- Email: [email protected]
The Lorentz transformation of heat and work
ANNALS
OF PHYSICS:
56,299-318
The Lorentz
(1970)
Transformation
of Heat and Work
P. T. LANDSBERG AND K. A. JOHNS Department
of AppIied Mathematics University College,
and Mathematical Cardiff, Wales
Physics,
A general analysis is given of the work dW done on a system as judged from an inertial frame 1. The novel features of the work are the following: (a) In the incremental process the velocity of the centre of mass of the system in I is allowed to change from w to w + du. (b) The surface elements of the system start to accelerate at different times in I even if they do so simultaneously in the initial rest frame I,, . This has to be taken into account. (c) The analysis allows for two definitions of force which have been in use and which are distinguished by a parameter p whose value is 0 or 1. (d) In order to enable the analysis to be applied to various types of systems, the momentum four-vector is taken in the form (cP, E + hpV) where h = 0 or I. With this notation we derive expressions for the compressive work dWc = -L&o
and the translational
dV +
(1 - A)po dV,/j31,
work dWt,
=w*
dP-p/3---dQ,
W
( c= 1 where /? = [1 - r@/~~]-l/~ and (/Q, is the heat transferred to the system in I, during the incremental process. The transformation of work dW = dWc + dWt, is given by dW,,,, = 0, and
(e) In the discussion of dW it is shown to what extent the Fitzgerald contraction can be said to contribute to the work in frame I. A fairly simple solution is proposed for this problem. (f) The transformation of heat dQ is finally derived as
The paper therefore reduces disagreements about this transformation to a disagreement about definitions (i.e., the value of p) in quite a general way. The temperature transformation is touched on only in passing, .‘ . . the conditions demanded to make possible the ordinary operations of calorimetry still hold under the general conditions assumed by the first law. But this is almost certainly not always the case. An examination of what is involved in classical calorimetry shows that its operations are performed in the absence of ordinary mechanical phenomena, whereas here we have to know the meaning of heat flow across a surface which may be in motion and across which mechanical work is simultaneously being performed. So general a situation is certainly not contemplated in the original operations
299 0 1970 by Academic Press, Inc. 595/5~/~-
1
300
LANDSBERG
AND
JOHNS
of calorimetry, and whether the original operations can be sufficiently broadened is to my mind very questionable. At least I have never seen any adequate discussion of the possibility.” From I?. W. Bridgman The Nature of Thermodynamics (Harvard 1941; Harper Torchbooks 1961, p. 29)
1. 1NTRODUCTION
Most authors have approached the problem of identifying the Lorentz transformation of heat dQ given to a system in an incremental process, by first considering the transformation of mechanical work dW in the same process. Since different answers have been obtained to this fundamental question, (Z-5), we consider it here in a generalised way which enables one to see the answers recently given in a single context. This generalised calculation of the work done on a moving system faces two main difficulties, both of which are solved in this paper. The first difficulty resides in the multiplicity of inertial frames, each with its own proper time, which enter into the calculation. The process of compression or expansion may start simultaneously for all surface elements of the system if the inertial frame used is the initial rest frame I,, of the system. But at the end of the process the system will in general be at rest in a different inertial frame. However, the calculation of dW is in any case required for a general inertial frame I and in such a frame the surface elements start to accelerate relative to the centre of mass at different times. The centre-of-mass frame has initially velocity w, in I, and when the last element has ceased to accelerate relative to the centre of mass the rest frame may have a different velocity, w + du, say, in I(& < w). The analysis required to deal with these questions is rather delicate, and will be found in the Appendix. The second difficulty is that the calculation of dW must hold for two different definitions of force which have been used. It is possible to handle this situation by operating throughout with a definition of force which is sufficiently general to accommodate both points of view. This is achieved by using a parameter p with value 0 or 1 for the two definitions (see Section 2). It is found, as one would expect, that the transformation of both dQ and dW depends on this parameter (see Section 4). An adequately general discussion must include a further complicating feature. Special relativity deals with systems for which the energy-momentum four-vector is PI” = (cP, E) and systems for which it is Pzw = (cP, E + pV). The former systems are the unconfined systems of particles, and thermodynamics cannot readily be applied to these, since they are not equilibrium systems. For a normal confined equilibrium system to which thermodynamics can be applied, P,@ is the relevant quantity. However, the simpler quantity Plu becomes again appropriate
RELATIVISTIC
HEAT
AND
301
WORK
for a confined system if it is agreed to add to it the energy and momentum arising from the stresses in the container, but not from its rest mass (6). Such systems may be called inclusive. To handle both confined and inclusive systems we use P$ f (c,P,E+ApV)
(1.1)
as the relevant four-vector, and allow the values X = 0 or h = 1. It is found that the transformation law of dW, but not of dQ, depends on h. This is satisfactory since it shows that whatever type of relativistic thermodynamics one may favour, at least it is the same for both classes of systems. It proves possible (Section 3) to separate the mechanical work dW done on the system in frame I into two parts: (a) The translational work dWt, has the form: total force acting on the system times velocity of the centre of mass times an increment of time in I. (b) The compressive work dW, takes one of the familiar forms -,pO dV or --p,, dV,,/fl for confined and inclusive systems respectively. Here
p = (1 - EL)-,
0.2)
and it should be noted that while V = V,,//3 if terms in du are neglected, dV and during the process considered. A further difficulty is to explain in what sense the Fitzgerald contraction effect is to be regarded as contributing to the mechanical work. It is explained in Section 3 that it contributes only to the compressive part, and then only in confined systems.. This paper therefore gives a unified discussion of the transformation of work and heat for systems which may suffer accelerations during the incremental process. considered. It covers the four cases generated by X = 0, 1 and p = 0, 1. All terms in the expression for the work dW in I are physically interpreted in this paper, and the only arbitrariness left is in the definition of force. This can be left to personal preference. The transformation of temperature is a separate problem and is touched on only once in this paper (Section 5). dV,//3 are different if the system suffers an acceleration
2. A GENERAL DONE
EXPRESSION ON NON-INERTIAL
FOR
AN INCREMENT SYSTEMS IN
OF SPECIAL
Using the notation of (1.1) and (1.2) the transformation
MECHANICAL RELATIVITY
WORK
from I, to 1 yields
E + AP,v = BWO+ bJ,~lll + w - PO), pw = B (pwo+ ; L% + h&l)~ where the suffix w indicates a component parallel to w, and p = p. .
(2.1) (2.2)
302
LANDSBERG
AND
JOHNS
The frame I, will be regarded as the frame in which the thermodynamic system of interest is initially at rest so that dP, can be non zero. The values of the volume, measured in I and I, , will therefore be related by the formula JL3-.
(2.3) P After a certain time the velocity of the system (in I) has increased by du, and the volume (in I,) by dV, . The volume observed in I will therefore be given by V + dV = (V, + dVo) (1 -
lw ;2du12 )1’2,
so that &$d!!$%.Vo~
(2.4)
The inverse of equation (2.1) is
Eo+ &Jovo= P([E + APO VI - w *P). Differentiation
(2.5)
of this leads to dE = f%’ + xp, --X ‘; B
+ w * dP - hpo dV,
In frame I,, , the first Law of thermodynamics dE, =
dQo-
(2.6)
is POdVo ,
(2.7)
whence Ap, dV - (1 - A) p. -dvo
dE=w*dP++$-
Similarly,
differentiation
B ’
(2.8)
of (2.2), and use of (2.7), gives
dP, = is (d&o, + s
VQo-
PO dVo1 + h s [PO dVo + Vo +,I),
(2.9)
These results should not be controversial since they are based on standard forms of the Lorentz transformation. The concepts of work and heat have been introduced only in the centre of mass frame I, , of the system, and their transformation properties have not yet been used. To obtain these properties it is first necessary to use a definition of force, and it is here that differences of opinion can arise. The orthodox definition takes as force, f, in any frame I, the rate of change of momentum II in that frame f=-&.
dn
(2.10)
RELATIVISTIC
HEAT
AND
303
WORK
This definition is equally applicable to the force acting on a particle, where dn is its increase in momentum, or to the force acting on a surface element of some larger body where &I must be interpreted as the momentum flowing across the surface in a prescribed direction. An alternative definition for the inertial frame I can be written, for a particle, as (2.11)
where 02, is the proper mass of the particle. This definition holds for an inertial frame in which the particle on which the force acts has velocity v. The identity (2.11) is equiv,alent to f alt -_ E - -=v d% -dt
dII --dt
41
-
LJ~/C~
dt
dr c2 41 - oZjc” dt V
__---
(2.12)
When the force is acting on a surface element, in, loses its meaning, and the last of the above expressions may be taken as the definition of force for an inertial frame in which the surface element or point of application has velocity v. It is to be emphasized that de is the energy increment transferred across the surface element measured in that frame of reference in which the surface element or point of application is at rest, not in the rest frame of the whole body enclosed by the surface. The two definitions given here can be combined in one formula by introducing a factor /L which takes the values 0 for the orthodox definition, 1 for the alternative definition, but which may even be assigned other values. Thus
f_Edt
__- de - v2jc” dt ’
V
’ c2 dl
(2.13)
The work done on a moving closed system can then be shown (see appendix) to be dW = @w . dP, - fipo dV,,) + (1 - pL)B ;
dQ,
where du is the change in velocity of the whole system, as observed in frame I. This equation incorporates the results for four different cases (h = 0, I ; p = 0, 1). Using Eqs. (2.4) and (2.9) it may be written dW=w.(dP-p/$dQ,j-[Ap,dV+(l
where P refers henceforth to the total momentum
-h)p,,y],
of the system.
