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 → ∞, 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 cMd(γvv) cMg sin α dt 0 0 −cMg cos α dt cN dt = + + 0 0 0 2 Md(γvc ) Mg sin α dx 0 −cµdNdt 0 0 0 + + . (7) 0 0 0 δQ

With N = Mg cos α, and fR = µdMg cos α, by integrating eq. (7) we have the equations

γv(t)Mv(t) = (Mg sin α − µdMg cos α ) t , (8)

a = g (sin α − µd cos α ) , (9) 2 M(γv(t) − 1)c = Mgx sin α + Q, (10)

5 ext where a = ΣjFj /M, is an acceleration parameter (the block’s linear acceleration in classical limit), and with height h = x sin α. From equation γv(t)v(t) = at, one gets the block’s velocity [13] at time t˜, with

−1/2 v(t˜) = at 1 + a2t˜2c−2 ⇒ at˜ (v/c → 0) . (11)

For t˜ → ∞, limt˜→∞ v(t˜) = c, so eq. (11) ensures that the block’s speed does not exceed c. By integrating dx = vdt from (11), one obtains the distance x(t˜) travelled by the block until time t˜,

1/2 1 x(t˜) = c2a−1 1 + a2t˜2c−2 − 1 ⇒ at˜2 (v/c → 0) . (12) 2

In the low speed limit v/c → 0, one returns to the classical expressions for x(t˜) and v(t˜) for this process. 2 From relationship d(γvc ) = vd(γvv) [4], and spatial components of eq. (7), we get Newton’s second law complementary dynamical relation- ship [14] (pseudo-work equation [15])

Mvd(γvv) = (Mg sin α − µdMg cos α)vdt , (13) 2 M(γv − 1)c = (Mg sin α − µdMg cos α)x , (14) with vdt = dx. Eq. (14) includes the work term WE/b = Mgx sin α and the pseudo-work term pWR = −µdMgx cos α. Thermal eﬀects equation. For heat exchanged between the block and the thermal reservoir, at temperature T during time interval [0, t˜], comparing eqs. (10) and (14), the following relationship has been ob- tained Q(t˜) = −µdMg x(t˜) cos α . (15) This magnitude quantiﬁes process’ dissipated mechanical energy [16]. For the entropy-of-the universe variation for this process ∆SU = −Q/T (the block does not vary its entropy and the thermal reservoir increases its entropy as ∆SF = −Q/T ), one has [9]

T ∆SU(t˜) = µdMg cos α x(t˜) > 0 . (16)

Classical thermodynamics results for Q(t˜) and ∆SU(t˜) are obtained. Although this problem has parts in common with the ball-wall isother- mal inelastic collision problem studied in Ref. [4], its analysis allows one frictional force characterisation into the EML formalism.

6 N v Fa/b x

F⇠/b Fb/⇠ h F G c/⇠ ↵

Figure 2: A ball, with inertia M, initial velocity vi = 0, ascends a frictionless cannon located on an incline with angle α, powered by a chemical reaction. The cannon exerts force Fc/ξ on chemicals and chem- icals exert force Fξ/b on the cannonball. When the ball has covered distance x(t), in time interval [0, t], its velocity is v(t) and its height is h(t). The x-axis advances up the incline.

3 Cannonball launched by a chemical reaction

Fig. 2 sketches a process in which a cannonball is launched by the force Fξ/b, taken constant, produced inside a cannon in which a chemical reaction is developing (a cannon is a kind of thermal engine [17, p. 24]). For a chemical reaction it is possible to combine a large decrease in the Gibbs free enthalpy function and an extent of reaction of almost zero. In a real process, this work will depend on how the chemical reaction is carried out, by using a catalyst, sparks, the reactants’ injection rate, for example. We assume that the process is designed in such a way that force Fξ/b is known. Cannonball plus chemical reaction is our system. Process is described in frame S in which forces are simultaneously applied and the cannon and thermal reservoir are at rest. Ball’s four-vector momentum-energy. The four-vector momen- tum–energy variation for the ball, with inertia M is

µ 2 dEb = cMd(γvv), 0, 0, Md(γvc ) . (17)

