aa r X i v : . [ m a t h - ph ] O c t Note on the Relativistic Thermodynamics of Moving Bodies by Geoffrey L. SewellDepartment of Physics, Queen Mary University of London, Mile End Road, London E1 4NS, UKe-mail: [email protected]
Abstract
We employ a novel thermodynamical argument to show that, at the macroscopic level,there is no intrinsic law of temperature transformation under Lorentz boosts. This resultextends the corresponding microstatistical one of earlier works to the purely macroscopicregime and signifies that the concept of temperature as an objective entity is restrictedto the description of bodies in their rest frames. The argument on which this result isbased is centred on the thermal transactions between a body that moves with uniformvelocity relative to a certain inertial frame and a thermometer, designed to measure itstemperature, that is held at rest in that frame.PACS numbers: 03.30.+p, 05.70.-a, 03.65.Bz1 . Introduction and Discussion
Classical thermodynamics has been extended to the special relativistic regime in anumber of different, logically consistent, ways, which have led to different formulae forthe relationship between the temperature, T , of a body in a rest frame, K , and itstemperature, T , in an inertial frame, K , that moves with velocity v relative to K . To bespecific, in the schemes of Planck [1] and Einstein [2], T = T (1 − v /c ) / ; whereas inthose of Ott [3] and Kibble [4], T = T (1 − v /c ) − / ; and in those of Landsberg [5], VanKampen [6] and Callen and Horowitz [7], T = T , i.e. temperature is a scalar invariant.The relationships between the conventions and assumptions behind these different formulaehas been lucidly exposited by Van Kampen [6].In fact, all the above works were based exclusively on relativistic extensions of the firstand second laws of classical thermodynamics. A different, quantum statistical, approachwas introduced by Costa and Matsas [8] and by Landsberg and Matsas [9], who investigatedthe action of black body radiation on a monopole that moved with uniform velocity relativeto the rest frame of the radiation and played the role of a thermometer or detector. Theresult they obtained was that the spectrum of the radiation, as registered by this detector,was non-Planckian, and therefore that it was only in a rest frame that the radiation hada well defined temperature.A much more general version of this result was obtained by the present author [10,11], who showed that the coupling of a moving macroscopic quantum system, Σ , to afixed finite probe, Σ, drives the latter to a terminal state that, generically, is non-thermal.This signifies that, at the microstatistical level, the concept of temperature, as measuredby any , possibly microscopic, probe is restricted to systems in their rest frames. Thereremain, therefore, the open questions of whether the temperature of a moving body, asregistered by macroscopic observables of a probe or thermometer, is well defined and, ifso, whether it transforms, under Lorentz boosts, according to some general law.These are the questions that we address in the present article by an argument basedon the classical thermodynamics of the composite, Σ c , of two macroscopic bodies Σ andΣ , subject to the following conditions. Σ is in thermal equilibrium at temperature T ina rest frame K and moves with uniform velocity v relative to a frame K in which Σ isclamped at rest. Here again Σ serves as a thermometer for Σ . We investigate whether thecoupling between Σ and Σ can drive Σ to equilibrium at a temperature T that dependson T and v only: if so, T would be interpreted as the temperature of the moving bodyΣ , relative to the frame K . In fact, we show that there is no such model-independenttemperature T . Hence, in the purely macroscopic picture, as well as in the quantummicrostatistical one of Refs. [8]-[ 11], the concept of temperature as an objective entity islimited to bodies in their rest frames.We formulate the thermodynamic description of Σ c = (Σ + Σ ) in Sec. 2, concludingthat Section with the observation that its entropy can increase indefinitely and thereforethat it cannot evolve into a true equilibrium state. This, however, does not preclude thepossibility that Σ might drive Σ into an equilibrium state, and in Sec. 3 we investigatethis possibility for a specific tractable model in which Σ and Σ interact via emission and2bsorption of radiation. This model is a variant of the one constructed by Van Kampen [6]for his treatment of heterotachic processes. We show that, for this model, Σ is indeed driveninto a thermal equilibrium state, but that the resultant temperature depends on variableparameters of this system. Accordingly we conclude in Sec. 4 that, since the temperatureattained by Σ is just that of the moving body Σ , as measured by a fixed thermometer,there is no intrinsic law of temperature transformations under Lorentz boosts. This resultextands those of [8]-[11] from the microstatistical picture to the purely macroscopic one.