(2.15)
304
LANDSBERG 3. COMPARISON
OF
THE
AND TWO
JOHNS DEFINITIONS
OF
FORCE
The total force fr acting on the centre of mass of the system in frame I can be obtained by summing (2.13) over all the surface elements. When p = 0, it immediately yields, P being now the total momentum of the system,
f =dp T
dt
(P = 0).
Similarly, when p = 1 and the system is not subject to compression or expansion, the velocity v of each element is always equal to w, and both terms in (2.13) can be summed to give dP w dE, fT=z-PFx
(p = 1, dV,, = 0),
or, by (2.7) (3.2) In the remaining case to be considered (p = 1, dl/, f 0) it is not possible to sum (2.13) directly since each surface element has, in general, a different velocity v. However, only quasistatic processes are considered here, and one can assume (see Appendix) that each value of v differs from w only by an infinitesimally small increment, It is then possible to neglect higher powers of these increments to find the total force fr which then depends only on dP and de,, . This is because in the rest frame of each surface element, where the energy increment de is measured, the point of application of any force acting on that element must have zero velocity and therefore such a force can do no work. The changes in energy de are thus due solely to the flow of heat (the effect of accretion of particles being excluded for simplicity). Since furthermore the differences in velocity between the centre of mass frame I, of the system and the rest frames of the individual elements are all infinitesimal, it is possible to equate, to first order accuracy, dQo and C de (the sum being taken over the surface of the system). It is reasonable, therefore, to take (3.2) as the total force on the system, even in the case p = 1, dV, f 0, and thus to combine (3.1) and (3.2) as (3.3) which is valid for both values of p, and in the presence of compression. The consequences of the choice TV= 1 are seen to be rather complicated. Another argument against this choice is the following: Consider a collision between two systems moving on the line joining their centres of mass. Let the time occupied
RELATIVISTIC
HEAT
AND
305
WORK
by the collision be 6t in some arbitrary inertial frame. Then by momentum conservation the increments of momentum gained are related in this frame by SP, = --6P, . For the conventional definition of force f as dP/dt we have at once as 6t -+ 0 that the force fi acting on System 1 and the force fi acting on System 2 are related by fi = -fi . Thus the third law of Newton, that action and reaction are opposite, holds. Alternative definitions will lead to a possible violation of this fundamental property and this seems to us a serious disadvantage of such alternatives. In particular, writing P = /3vM,, in a usual notation, we have
KN WlVl) + Ah wfo,) = -42 W2v2) - B2v2S(%,). Hence if the rest mass of either or both systems changes
Thus the “alternative” definition of force (2.11) can lead to fi # f2 . These considerations do not rule out the choice TV= 1, but they make it appear inconvenient.
4. AN
INTERPRETATION
OF
THE
EXPRESSION
FOR
THE
WORK
Since the centre of mass of the system has velocity w, it is now possible to define a quantity of work dWt, , done owing to the translational motion of the whole system, by dWt, = fT w dt.
(4.1)
l
Then at once, from (3.3) and (3.4) dwtr
= w - (dP -
,u/3 f
d&j,
(4.2)
which is the first term in the expression (2.15) for the total work done, The p-dependence of the translational work shows that it depends on the definition of force. It leads by (2.9) and (4.2) to the result that a flow of heat de,, in I, implies translational work (wz/c2)~ dQ, in I only for the orthodox definition of force (p = 0). If the definition p = 1 is adopted this term is cancelled by the second term in (4.2) and dWt, in I depends solely on the action of forces which already exist in I,, . The second term of (2.15) can be explained as follows: Consider an incremental compression of a conjined system which is (in its rest frame) initially in equilibrium
306
LANDSBERG
AND
JOHNS
with a uniform pressure pO. During the process the pressure remains uniform (except for terms of order dp,,) in any inertial frame in which the system is observed. It is thus clear that any change in volume dV in frame I can be said to be accompanied by an amount of mechanical compressive work --pO dV (which includes the effect of the Lorentz contraction). In the case of an inclusive system the container (considered as part of the system) requires no external force to keep the system in equilibrium. Only for a specific time interval do external forces act to compress the system by an amount dV,, (in frame I,,) and dV,//3 (in frame I), giving rise to work --pO dV, and --pO &‘,//I respectively. Equation (2.4) shows that an additional change of volume remains, namely dV - dV,,/P = -/3V,, ;
. du.
This is due solely to the change in magnitude of the Lorentz contraction factor as the system increases its velocity by du; it requires no external forces, other than those which produce the acceleration of the whole body, and can take place even when no external compressive forces act at all. This volume change is thus not accompanied by any compressive work, and the total work involved in the compression of an inclusive system is therefore --pO dV,,/p in frame I. These two results can be combined to give the compressive work in frame I dWc = -[APO dV + (1 - 4po dVoIP1.
(4.3)
The compressive work is independent of CL,and hence of the definition of because the point of application of each compressive force is at rest (v = 0) rest frame of the element on which it acts. Hence Eq. (2.13) shows that forces have no p-dependent term in frame I. We have therefore split (2.15) the mechanical work done on the system inertial frame I, into two components
force, in the these in an
dW = dW, + dWtr ,
each of which depends on a parameter. Whether a given system is treated as confined or inclusive affects dW, , and the choice of force definition affects dWt, . Some special cases of the general results (2.14), (2.15) are of interest. Case 1. Conventional
Equation
definition
of force,
confined
system
(2.15) yields dW=w.dP-podV.
This is the basic result used previously [Eq. (5.1) in Ref. (6).
(II = 0, X = 1).
RELATIVISTIC
HEAT
AND
WORK
307
Case 2. Conventional dejinition of force. (dP, = du = 0). (In this case the system remains at rest in I0 throughout the process). Equations (2.7) and (2.14) yield: dW = P[dWo + (tic)” Wo - dWJ + 4W2 d(p,,KJl = dWo/P + (NcY ,M& + h,~,).
This is another form of the work transformation obtained before [Eq. (5.3) in Ref. (6)]. It was shown in this reference that if there is no heat flow in I, it yields d W = PdW,
+ 44~)’
Bd(l?o v,,),
WQo = Oh
(4.4)
and hence dW = ,kldWo
(inclusive system, dQ, = 0),
(4.5)
(confined system, dpO= 0, dQ, = 0).
(4.6)
and dW = dW,//3
Case 3. Alternative definition of force (p = 1, dP, = clu = 0). Under these conditions (4.4) holds even in the presence of heat flow in I,, , and hence (4.5) and (4.6) become valid even if dQ, f 0: Comparison of Cases 2 and 3 shows the extent of the confusion which can result if the distinction between confined and inclusive s,ystems and between the various definitions of force is not clearly made.
5. TRANSFORMATION
Turning to the transformation dynamics in the form
OF
HEAT:
DISCUSSION
of heat, assume a relativistic first law of thermodE = dW + dQ.
(5.1)
Hence by (2.8) and (2.15) (5.2) This is the general transformation In particular, dQ = dQo/P dQ = PdQo
of heat for a whole range of force definitions.
(conventional force definition, p = 0) (force definition with p = 1).
(5.3) (5.4)
Note that these transformations are the same for inclusive and confined systems. The choice p = l/(1 + p) renders dQ invariant, but this is only a noteworthy curiosity and such a choice has not been suggested.
308
LANDSBERG
AND
JOHNS
In this paper we only make a passing reference to the temperature transformation which we have suggested should be T = T,, (6, 7), and which had been discussed by several other authors (8-13). Our remark is that if temperature is regarded as invariant the relativistic second law takes the form TdS >,
1-
P P + PP2
dQ.
(5.5)
If, on the other hand, the relativistic second law is assumed to have the form transformation becomes
T dS > dQ, the temperature
We advocate (5.5) with p = 0. One can also obtain the heat transformation (5.2) by four-vector methods. These give little additional physical insight in the present case, but we give the argument for the sake of completeness. One can start by extending (2.13) to give the Minowski force in a general frame I (m, = E/C”)
Here de is the energy-momentum in I passing through a surface element of the system during the process, uy is the four-velocity in I of the element during the process (assumed uniform for the short period of time involved), and dT is the proper time taken by the process at the particular element being considered. In fact the velocity of each surface element is small in the centre-of-mass frame I, (which has velocity w in I), so that the time taken by the process in I, is
where the approximate considered. The invariant
equality is valid to the first order of the small quantities
dx E c Kw, d7
(5.9)
(the summation is over all surface elements), is a generalised quantity analogous to the work. Then dx = C (f u, de’ -
pu,u” dm,)
= (1 - ,u)c2Cdm,.