We assume massless chemical reaction components and, for simplic- ity, that its internal energy does not contribute to its inertia (relativistic fuel). Chemical reaction’s four-vector momentum-energy. Through- out the chemical reaction, there will be a consumption of n mol reactants (fuel), with chemical internal energy change dUξ = nduξ < 0; chemical

7 reaction four-vector internal energy change is µ dUξ = (0, 0, 0, dUξ) . (18)

Heat four-vector. Chemical reaction entropy change is dSξ = min ndsξ. When this entropy change is negative, heat δQξ = T dSξ is the minimum amount of heat that must be provided to the thermal reservoir to ensure that dSU(ξ) = 0. We assume that this is the only heat exchanged ant that it is emitted as an ensemble of photons with total zero linear momentum (thermal photons). The four-vector heat µ δQξ for this process is µ δQξ = (0, 0, 0,T dSξ) . (19)

When dSξ < 0, δQξ < 0 and the thermal reservoir increases its entropy. For the ball–chemicals system, the external forces are: Fc/ξ (cannon on chemicals), Fa/b (atmospheric pressure on the ball), FE/b (gravita- tional force on the ball), and N (incline normal force on the ball). For inertialess chemicals, force Fc/ξ (cannon on chemicals) equals force Fξ/b (chemicals on the ball). Forces Fξ/b (chemicals on the ball) and Fb/ξ (ball on chemicals) are internal forces for the ball–chemicals system. Forces four-vector impulse-work. The four-vector impulse– µ work δWc/ξ is given by cFc/ξdt −cPextAdt µ 0 µ 0 δWc/ξ = ; δWa/b = . (20) 0 0 0 −PextAdx

For the eﬀect of the external atmospheric pressure Pext force on the ball µ we have the four-vector δWa/b. For this process reaction volume change is dVξ = Adx = ndvξ, where A is the section of cannon and ball, and dx is the ball’s displacement. µ Four-vector impulse–work for the gravitational force on the ball δWE/b is −cMg sin αdt 0 µ −cMg cos αdt µ Ndt δWE/b = ; δWN/b = . (21) 0 0 −Mg sin αdx 0 µ For the normal force eﬀect on the ball, we have the four-vector δWN/b. The four-vector fundamental equation for the ball–chemicals system and this process is µ µ µ µ µ µ µ dE + dEξ = δWc/ξ + δWE/b + δWN + δWa/b + δQξ . (22)

8 In its matrix form, cMd(γvv) 0 cFξ/bdt −cMgdt sin α 0 0 0 −cMg dt cos α + = + + 0 0 0 0 2 Md(γvc ) dUξ 0 −Mgdx sin α 0 −cPextAdt 0 cNdt 0 0 + + + . (23) 0 0 0 0 −PextdVξ T dSξ

For this system and process, we have the impulse–linear momentum change equation (Newton’s second law) (N = Mg cos α) and the energy equation (ﬁrst law of thermodynamics)

Md(γvv) = [Fξ/b − (Mg sin α + PextA)]dt , (24) 2 Md(γvc ) + dUξ = −Mgdx sin α − PextdVξ − T dSξ . (25)

Force Fξ/b > (Mg sin α + PextA) must be greater than the sum of the incline component of the gravitational force plus the force exerted by the external pressure on the ball. For Newton’s second law complementary dynamical relationship we obtain 2 Md(γvc ) = [Fξ/b − (Mg sin α + PextA)]dx , (26) where vdt = dx is the ball’s displacement. By integrating these equations over time interval [0, t˜], we have

γv(t˜)Mv(t˜) = [Fξ/b − (Mg sin α + PextA)]t˜ , (27) 2 M(γv(t˜) − 1)c = [Fξ/b − (Mg sin α + PextA)]x(t˜) , (28) 2 M(γv(t˜) − 1)c + n∆uξ = −Mg sin α x(t˜) − nPext∆vξ − nT ∆sξ ,(29) where v(t˜) is the ball’s ﬁnal velocity, Lcn ≡ x(t˜) is the cannon’s length and h(t˜) = x(t˜) sin α is height reached by the ball. With the (constant) acceleration parameter a,

