dynamics) is untestable in the lab- oratory, since it requires the absence of accelerations in all directions, a modification of Newtonian dynam- ics provides a simple explanation of the galactic rota- tion curves. Milgrom suggested that Newton's second law would smoothly transition from F ∝ a to F ∝ a2/a0 at a ≈ a0. Hence for a ≪ a0 a force would yield a larger acceleration as compared to standard Newtonian dynamics. The functional form of the transition between