(5.10)
RELATIVISTIC
= /3 2 de4 -
HEAT
AND
309
WORK
pp2c2 c dm, + ,8 c fi dxi
= p(dE - dW) -
(5.11)
p/3%? c dm,
(Summations over the index i are from 1 to 3; f i, by analogy with (2.13), is the three-force on a surface element; dxi is the displacement Ui d7, of the element and incorporates a negative sign). Thus, by (5.1), (5.10) and (5.11) dQ = dE - dW = [c;A
+ pfl] c2 c dm, .
(5.12)
By considering frame T, , where /3 = 1, it is clear that c2 1 dm, = de,,,
and hence the generalised transformation (5.2) is obtained again. If one adopts (5.5) with p = 0, as we advocate, (5.11) shows that the invariant formulation, TdS>CKYuydr=dx,
(5.13)
of the second law is available. Alternatively (5.13) can be taken as a definition of temperature in such a theory. It should be noted that the work dW in the above equations does nut include the amounts of work which, in the case of a confined system, may be done in I on elements for which, in I,, , the process has yet to begin or has already ended. It is thus equivalent to rl WA of the appendix. However, the purpose of the above argument is to determine the transformation of dQ, and its validity is unaffected by the omission of this additional work, since it is unaccompanied by any flow of heat in all frames of reference.
6. CONNECTION
WITH
PREVIOUS
WORK
Some authors derived variations of special cases of (2.14) and (2.15) for the transformation of work, (Z-3). Sometimes implicitly, they adopted a definition of force equivalent to (2.1 l), and their results are essentially represented by our Case 3 of Section 4 and Eq. (5.4). They can be summarized by dQ E dE - dW = p(dE,, - d W,,) = /I de,,
(6.1)
310
LANDSBERG
AND
JOHNS
Moller (4, 14) considers the transformations of work and heat in essentially the way of Ott, Arzelies and Kibble, but analyses in greater depth the changes in momentum brought about by various processes. In particular, the total change in momentum, dP, is separated [Eq. (56), in (4)] according to1 dP = dP’“’ + dP’“’ 7 where “dPCh) is the increase of the momentum due to the conduction of heat to the system,” and “dPcm) is that increase of the momentum which is due to the action of mechanical forces.” The dP(“) is obtained by the Lorentz-transformation of the energy-momentum produced by forces of compression in the rest frame of the system. The work is then stated to be (4) d W = w * dPfnL) + d W&3
[(63)
in (41
instead of dW = w * dP + dW,//3
t(64)
in WI,
the difference dPth) s w being neglected since momentum dPfh) is not regarded as being produced by real forces capable of doing work. This is just the term /3(wz/cz) dQo of Eq. (4.2). Indeed, Meller’s view is clearly expressed by his comment on dPch’ [his eq. (70)]. ~ dt = f? “The quantity fkh’ is therefore not a real force, it just describes the rate at which momentum grows on account of the conduction of heat to the fluid.” This departure from the conventional definition of force has been fully discussed in Sections 2 and 3. Note that Mgller’s Eq. (63) is contained in (2.15) if one chases p = X = 1 and treats the velocity of the system as constant (du = 0). His equation (64) is obtained by putting h = 1, p = du = 0. In the paper (f4), a relativistic heat engine is considered: first according to Ott’s formulation, and then according to that of Planck and Einstein. A discrepancy is found in the latter, which “clearly shows the inconsistency of the Planck formulation.” Actually, the discrepancy is due not to a fault in theory, but, as pointed out by Bicak (IS) since this paper was written, it is due to the omission by Merller an an amount of work
required by Planck, done on the fluid during process 1 -F 2. If this work is included, 1 Mder’s
equations are rewritten in our notation,
v --f w, ,3 = v/c -+ w/c, K 4 f, .
the
replacements being
dA
-+ dW,
G --t P,
RELATIVISTIC
HEAT
AND
WORK
311
both formulations become self-consistent, and the example of the heat engine cannot be used to distinguish between them. The heat transformation (5.2) with p = 0, which we advocate, belongs to that part of the orthodox (or Planck) theory which we accept. It has recently been criticised by van Kampen (IO, 97), by the remark that the theory yields dQ = 0 for pure acceleration “only by a clever cancellation” in his Eq. (15) [broadly our Eq. (5.1) with p = 0, A = I] dQ = dE-w.dP+p,dV.
But, as has been shown in Sections 4 and 5, this is a satisfactory equation whose terms have simple interpretations, the last two terms being translational and compressive work, even though dV involves a velocity increment [cf. Eq. (2.4).] Nor do we regard as criticisms that the orthodox theory contemplates system of uniform pressure (this is needed in equilibrium thermodynamics), or that it considers as the thermodynamic system the gas in the box alone: this merely refers to the distinction which we introduced (6) between inclusive and confined systems, and which is only incidental to the main thermodynamic points at issue. Accordingly van Kampen considers the case X = 0, and he introduces [Eq. (16)] a force concept whichis not the orthodox Minkowski force, but amounts to an implicit choice p = 1. It is this implicit choice rather than his explicit use of the h = 0 theory that determines the transformation of heat, as is clear from Section 5 above. Dewan and Beran (16) and Dewan (17) considered two rockets accelerated at the same rate from an initial rest frame I, to a final rest frame I’, so that their distance apart is always the samein I, . The proper length of a string connecting the rockets varies, and they consider conditions for the breaking of the string. This experiment shows with attractive clarity the reality of the forces which enter into the calculation of the work dW in Section 2. If the rockets are replaced by accelerated points and the string by a spring accelerated along its length, we have a typical cor$ned system. Let its initial compressive force and length in I, bef:“, L$,‘)respectively, and their final values in I’ beff2)‘, ur2)’ respectively. In a general frame I these forces give rise to a momentum flow, which can be cancelled by the tensile forces in the walls of a box to which the ends of the spring might be connected. This arrangement would be a inclusive system. In order to gain insight into the stress in the spring, we revert, however, to the confined system. From standard transformations one then finds for the work done on the systemin a general frame I W
=
EC"'
_
E'l'
= j& [ &, ( 1 + 5)
E@” - E;” (6.2)
312
LANDSBERG
AND
JOHNS
Here the velocity of I, in I is W, of I’ in I, is u, rS, = (1 - w~/c~)-~/~ and /3U = (1 - +/c~)-~/~. Various special cases may be considered, notably the cases (i) a(2)’ = TV@, and (ii) a(2)’ = ah”. In the first case the spring has constant length in I,, and the compression stress must be relaxed during the acceleration to counteract the Lorentz contraction. In the second case the proper length and the stress are constant. In both cases the work done can be computed from (6.2). For incremental changes in length, force and velocity, (6.2) can be written in the form (2.15) with X = 1 and dQ, = 0: dW = w dP -f”’
da(l).
The compressive work may be written (note that dv = &,” du) dW, = dWc,,/jIw
+ (w/c”)
/&ap’fy)
du,
as one would expect from (4.3) with h = 1 using (2.4). This type of consideration is therefore in agreement with our general results. A short summary of some of the above work is available in Ref. 18.
APPENDIX: A.
CHANGES
IN
THE
DERIVATION
CENTRE-OF-MASS
FRAME
OF FOR
EQ. (2.14) A QUASISTATIC
PROCESS
Consider a thermodynamic system, the surface of which is divided into small elements Au,, each impervious to matter but not to energy or momentum, and each with unit outward normal ni . During a thermodynamic process each element moves with a velocity &vi in the rest frame I, of the system as a whole. The prefix 6 implies here and in the rest of the appendix that the quantity is small enough for the system to be considered to be in equilibrium throughout the process (quasistatic process). Let dai 6g, be the energy which passes through the ith surface element during a short period of time judged in its instantaneous rest frame Ii. The surface then receives energy at the rate hi d@qi)/dt, . Let pO be the initial uniform pressure of the system and let Sfi be the force increment in excess of -p& , which is needed to accelerate the element, thus contributing to the change in volume. Then the element is acted upon by a force (-PO%
+ %) da, .
Since it is the rest frame of the surface element which is being considered, this force is equal to the rate of flow of momentum through the surface whichever definition of force is used.
RELATIVISTIC
HEAT
AND
313
WORK
Applying a Lorentz transformation, the rates of flow of energy and momentum through dai in the centre of mass frame I, are d&o __
xzz
dto G .
dpio --Z-Z dto
-- 4W I
poni . 6vi Aai . I
dti
-.--__ i d/1 - kiajc2
[
(-p
(A. 1)
.* + sf.*) On’ ’
j 6% d(W c2 dti
I
+ (-poni+ + Sf,+)1 Aai $ + (-poni + SfJ Aa, .