−1 a = M [Fξ/b − (Mg sin α + PextA)] , (30) and by using (11) and (12) for time t˜, we obtain v(t˜) and x(t˜) for the ball in this process. By integrating eq. (25), one has

2 M(γv(t˜) − 1)c + Mgx(t˜) sin α = −∆Λξ , (31)

9 where ∆Λξ = ∆Uξ + Pext∆Vξ − T ∆Sξ is exergy (or availabilty) [18, Ch. 19] chemical reaction change. Assuming that during the process chemical pressure is close to the external pressure (but diﬀerent enough 5 5 to move the ball; P ≈ 1, 1 · 10 Pa and Pext ≈ 1, 0 · 10 Pa, for example),

2 M(γv(t˜) − 1)c + Mgx(t˜) sin α ≈ −n∆gξ , (32) where ∆gξ = ∆uξ + P ∆vξ − T ∆sξ is Gibbs’ free enthalpy chemical reaction change per mol. By comparing equations

2 M(γv(t˜) − 1)c + Mgx(t˜) sin α = Fξ/bx(t˜) − nP ∆vξ , (33) 2 M(γv(t˜) − 1)c + Mgx(t˜) sin α = −n∆gξ , (34) we have

Fξ/bx(t˜) = −n∆gξ + nP ∆vξ = −n∆fξ , (35) where ∆fξ = ∆uξ − T ∆sξ is Helmholtz free energy function variation. The chemical reaction Helmholtz free energy function decrease ∆Fξ = −n∆fξ (internal energy released minus the energy transferred to the thermal reservoir at T ) is the chemical reaction maximum available work, with max W = Fξ/bx(t˜) = −∆Fξ . (36) Gibbs’ function decreasing is the maximum mechanical energy available from chemical reaction [19, pp 65–70], with

max 2 E = Md(γv − 1)c + Mgx(t˜) sin α = −∆Gξ . (37)

These results are classical thermodynamic results for work (W ≤ −∆Fξ) and produced mechanical energy (E ≤ −∆Gξ) in a real process related with thermodynamic potentials decreasing [20].

4 Conclusions

To introduce Einstein’s relativity in a sophomore course, we have pre- sented a resume of the four-vector fundamental equation formalism (a matrix algebra). This EML formalism allows one to analyse any pro- cess (with mechanical energy conservation, dissipation or production) compactly, by simultaneously considering the process’ mechanical and thermodynamic aspects. By using EML four-vector fundamental equation, we have solved a sophomore course exercises in which a solid body moves on an inclined plane.

10 For a process with no dissipative forces (in its case, with µd = 0), mechanical energy is conserved, with zero entropy-of-the-universe in- creasing. The energy equation (ﬁrst law of thermodynamics) and the Newton’s second law complementary dynamical relationship (in general, the pseudo-work equation) are the same. The system is point-like. For a process involving dissipative forces, thermal eﬀects appear, and the process is irreversible. The energy equation and the comple- mentary dynamical relationship (pseudo-work equation) are diﬀerent, and they contain diﬀerent and complementary information about the process. The mechanical energy dissipated relates to the entropy-of-the- universe increment. For a process with mechanical energy production by chemical reac- tions, the decreasing in Helmholtz free energy is related to the (max- imum) work that the n mol chemical reaction can provide, and the decreasing in Gibbs free enthalpy function for the chemical reaction is related to the (maximum) mechanical energy that can be produced. The problems solved here are intended demonstrate that Einsteinian STR can be developed by integrating mechanics and thermodynamics. Relativistic mechanics applied to processes involving extended bodies re- quires the use the principle of inertia of energy, the relationship between body’s inertia and its internal energy content. Relativistic thermody- namics requires the thermodynamic magnitudes mechanical characteri- sation, i.e. by attaching to them linear momentum. The EML formalism distinguishes between the ﬁrst law of thermo- dynamics and the complementary dynamical relationship for Newton’s second law [21], equations which are sometimes confused. The compari- son between these two equations allows one to describe thermal eﬀects in terms of a force (dissipative or related with a thermodynamic potential) times a displacement product. A didactic advantage of using EML formalism, mathematically more demanding than the Newtonian formalism, is that its user must be pre- pared to build correct four-vectors (thinking vertically) and to propose an equation that is correct in every component (thinking horizontally), with all the pieces of the four-vector fundamental equation puzzle ﬁt- ting together. The possibility of making a mistake in the description of a process using the four-vector formalism is less than when each equation and each term must be written by hand, because in the EML formalism the pieces must ﬁt by themselves both vertically and horizontally. Interested students, once they notice that EML formalism is a not particularly complicated matrix algebra, will be able to recognise that Einstein’s STR is an excellent candidate to obtain a panoramic view of sophomore physics, at least, at the level it has been treated here.