2. The Thermodynamic Description
Let Σ be a macroscopic system that moves with velocity v relative to an inertial frame K and that is in equilibrium at temperature T relative to a rest frame K . In order toformulate its thermodynamics relative to K , we consider the situation in which it is placedin diathermic interaction with a macroscopic probe, Σ, that is clamped at rest relative to K . We assume that the clamp is infinitely massive, and therefore immovable, and thatits action on Σ is adiabatic. Under these conditions, there is no thermal or mechanicalexchange of energy, relative to K , between Σ and the clamp. Further, we assume thatthe systems Σ and Σ are spatially separated, so that they do not exchange energy bymechanical means.The transactions between Σ and Σ constitute a heterotachic process, as defined byVan Kampen [6], but with the crucial constraint that the momentum of Σ, relative to K , is held at the value zero. In this process, the energy relative to K of the compositeΣ c = (Σ + Σ ) is conserved, but its momentum is not: any momentum received by Σ isimmediately discharged into the immovable clamp.We assume that, although both Σ and Σ are macroscopic, the former is of muchsmaller size than the latter in that, if Ω and Ω are dimensionless extensivity parameters(e.g. particles numbers) that provide measures of their respective sizes, then Ω >> Ω >>
1. In order to sharpen our formulation, we take Σ to be an infinite system, as in [10, 11],so that Ω = ∞ . Thus, Σ serves as a thermal reservoir whose temperature and pressureremain constant during its transactions with Σ.We assume, for simplicity, that the energy E and volume V of Σ, relative to the restframe K , constitute a complete set of its extensive thermodynamical variables*. In fact, V is merely constant during the transactions between this system and Σ since, as stipulatedabove, no mechanical work is done on it relative to its rest frame. As for Σ , we assumethat its temperature T and pressure Π , relative to K , together with its velocity v relativeto K , constitute a complete set of its intensive thermodynamic control variables. Finitechanges from the equilibrium state of this system are given by increments E and P ofits energy and momentum, respectively, relative to K . Hence, by Lorentz transformation,the increment in its energy relative to K is (1 − v /c ) − / ( E + v.P ) and therefore the* A general quantum statistical characterisation of a complete set of extensive thermo-dynamical variables is provided in [12, Sec. 6.4].3onservation of energy condition for Σ c , relative to K , is . E + γ ( E + v.P ) = const., (2 . γ = (1 − v /c ) − / . (2 . K , since energyin this frame is a linear combination of energy and momentum in K , and the clampingcondition destroys the conservation of momentum of Σ c relative to the latter frame.The entropy of Σ is a function S of E and V , which is jointly concave in its arguments[13, Sec. 1.10], and its value is Lorentz invariant [6; 14, Sec. 46]. The temperature T of Σis related to S by the standard formula T − = ∂S ( E, V ) ∂E . (2 . K is a rest frame for Σ , the incremental entropy of this system, due to modificationof its equilibrium state by changes E and P of its energy and momentum relative to thisframe, is simply S ( E ) = T − E . (2 . c , as measured relative to the specified equilibriumstate of Σ , is just the sum of those of Σ and Σ , which, by Eqs. (2.1) and (2.4), is equalto S ( E, V ) − T − ( γ − E + v.P ), plus a constant. Hence, defining˜ T = γT (2 . S ( E, V ) = S ( E, V ) − ˜ T − E, (2 . c is S c ( E, V ; P ) = ˜ S ( E, V ) − T − v.P + const.. (2 . S that ˜ S is maximisedat the value of E for which ∂S ( E, V ) /∂E = ˜ T − and that the resultant value of ˜ S is thefinite quantity given by − ˜ T − times the Helmholtz free energy of Σ at temperature ˜ T andvolume V [13, Sec.5.3]. On the other hand, the second term on the r.h.s. of Eq. (2.7)increases indefinitely with the modulus of P when the direction of this excess momentumopposes that of v . Hence, S c has no finite upper bound and so we reach the followingconclusion.(I) Under the prescribed conditions, the composite system Σ c does not support anyequilibrium state, as defined by the maximum entropy condition.
4f course this does not rule out the possibility that Σ might be driven into a thermalstate, with well defined temperature, as a result of its interaction with Σ . In the followingSection, we shall show that this possibility is realised by a tractable model, but that theresultant temperature varies with the parameters of the model.
3. The Radiative Transfer Model