(A.3
Superhces* and + indicate components respectively parallel and perpendicular to 6vi . Terms in 6 of higher order than the first, such as (d&l&,), have been neglected.
Therefore, from the definition 8ai is, in frame I,, equal to
(2.13) the force acting on the surface element
independently of p, again with second- and higher-order terms neglected. The rate at which work is done on Aai is therefore, to first order: -dwio
= (-poni + Sf;l) Aad * Svi
dto
+ -poAaini From (A. l), (A.3), and the relativistic the ith surface element at the rate -zdQ
* hi .
(A-3)
first law (5.1), heat is transported across
d&o ---= dto
dwio dto
$!%&Ja,. z
(A.4)
Suppose the force increments Sfi are nonzero for a time interval 7. in I, . The momentum gained by the system during this time in I, is then, by (A.2) APO = J” C Sfi Aai dto *
70 i
(A.3
314
LANDSBERG AND JOHNS
since Ci ni dai = 0, for a system with a closed surface. The work done on the system is similarly by (A.3)
AW,=j 70
* Svj Aa,1 dt, = -p. A V, ,
64.6)
when d V, is the change of volume in the process. The heat gained in I,, is
AQ”= @,--d@q’) Aai
dt, .
(A-7)
B. CORRESPONDING CHANGES IN A FRAME I For an inertial frame I in which the centre of mass of the system has velocity w one finds from (A.l) and (A.2) by a Lorentz transformation %
= ifi [(-p&*
+ Sf,*) + 5 (*
- poni . hi)]
dt ’ dt< ’
+ 6f,t)i Au. -Z
+ (-poni+
1
(A4
where superfices * and + here indicate components respectively parallel and per pendicular to w. To first order in 6, the velocity ui of frame Ii in frame I has components w + Sv,”
ui* =
1$-d
+ w + -L SVj”
w . sv.
B"
C2
Hence lli
+
W +
’ P”
6Vi”
+
’
6Vit B
64.9)
The force acting on Au, in I is obtainable from (2.13), noting that the velocity required is calculated with respect to the frame Ii , whence v = ui in this case. The energy E gained by the system refers to its proper frame, and is the quantity given at the beginning of this Appendix. Hence this force is dpi dt
Ui Aa, 4%) ’ c2 41 -____ - tqc2 “-iii-’
RELATIVISTIC
HEAT
AND
315
WORK
To the first order in 8, this force is equal to
+
( -poni+
+
The rate at which work T
=
)/3 [(--pani*
+
=:
+
( -poni+
+
i/3 [W * (-p&
Aai 2
sf,+)/
.
(A. 10)
is done on Aai is therefore, Sfi*)
Sf,i)(
+
+
5
([l
Aa, 2 Sfi)
+
-
~1 F
. (w + $$ $
([1
-
by (A.9) and (A.lO):
-
I
poni
. hi)]
+ y)
jh] y
-
poni
. &vi)]
1 +
7
- (-p&l
Aa, 2 -
poni
* hi
+
w * (-poni
+
Sf,)]
Aa, 2
(A. 11)
In I the total work done during the process (which occurred in time interval TV in I,) is obtained by summing Eq. (A.ll) over Aai and integrating over dt. In this integration the time limits for each element i are the Lorentz transforms to I of the end points of the interval TV. These transforms involve not only the velocity w, but also the position coordinates (the initial coordinates for the lower limit, the final coordinates for the upper limit) of each surface element. Thus one has a sum of integrals with different end points on the time scale of frame I. From (A. 11) one can clearly express this result as an integral over the single time interval 7(, : Aw,=~J Comparison
C[(l-i”)~~-~~“~.Sv~+w.~f]Aa~dt,. 70 i of (A.12) with (A.$ AW,=/3([1
(A.12)
(A.6), and (A.7) gives:
-&AQ,-p,dV/,+w~AP,).
(A.13)
316
LANDSBERG AND JOHNS
C. APPLICATION
TO INCLUSIVE
AND CONFINED SYSTEMS
Following (A. 11) it has been noted that there will be in general a surface element for which the Lorentz transform to I of the starting point of T,, occurs earliest in time, say at t = tl . Similarly there will be a surface element for which the Lorentz transform to I of the endpoint of TV occurs latest in time, say at t = t, . It follows that in I the process which corresponds to the one considered in I, must be extended to cover the period from t, to t, for all surface elements. The work done on the system is therefore Aw= AW,+ AW,, (A.14) where A W, has to be calculated next. For a confined system and external pressure pO acts over the whole surface of the system, and gives rise to A W, , the effect of the Sfi being fully included in d W, . Imagine the system divided into cylindrical elements parallel to w, of crosssectional areas &4, and lengths lj . During the process let these lengths increase by Alj in I and let the velocity of the centre of mass of the system in I increase from w to (w + Au). In frame I the start of the compression at the smaller coordinate end (as resolved parallel to w) of a typical cylinder precedes that at the end of larger coordinate by a time (/3w/c2)fj , During this time interval the
equilibrium pressure pO acts on this second end which has velocity w. This contributes work which is not already included in A W, , to the system. Similarly the termination of compression at the first end precedes termination at the second end by a time @w/c”) (& + AlJ. During this time interval the new equilibrium pressure (p,, + Ap,) acts on the first end which now has velocity (w + Au). This contributes work
[P(w2+ w - Au)/c”l(& + AMP, + 4,) 6-4 . To first order in A these work increments sum to p SAj[(w2/c2) lj Ape + (w”/c”) Al,p,, + (w . Au/c2) ljpo]. Summing
over j and noting that v, = C lj aAj , i
AV,, = C Alj 6Aj
j
one finds that for the confined system as a whole Aw,=&‘odp,+$,A&,+
w - Au c,PoKl
1.
(A.15)
RELATIVISTIC
HEAT
AND
317
WORK
For an inclusive system external forces act only for the time interval 7O(in I,,), and therefore all the work they do in I is included in d W, . The equilibrium pressure, which caused work A W, to be done on a confined system, is in this case exerted by the container alone. Thus all the work done during this process is done by the stresses in the container, and since the total energy here includes the energy of those stresses, (as discussed in Sections 3 and 4 of Ref. (6)) the total work done on the system is ROW zero. Hence for an inclusive system A W, = 0. The general expression for A W, is therefore given by (A.15) multiplied by X. For a general (inclusive or confined) system the total work is accordingly, by (A.13) to (A.15)
In the notation of the main body of this paper, the symbol “A” is replaced by “d”, Equation (2.14) is then obtained at once from (A. 16).
ACKNOWLEDGMENT
One of us (K. A. J.) is indebted RECEIVED:
to the Science
Research
Council
for a Studentship.
April 4, 1969
REFERENCES I. H. OTT, Z. Phys. 2. H.
ARZELI~S,
175, 70-104
NUOVO
C’irnttznto
(1963). 35,
792-804
(1965).
3. T. W. B. KIBBLE, Nuoco Cimento 41B, 72-78, 83-85 (1966). 4. C. MBLLER, Mat. Fys. Medd. Dan. Vid. Selsk. 36 (1967). 5. I. BREVIK, Mat. Fys. Medd. Dan. Vid. Selsk. 36 (1967). 6. P. T. LANDSBERG 7. P. T. LANDSBERG
AND AND
K. K.
A. JOHNS, A. JOHNS,
Nuovo Cimento 52B, 28-44 (1967). Proc. Roy. Sot. A.306, 477-486 (1968).
8. J. L.
REDDING, Nature 215, 1160-1161 (1967). 9. J. LINDHARD, Physica 38, 635-640 (1968). 10. N. G. VAN KAMPEN, Phys. Rev. 173, 295-301 (1968). II. N. G. VAN KAMPEN, J. Phyx Sot. Japan 26 (suppl.), 12. R. BAIESCU, Physfcu 40, 309-338 (1968). 13.
R. BAI.ESCU,
J. Phys. Sot. Japan 26 (suppl.),
313-315
316-321 (1969).
(1969).
318 14.
LANDSBERG C.
AND
JOHNS
MBLLER, Thermodynamics in the special and the general theory of relativity, in “Old and New Problems in Elementary Particles,” pp. 202-221 (Bemadini Festschrift, G. Puppi, Ed., Academic Press), New York, 1968.
J. BICAK, Lett. Nuooo Cimento 1, 302-304 (1969). E. M. DEWAN AND M. BERAN, Am. J. Phys. 27, 517-518 17. E. M. DEWAN, Am. J. Phys. 31, 383-386 (1963). 15. 16. 18.
P. T. LANDSBERG
AND
K. A.
JOHNS,
(1959).
J. Phys. Sot. Japan 26 (suppl.), 310-312 (1969).
OF PHYSICS:
56,299-318
The Lorentz
(1970)
Transformation
of Heat and Work
P. T. LANDSBERG AND K. A. JOHNS Department
of AppIied Mathematics University College,
and Mathematical Cardiff, Wales
Physics,
A general analysis is given of the work dW done on a system as judged from an inertial frame 1. The novel features of the work are the following: (a) In the incremental process the velocity of the centre of mass of the system in I is allowed to change from w to w + du. (b) The surface elements of the system start to accelerate at different times in I even if they do so simultaneously in the initial rest frame I,, . This has to be taken into account. (c) The analysis allows for two definitions of force which have been in use and which are distinguished by a parameter p whose value is 0 or 1. (d) In order to enable the analysis to be applied to various types of systems, the momentum four-vector is taken in the form (cP, E + hpV) where h = 0 or I. With this notation we derive expressions for the compressive work dWc = -L&o
and the translational
dV +
(1 - A)po dV,/j31,
work dWt,
=w*
dP-p/3---dQ,
W
( c= 1 where /? = [1 - r@/~~]-l/~ and (/Q, is the heat transferred to the system in I, during the incremental process. The transformation of work dW = dWc + dWt, is given by dW,,,, = 0, and
(e) In the discussion of dW it is shown to what extent the Fitzgerald contraction can be said to contribute to the work in frame I. A fairly simple solution is proposed for this problem. (f) The transformation of heat dQ is finally derived as
The paper therefore reduces disagreements about this transformation to a disagreement about definitions (i.e., the value of p) in quite a general way. The temperature transformation is touched on only in passing, .‘ . . the conditions demanded to make possible the ordinary operations of calorimetry still hold under the general conditions assumed by the first law. But this is almost certainly not always the case. An examination of what is involved in classical calorimetry shows that its operations are performed in the absence of ordinary mechanical phenomena, whereas here we have to know the meaning of heat flow across a surface which may be in motion and across which mechanical work is simultaneously being performed. So general a situation is certainly not contemplated in the original operations
299 0 1970 by Academic Press, Inc. 595/5~/~-
1
300
LANDSBERG
AND
JOHNS
of calorimetry, and whether the original operations can be sufficiently broadened is to my mind very questionable. At least I have never seen any adequate discussion of the possibility.” From I?. W. Bridgman The Nature of Thermodynamics (Harvard 1941; Harper Torchbooks 1961, p. 29)
1. 1NTRODUCTION
Most authors have approached the problem of identifying the Lorentz transformation of heat dQ given to a system in an incremental process, by first considering the transformation of mechanical work dW in the same process. Since different answers have been obtained to this fundamental question, (Z-5), we consider it here in a generalised way which enables one to see the answers recently given in a single context. This generalised calculation of the work done on a moving system faces two main difficulties, both of which are solved in this paper. The first difficulty resides in the multiplicity of inertial frames, each with its own proper time, which enter into the calculation. The process of compression or expansion may start simultaneously for all surface elements of the system if the inertial frame used is the initial rest frame I,, of the system. But at the end of the process the system will in general be at rest in a different inertial frame. However, the calculation of dW is in any case required for a general inertial frame I and in such a frame the surface elements start to accelerate relative to the centre of mass at different times. The centre-of-mass frame has initially velocity w, in I, and when the last element has ceased to accelerate relative to the centre of mass the rest frame may have a different velocity, w + du, say, in I(& < w). The analysis required to deal with these questions is rather delicate, and will be found in the Appendix. The second difficulty is that the calculation of dW must hold for two different definitions of force which have been used. It is possible to handle this situation by operating throughout with a definition of force which is sufficiently general to accommodate both points of view. This is achieved by using a parameter p with value 0 or 1 for the two definitions (see Section 2). It is found, as one would expect, that the transformation of both dQ and dW depends on this parameter (see Section 4). An adequately general discussion must include a further complicating feature. Special relativity deals with systems for which the energy-momentum four-vector is PI” = (cP, E) and systems for which it is Pzw = (cP, E + pV). The former systems are the unconfined systems of particles, and thermodynamics cannot readily be applied to these, since they are not equilibrium systems. For a normal confined equilibrium system to which thermodynamics can be applied, P,@ is the relevant quantity. However, the simpler quantity Plu becomes again appropriate
RELATIVISTIC
HEAT
AND
301
WORK
for a confined system if it is agreed to add to it the energy and momentum arising from the stresses in the container, but not from its rest mass (6). Such systems may be called inclusive. To handle both confined and inclusive systems we use P$ f (c,P,E+ApV)
(1.1)
as the relevant four-vector, and allow the values X = 0 or h = 1. It is found that the transformation law of dW, but not of dQ, depends on h. This is satisfactory since it shows that whatever type of relativistic thermodynamics one may favour, at least it is the same for both classes of systems. It proves possible (Section 3) to separate the mechanical work dW done on the system in frame I into two parts: (a) The translational work dWt, has the form: total force acting on the system times velocity of the centre of mass times an increment of time in I. (b) The compressive work dW, takes one of the familiar forms -,pO dV or --p,, dV,,/fl for confined and inclusive systems respectively. Here
p = (1 - EL)-,
0.2)
and it should be noted that while V = V,,//3 if terms in du are neglected, dV and during the process considered. A further difficulty is to explain in what sense the Fitzgerald contraction effect is to be regarded as contributing to the mechanical work. It is explained in Section 3 that it contributes only to the compressive part, and then only in confined systems.. This paper therefore gives a unified discussion of the transformation of work and heat for systems which may suffer accelerations during the incremental process. considered. It covers the four cases generated by X = 0, 1 and p = 0, 1. All terms in the expression for the work dW in I are physically interpreted in this paper, and the only arbitrariness left is in the definition of force. This can be left to personal preference. The transformation of temperature is a separate problem and is touched on only once in this paper (Section 5). dV,//3 are different if the system suffers an acceleration
2. A GENERAL DONE
EXPRESSION ON NON-INERTIAL
FOR
AN INCREMENT SYSTEMS IN
OF SPECIAL
Using the notation of (1.1) and (1.2) the transformation
MECHANICAL RELATIVITY
WORK
from I, to 1 yields
E + AP,v = BWO+ bJ,~lll + w - PO), pw = B (pwo+ ; L% + h&l)~ where the suffix w indicates a component parallel to w, and p = p. .
(2.1) (2.2)
302
LANDSBERG
AND
JOHNS
The frame I, will be regarded as the frame in which the thermodynamic system of interest is initially at rest so that dP, can be non zero. The values of the volume, measured in I and I, , will therefore be related by the formula JL3-.
(2.3) P After a certain time the velocity of the system (in I) has increased by du, and the volume (in I,) by dV, . The volume observed in I will therefore be given by V + dV = (V, + dVo) (1 -
lw ;2du12 )1’2,
so that &$d!!$%.Vo~
(2.4)
The inverse of equation (2.1) is
Eo+ &Jovo= P([E + APO VI - w *P). Differentiation
(2.5)
of this leads to dE = f%’ + xp, --X ‘; B
+ w * dP - hpo dV,
In frame I,, , the first Law of thermodynamics dE, =
dQo-
(2.6)
is POdVo ,
(2.7)
whence Ap, dV - (1 - A) p. -dvo
dE=w*dP++$-
Similarly,
differentiation
B ’
(2.8)
of (2.2), and use of (2.7), gives
dP, = is (d&o, + s
VQo-
PO dVo1 + h s [PO dVo + Vo +,I),
(2.9)
These results should not be controversial since they are based on standard forms of the Lorentz transformation. The concepts of work and heat have been introduced only in the centre of mass frame I, , of the system, and their transformation properties have not yet been used. To obtain these properties it is first necessary to use a definition of force, and it is here that differences of opinion can arise. The orthodox definition takes as force, f, in any frame I, the rate of change of momentum II in that frame f=-&.
dn
(2.10)
RELATIVISTIC
HEAT
AND
303
WORK
This definition is equally applicable to the force acting on a particle, where dn is its increase in momentum, or to the force acting on a surface element of some larger body where &I must be interpreted as the momentum flowing across the surface in a prescribed direction. An alternative definition for the inertial frame I can be written, for a particle, as (2.11)
where 02, is the proper mass of the particle. This definition holds for an inertial frame in which the particle on which the force acts has velocity v. The identity (2.11) is equiv,alent to f alt -_ E - -=v d% -dt
dII --dt
41
-
LJ~/C~
dt
dr c2 41 - oZjc” dt V
__---
(2.12)
When the force is acting on a surface element, in, loses its meaning, and the last of the above expressions may be taken as the definition of force for an inertial frame in which the surface element or point of application has velocity v. It is to be emphasized that de is the energy increment transferred across the surface element measured in that frame of reference in which the surface element or point of application is at rest, not in the rest frame of the whole body enclosed by the surface. The two definitions given here can be combined in one formula by introducing a factor /L which takes the values 0 for the orthodox definition, 1 for the alternative definition, but which may even be assigned other values. Thus
f_Edt
__- de - v2jc” dt ’
V
’ c2 dl
(2.13)
The work done on a moving closed system can then be shown (see appendix) to be dW = @w . dP, - fipo dV,,) + (1 - pL)B ;
dQ,
where du is the change in velocity of the whole system, as observed in frame I. This equation incorporates the results for four different cases (h = 0, I ; p = 0, 1). Using Eqs. (2.4) and (2.9) it may be written dW=w.(dP-p/$dQ,j-[Ap,dV+(l
where P refers henceforth to the total momentum
-h)p,,y],
of the system.
