Final Project Report : Tests of Mach ' s Principle with a Mechanical Oscillator *

James F. Woodward has made a prediction, based on Sciama's formulation of Mach's Principle in the framework of general relativity, that in the presence of an energy flow the inertial mass of an object may undergo sizable variations, changing as the second time derivative of the energy. We describe an attempt to test for the predicted effect with a charging capacitor, using a technique that does not require an unbalanced force or any local violation of Newton's 3 law of motion. We attempt to observe (1) the gravitational effect of the varying mass and (2) the effect of the mass variation on a driven harmonic oscillator with the charging capacitor as the oscillating mass. We report on the predicted effect, the design and implementation of the measurement apparatus, and initial experience with the apparatus. At this time, however, we will not report on observations of the presence or absence of the Woodward effect. * Supported in part by the National Aeronautics and Space Administration. † E-mail address: cramer@phys.washington. edu Introduction While the principle of equivalence, the exact universal identity of inertial mass and gravitational mass, is a cornerstone of general relativity, our standard model of gravitation, the origins of inertia remain obscure. Mach's Principle, the idea that inertia originates in the gravitational interaction of massive objects with the distant matter of the universe, is an attempt to unify gravitational and inertial mass, but it is not a part of general relativity. Dennis Sciama [1,2] attempted to improve this situation by showing that, for sufficiently symmetric and homogeneous universes, the gravitational interaction of massive objects with distant matter leads to an acceleration-dependent force, i.e., inertia. James F. Woodward extended Sciama's calculations by introducing an energy flow (e.g., the changing energy in a charging capacitor) into the gravitating system, and demonstrated that the formulation acquires extra transient contributions in Sciama's inertia term that are quite large. The implications of this work are: (a) that it may be possible to modify inertia, and (b) that it may be possible to demonstrate the validity of Mach's Principle with a tabletop experiment. Woodward and his students [3,4] have attempted to observe the predicted effect by producing an unbalanced force (say, to the right) that is expected to arise when the inertia varying test mass is accelerated to the right when it has low inertia and to the left when it has high inertia. They report having observed small unbalanced forces, near the limits of their sensitivity, which are about five orders of magnitude smaller than the predicted effect. Unfortunately, this scheme for observing the inertia variation appears to be at odds with the relativistically invariant form of Newton's 2 law of motion: dm/dt v /dt v d m /dt p d F     + = = (1) Since the inertial mass m of the test body is expected to vary with time, the last term of Equation (1) cannot be ignored. It is not surprising, in view of Newton's 3 law of motion, that for any sinusoidal variations of the mass around a central value, the force contribution from the v dm/dt term is found to precisely cancel the supposed "unbalanced force" arising from the m dv/dt term, leading to a timeaveraged net force of zero on the overall system. From this simple calculation, it appears that unbalanced force searches are not good tests of the proposed effect. There remains the question of whether the Woodward inertia variation is present in a system having an energy flow. In the present work, we have devised two tests of this effect that do not require the presence of an unbalanced inertial force.