Relativistic mechanics and thermodynamics. I. Linear translation four-vector formalism J G¨u´emez Applied Physics Department University of Cantabria (Spain) July 13, 2020 Abstract Einstein's special theory of relativity is presented in a Minkowski's four-vector formalism, integrating mechanics and thermodynam- ics at sophomore level, to solving undergraduate exercises in lin- ear translation requiring both. This relativistic formalism directly incorporates, in a four-vector fundamental matrix equation, the mechanics (Newton's second law) and thermodynamics (first law of thermodynamics) for a process. This four-vector formalism is used to analyse two processes: a block descending an inclined plane with friction, a mechanical energy dissipation process, with entropy-of-the-universe increas- ing (frictionless, a mechanical energy conservation process), and a cannonball ascending on an incline, moved by a force exerted by a chemical reaction, a mechanical energy production process, with Gibbs' free enthalpy function decreasing. 1 Introduction In the sophomore course, students learn mechanics and thermodynam- ics, considering that Newton's second law and the first law of thermody- namics are unrelated equations. However, Newton's second law and the (generalised) first law of thermodynamics [1] have a deep relationship, around the system's centre-of-mass acceleration, not considered in text- books and scarcely in papers [2]. In analytical mechanics, processes involving dissipative forces are not usually studied, and when they are, 1 with frictional forces, for example, concepts such as `friction-work' are introduced [3, p. 143], avoiding thermodynamics [4]. In thermodynamics courses, processes in which the system's centre- of-mass accelerates are not considered; so Newton's laws applied to them provide no relevant information. Moreover, as thermodynamic processes are not studied in a different coordinate reference frame (frame in short) that where the system is at rest, the relation between Newton's second law and the first law of thermodynamics when the process' description is transformed from one frame to another [5], is not shown. Manage magnitudes, equations, descriptions, correctly is not easy when one has a segmented view of sophomore physics (divided into me- chanics and thermodynamics, for example) instead of a panoramic, in- tegrated, view. In our opinion, Einstein's special theory of relativity ( STR in short) deserves an opportunity demonstrating that it provides a global view when developed with four-vectors. We have recently shown the way to applying Einstein's STR by a Minkowski's four-vector fundamental equation to extended bodies in processes with mechanical and thermodynamic aspects, such as solar sails [6] or the photoelectric effect [7]. The solar sail's speed or the emitted electrons' velocity, do not are close to the speed of light, c. But both processes involve photons, justifying the use of STR in their description. The STR four-vector description of these processes allows complementary analyses in terms of energies (in the case of the solar sail only the linear momentum equation is considered) or in terms of linear momentum (in the photoelectric effect case only the energy equation is considered), which are not in textbooks analyses of these phenomena. Moreover, the STR four-vector formalism is useful when the process must be described in a different frame, goal which is achieved by apply- ing the Lorentz transformation to the four-vector equation, allowing to demonstrate that the process' description is by the principle of relativity. University physics teachers usually assume that Einstein's STR should be used only in those processes involving speeds approaching the speed of light. Consider that such a teacher resolves the exercise of the block on a friction incline (Fig. 1), by giving acceleration a = (sin α − µd cos α)g and speed v(t) = at. How is it ensured that in the long-time limit, t ! 1, the block's speed v(t) does not exceed c? Mechanical energy dissipated by friction produces the incline's surface temperature increasing and energy ex- changed as heat with its surroundings. How does this thermal effect be related to the process' mechanical description? How does another observer, moving at almost speed c with respect to the incline, describe the process? How is it ensured that both descriptions are in agreement 2 with the principle of relativity? On the one hand, although these subjects can be considered formal questions, as no one expects to observ blocks moving at near c speed, previous questions deserve an answer. On the other hand, although sophomore students learn the STR in mechanics, they will not see any relativistic formulation of thermodynamics [8], a conspicuous asymme- try. Moreover mechanical energy conservation processes, some processes evolve with mechanical energy dissipation, with