AN EVALUAFION 0F RELATIVESTIC
THERMODYNAMICS

 

Thesis for the Degree of Ph. D. ,
WCHIGAN STATE UNEVERSIW
DARRYL LEONARD STEINERT

1959

LIBRARY

THESIS

Michigan State
University

 

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ABSTRACT

AN EVALUATION OF RELATIVISTIC
THERMODYNAMICS

BY

Darryl Leonard Steinert

The problem of which of the formulations of
relativistic thermodynamics that have been proposed by
Planck, Eckart, Ott, and Landsberg is correct is exam-
ined. Due to the lack of experimental data on relativ-
istic thermodynamic systems, it is not possible to com-
pare predictions made by the various formulations with
experimental data. But, using as a model the process of
evaporation, I found that it is possible to study the
consistency between the transformation laws for temper-
ature and for mechanical energy. The result obtained is
that only the formulation by Ott is free of contradic—
tion.

Ott's proposed transformation laws are further
evaluated in terms of their compatibility with relativis-
tic formulations of fluid dynamics and statistical
mechanics. Compatibility is to be expected because
thermodynamics, fluid dynamics, and statistical mechanics

are compatible in their non-relativistic formulations.

Darryl Leonard Steinert

The lack of contradiction between Ott's formula-
tion and the transformation law for mechanical energy,
and its compatibility with formulations of relativistic
fluid dynamics and statistical mechanics, give support
to a conclusion that the Ott formulation of relativistic

thermodynamics is correct.

AN EVALUATION OF RELATIVISTIC

THERMODYNAMICS

BY

Darryl Leonard Steinert

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1969

DEDICATION

I want to dedicate this thesis to my wife Beth for

giving me the support needed during this work.

ii

ACKNOWLEDGEMENT

I wish to express my gratitude to Professor
Richard Schlegel for his suggestion of the problem of
relativistic thermodynamics, his help throughout the
course of this research, and the overall concern that

he has shown in my interest in physics.

iii

 

Chapter

I. HISTORY . .

TABLE OF

9

0

CONTENTS

A. Planck-Einstein . . . . . .

B. Eckart . . . . . . . . . .

C O Ott O O O O O O O O O O O

D. Landsberg . . . . . . . . .

E. van Kampen . . . . . . . .

F. Historical Resume . . . . .

II. THE PROBLEM . . . . . . . . . . .
A. Consistency . . . . . . . .

B. Covariance . . . . . . . .

C. Compatability with Related
Theories . . . . . . . .

1. Statistical Mechanics

2. Fluid Dynamics . . .

III. CONCLUSION . . . . . . . . . . .
REFERENCES . . . . . . . . . . .

iv

Page

15
23
28
29
33

34
41

42

43
47

55

57

LIST OF FIGURES

Page

A pictorhl comparison of the Planck-

Einstein and Ott formulation . . . . . . 17
A review of the proposed transforma-

tions . . . . . . . . . . . . . . . . . 32
Ratio of entropy and temperature

transformations for the formulations

of relativistic thermodynamics

being considered . . . . . . . . . . . . 40

I. HISTORY

A. Planck-Einstein

 

In 1907 and 1908 Max Planckl’z

published papers on
relativity theory in which a formulation of relativistic
thermodynamics was developed. He approached the problem

using a variational method that was a generalization of

the work by Helmholtz. He generalized Helmholtz's

Lagrangian,
L = %m(g%-2-F, (l)
where
F = U-TS (2)

is the Helmholtz free-energy, to
l
L = —(mc2+F°)[l-(g%)2/c2]7 (3)

Equations (1) and (3) are for one-dimensional motion and
F° is the rest-frame free energy. This is the same
generalization that he used in 19063 to develOp a rela—
tivistic Lagrangian formulation of mechanics.

The transformation law for the temperature was
calculated as follows. Let T° be the rest-frame tempera-
ture and T the temperature in the moving frame of refer—

ence. Now write

T = DT°, (4)

where D is an undetermined function of the relative veloc-
ities between the two frames of reference. This function
must be even with respect to the velocity 1 and become
unity When z_approaches zero. The entropy is defined to
be S = 3%, where T is the temperature. Thus, taking the

partial derivative of Eqn (2) with respect to the temper-

ature T and using Eqn (4), one gets

-1 -l
EE—_'13F°__Y 3F°_y__o
8T_Y fi‘ Da—rfo‘ns' ‘5’
-1_ 2-1- _ 4
where y — [l-B ]2 and B — V/c. Planck argued , as fol-

lows below, that a reversible adiabatic acceleration would
leave the entropy constant. Consider a thermodynamic
system at rest in the frame 2°. Reversibly and adiabat-
ically accelerate the system until it has the same velocity
X as the system 2 relative to 2°. Since the acceleration
was adiabatic, the system still has the same entropy (S°)
relative to 2°. However, the system now has zero velocity
with respect to 2. Thus it is now in the same situation
with respect to 2 as it previously was to 2°. Hence, the

entropy (S) of the system relative to 2 must be equal to

the entropy (S°) of the system relative to 2°. Therefore,
the entropy of a system is an invariant. Then, since S=S°
8L = 8L° ,