The model presented here is a variant of Van Kampen’s [6] system of two bodies thatinteract by radiation through a small hole in a metallic sheet placed between them. In thepresent context, these systems are the above described ones Σ and Σ . We assume thattheir respective boundaries facing the sheet are plane surfaces, F and F , that are parallelboth to it and to the velocity v . We assume that the sheet and the face F are unboundedand that the sheet is at rest relative to K . Further, we assume that the hole is in the partof the sheet given by the orthogonal projection of F onto it and that both the linear spanof the hole and its distance from F * are negligibly small by comparison with its distancefrom the boundary of that face.The modifications of Van Kampen’s model that we introduce here are the following. • Only Σ , but not Σ, is a black body. We denote by A ( ω ) the absorption coefficient ofΣ for radiation of frequency ω . By Kirchoff’s law [15, Sec. 60], it is also the emissioncoefficient of this system, and it necessarily lies in the interval [0,1]. • Σ is clamped at rest in K . • No radiation emanating from Σ falls on the clamp: this can be achieved by placing Σbetween the hole and the clamp. Our treatment of the transactions between Σ andΣ will be basd on a calculation of the increment in the energy, ∆ E , of Σ relative to K intime ∆ t . Evidently this may be expressed in the form∆ E = ∆ E − ∆ E , (3 . E (resp. ∆ E ) is the energy transferred from Σ to Σ (resp. Σ to Σ) in thattime. These energy transfers are achieved by leaks of the radiations emanating from Σand Σ through the hole in the metallic sheet. Since both the linear span of the hole andits distance from F are negligible by comparison with its distance from the boundary of F , we may assume, for the purpose of calculating ∆ E , that the face F , as well as F , isinfinitely extended. We denote by Γ (resp. Γ ) the region bounded by F (resp. F ) andthe sheet. Thus Γ and Γ are are filled with the thermal radiation emanating from Σ andΣ , respectively, as modified by the leakages through the hole.In order to calculate ∆ E , we first note that the energy density of the radiation in Γthat lies in the infinitesimal frequency range [ ω, ω + dω ] and whose direction lies in a solid* The distance of the hole from F has to be so small in order to suppress end effects atthe boundary of that surface. 5ngle d Ω is A ( ω ) ω [exp(¯ hw/kT ) − − dωd Ω, times a universal constant. Hence, denotingthe area of the hole by ∆ a , the energy transferred by this pencil of radiation from Γ to Γ in time ∆ t is C ∆ a ∆ tA ( ω ) ω [exp(¯ hω/kT ) − − cos( ψ ) dωd Ω , where C is a universal constant and ψ is the angle between the pencil and the outwarddrawn normal to the sheet. It is convenient to express d Ω and cos( ψ ) in terms of sphericalpolar coordinates θ ( ∈ [0 , π ]) and φ ( ∈ [ − π/ , π/ v and the latter is the azimuthal angle of rotation of thepencil about the line of v . Specifically, d Ω = sin( θ ) dθdφ and cos( ψ ) = sin( θ )cos( φ )and therefore the above expression for the energy transferred across the hole from Γ maybe re-expressed as C ∆ a ∆ tA ( ω ) ω [exp(¯ hω/kT ) − − sin ( θ )cos( φ ) dωdθdφ, (3 . is a black body, the total energy ∆ E , relative to K , that is transferred fromΣ to Σ in time ∆ t is obtained by integration of this quantity over the ranges [0 , ∞ ] for ω, [0 , π ] for θ and [ − π/ , π/
2] for φ . Thus∆ E = C Φ( T )∆ a ∆ t, (3 . T ) = π Z ∞ dωA ( ω ) ω [exp(¯ hω/kT ) − − . (3 . A is measurable: oth-erwise the integral in Eq. (3.4) would not be well defined. However, from the physicalstandpoint, this assumption is very mild, as it is satisfied if A is piecewise continuous. Itfollows from Eq. (3.4) that Φ( T ) is a continuous and monotonically increasing function of T whose range is [0 , ∞ ].The calculation of ∆ E proceeds along similar lines, with modifications due to themotion of Σ relative to K . To effect this calculation we first note that the radiationemanating from the black body Σ is Planckian, and therefore isotropic, relative to K .Wethen define ω , θ and φ to be the natural counterparts of ω, θ and φ , respectively, for thedescription of Σ relative to K , and we denote by P the pencil of radiation emanatingfrom Σ for which these variables lie in the infinitesimal ranges [ ω , ω + dω ] , [ θ , θ + dθ ]and [ φ , φ + dφ ]. We then note that ∆ a ∆ t is Lorentz invariant, i.e. it is equal to theproduct of the counterparts ∆ a and ∆ t of ∆ a and ∆ t relative to the frame K . It nowfollows by simple analogy with the derivation of (3.2) that the energy, relative to K , thatis transferred by this pencil through the hole from the in time ∆ t is given by the canonicalanalogue of the expression (3.2), but with the term A ( ω ) omitted, since Σ is a black body.Hence, in view of the Lorentz invariance of ∆ a ∆ t , the energy relative to K transmittedby the pencil P through the hole in time ∆ t is C ∆ a ∆ tω [exp(¯ hω /kT ) − − sin ( θ )cos( φ ) dω dθ dφ . v of the momentum of P , relative to K , thatis transferred from Γ to Γ in time ∆ t is just c − cos( θ ) times this quantity. Hence, byLorentz transformation, the energy of this pencil, relative to K , that is transferred to Σin time ∆ t is Cγ ∆ a ∆ tω [exp(¯ hω /kT ) − − (cid:0) | v | /c )cos( θ ) (cid:1) sin ( θ )cos( φ ) dω dθ dφ . (3 . K , is ω = γ (cid:0) | v | /c )cos( θ ) (cid:1) ω . (3 . K , the energy transferred by the pencil P from Σ to Σ in time∆ t is just γ times the expression (3.5), but with ω replaced by γ − (cid:0) | v | /c )cos( θ ) (cid:1) − ω .Moreover, the resultant energy absorbed by Σ from the pencil is just the absorption coef-ficient A ( ω ) times this quantity. The total energy ∆ E absorbed by Σ in time ∆ t is thenobtained by integration and takes the form∆ E = C Φ ( T )∆ a ∆ t, (3 . ( T ) = 2 γ − Z ∞ dω Z π dθ A ( ω ) ω sin ( θ ) × (cid:0) | v | /c )cos( θ ) (cid:1) − (cid:2) exp (cid:0) (¯ hω/γkT ) (cid:0) | v | /c )cos( θ ) (cid:1) − (cid:1) − (cid:3) − . (3 . is a continuous, monotonically increasingfunction of T whose range is [0 , ∞ ).We now infer from Eqs. (3.1), (3.3) and (3.7) that the net energy increment in theenergy of Σ, relative to K , in time ∆ t is∆ E = C [Φ ( T ) − Φ( T )]∆ a ∆ t. (3 . t →
0, the rate of change of the energy E of Σ is dEdt = C [Φ ( T ) − Φ( T )]∆ a. (3 . Σ . Since the functions Φand Φ are continuous and monotonically increasing, with range [0 , ∞ ), it follows fromEq. (3.10) that there is precisely one value, T , of T for which E is stationary. Thus T isdetermined by the equation Φ( T ) = Φ ( T ) . (3 . , as well as Φ, increases monotonically and continuously with its argu-ment, this formula implies that T is an increasing function of T .7n order to show that the temperature of Σ evolves irreversibly to the value T , weintroduce the free energy function F ( E, V ) = E − T S ( E, V ) (3 . ddt F ( E, V ) = C [1 − T /T ][Φ ( T ) − Φ( T )]∆ a and consequently, by Eq. (3.11), that ddt F ( E, V ) = C [1 − T /T ][Φ( T ) − Φ( T )]∆ a. (3 . dF/dt is negative except at T = T , where it is zero. Thisleads us to the following result.(II) F serves as a Lyapounov function whose monotonic decrease with time ensuresthat the temperature of Σ evolves irreversibly to a stable terminal value T , which is the tem-perature of the moving system Σ , as registered by the thermometer fixed in K . Moreover,as noted following Eq. (3.9), this temperature is an increasing function of T . T on the Parameters of the Model. We now remark that,by Eqs. (3.4) and (3.8), the functions Φ and Φ depend on the form of the absorptioncoefficient A ( ω ), and therefore, by Eq. (3.11), so too does the temperature T . In orderto establish that this dependence is non-trivial, we consider the case where A ( ω ) is unitywhen ω lies in a narrow interval [ f, f + ∆ f ] and is otherwise zero. In this case, it followsfrom Eqs. (3.4), (3.8) and (3.11) that[exp(¯ hf /kT ) − − = 2 π − γ − Z π dθ sin ( θ ) × (cid:0) | v | /c )cos( θ ) (cid:1) − (cid:2) exp (cid:0) (¯ hf /γkT ) (cid:0) | v | /c )cos( θ ) (cid:1) − (cid:1) − (cid:3) − . (3 . T depends non-trivially on the frequency f , i.e. that it is notjust a constant, we show T tends to different limits as f tends to zero and infinity. Thus,in the case of small f , we may approximate the quantities in the square brackets on theleft and right hand sides of Eq. (3.14) by the exponents occurring there. Thus we findthat T → π − γ − T Z π dθ sin ( θ ) (cid:0) | v | /c )cos( θ ) (cid:1) − as f → . (3 . f , we may discount the terms − − ¯ hf /kT ) = 2 π − γ − × π dθ sin ( θ ) (cid:0) | v | /c )cos( θ ) (cid:1) − exp (cid:0) − (¯ hf /γkT ) (cid:0) | v | /c )cos( θ ) (cid:1) − (cid:1) . For large f , the r.h.s. of this equation is dominated by the exponential term occurringtherein and its logarithm reduces to the maximum value of the exponent for θ ∈ [0 , π ].Hence, using Eq. (2.2), we find that T → T (cid:16) | v | /c − | v | /c (cid:17) / as f →∞ . (3 . T (cid:0) O ( v /c ) (cid:1) . Hence the temperature T must bea nontrivial, i.e. non-constant, function of f . It follows that this temperature depends onthe parameter of the model and therefore we arrive at the following general conclusion.(III) According to the purely macroscopic picture, there is no intrinsic law of temper-ature transformations under Lorentz boosts.
4. Conclusion
Our essential results are encapsulated by the assertions (I) of Sec. 2 and (II) and(III) of Sec. 3. The first of these is that, under the prescribed conditions, the composite of(Σ+Σ ) cannot evolve to an equilibrium state, as given by the maximum entropy condition.However, as in the case of Sec. 3, where these systems interact via radiative transfer, theircoupling can drive Σ into an equilibrium state whose temperature T varies not only with T but also with the parameters of the thermometer Σ. From this we conclude that in thepurely macroscopic picture, as in the microstatistical one of [8]-[11], there is no intrinsiclaw of temperature transformation under Lorentz boosts.bf Acknowledgment. The author wishes to thank a referee for correcting some mis-takes in an earlier draft of this article. References [1] Planck M.1907
Sitzber. K1. Preuss. Akad. Wiss. , 542[2] Einstein A. 1907
Jahrb. Radioaktivitaet Elektronik , 411[3] Ott H. 1963 Z. Phys. , 70[6] Kibble T. W. B. 1966
Nuov. Cim.