11 5 Appendix. Four-vector fundamental equation

Minkowski’s words indicating that after Einstein’s STR, the classical, separate, concepts of space and time must overcome and integrated into the space-time concept, i.e., in a four-vector space-time, can be gener- alised. The concepts of linear momentum and kinetic energy must be overcome and substituted by a four-vector momentum-energy, and the linear impulse and work by a four-vector impulse-work [4]. The classical concept of heat, with no linear momentum associated in classical ther- modynamics, must be relativistically characterised by attaching it with linear momentum [8]. Then, a spatial-temporal four-vector fundamen- tal equation includes both Newton’s second law (a vectorial equation with three components, the spatial components) and the ﬁrst law of thermodynamics (a scalar equation, the temporal component). Four-vector space-time. Given an event (x, y, x, t) in frame S, the event’s coordinates (¯x, y,¯ x,¯ t¯) in frame S,¯ in standard conﬁguration, moving with speed V = (V, 0, 0), with respect to S, ensuring Einstein’s STR second postulate fulﬁlment are

x¯ = γV (x − βV ct) , (38) y¯ = y , z¯ = z , (39)

ct¯ = γV (ct − βV x) , (40)

2 −1/2 where γV = (1 − βV ) (Lorentz factor) and βV = V/c. The events’ description and the way in which coordinates transform, indicates, ac- cording to Minkowski, that an space-time entity be constructed, trans- forming as a whole from frame to frame, such that x¯ γV 0 0 −βV γV x¯ y¯ 0 1 0 0 y¯ = (41) z¯ 0 0 1 0 z¯ ct¯ −βV γV 0 0 γV ct¯

Eq. (41) means that it is the mathematical structure

x x¯ µ y µ y¯ x ≡ , x¯ ≡ . (42) z z¯ ct ct¯ which has a complete physical meaning (and not space and time sepa- rately). In frame S, xµ is the four-vector space-time for the event and x¯µ is the four-vector space-time for the event in frame S¯ coordinates.

12 Time t (and t¯) is multiplied by c so that components of column matrix xµ (andx ¯µ) be dimensionally homogeneous. µ Lorentz transformation. The transformation 4×4 matrix Lν (V )

µ µ µ x¯ = Lν (V )x , (43) γV 0 0 −βV γV µ 0 1 0 0 Lν (V ) ≡ , (44) 0 0 1 0 −βV γV 0 0 γV µ µ µ x = Lν (−V )¯x , (45) is the Lorentz transformation [4]. By this transformation,x ¯ in S¯ is a linear combination of x and ct in S, and the same for ct¯. And vice-versa by changing S¯ ←→ S. Einstein’s inertia of energy principle. According to Einstein’s principle of inertia of energy, a system’s internal energy E0(T, ξ, ω), i.e. the system’s total energy in reference frame S0 in which its total linear momentum is zero, depending on temperature, chemical composition, or angular velocity [22], contributes to its inertia M as [23]

−2 M(T, ξ, ω) ≡ c E0(T, ξ, ω) . (46)

Internal energy four-vector. In relativity, the block’s state is given by its four-vector momentum-energy, in which linear momentum and energy E, sum of energy at rest E0 ≡ U(T ) and kinetic energy K, enter. Linear momentum and energy are not independent, with 2 2 2 2 E = p c + E0 . The four-vector Eµ is obtained by combining the linear momentum vector p = (px, py, pz) (multiplied by c) and the total energy E = E0+K. In the zero-momentum reference frame S0, the system’s state, at µ time t, is given by the zero-momentum–energy-at-rest four-vector E0 0 0 µ 0 0 E0 ≡ = . (47) 0 0 2 E0 Mc