 

3 8T°

and therefore

Y = l!
D
or,
D = y-l. (6)
Hence, from Eqn (4);
T = y—1 T°. (7)

From this transformation law and the invariance of the en-
tropy, it follows that the transformation law for heat (Q)
is the same as for the temperature.

Therefore, the transformation laws for the thermo-

dynamic quantities are, in the Planck formulation.

S = 8°, (8.1)
Q = y”1 0°. (8.2)
T = y’1 T°, (8.3)
P = P°, (9.1)
v = y_1 V°. (9.2)

The transformation laws for the pressure (P) and
the volume (V) were determined from mechanical considera—
tions and thus are numbered differently.

An equivalent approach to the transformation law
for heat (and hence for the temperature) follows. It is

used to show the need, in the Planck formulation, for an

arbitrary force called the Ffihrungskraft. Planck defined

 

heat to be pure energy. Hence it made no sense to talk
about the velocity of heat. This meant that to all ob-
servers the heat reservoir and the transferred heat have
no momentum, effectively keeping them at rest. (This
might now be strange because of the equivalence between
mass and energy; however, in 1907 this equivalence was
probably not completely appreciated.)

We examine, in the rest frame, the process of the
transfer of an amount of heat (AQ°) from the reservoir
to a body. Since heat is pure energy, the body increases

its rest-energy by the amount

AE° = AQ°. (10)
This implies an increase in the rest mass of

Am° = AQ°/C2- (11)

We now examine this process from a frame of refer-
ence in which the body is at rest, but has a velocity X
with respect to the reservoir. Since the heat is energy,

the body increases its energy by an amount,
AB = yAE°. (12)

This energy carries no momentum, in accordance with the
postulate about heat, but it does increase the mass of the

body by an amount,

Am = yAm°. (13)

Thus, the rest frame observer would see the body slow down
due to the increase in its mass. However, to an observer

in the body's frame of reference there would be no change

because the body had no momentum in the beginning and the

heat transferred into the body carried no momentum. This

paradox is resolved by creating a force, called the

Ffihrungskraft, that increases the body's momentum such

 

that its velocity y_remains constant with respect to the

rest observer. This force is

or,

dQ°

dt , (15)

EF = (X Y/CZ)

 

The work done on the body (-AW) by this force in the time

At is

-AW = y'F At = 82yA0°. (16)

—F

Hence, from the first law of thermodynamics,
AQ = AE + AW:

we have,

AQ = yAQ° - yBZAQ° = y(1-82)AQ°.

or,

AQ = Y AQ°. (17)

This is the same as Eqn (8.2).

In a 1907 papers, Einstein attacked the problem of
relativistic thermodynamics. Instead of Planck's varia-
tional approach, he examined contributions to the thermo-
dynamic quantities that were simultaneous in the rest frame
of the system (2). He then compared these results with
those obtained by similar calculations made from a moving
frame of reference (2). His results for the transforma-
tion laws of the thermodynamic quantities agreed with
Planck's, and the agreement has been used in recent liter-
ature6 to add credence to this formulation, which today is
commonly known as the Planck—Einstein formulation. How-
ever, it is never mentioned that when Einstein wrote the
first law of thermodynamics in his paper, he included the

work contributed by the Ffihrungskraft (16). This inclu-

 

sion guaranteed that Einstein's solution agree with
Planck's. But, because of the inability of Planck's heat
to carry momentum, his and Einstein's formulation is non-
covariant. This is not surprising, because the develop-
ment of four-dimensional tensors was just beginning in
1908. This development, started by Minkowski7, was
finished after his death by Sommerfeld with the publica-
tion of two papers8 in 1910. The power and significance
of their work were only appreciated by physicists after

Einstein's proposal of the General Theory of Relativity in

1916 and the develOpment of Relativistic Quantum Mechanics
at the end of the next decade.