41 B , 72[5] Landsberg P. T. 1966
Nature , 571 ; and 1967
Nature , 903[6] Van Kampen N. G. 1968
Phys. Rev. , 295[7] Callen H. and Horowitz G. 1971
Am. J. Phys. , 938[8] Costa S. S. and Matsas G. E. A. 1995 Phys. Lett. A , 155.99] Landsberg P. T. and Matsas G. E. A. 1996
Phys. Lett. A , 401[10] Sewell G. L. 2008
J. Phys. A: Math. Theor. , 382003[11] Sewell G. L. 2009 Rep. Math. Phys. , 285[12] Sewell G. L. 2002 Quantum Mechanics and its Emergent Macrophysics (Princeton NJ:Princeton University Press)[13] Callen H. B. 1987
Thermodynamics and an Introduction to Thermostatistics (JohnWiley; New York)[14] Pauli W. 1958
Theory of Relativity , (London: Pergamon)[15] Landau L. D. and Lifshitz E. M. 1959
Statistical Physics (Pergamon; London, Paris)10 r X i v : . [ m a t h - ph ] O c t Note on the Relativistic Thermodynamics of Moving Bodies by Geoffrey L. SewellDepartment of Physics, Queen Mary University of London, Mile End Road, London E1 4NS, UKe-mail: [email protected]
Abstract
We employ a novel thermodynamical argument to show that, at the macroscopic level,there is no intrinsic law of temperature transformation under Lorentz boosts. This resultextends the corresponding microstatistical one of earlier works to the purely macroscopicregime and signifies that the concept of temperature as an objective entity is restrictedto the description of bodies in their rest frames. The argument on which this result isbased is centred on the thermal transactions between a body that moves with uniformvelocity relative to a certain inertial frame and a thermometer, designed to measure itstemperature, that is held at rest in that frame.PACS numbers: 03.30.+p, 05.70.-a, 03.65.Bz1 . Introduction and Discussion
Classical thermodynamics has been extended to the special relativistic regime in anumber of different, logically consistent, ways, which have led to different formulae forthe relationship between the temperature, T , of a body in a rest frame, K , and itstemperature, T , in an inertial frame, K , that moves with velocity v relative to K . To bespecific, in the schemes of Planck [1] and Einstein [2], T = T (1 − v /c ) / ; whereas inthose of Ott [3] and Kibble [4], T = T (1 − v /c ) − / ; and in those of Landsberg [5], VanKampen [6] and Callen and Horowitz [7], T = T , i.e. temperature is a scalar invariant.The relationships between the conventions and assumptions behind these different formulaehas been lucidly exposited by Van Kampen [6].In fact, all the above works were based exclusively on relativistic extensions of the firstand second laws of classical thermodynamics. A different, quantum statistical, approachwas introduced by Costa and Matsas [8] and by Landsberg and Matsas [9], who investigatedthe action of black body radiation on a monopole that moved with uniform velocity relativeto the rest frame of the radiation and played the role of a thermometer or detector. Theresult they obtained was that the spectrum of the radiation, as registered by this detector,was non-Planckian, and therefore that it was only in a rest frame that the radiation hada well defined temperature.A much more general version of this result was obtained by the present author [10,11], who showed that the coupling of a moving macroscopic quantum system, Σ , to afixed finite probe, Σ, drives the latter to a terminal state that, generically, is non-thermal.This signifies that, at the microstatistical level, the concept of temperature, as measuredby any , possibly microscopic, probe is restricted to systems in their rest frames. Thereremain, therefore, the open questions of whether the temperature of a moving body, asregistered by macroscopic observables of a probe or thermometer, is well defined and, ifso, whether it transforms, under Lorentz boosts, according to some general law.These are the questions that we address in the present article by an argument basedon the classical thermodynamics of the composite, Σ c , of two macroscopic bodies Σ andΣ , subject to the following conditions. Σ is in thermal equilibrium at temperature T ina rest frame K and moves with uniform velocity v relative to a frame K in which Σ isclamped at rest. Here again Σ serves as a thermometer for Σ . We investigate whether thecoupling between Σ and Σ can drive Σ to equilibrium at a temperature T that dependson T and v only: if so, T would be interpreted as the temperature of the moving bodyΣ , relative to the frame K . In fact, we show that there is no such model-independenttemperature T . Hence, in the purely macroscopic picture, as well as in the quantummicrostatistical one of Refs. [8]-[ 11], the concept of temperature as an objective entity islimited to bodies in their rest frames.We formulate the thermodynamic description of Σ c = (Σ + Σ ) in Sec. 2, concludingthat Section with the observation that its entropy can increase indefinitely and thereforethat it cannot evolve into a true equilibrium state. This, however, does not preclude thepossibility that Σ might drive Σ into an equilibrium state, and in Sec. 3 we investigatethis possibility for a specific tractable model in which Σ and Σ interact via emission and2bsorption of radiation. This model is a variant of the one constructed by Van Kampen [6]for his treatment of heterotachic processes. We show that, for this model, Σ is indeed driveninto a thermal equilibrium state, but that the resultant temperature depends on variableparameters of this system. Accordingly we conclude in Sec. 4 that, since the temperatureattained by Σ is just that of the moving body Σ , as measured by a fixed thermometer,there is no intrinsic law of temperature transformations under Lorentz boosts. This resultextands those of [8]-[11] from the microstatistical picture to the purely macroscopic one.