We assume that the block’s chemical structure –the internal chemical bonds– does not vary. In reference frame S, in the standard conﬁguration with respect to S0 with velocity v = (−v, 0, 0) (the frame S observer describes a body µ moving with speed v), the four-vector E0 transforms into the four-vector

13 momentum-energy Eµ [24] by using the Lorentz-transformation operator µ Lν (−v) as µ µ ν E = Lν (−v)E0 , (48) γv 0 0 βvγv 0 cγvMv µ 0 1 0 0 0 0 E = = . (49) 0 0 1 0 0 0 βvγv 0 0 γv E0 γvE0

Block’s linear momentum and energy. When the block moves as a whole with velocity v in frame S the linear momentum p, energy E and kinetic energy K are

p = γvMv , (50) 2 2 2 2 4 1/2 E = γvE0 = γvMc = (p c + M c ) , (51) 2 K = E − E0 = (γv − 1)Mc . (52)

These expressions are used to obtain the linear-momentum–total- energy four-vector Eµ, which characterises in frame S the system’s state, and the diﬀerential four-vector dEµ for the system’s state change, as Eµ ≡ (cp, 0, 0,E), with dEµ = (cdp, 0, 0, dE). Impulse–work four-vector. Newton’s second law does not distin- guish between conservative and non-conservative force: they all exert impulse. The ﬁrst law of thermodynamics considers just conservative forces, those that do work. To construct a force’s correct four-vector impulse-work requires taking into account this circumstance. The four-vector δW for force F = (Fx,Fy,Fz), with application point ve- locity v = (vx, vy, vz), is obtained by combining (vector) impulse I = (Fxdt, Fydt, Fzdt) (times c) and (scalar) work δW = F · vdt = F · dx. For instance, for the force Fk exerting an impulse Ik = Fkdt and performing work δWk = Fkdxk on a body, we have the impulse–work µ four-vector δWk [4] , µ δWk ≡ (cFkdt, 0, 0,Fkdxk) . (53)

For a friction force fR, no work is stored in any mechanical potential or work reservoir (a mass of water raised in Earth’s gravitational ﬁeld by Watt’s engine, for example) [25]. For a friction force, δWR = 0. Heat four-vector. In classical thermodynamics heat has no associ- ated linear momentum (not even to conclude that it is zero). In STR, every form of energy must have associated linear momentum. Heat, electromagnetic radiation, is energy exchanged that is described as a set

14 of photons with zero total linear momentum or thermal radiation with maximum entropy (thermal photons) [4]. The four-vector δQµ for an ensemble of thermal photons interchanged by the system with its envi- ronment (thermal reservoir), in frame S, in which the thermal reservoir is at rest, is given by, c(hνi/c)uxi 0 µ N c(hνi/c)uyi 0 δQ = Σi=1 ≡ . (54) c(hνi/c)uzi 0 hνi δQ

Four-vector fundamental equation. Four-vectors involved in a process are organised into a four-vector fundamental equation. We postulate a fundamental four-vector equation [4]

µ µ µ dE = ΣkδWk + δQ , (55) µ µ µ µ Ef − Ei = ΣkWk + Q . (56)

This matrix equation allows one to obtain variations in the system’s state, characterised by its linear momentum and its energy, as a function of impulses, work and heat of the diﬀerent interactions (mechanical or thermodynamic) that the system performs with its surroundings. Four-vector fundamental equation structure demands that its trans- formation from frame S to frame S¯ be carried out by applying the Lorentz transformation, µ ν ν ν ν ¯µ ¯µ ¯ µ ¯µ Lν (V )[Ef − Ei = ΣkWk + Q ] → Ef − Ei = ΣkWk + Q , straight ensuring principle of relativity fulﬁlment; the ﬁrst law of ther- modynamics equation in S¯ is a linear combination of Newton’s second law and the ﬁrst law of thermodynamics in S, and the same for Newton’s second law in S.¯ And vice-versa by changing S¯ ←→ S.

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