An important aspect of the work in General Rela-
tivity and Relativistic Quantum Mechanics is that the
Relativity Principle was reformulated. Instead of saying
that all the laws of physics must be of the same form in
all inertia frames, the more general statement is that the
laws of physics belong to an irreducible representation of
the Lorentz Group. Thus, all the operators and variables
in the equation stating a physical law are such that each
term in the equation belongs to the same irreducible rep-
resentation of the Lorentz Group. This means that in
every frame it satisfies the same irreducibility condition.
Because the Planck definition of heat does not allow the
heat to ever carry momentum, the heat only appears in the
time-like component of a four-vector. This means that
this formulation of relativistic thermodynamics does not
satisfy this condition of covariance or irreducibility.

The fact that relativistic thermodynamics was ig-
nored during the 1920's is not surprising. The major
thrust of research was in the development of Quantum
Mechanics. In areas related to relativistic thermodynam-
ics, relativistic fluid dynamics and cosmology, the prob—
lem of relativistic thermodynamics was of secondary impor-
tance. Relativistic fluid dynamics was concerned with

adiabatic flow and barotropic flow. Eddington's9 View of

the role of relativistic thermodynamics in cosmology was
that " . . . the transformation to moving axes introduces
great complications without any evident advantages, and is

of little interest except as an analytical exercise.“

B. Eckart

 

In 1940 C. Eckart10

examined the possibility of
constructing a systematic theory of irreversible proces-
ses. In this development he used what he called an s-sub—

stitution, where e is the internal energy per unit mass of

substance, to verify certain equations. In this method

N

one replaces e by e + f(2K=l

Mka), where f is arbitrary,

MR is the molecular weight of the substance k and Ck is

its concentration. Because 211::1Mkck = l and the zero
point of the energy is arbitrary, the equations must be
invariant when the e-substitution is made. In a footnote11
he voiced concern that the equivalence of mass and energy
made the status of the e-substitution unclear.

Because of this concern he12 did develop a rela—
tivistically invariant theory of the simple fluid. This
formulation is completely different from the Planck—

Einstein formulation and hence will be presented.

Eckart assumed a Galilean metric of the form,

908 = (18)

COOP
COP-'0
OHOO
HOOO

with (x° ct, x1, x2, x3). (Greek indices range from 0

to 3 and Latin indices range from 1 to 3.) He used the
four-vector ma to represent matter. It has units of

gm/cc. He then defined two projection operators,

1
U“ = ma/(mYmY)2 (l9)
and
a a a
= + 20
SB 58 U U8, ( )
such that for any vector F“,
f = -U P“ (21)
a
is the projection of Pa on the proper-time axis, and
f“ = s a F8 (22)
B
is the projection of F“ into proper-space. He then de-
fined the proper-rate of change operator (D) as
D 2 0° —3— . (23)
3x”

He used these quantities in the law of conserva-

tion of energy-momentum,

awaB
Bxa

 

= 0, W“8 = wBa, (24)

where W“8 has units of erg/cm3, to derive a statement of

the first law of thermodynamics. He used the proper com—

ponents of was,

w = W Ua UB' (25)
a _ _ a BY
and
wo‘B = s a s B wY3 (27)

to define the internal energy, 6, and the heat flow, q“,

as follows:

m(€ + a) = w, (28)

where 8 has units of erg/gm and a is an arbitrary constant;

q“ = cwa (29)

and has units of erg/cm2 sec. Then by taking the scalar
product,

awaB
a

 

_U (

B ) = Ol (30)

3x

B

and using the proper-components of Wu , the definitions

of the internal energy, a, and the heat flow, qa, he de-

rived a statement of the first law of the form,
mDe + %[(3qa/3xa) + anUa] + w°B(aUB/ax°) = o. (31)

This is quite similar to the following classical form of

the first law

m(D€/Dt) + V~q - (P-V)-Z = 0,

11

where q_is the classical heat flow, 2 the velocity, and P
the total stress. He interpreted the term q: DU“ in Eqn
(31) as the work done by a flow of heat thrgugh acceler—
ated matter.

To introduce the quantities temperature and en-

tropy he defined the hydrostatic pressure (p) as
p = %-w a, (32)

and the viscous stress tensor (PaB) as

PaB

= -w°‘B + ps°B. (33)

08

By solving Eqn (33) for w and substituting this into Eqn

(31), the first law can be rewritten as

m(D€ + pDv) + % [(aqa/axa) + anUa] +

-P°B(aUB/ax°) o.

The term mpDv appears because

s°B(8UB/ax°) = (aua/axa) mDv,

where v E l/m is the invariant specific volume.
Since a is only a function of p and v for a simple
fluid, Eckart used a standard argument to show that there

are two functions 6 and n such that

De + pDV = GDn. (34)

12

He then identified 6 as being the temperature and n as
being the specific entropy. The significant property of
9 and n is that they are true scalars because 6, p and v
are true scalars. In fact, his entire approach was such
that he formulated a relativistically invariant scalar
thermodynamics.