2. The Thermodynamic Description
Let Σ be a macroscopic system that moves with velocity v relative to an inertial frame K and that is in equilibrium at temperature T relative to a rest frame K . In order toformulate its thermodynamics relative to K , we consider the situation in which it is placedin diathermic interaction with a macroscopic probe, Σ, that is clamped at rest relative to K . We assume that the clamp is infinitely massive, and therefore immovable, and thatits action on Σ is adiabatic. Under these conditions, there is no thermal or mechanicalexchange of energy, relative to K , between Σ and the clamp. Further, we assume thatthe systems Σ and Σ are spatially separated, so that they do not exchange energy bymechanical means.The transactions between Σ and Σ constitute a heterotachic process, as defined byVan Kampen [6], but with the crucial constraint that the momentum of Σ, relative to K , is held at the value zero. In this process, the energy relative to K of the compositeΣ c = (Σ + Σ ) is conserved, but its momentum is not: any momentum received by Σ isimmediately discharged into the immovable clamp.We assume that, although both Σ and Σ are macroscopic, the former is of muchsmaller size than the latter in that, if Ω and Ω are dimensionless extensivity parameters(e.g. particles numbers) that provide measures of their respective sizes, then Ω >> Ω >>
1. In order to sharpen our formulation, we take Σ to be an infinite system, as in [10, 11],so that Ω = ∞ . Thus, Σ serves as a thermal reservoir whose temperature and pressureremain constant during its transactions with Σ.We assume, for simplicity, that the energy E and volume V of Σ, relative to the restframe K , constitute a complete set of its extensive thermodynamical variables*. In fact, V is merely constant during the transactions between this system and Σ since, as stipulatedabove, no mechanical work is done on it relative to its rest frame. As for Σ , we assumethat its temperature T and pressure Π , relative to K , together with its velocity v relativeto K , constitute a complete set of its intensive thermodynamic control variables. Finitechanges from the equilibrium state of this system are given by increments E and P ofits energy and momentum, respectively, relative to K . Hence, by Lorentz transformation,the increment in its energy relative to K is (1 − v /c ) − / ( E + v.P ) and therefore the* A general quantum statistical characterisation of a complete set of extensive thermo-dynamical variables is provided in [12, Sec. 6.4].3onservation of energy condition for Σ c , relative to K , is . E + γ ( E + v.P ) = const., (2 . γ = (1 − v /c ) − / . (2 . K , since energyin this frame is a linear combination of energy and momentum in K , and the clampingcondition destroys the conservation of momentum of Σ c relative to the latter frame.The entropy of Σ is a function S of E and V , which is jointly concave in its arguments[13, Sec. 1.10], and its value is Lorentz invariant [6; 14, Sec. 46]. The temperature T of Σis related to S by the standard formula T − = ∂S ( E, V ) ∂E . (2 . K is a rest frame for Σ , the incremental entropy of this system, due to modificationof its equilibrium state by changes E and P of its energy and momentum relative to thisframe, is simply S ( E ) = T − E . (2 . c , as measured relative to the specified equilibriumstate of Σ , is just the sum of those of Σ and Σ , which, by Eqs. (2.1) and (2.4), is equalto S ( E, V ) − T − ( γ − E + v.P ), plus a constant. Hence, defining˜ T = γT (2 . S ( E, V ) = S ( E, V ) − ˜ T − E, (2 . c is S c ( E, V ; P ) = ˜ S ( E, V ) − T − v.P + const.. (2 . S that ˜ S is maximisedat the value of E for which ∂S ( E, V ) /∂E = ˜ T − and that the resultant value of ˜ S is thefinite quantity given by − ˜ T − times the Helmholtz free energy of Σ at temperature ˜ T andvolume V [13, Sec.5.3]. On the other hand, the second term on the r.h.s. of Eq. (2.7)increases indefinitely with the modulus of P when the direction of this excess momentumopposes that of v . Hence, S c has no finite upper bound and so we reach the followingconclusion.(I) Under the prescribed conditions, the composite system Σ c does not support anyequilibrium state, as defined by the maximum entropy condition.
4f course this does not rule out the possibility that Σ might be driven into a thermalstate, with well defined temperature, as a result of its interaction with Σ . In the followingSection, we shall show that this possibility is realised by a tractable model, but that theresultant temperature varies with the parameters of the model.
3. The Radiative Transfer Model
