Quantum fluctuations for drag free geodesic motion

The Galilean principle of the universality of free fall has for a long time had a limited accuracy due to technical difficulties. The effect of air drag on falling bodies, the main of these difficulties, was already discussed at length by Galileo and Newton [1,2]. Nowadays this effect may be mastered either by an active control of the falling bodies or by letting the fall take place in a vacuum drop tower. Using an active control greatly reduces the vacuum requirement while allowing a test accuracy of 5×10−10 [3]. This accuracy does not reach the level of torsion balance experiments [4–6] but it may do so in the future with the additional use of a vacuum drop chamber [7]. The accuracy of the measurement of relative acceleration of freely falling bodies may thus reach 10−12 with a measurement time of 4.7 s corresponding to a fall of 109 m in the Bremen drop tower [8]. The use of drag free technique has also been proposed for ultra high accuracy satellite tests of the equivalence principle [9,10]. When such a technique is used, the freely falling proof mass is protected from the non gravitational perturbations like residual air drag, radiation pressure, et cetera by the satellite and its relative motion with respect to the satellite is monitored by a high sensitivity position sensor. The effect of these environmental perturbations on the satellite motion is then finely compensated through the action of thrusters. The aim of the present paper is to evaluate the ultimate limits of the drag free technique. We study the residual motion of a proof mass protected from environmental perturbations by a cage, the motion of which is itself actively controlled to follow the geodesic motion of