(2.15)
304
LANDSBERG 3. COMPARISON
OF
THE
AND TWO
JOHNS DEFINITIONS
OF
FORCE
The total force fr acting on the centre of mass of the system in frame I can be obtained by summing (2.13) over all the surface elements. When p = 0, it immediately yields, P being now the total momentum of the system,
f =dp T
dt
(P = 0).
Similarly, when p = 1 and the system is not subject to compression or expansion, the velocity v of each element is always equal to w, and both terms in (2.13) can be summed to give dP w dE, fT=z-PFx
(p = 1, dV,, = 0),
or, by (2.7) (3.2) In the remaining case to be considered (p = 1, dl/, f 0) it is not possible to sum (2.13) directly since each surface element has, in general, a different velocity v. However, only quasistatic processes are considered here, and one can assume (see Appendix) that each value of v differs from w only by an infinitesimally small increment, It is then possible to neglect higher powers of these increments to find the total force fr which then depends only on dP and de,, . This is because in the rest frame of each surface element, where the energy increment de is measured, the point of application of any force acting on that element must have zero velocity and therefore such a force can do no work. The changes in energy de are thus due solely to the flow of heat (the effect of accretion of particles being excluded for simplicity). Since furthermore the differences in velocity between the centre of mass frame I, of the system and the rest frames of the individual elements are all infinitesimal, it is possible to equate, to first order accuracy, dQo and C de (the sum being taken over the surface of the system). It is reasonable, therefore, to take (3.2) as the total force on the system, even in the case p = 1, dV, f 0, and thus to combine (3.1) and (3.2) as (3.3) which is valid for both values of p, and in the presence of compression. The consequences of the choice TV= 1 are seen to be rather complicated. Another argument against this choice is the following: Consider a collision between two systems moving on the line joining their centres of mass. Let the time occupied
RELATIVISTIC
HEAT
AND
305
WORK
by the collision be 6t in some arbitrary inertial frame. Then by momentum conservation the increments of momentum gained are related in this frame by SP, = --6P, . For the conventional definition of force f as dP/dt we have at once as 6t -+ 0 that the force fi acting on System 1 and the force fi acting on System 2 are related by fi = -fi . Thus the third law of Newton, that action and reaction are opposite, holds. Alternative definitions will lead to a possible violation of this fundamental property and this seems to us a serious disadvantage of such alternatives. In particular, writing P = /3vM,, in a usual notation, we have
KN WlVl) + Ah wfo,) = -42 W2v2) - B2v2S(%,). Hence if the rest mass of either or both systems changes
Thus the “alternative” definition of force (2.11) can lead to fi # f2 . These considerations do not rule out the choice TV= 1, but they make it appear inconvenient.
4. AN
INTERPRETATION
OF
THE
EXPRESSION
FOR
THE
WORK
Since the centre of mass of the system has velocity w, it is now possible to define a quantity of work dWt, , done owing to the translational motion of the whole system, by dWt, = fT w dt.
(4.1)
l
Then at once, from (3.3) and (3.4) dwtr
= w - (dP -
,u/3 f
d&j,
(4.2)
which is the first term in the expression (2.15) for the total work done, The p-dependence of the translational work shows that it depends on the definition of force. It leads by (2.9) and (4.2) to the result that a flow of heat de,, in I, implies translational work (wz/c2)~ dQ, in I only for the orthodox definition of force (p = 0). If the definition p = 1 is adopted this term is cancelled by the second term in (4.2) and dWt, in I depends solely on the action of forces which already exist in I,, . The second term of (2.15) can be explained as follows: Consider an incremental compression of a conjined system which is (in its rest frame) initially in equilibrium
306
LANDSBERG
AND
JOHNS
with a uniform pressure pO. During the process the pressure remains uniform (except for terms of order dp,,) in any inertial frame in which the system is observed. It is thus clear that any change in volume dV in frame I can be said to be accompanied by an amount of mechanical compressive work --pO dV (which includes the effect of the Lorentz contraction). In the case of an inclusive system the container (considered as part of the system) requires no external force to keep the system in equilibrium. Only for a specific time interval do external forces act to compress the system by an amount dV,, (in frame I,,) and dV,//3 (in frame I), giving rise to work --pO dV, and --pO &‘,//I respectively. Equation (2.4) shows that an additional change of volume remains, namely dV - dV,,/P = -/3V,, ;
. du.
This is due solely to the change in magnitude of the Lorentz contraction factor as the system increases its velocity by du; it requires no external forces, other than those which produce the acceleration of the whole body, and can take place even when no external compressive forces act at all. This volume change is thus not accompanied by any compressive work, and the total work involved in the compression of an inclusive system is therefore --pO dV,,/p in frame I. These two results can be combined to give the compressive work in frame I dWc = -[APO dV + (1 - 4po dVoIP1.
(4.3)
The compressive work is independent of CL,and hence of the definition of because the point of application of each compressive force is at rest (v = 0) rest frame of the element on which it acts. Hence Eq. (2.13) shows that forces have no p-dependent term in frame I. We have therefore split (2.15) the mechanical work done on the system inertial frame I, into two components
force, in the these in an
dW = dW, + dWtr ,
each of which depends on a parameter. Whether a given system is treated as confined or inclusive affects dW, , and the choice of force definition affects dWt, . Some special cases of the general results (2.14), (2.15) are of interest. Case 1. Conventional
Equation
definition
of force,
confined
system
(2.15) yields dW=w.dP-podV.
This is the basic result used previously [Eq. (5.1) in Ref. (6).
(II = 0, X = 1).
RELATIVISTIC
HEAT
AND
WORK
307
Case 2. Conventional dejinition of force. (dP, = du = 0). (In this case the system remains at rest in I0 throughout the process). Equations (2.7) and (2.14) yield: dW = P[dWo + (tic)” Wo - dWJ + 4W2 d(p,,KJl = dWo/P + (NcY ,M& + h,~,).
This is another form of the work transformation obtained before [Eq. (5.3) in Ref. (6)]. It was shown in this reference that if there is no heat flow in I, it yields d W = PdW,
+ 44~)’
Bd(l?o v,,),
WQo = Oh
(4.4)
and hence dW = ,kldWo
(inclusive system, dQ, = 0),
(4.5)
(confined system, dpO= 0, dQ, = 0).
(4.6)
and dW = dW,//3
Case 3. Alternative definition of force (p = 1, dP, = clu = 0). Under these conditions (4.4) holds even in the presence of heat flow in I,, , and hence (4.5) and (4.6) become valid even if dQ, f 0: Comparison of Cases 2 and 3 shows the extent of the confusion which can result if the distinction between confined and inclusive s,ystems and between the various definitions of force is not clearly made.
5. TRANSFORMATION
Turning to the transformation dynamics in the form
OF
HEAT:
DISCUSSION
of heat, assume a relativistic first law of thermodE = dW + dQ.
(5.1)
Hence by (2.8) and (2.15) (5.2) This is the general transformation In particular, dQ = dQo/P dQ = PdQo
of heat for a whole range of force definitions.
(conventional force definition, p = 0) (force definition with p = 1).
(5.3) (5.4)
Note that these transformations are the same for inclusive and confined systems. The choice p = l/(1 + p) renders dQ invariant, but this is only a noteworthy curiosity and such a choice has not been suggested.
308
LANDSBERG
AND
JOHNS
In this paper we only make a passing reference to the temperature transformation which we have suggested should be T = T,, (6, 7), and which had been discussed by several other authors (8-13). Our remark is that if temperature is regarded as invariant the relativistic second law takes the form TdS >,
1-
P P + PP2
dQ.
(5.5)
If, on the other hand, the relativistic second law is assumed to have the form transformation becomes
T dS > dQ, the temperature
We advocate (5.5) with p = 0. One can also obtain the heat transformation (5.2) by four-vector methods. These give little additional physical insight in the present case, but we give the argument for the sake of completeness. One can start by extending (2.13) to give the Minowski force in a general frame I (m, = E/C”)
Here de is the energy-momentum in I passing through a surface element of the system during the process, uy is the four-velocity in I of the element during the process (assumed uniform for the short period of time involved), and dT is the proper time taken by the process at the particular element being considered. In fact the velocity of each surface element is small in the centre-of-mass frame I, (which has velocity w in I), so that the time taken by the process in I, is
where the approximate considered. The invariant
equality is valid to the first order of the small quantities
dx E c Kw, d7
(5.9)
(the summation is over all surface elements), is a generalised quantity analogous to the work. Then dx = C (f u, de’ -
pu,u” dm,)
= (1 - ,u)c2Cdm,.