The model presented here is a variant of Van Kampen’s [6] system of two bodies thatinteract by radiation through a small hole in a metallic sheet placed between them. In thepresent context, these systems are the above described ones Σ and Σ . We assume thattheir respective boundaries facing the sheet are plane surfaces, F and F , that are parallelboth to it and to the velocity v . We assume that the sheet and the face F are unboundedand that the sheet is at rest relative to K . Further, we assume that the hole is in the partof the sheet given by the orthogonal projection of F onto it and that both the linear spanof the hole and its distance from F are negligibly small* by comparison with its distancefrom the boundary of that face.The modifications of Van Kampen’s model that we introduce here are the following. • Only Σ , but not Σ, is a black body. We denote by A ( ω ) the absorption coefficient ofΣ for radiation of frequency ω . By Kirchoff’s law [15, Sec. 60], it is also the emissioncoefficient of this system, and it necessarily lies in the interval [0,1]. • Σ is clamped at rest in K . • No radiation emanating from Σ falls on the clamp: this can be achieved by placing Σbetween the hole and the clamp. Our treatment of the transactions between Σ andΣ will be basd on a calculation of the increment in the energy, ∆ E , of Σ relative to K intime ∆ t . Evidently this may be expressed in the form∆ E = ∆ E − ∆ E , (3 . E (resp. ∆ E ) is the energy transferred from Σ to Σ (resp. Σ to Σ) in thattime. These energy transfers are achieved by leaks of the radiations emanating from Σand Σ through the hole in the metallic sheet. Since both the linear span of the hole andits distance from F are negligible by comparison with its distance from the boundary of F , we may assume, for the purpose of calculating ∆ E , that the face F , as well as F , isinfinitely extended. We denote by Γ (resp. Γ ) the region bounded by F (resp. F ) andthe sheet. Thus Γ and Γ are are filled with the thermal radiation emanating from Σ andΣ , respectively, as modified by the leakages through the hole.In order to calculate ∆ E , we first note that the energy density of the pencil ofradiation in Γ that lies in the infinitesimal frequency range [ ω, ω + dω ] and whose direction* The distance of the hole from F has to be so small in order to suppress end effects atthe boundary of that surface. 5ies in a solid angle d Ω is A ( ω ) ω [exp(¯ hw/kT ) − − dωd Ω, times a universal constant.Hence, denoting the area of the hole by ∆ a , the energy transferred by this pencil ofradiation from Γ to Γ in time ∆ t is C ∆ a ∆ tA ( ω ) ω [exp(¯ hω/kT ) − − cos( ψ ) dωd Ω , where C is a universal constant and ψ is the angle between the pencil and the outwarddrawn normal to the sheet. It is convenient to express d Ω and cos( ψ ) in terms of sphericalpolar coordinates θ ( ∈ [0 , π ]) and φ ( ∈ [ − π/ , π/ v and the latter is the azimuthal angle of rotation of thepencil about the line of v . Specifically, d Ω = sin( θ ) dθdφ and cos( ψ ) = sin( θ )cos( φ )and therefore the above expression for the energy transferred across the hole from Γ maybe re-expressed as C ∆ a ∆ tA ( ω ) ω [exp(¯ hω/kT ) − − sin ( θ )cos( φ ) dωdθdφ, (3 . is a black body, the total energy ∆ E , relative to K , that is transferred fromΣ to Σ in time ∆ t is obtained by integration of this quantity over the ranges [0 , ∞ ] for ω, [0 , π ] for θ and [ − π/ , π/
2] for φ . Thus∆ E = C Φ( T )∆ a ∆ t, (3 . T ) = π Z ∞ dωA ( ω ) ω [exp(¯ hω/kT ) − − . (3 . A is measurable: oth-erwise the integral in Eq. (3.4) would not be well defined. However, from the physicalstandpoint, this assumption is very mild, as it is satisfied if the function A is piecewise con-tinuous. It follows from Eq. (3.4) that Φ( T ) is a continuous and monotonically increasingfunction of T whose range is [0 , ∞ ].The calculation of ∆ E proceeds along similar lines, with modifications due to themotion of Σ relative to K . To effect this calculation we first note that the radiationemanating from the black body Σ is Planckian, and therefore isotropic, relative to K .Wethen define ω , θ and φ to be the natural counterparts of ω, θ and φ , respectively, for thedescription of Σ relative to K , and we denote by P the pencil of radiation emanatingfrom Σ for which these variables lie in the infinitesimal ranges [ ω , ω + dω ] , [ θ , θ + dθ ]and [ φ , φ + dφ ]. We then note that ∆ a ∆ t is Lorentz invariant, i.e. it is equal to theproduct of the counterparts ∆ a and ∆ t of ∆ a and ∆ t relative to the frame K . It nowfollows by simple analogy with the derivation of (3.2) that the energy, relative to K , thatis transferred by this pencil through the hole from Γ to Γ in time ∆ t is given by thecanonical analogue of the expression (3.2), but with the term A ( ω ) omitted, since Σ is a6lack body. Hence, in view of the Lorentz invariance of ∆ a ∆ t , the energy relative to K transmitted by the pencil P through the hole in time ∆ t is C ∆ a ∆ tω [exp(¯ hω /kT ) − − sin ( θ )cos( φ ) dω dθ dφ . Correspondingly, the component parallel to v of the momentum of P , relative to K , thatis transferred from Γ to Γ in time ∆ t is just c − cos( θ ) times this quantity. Hence, byLorentz transformation, the energy of this pencil, relative to K , that is transferred to Σin time ∆ t is Cγ ∆ a ∆ tω [exp(¯ hω /kT ) − − (cid:0) | v | /c )cos( θ ) (cid:1) sin ( θ )cos( φ ) dω dθ dφ . (3 . K , is ω = γ (cid:0) | v | /c )cos( θ ) (cid:1) ω . (3 . K , the energy transferred by the pencil P from Σ to Σ in time∆ t is just γ times the expression (3.5), but with ω replaced by γ − (cid:0) | v | /c )cos( θ ) (cid:1) − ω .Moreover, the resultant energy absorbed by Σ from the pencil is just the absorption coef-ficient A ( ω ) times this quantity. The total energy ∆ E absorbed by Σ in time ∆ t is thenobtained by integration and takes the form∆ E = C Φ ( T )∆ a ∆ t, (3 . ( T ) = 2 γ − Z ∞ dω Z π dθ A ( ω ) ω sin ( θ ) × (cid:0) | v | /c )cos( θ ) (cid:1) − (cid:2) exp (cid:0) (¯ hω/γkT ) (cid:0) | v | /c )cos( θ ) (cid:1) − (cid:1) − (cid:3) − . (3 . is a continuous, monotonically increasingfunction of T whose range is [0 , ∞ ).We now infer from Eqs. (3.1), (3.3) and (3.7) that the net energy increment in theenergy of Σ, relative to K , in time ∆ t is∆ E = C [Φ ( T ) − Φ( T )]∆ a ∆ t. (3 . t →