His scalar temperature (6) is related to Planck's

temperature (T) by the following equation,
T = 6U°, (35)

and his definition of heat flow (qa = cw“) is such that it
does involve momentum terms, in contrast to Planck's heat
which is not able to "carry" momentum. That Eckart's
definition of heat flow "carries" momentum is easily
shown. Let the fluid be at rest in a Galilean frame x0t

whose coordinates are parallel to the prOper time coor-

dinates at the point 0. Thus, at this point,

11° (1, o. o. 0). (36)

and

s°‘B = (37)

0000
ODE-‘0
Ol—‘OO
l—‘OOO

This means that the conservation of matter equation,

Ema/axa = 0,

becomes

13

Bm/Bt = 0, (38)
2 3

so that m is only a function of X1! x , x . The components

of the energy-momentum tensor (was) are,

WOO

m(e + a),

0' , '0
WJ == g; = wJ , j = 1,2,3
C

WJu = p53u.

With these components, the energy-momentum principle, Eqn

(24): becomes

mas/3t + (aqj/axj) = o, (39)
and

3 (qj/cz) + zap/axJ = o. (40)

FE

Eqn (39) is the law of conservation of energy and Eqn (40)
are the conservation of momentum equations. Since the
time rate of increase of (qj/cz) is equal to the negative
'gradient of the pressure, it is seen that (qj/cz) is a
momentum.

The significance of Eckart's work is that he
showed that the presence of heat flux can be accounted for
by the addition of a symmetric tensor, QaB, to the usual

stress-energy tensor, TaB. Thus, in his work

w°‘B = T93 + Q95, (41)

where

14
T0‘8 = m(l + h/c2) UaUB - pgaB,

with h being the specific enthalpy and p the pressure.

So, we have the heat flux, qa, as

q“ = US QB“. (42)

In 1951 Boris Leafl3

published a paper in which he
showed that Eckart's formulation of a relativistic thermo-
dynamics was related to the time components of the energy-

momentum principle,

awaB

= 0! (24)
a

 

3x
and that the laws of dynamics were related to the spatial
components.

In the 1952 edition of his book on relativity
theory, von Laue was probably aware of Eckart's work (al-
though perhaps not Leaf's, because of the time delay be-
tween writing and printing), since in the sections on
relativistic thermodynamics, in which the Planck-Einstein
formulation is develOped, he comments in a footnote14 on
the possibilities of formulating a relativistic thermody-
namics with an invariant temperature. However, after
doing this he adds the following statement: " . . . this

change, if one attempts to carry it out, so deeply affects

our perception of and methods of expressing things that

15

one would be wise to use the definition of temperature

given in the text."
C. Ott

Other than von Laue's footnote, I have found
nothing further in the literature until 1963 when H. Ott
published a paper15 in which he proposed a set of trans-
formation laws for the thermodynamic quantities that were
different than those in either the Planck—Einstein or
Eckart formulations.

Ott agreed with the Planck-Einstein formulation
in that he also felt that the relativistically transformed
thermodynamic variables should be related by the same laws
of thermodynamics that had been established for the non-
relativistic variables. In his paper he used many differ-
ent phenomena to argue for the following transformation

laws for the thermodynamic quantities:

Q = YQOI (41.1)
T = YTo, (41.2)
S = S°, (41.3)

where the ° superscript is for the rest observer.
Because of the invariance of the entropy, the tem—
perature and heat must transform in the same manner. This

argument is the same as in the Planck—Einstein formulation.

16

However, in the Ott formulation the heat is allowed to
"carry" momentum. Thus, if the heat arrives from a reser-
voir that is traveling at a velocity 2, the heat carries
the momentum (yAQ°/c2)y_with it. Hence, there is no need
for the Fuhrungskraft, as in the Planck-Einstein formula—
tion, if the body receiving the heat and the reservoir are
traveling with the same velocity. Therefore, an observer
moving uniformly at a velocity least the reservoir-body

system would write the first law of thermodynamics as

AQ AB, (42)

and, since from Eqn (10) AE yAE° = yAQ°:

AQ = yAQ°. (43)

Figure l on page 17 shows the differences between the
Planck-Einstein results and Ott's.

