Trouton-Noble paradox revisited

An apparent paradox is obtained in all previous treatments of the Trouton-Noble experiment; there is a three-dimensional (3D) torque T in an inertial frame S in which a thin parallel-plate capacitor is moving, but there is no 3D torque T in S, the rest frame of the capacitor. Different explanations are offered for the existence of another 3D torque, which is equal in magnitude but of opposite direction giving that the total 3D torque is zero. In this paper instead of using 3D quantities, e.g., E, B, F, L, T, .. and their “apparent” transformations we deal with 4D geometric quantities E, B, K, M , N , ... their Lorentz transformations and equations with them. It is considered in our approach that 4D geometric quantities and not the usual 3D quantities are well-defined both theoretically and experimentally in the 4D spacetime. In analogy with the decomposition of the electromagnetic field F (bivector) into two 1-vectors E and B we introduce decompositions of the torque N and the angular momentum M (bivectors) into 1-vectors Ns, Nt and Ms, Mt respectively. The torques Ns, Nt (the angular momentums Ms, Mt), taken together, contain the same physical information as the bivector N (the bivector M). It is shown that in the frame of “fiducial” observers, in which the observers who measure Ns and Nt are at rest, and in the standard basis, only the spatial components N i s and N i t remain, which can be associated with components of two 3D torques T and Tt. In such treatment with 4D geometric quantities the mentioned paradox does not appear. The presented explanation is in a complete agreement with the principle of relativity and with the Trouton-Noble experiment without the introduction of any additional torque.