Equilibrium in special relativity

Abstract

It is shown that there are no energy currents perpendicular to the forces on an object. Many reasons for the erroneous nature of the von Laue energy current are given. This energy current is of no help in formulating equilibrium conditions in the conventional synchronous formulation, where additionalad hoc momenta due to the nonconservation of simultaneity have to be added. Equilibrium maintained by continuous time-dependent forces cannot be defined by the simultaneous vanishing of the forces. On the other hand, in the asynchronous formulation of Cavalleri and Salgarelli, equilibriumis given in terms of only the forces and torques. The relevant torque tensor ℐΜΝ is defined. This tensor is the sum of quantities evaluated at different times, corresponding to the Lorentz transformation of simultaneous measurements in the proper inertial frameS. Consequently, if ℐΜΝ= 0, then ℐΜΝ1, the torque in another inertial frameS′, also vanishes. Numerous objections are fully discussed. It is shown that there is no paradox in the Trouton-Noble experiment. Finally, it is shown why it is impossible to formulate the usual Hamiltonian for interacting particles in special relativity in terms of the co-ordinate timet; instead, one must use individual proper timesΤ.