The Lewis-Tolman Lever Paradox

Abstract

When a rigid right-angled lever experiences equal and opposite torques as measured by an observer O′ at rest with respect to the lever in frame S′, the usual transformation of these torques to another Lorentz frame S makes it appear that the torques are not balanced for an observer 0 at rest in S. This paradox was first pointed out by Lewis and Tolman. The consensus among textbook authors is that a net torque does exist on the lever according to the general Lorentz observer. The present paper is a basic reexamination of both the Lewis-Tolman lever paradox and another, closely related, paradox: equal and opposite torques applied at the same point on a rigid square. The basic approach herein is to construct for observer O a four-tensor Iμν to represent the relative four-torque about a point P′ fixed in frame S′ which has a constant speed ν = βc along the +x axis of S. For some unknown reason, this tensor seems never to have been defined before. The space part of Iμν reduces to the corresponding classical definition of torque in three-space if ν = 0. Based on Iμν a comparison of the torque components for two different Lorentz observers shows that no torque exists on the square or on the lever for any Lorentz observer if no net torque exists for the observer at rest with respect to the square or lever. This analysis suggests a new (covariant) definition for the torque I about a point P′a in three-space: I = l × f, where l is the space part of the four-vector lμ = (γ2lx, ly, lz, iγ2βlx) and where f is the space part of the four-vector fμ = (γF, iγβFx). Here, a force F is applied at a point P′b (stationary in S′) whose displacement from P′a (also stationary in S′) is 1. Both the force F and the displacement 1 are as measured by observer O at rest in S; and I is the torque about P′a according to O.