Relativistic pendulum and the weak equivalence principle

Abstract

This paper derives equations for the relativistic proper period of oscillations of a pendulum driven by the electrical forces and for a pendulum driven by the gravitational forces. The derivations are based on the Einstein’s Special Relativity Theory and in particular on the Lorentz coordinate transformation, which has been experimentally verified many times and which is a well-recognized principle for all the modern physics. Since the pendulum proper period of oscillations is an absolute inertial motion invariant the derived formulas may be used to study the motion dependence of the inertial and gravitational masses. It is found that the well-publicized equivalence between these two masses, which is assumed independent of any inertial motion, cannot be sustained and a new mass equivalence principle must be considered where the equivalence of these two masses holds only at rest. INTRODUCTION The pendulum is an ages proven device that has attracted attention of many researchers in the past for its simplicity of operation, its accuracy to measure time, and for its ability to study the gravitational or electrical fields. One can only wonder why it was not studied in modern times in more detail, since it offers some clues for resolving the “mystery” of the inertial and gravitational mass equivalence, the so called Einstein’s week equivalence principle . Recently an interesting article was published [2] where the author derived relativistic equations of motion for the pendulum starting from a simple relativistic Lagrangian and the formula for the relativistic conservation of energy. This paper will also focus its attention of the relativistic equations of motion of the simple pendula, one that is driven by electrical forces, and the second one that is driven by gravitational forces and will compare how these pendula behave when they undergo an inertial motion relative to the laboratory coordinate system. The key idea of this work is to derive formulas for the proper period of oscillations of the particular pendulum in terms of the pendulum physical parameters such as the mass of the bob, the length of the pendulum string, and the remaining parameters of the experimental setup. Since the proper period of oscillations is an inertial motion invariant, the derived formulas must thus also be inertial motion invariants and * jhynecek@netscape.net 4/4/2006