(5.10)
RELATIVISTIC
= /3 2 de4 -
HEAT
AND
309
WORK
pp2c2 c dm, + ,8 c fi dxi
= p(dE - dW) -
(5.11)
p/3%? c dm,
(Summations over the index i are from 1 to 3; f i, by analogy with (2.13), is the three-force on a surface element; dxi is the displacement Ui d7, of the element and incorporates a negative sign). Thus, by (5.1), (5.10) and (5.11) dQ = dE - dW = [c;A
+ pfl] c2 c dm, .
(5.12)
By considering frame T, , where /3 = 1, it is clear that c2 1 dm, = de,,,
and hence the generalised transformation (5.2) is obtained again. If one adopts (5.5) with p = 0, as we advocate, (5.11) shows that the invariant formulation, TdS>CKYuydr=dx,
(5.13)
of the second law is available. Alternatively (5.13) can be taken as a definition of temperature in such a theory. It should be noted that the work dW in the above equations does nut include the amounts of work which, in the case of a confined system, may be done in I on elements for which, in I,, , the process has yet to begin or has already ended. It is thus equivalent to rl WA of the appendix. However, the purpose of the above argument is to determine the transformation of dQ, and its validity is unaffected by the omission of this additional work, since it is unaccompanied by any flow of heat in all frames of reference.
6. CONNECTION
WITH
PREVIOUS
WORK
Some authors derived variations of special cases of (2.14) and (2.15) for the transformation of work, (Z-3). Sometimes implicitly, they adopted a definition of force equivalent to (2.1 l), and their results are essentially represented by our Case 3 of Section 4 and Eq. (5.4). They can be summarized by dQ E dE - dW = p(dE,, - d W,,) = /I de,,
(6.1)
310
LANDSBERG
AND
JOHNS
Moller (4, 14) considers the transformations of work and heat in essentially the way of Ott, Arzelies and Kibble, but analyses in greater depth the changes in momentum brought about by various processes. In particular, the total change in momentum, dP, is separated [Eq. (56), in (4)] according to1 dP = dP’“’ + dP’“’ 7 where “dPCh) is the increase of the momentum due to the conduction of heat to the system,” and “dPcm) is that increase of the momentum which is due to the action of mechanical forces.” The dP(“) is obtained by the Lorentz-transformation of the energy-momentum produced by forces of compression in the rest frame of the system. The work is then stated to be (4) d W = w * dPfnL) + d W&3
[(63)
in (41
instead of dW = w * dP + dW,//3
t(64)
in WI,
the difference dPth) s w being neglected since momentum dPfh) is not regarded as being produced by real forces capable of doing work. This is just the term /3(wz/cz) dQo of Eq. (4.2). Indeed, Meller’s view is clearly expressed by his comment on dPch’ [his eq. (70)]. ~ dt = f? “The quantity fkh’ is therefore not a real force, it just describes the rate at which momentum grows on account of the conduction of heat to the fluid.” This departure from the conventional definition of force has been fully discussed in Sections 2 and 3. Note that Mgller’s Eq. (63) is contained in (2.15) if one chases p = X = 1 and treats the velocity of the system as constant (du = 0). His equation (64) is obtained by putting h = 1, p = du = 0. In the paper (f4), a relativistic heat engine is considered: first according to Ott’s formulation, and then according to that of Planck and Einstein. A discrepancy is found in the latter, which “clearly shows the inconsistency of the Planck formulation.” Actually, the discrepancy is due not to a fault in theory, but, as pointed out by Bicak (IS) since this paper was written, it is due to the omission by Merller an an amount of work
required by Planck, done on the fluid during process 1 -F 2. If this work is included, 1 Mder’s
equations are rewritten in our notation,
v --f w, ,3 = v/c -+ w/c, K 4 f, .
the
replacements being
dA
-+ dW,
G --t P,
RELATIVISTIC
HEAT
AND
WORK
311
both formulations become self-consistent, and the example of the heat engine cannot be used to distinguish between them. The heat transformation (5.2) with p = 0, which we advocate, belongs to that part of the orthodox (or Planck) theory which we accept. It has recently been criticised by van Kampen (IO, 97), by the remark that the theory yields dQ = 0 for pure acceleration “only by a clever cancellation” in his Eq. (15) [broadly our Eq. (5.1) with p = 0, A = I] dQ = dE-w.dP+p,dV.
But, as has been shown in Sections 4 and 5, this is a satisfactory equation whose terms have simple interpretations, the last two terms being translational and compressive work, even though dV involves a velocity increment [cf. Eq. (2.4).] Nor do we regard as criticisms that the orthodox theory contemplates system of uniform pressure (this is needed in equilibrium thermodynamics), or that it considers as the thermodynamic system the gas in the box alone: this merely refers to the distinction which we introduced (6) between inclusive and confined systems, and which is only incidental to the main thermodynamic points at issue. Accordingly van Kampen considers the case X = 0, and he introduces [Eq. (16)] a force concept whichis not the orthodox Minkowski force, but amounts to an implicit choice p = 1. It is this implicit choice rather than his explicit use of the h = 0 theory that determines the transformation of heat, as is clear from Section 5 above. Dewan and Beran (16) and Dewan (17) considered two rockets accelerated at the same rate from an initial rest frame I, to a final rest frame I’, so that their distance apart is always the samein I, . The proper length of a string connecting the rockets varies, and they consider conditions for the breaking of the string. This experiment shows with attractive clarity the reality of the forces which enter into the calculation of the work dW in Section 2. If the rockets are replaced by accelerated points and the string by a spring accelerated along its length, we have a typical cor$ned system. Let its initial compressive force and length in I, bef:“, L$,‘)respectively, and their final values in I’ beff2)‘, ur2)’ respectively. In a general frame I these forces give rise to a momentum flow, which can be cancelled by the tensile forces in the walls of a box to which the ends of the spring might be connected. This arrangement would be a inclusive system. In order to gain insight into the stress in the spring, we revert, however, to the confined system. From standard transformations one then finds for the work done on the systemin a general frame I W
=
EC"'
_
E'l'
= j& [ &, ( 1 + 5)
E@” - E;” (6.2)
312
LANDSBERG
AND
JOHNS
Here the velocity of I, in I is W, of I’ in I, is u, rS, = (1 - w~/c~)-~/~ and /3U = (1 - +/c~)-~/~. Various special cases may be considered, notably the cases (i) a(2)’ = TV@, and (ii) a(2)’ = ah”. In the first case the spring has constant length in I,, and the compression stress must be relaxed during the acceleration to counteract the Lorentz contraction. In the second case the proper length and the stress are constant. In both cases the work done can be computed from (6.2). For incremental changes in length, force and velocity, (6.2) can be written in the form (2.15) with X = 1 and dQ, = 0: dW = w dP -f”’
da(l).
The compressive work may be written (note that dv = &,” du) dW, = dWc,,/jIw
+ (w/c”)
/&ap’fy)
du,
as one would expect from (4.3) with h = 1 using (2.4). This type of consideration is therefore in agreement with our general results. A short summary of some of the above work is available in Ref. 18.
APPENDIX: A.
CHANGES
IN
THE
DERIVATION
CENTRE-OF-MASS
FRAME
OF FOR
EQ. (2.14) A QUASISTATIC
PROCESS
Consider a thermodynamic system, the surface of which is divided into small elements Au,, each impervious to matter but not to energy or momentum, and each with unit outward normal ni . During a thermodynamic process each element moves with a velocity &vi in the rest frame I, of the system as a whole. The prefix 6 implies here and in the rest of the appendix that the quantity is small enough for the system to be considered to be in equilibrium throughout the process (quasistatic process). Let dai 6g, be the energy which passes through the ith surface element during a short period of time judged in its instantaneous rest frame Ii. The surface then receives energy at the rate hi d@qi)/dt, . Let pO be the initial uniform pressure of the system and let Sfi be the force increment in excess of -p& , which is needed to accelerate the element, thus contributing to the change in volume. Then the element is acted upon by a force (-PO%
+ %) da, .
Since it is the rest frame of the surface element which is being considered, this force is equal to the rate of flow of momentum through the surface whichever definition of force is used.
RELATIVISTIC
HEAT
AND
313
WORK
Applying a Lorentz transformation, the rates of flow of energy and momentum through dai in the centre of mass frame I, are d&o __
xzz
dto G .
dpio --Z-Z dto
-- 4W I
poni . 6vi Aai . I
dti
-.--__ i d/1 - kiajc2
[
(-p
(A. 1)
.* + sf.*) On’ ’
j 6% d(W c2 dti
I
+ (-poni+ + Sf,+)1 Aai $ + (-poni + SfJ Aa, .