0, the rate of change of the energy E of Σ is dEdt = C [Φ ( T ) − Φ( T )]∆ a. (3 . Σ . Since the functions Φand Φ are continuous and monotonically increasing, with range [0 , ∞ ), it follows from7q. (3.10) that there is precisely one value, T , of T for which E is stationary. Thus T isdetermined by the equation Φ( T ) = Φ ( T ) . (3 . , as well as Φ, increases monotonically and continuously with its argu-ment, this formula implies that T is an increasing function of T .In order to show that the temperature of Σ evolves irreversibly to the value T , weintroduce the free energy function F ( E, V ) = E − T S ( E, V ) (3 . ddt F ( E, V ) = C [1 − T /T ][Φ ( T ) − Φ( T )]∆ a and consequently, by Eq. (3.11), that ddt F ( E, V ) = C [1 − T /T ][Φ( T ) − Φ( T )]∆ a. (3 . dF/dt is negative except at T = T , where it is zero. Thisleads us to the following result.(II) F serves as a Lyapounov function whose monotonic decrease with time ensuresthat the temperature of Σ evolves irreversibly to a stable terminal value T , which is the tem-perature of the moving system Σ , as registered by the thermometer fixed in K . Moreover,as noted following Eq. (3.9), this temperature is an increasing function of T . T on the Parameters of the Model. We now remark that,by Eqs. (3.4) and (3.8), the functions Φ and Φ depend on the form of the absorptioncoefficient A ( ω ), and therefore, by Eq. (3.11), so too does the temperature T . In orderto establish that this dependence is non-trivial, we consider the case where A ( ω ) is unitywhen ω lies in a narrow interval [ f, f + ∆ f ] and is otherwise zero. In this case, it followsfrom Eqs. (3.4), (3.8) and (3.11) that[exp(¯ hf /kT ) − − = 2 π − γ − Z π dθ sin ( θ ) × (cid:0) | v | /c )cos( θ ) (cid:1) − (cid:2) exp (cid:0) (¯ hf /γkT ) (cid:0) | v | /c )cos( θ ) (cid:1) − (cid:1) − (cid:3) − . (3 . T depends non-trivially on the frequency f , i.e. that it is notjust a constant, we show T tends to different limits as f tends to zero and infinity. Thus,in the case of small f , we may approximate the quantities in the square brackets on the8eft and right hand sides of Eq. (3.14) by the exponents occurring there. Thus we findthat T → π − γ − T Z π dθ sin ( θ ) (cid:0) | v | /c )cos( θ ) (cid:1) − as f → . (3 . f , we may discount the terms − − ¯ hf /kT ) = 2 π − γ − × Z π dθ sin ( θ ) (cid:0) | v | /c )cos( θ ) (cid:1) − exp (cid:0) − (¯ hf /γkT ) (cid:0) | v | /c )cos( θ ) (cid:1) − (cid:1) . For large f , the r.h.s. of this equation is dominated by the exponential term occurringtherein and its logarithm reduces to the maximum value of the exponent for θ ∈ [0 , π ].Hence, using Eq. (2.2), we find that T → T (cid:16) | v | /c − | v | /c (cid:17) / as f →∞ . (3 . T (cid:0) O ( v /c ) (cid:1) . Hence the temperature T must bea nontrivial, i.e. non-constant, function of f . It follows that this temperature depends onthe parameter of the model and therefore we arrive at the following general conclusion.(III) According to the purely macroscopic picture, there is no intrinsic law of temper-ature transformations under Lorentz boosts.
4. Conclusion
Our essential results are encapsulated by the assertions (I) of Sec. 2 and (II) and (III)of Sec. 3. The first of these is that, under the prescribed conditions, the composite of Σand Σ cannot evolve to an equilibrium state, as given by the maximum entropy condition.However, as in the case of Sec. 3, where these systems interact via radiative transfer, theircoupling can drive Σ into an equilibrium state whose temperature T varies not only with T but also with the parameters of the thermometer Σ. From this we conclude that in thepurely macroscopic picture, as in the microstatistical one of [8]-[11], there is no intrinsiclaw of temperature transformation under Lorentz boosts. Acknowledgment.
The author wishes to thank a referee for correcting some mistakesin an earlier draft of this article.
References [1] Planck M.1907
Sitzber. K1. Preuss. Akad. Wiss. , 542[2] Einstein A. 1907
Jahrb. Radioaktivitaet Elektronik , 41193] Ott H. 1963 Z. Phys. , 70[6] Kibble T. W. B. 1966
Nuov. Cim.
41 B , 72[5] Landsberg P. T. 1966
Nature , 571 ; and 1967
Nature , 903[6] Van Kampen N. G. 1968
Phys. Rev. , 295[7] Callen H. and Horowitz G. 1971
Am. J. Phys. , 938[8] Costa S. S. and Matsas G. E. A. 1995 Phys. Lett. A , 155.[9] Landsberg P. T. and Matsas G. E. A. 1996
Phys. Lett. A , 401[10] Sewell G. L. 2008
J. Phys. A: Math. Theor. , 382003[11] Sewell G. L. 2009 Rep. Math. Phys. , 285[12] Sewell G. L. 2002 Quantum Mechanics and its Emergent Macrophysics (Princeton NJ:Princeton University Press)[13] Callen H. B. 1987
Thermodynamics and an Introduction to Thermostatistics (JohnWiley; New York)[14] Pauli W. 1958
Theory of Relativity , (London: Pergamon)[15] Landau L. D. and Lifshitz E. M. 1959