On the relativistic conveyor belt

Abstract

A relativistic four-vector fundamental equation formalism is used to analyse processes that are carried out on a conveyor belt, in reference frames S (with the conveyor belt at rest, ground) and ${\bar {\rm S}}$ (moving conveyor belt frame); these processes involve thermal effects. Examples solved are: a block thrown with initial speed on an inclined plane conveyor belt, and a ring launched with initial linear velocity on a moving belt. For each process, a four-vector fundamental equation is obtained, first in frame S, and then it is transformed by the Lorentz transformation to the process' four-vector fundamental equation in ${\bar {\rm S}}$. It is shown that Newton's second law and the first law of thermodynamics are no-independent equations. In this description, Newton's second law in frame ${\bar {\rm S}}$, in which forces are non simultaneously applied, includes terms related to linear momentum for heat (relativistic Doppler effect). The first law of thermodynamics in ${\bar {\rm S}}$ includes the motor's performed work to keep the belt moving at a constant speed (conveyor belt effect). Classical descriptions of processes are obtained by taking the relativistic equations' low-speed limit, keeping the conveyor belt effect.

Relativistic thermodynamics on conveyor belt

“…Fuel system. The four-vector fundamental equation for the fuel in frame S ¯is obtained by applying the Lorentz transformation to the corresponding equation in S [equation (16)]:…”

“…Throughout this work, relativistic thermodynamics will be understood as the description of a thermodynamic process carried out on a conveyor belt, or wagon, in motion relative to an observer S ¯, at rest on the ground, by means of Einstein's special theory of relativity equations and methods: in the asynchronous formulation [7], by using a four-vector formalism [2]. The description of the process in frame S ¯will be obtained from that in S, at rest with respect to the conveyor belt and transforming the corresponding four-vector equation by application of the Lorentz transformation [15,16].…”

“…The principle of relativity (Huygens-Galileo principle [20], Ch. 16) does not play a significant role in Newtonian physics [21], p. 451 and even less in thermodynamics. Although the change of reference frame is considered in describing processes involving a conveyor belt [22], the thermodynamic aspects of such processes where mechanical energy dissipation is present, are hardly considered [23].…”

“…The remainder of the article is organized as follows. In section 2, a process with mechanical energy production-launching of a projectile by a chemical load inside a cannon placed on a moving wagon -, is described (by using the same formalism, mechanical energy dissipation processes have been described in [16]). In section 3, the Joule Kelvin process for a Van der Waals gas is described when the whole experimental device is located on a moving conveyor belt.…”