(A.3
Superhces* and + indicate components respectively parallel and perpendicular to 6vi . Terms in 6 of higher order than the first, such as (d&l&,), have been neglected.
Therefore, from the definition 8ai is, in frame I,, equal to
(2.13) the force acting on the surface element
independently of p, again with second- and higher-order terms neglected. The rate at which work is done on Aai is therefore, to first order: -dwio
= (-poni + Sf;l) Aad * Svi
dto
+ -poAaini From (A. l), (A.3), and the relativistic the ith surface element at the rate -zdQ
* hi .
(A-3)
first law (5.1), heat is transported across
d&o ---= dto
dwio dto
$!%&Ja,. z
(A.4)
Suppose the force increments Sfi are nonzero for a time interval 7. in I, . The momentum gained by the system during this time in I, is then, by (A.2) APO = J” C Sfi Aai dto *
70 i
(A.3
314
LANDSBERG AND JOHNS
since Ci ni dai = 0, for a system with a closed surface. The work done on the system is similarly by (A.3)
AW,=j 70
* Svj Aa,1 dt, = -p. A V, ,
64.6)
when d V, is the change of volume in the process. The heat gained in I,, is
AQ”= @,--d@q’) Aai
dt, .
(A-7)
B. CORRESPONDING CHANGES IN A FRAME I For an inertial frame I in which the centre of mass of the system has velocity w one finds from (A.l) and (A.2) by a Lorentz transformation %
= ifi [(-p&*
+ Sf,*) + 5 (*
- poni . hi)]
dt ’ dt< ’
+ 6f,t)i Au. -Z
+ (-poni+
1
(A4
where superfices * and + here indicate components respectively parallel and per pendicular to w. To first order in 6, the velocity ui of frame Ii in frame I has components w + Sv,”
ui* =
1$-d
+ w + -L SVj”
w . sv.
B"
C2
Hence lli
+
W +
’ P”
6Vi”
+
’
6Vit B
64.9)
The force acting on Au, in I is obtainable from (2.13), noting that the velocity required is calculated with respect to the frame Ii , whence v = ui in this case. The energy E gained by the system refers to its proper frame, and is the quantity given at the beginning of this Appendix. Hence this force is dpi dt
Ui Aa, 4%) ’ c2 41 -____ - tqc2 “-iii-’
RELATIVISTIC
HEAT
AND
315
WORK
To the first order in 8, this force is equal to
+
( -poni+
+
The rate at which work T
=
)/3 [(--pani*
+
=:
+
( -poni+
+
i/3 [W * (-p&
Aai 2
sf,+)/
.
(A. 10)
is done on Aai is therefore, Sfi*)
Sf,i)(
+
+
5
([l
Aa, 2 Sfi)
+
-
~1 F
. (w + $$ $
([1
-
by (A.9) and (A.lO):
-
I
poni
. hi)]
+ y)
jh] y
-
poni
. &vi)]
1 +
7
- (-p&l
Aa, 2 -
poni
* hi
+
w * (-poni
+
Sf,)]
Aa, 2
(A. 11)
In I the total work done during the process (which occurred in time interval TV in I,) is obtained by summing Eq. (A.ll) over Aai and integrating over dt. In this integration the time limits for each element i are the Lorentz transforms to I of the end points of the interval TV. These transforms involve not only the velocity w, but also the position coordinates (the initial coordinates for the lower limit, the final coordinates for the upper limit) of each surface element. Thus one has a sum of integrals with different end points on the time scale of frame I. From (A. 11) one can clearly express this result as an integral over the single time interval 7(, : Aw,=~J Comparison
C[(l-i”)~~-~~“~.Sv~+w.~f]Aa~dt,. 70 i of (A.12) with (A.$ AW,=/3([1
(A.12)
(A.6), and (A.7) gives:
-&AQ,-p,dV/,+w~AP,).
(A.13)
316
LANDSBERG AND JOHNS
C. APPLICATION
TO INCLUSIVE
AND CONFINED SYSTEMS
Following (A. 11) it has been noted that there will be in general a surface element for which the Lorentz transform to I of the starting point of T,, occurs earliest in time, say at t = tl . Similarly there will be a surface element for which the Lorentz transform to I of the endpoint of TV occurs latest in time, say at t = t, . It follows that in I the process which corresponds to the one considered in I, must be extended to cover the period from t, to t, for all surface elements. The work done on the system is therefore Aw= AW,+ AW,, (A.14) where A W, has to be calculated next. For a confined system and external pressure pO acts over the whole surface of the system, and gives rise to A W, , the effect of the Sfi being fully included in d W, . Imagine the system divided into cylindrical elements parallel to w, of crosssectional areas &4, and lengths lj . During the process let these lengths increase by Alj in I and let the velocity of the centre of mass of the system in I increase from w to (w + Au). In frame I the start of the compression at the smaller coordinate end (as resolved parallel to w) of a typical cylinder precedes that at the end of larger coordinate by a time (/3w/c2)fj , During this time interval the
equilibrium pressure pO acts on this second end which has velocity w. This contributes work which is not already included in A W, , to the system. Similarly the termination of compression at the first end precedes termination at the second end by a time @w/c”) (& + AlJ. During this time interval the new equilibrium pressure (p,, + Ap,) acts on the first end which now has velocity (w + Au). This contributes work
[P(w2+ w - Au)/c”l(& + AMP, + 4,) 6-4 . To first order in A these work increments sum to p SAj[(w2/c2) lj Ape + (w”/c”) Al,p,, + (w . Au/c2) ljpo]. Summing
over j and noting that v, = C lj aAj , i
AV,, = C Alj 6Aj
j
one finds that for the confined system as a whole Aw,=&‘odp,+$,A&,+
w - Au c,PoKl
1.
(A.15)
RELATIVISTIC
HEAT
AND
317
WORK
For an inclusive system external forces act only for the time interval 7O(in I,,), and therefore all the work they do in I is included in d W, . The equilibrium pressure, which caused work A W, to be done on a confined system, is in this case exerted by the container alone. Thus all the work done during this process is done by the stresses in the container, and since the total energy here includes the energy of those stresses, (as discussed in Sections 3 and 4 of Ref. (6)) the total work done on the system is ROW zero. Hence for an inclusive system A W, = 0. The general expression for A W, is therefore given by (A.15) multiplied by X. For a general (inclusive or confined) system the total work is accordingly, by (A.13) to (A.15)
In the notation of the main body of this paper, the symbol “A” is replaced by “d”, Equation (2.14) is then obtained at once from (A. 16).
ACKNOWLEDGMENT
One of us (K. A. J.) is indebted RECEIVED:
to the Science
Research
Council
for a Studentship.
April 4, 1969
REFERENCES I. H. OTT, Z. Phys. 2. H.
ARZELI~S,
175, 70-104
NUOVO
C’irnttznto
(1963). 35,
792-804
(1965).
3. T. W. B. KIBBLE, Nuoco Cimento 41B, 72-78, 83-85 (1966). 4. C. MBLLER, Mat. Fys. Medd. Dan. Vid. Selsk. 36 (1967). 5. I. BREVIK, Mat. Fys. Medd. Dan. Vid. Selsk. 36 (1967). 6. P. T. LANDSBERG 7. P. T. LANDSBERG
AND AND
K. K.
A. JOHNS, A. JOHNS,
Nuovo Cimento 52B, 28-44 (1967). Proc. Roy. Sot. A.306, 477-486 (1968).
8. J. L.
REDDING, Nature 215, 1160-1161 (1967). 9. J. LINDHARD, Physica 38, 635-640 (1968). 10. N. G. VAN KAMPEN, Phys. Rev. 173, 295-301 (1968). II. N. G. VAN KAMPEN, J. Phyx Sot. Japan 26 (suppl.), 12. R. BAIESCU, Physfcu 40, 309-338 (1968). 13.
R. BAI.ESCU,
J. Phys. Sot. Japan 26 (suppl.),
313-315
316-321 (1969).
(1969).
318 14.
LANDSBERG C.
AND
JOHNS
MBLLER, Thermodynamics in the special and the general theory of relativity, in “Old and New Problems in Elementary Particles,” pp. 202-221 (Bemadini Festschrift, G. Puppi, Ed., Academic Press), New York, 1968.
J. BICAK, Lett. Nuooo Cimento 1, 302-304 (1969). E. M. DEWAN AND M. BERAN, Am. J. Phys. 27, 517-518 17. E. M. DEWAN, Am. J. Phys. 31, 383-386 (1963). 15. 16. 18.
P. T. LANDSBERG
AND
K. A.
JOHNS,
(1959).
J. Phys. Sot. Japan 26 (suppl.), 310-312 (1969).