thermal effects [9], and there are mechanical energy production processes, by a thermodynamic potential decreasing [10]. As in the case of electricity and magnetism unified in a four-vector formalism [11, PART IV.], we think that, to responding questions similar to the previously posed, the best way analysing processes in between mechanics and thermodynamics is by using Einstein's STR. In this paper, which resumes in the Appendix the relativistic Einstein- Minkowski-Lorentz (EML in short) formalism previously developed in Refs. [4] (for collisions) and [8] (in thermodynamics), it is intended to give a unified STR overview of processes evolving with mechanical en- ergy conservation, dissipation or production, applied to solid bodies linear translation, i.e., with accelerating centre-of-mass. The last two processes include thermodynamic aspects that cannot be ignored. By generalising Minkowski's brilliant suggestion about four-vectors (space and time separately disappear and are integrated into the space-time four-vector), Newton's second law and the first law of thermodynam- ics are integrated into a four-vector fundamental equation, providing all physical information about a process. This paper is organized as follows. In section 2, a descending block process on an incline with friction, mechanical energy being dissipated (increasing the entropy-of-the-universe), is analysed by the EML formal- ism. In section 3, the process in which a cannonball moves up an incline due to being subjected to a force originated by chemical reactions, a process with mechanical energy production (by decreasing Gibbs' free enthalpy function), is analysed. Some conclusions are drawn in section 4 concerning the usefulness, conceptual and pedagogical, of this formal- ism. In the Appendix the EML formalism is briefly compiled. 2 Block descending an incline with friction For this process (Fig. 1) the friction force is Amontons-Coulomb kind, with fR = µdN, where, N = Mg cos α is the normal force on the block 3 T N y fR x(t) µ T (t) d G v(t) ↵ x Figure 1: A block, with initial velocity vi = 0, descends an incline, with block{incline (dynamic) coefficient of friction µd. When it has covered a distance x(t), during the time interval [0; t], its linear velocity is v(t). The x-axis advances down the incline. and µd a constant dynamical coefficient of friction. This block's (the block is our system) friction descending process is described in frame S, in which the incline is at rest, with Cartesian axes x − y, with x-axis parallel to the incline, increasing in its descending way, and y-axis perpendicular to the incline. Acting on the block are the gravitational G, the normal N and the frictional fR = (−µdN; 0; 0) forces. In frame S forces are simultaneously applied and the thermal reservoir surrounding the block is at rest. By doing µd = 0, one describes a block sliding on a frictionless incline (mechanical energy conservation), and no thermal effects. 1. Block linear momentum{(total){energy four-vector. The four- vector dEµ characterises the change of the four-vector Eµ, for a constant inertia M block, with (for typographical reasons, a contra-variant four-vector can be written as a row matrix instead of a column) µ 2 dE = cMd(γvv); 0; 0; Md(γvc ) ; (1) When block's temperature changes, its inertia changes too [4]; we assume that block's temperature remains constant. 2. Gravitational force impulse{work four-vector. The gravitational force performs work δWE=b = Mg sin αdx ; positive for the de- scending block. The gravitational force FE=b = Mg four-vector µ δWE=b, is µ δWE=b = (cMg sin α dt; −cMg cos α dt; 0; Mg sin α dx) : (2) 3. Normal force impulse{work four-vector This restrain force ex- erts impulse, it does not do work, on the block. The four-vector 4 µ δWN, for this normal force{block interaction, is µ δWN = (0; cNdt; 0; 0) : (3) 4. Friction force impulse{work four-vector The frictional force ex- erts impulse on the block, and this force does not perform work µ [12], on the system [4]. The four-vector δWR, is µ δWR = (−cµdNdt; 0; 0; 0) : (4) 5. Heat four-vector The four-vector δQµ, is δQµ = (0; 0; 0; δQ) ; (5) characterising the block's thermal interaction with its surround- ings at rest [8]. Four-vector fundamental equation. For a frictionless process µ µ (µd = 0) one has the four-vector fundamental equation dE = δWE=b + µ δWN. By introducing the frictional force (µd =6 0) impulse{work four- µ µ vector WR and the thermal effects four-vector δQ , one has the four- vector fundamental equation for this process µ µ µ µ µ dE = δWE=b + δWN + δWR + δQ : (6) In its matrix form, one has in frame S 0 1 0 1 0 1 cMd(γvv) cMg sin α dt 0 B C B C B C B 0 C B −cMg cos α dt C B cN dt C B C = B C + B C + @ 0 A @ 0 A @ 0 A 2 Md(γvc ) Mg sin α dx 0 0 1 0 1 −cµdNdt 0 B C B C B 0 C B 0 C + B C + B C : (7) @ 0 A @ 0 A 0 δQ With N = Mg cos α, and fR = µdMg cos α, by integrating eq.