Ott was very critical of the Planck-Einstein for—
mulation. He was very disturbed by two aspects of their
formulation. First, he argued that heat cannot be
uniquely defined in terms of only the first law of thermo-
dynamics, since the law only says that the heat is a form
of energy. Second, he felt that the Planck~Einstein form
of the equation of motion for a body of variable rest mass,
M, that is absorbing heat from a reservoir at a rate
-d'Qr/dIy measured in the rest frame of M, or at the rate

-D'Qr E -(d'Qr/dT)/Y measured in the observers frame, is

l7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PLANCK-EINSTEIN OTT
RESERVOIR RESERVOIR
REST
FRAME
+ AQ° 1 AO°
BODY ‘ BODY
RESERVOIR RESERVOIR + X
(at rest)
MOVING
FRAME

 

 

 

J.
+ AQ=[l-v2/c2]2AQ°

 

F + BODY + v

 

 

 

 

-1
+ AQ=[l-V2/02] 2AQ°

 

 

BODY + v

 

 

 

 

Figure l.--A pictoral comparison of the Planck-Einstein

and Ott formulation.

 

18

incorrect. (d' is an operator that gives a small change
in a quantity and is not a perfect differential.) This

differential equation of motion has the following form:

d(Mvu)/dT = F“, (44a)

where

u 1

F = vtc' (3;: - D'Qr), :1. (44b)

U (F°) contains

Note that the timelike component of F
'(Y/c)D'Qr. M¢llerl6 shows that when the heat transfer
is taken into account in this manner the Planck-Einstein
transformation law for the heat, Eqn (8.2), follows.

In the Ott formulation the equation of motion has

the following form:

d(Mv“)/dr = F“ - d'Qr /dT, (45a)
where
F“ = y[(y;§/c), 5]. (45b)
and
Qu = Qtl. y/CJ (45c)

Whereas Ott thinks that Eqn (44a) is wrong and
that Eqn (45a) is correct, it is obvious that the "correct"
form of the equation of motion is dependent on the defi—
nition of heat used. This is easily seen when one real—
izes that the Planck-Einstein definition of heat does not
allow the heat to carry momentum and therefore heat can

only appear in the timelike component, while the Ott

l9

definition allows the heat to carry momentum and therefore
the heat forms a four-momentum.
In Ott's formulation'the temperature, along with

the heat, is also written as a four-vector:
T“ = T(1, y_/c). (46)

Thus, the second law of thermodynamics can be written as

a four-equation:
Tuds = 89“. (47)

Hence, the Ott formulation forms a covariant representa~
tion of thermodynamics, as was discussed earlier.

In 1965, H. Arzeliés17

published a paper whose re-
sults agree with Ott's. In this paper he also argued that
an observer in relative motion with respect to a Hohlraum

would observe as the total energy (U),
U = YU , (48)

0

where U0 is the total internal energy as observed by an

observer at rest with respect to the Hohlraum, rather than

1
1+3Bu, (49)

which is just the usual18

 

20

with POVo = % Uo for a Hohlraum. Instead of using
Arzeliés' argument, which is rather obscure, it will be
easier to repeat the method used in an article by A.
Gamba19 that appeared shortly after Arzeliés' and is sub-
titled "Beware of Jacobians!"

Gamba's method is to pair photons of equal and
opposite momenta. The sum over all these pairs, in the
rest system of the Hohlraum, yields a total energy E and
a null momentum, P0 = 0. Then, the energy transformation
would yield Eqn (48) rather than Eqn (49). Gamba showed

that the extra term (% 82) came from an integration over

a solid angle (d9) of the type

[(1- Boos 0))de 21TI(1— Bcos 0L)2d(cos a)

411(1 + i 82).
3
where the integrand is the result of DOppler effects (D),

l - 8 cos a

'1—82

 

However, there is still an integration over the volume to
get the total internal energy. For the moving observer,

one writes

dV' = dV 1 - B ;

however, Gamba claimed that the correct volume transforma-

tion is

21

av, _ dv V 1 - 82
_ l - 8 cos d '

 

When this is incorporated into the solid angle integral

one gets
[(1 - Bcos (1)2le = 211f(l - Bcos on)2d(cos on) = O

which removes the "unwanted term 8/3."

T. Kibble20 pointed out that Gamba's approach is
such that the observer moving with respect to the Hohlraum
is required to integrate over the rest observer's hyper-
surface of simultaneity rather than the hypersurface of
simultaneity in his frame. Because Gamba requires that
the contributions, in the frame of the moving observer,
to the total energy be simultaneous in the rest frame of
the Hohlraum, this total energy is not necessarily an ad—
ditive constant of the motion for the moving observer.
Whereas, if the moving observer adds all the contributions
to the energy on his hypersurface of simultaneity the
total energy is an additive constant of the motion in his
frame. Thus, if one wishes to keep the total energy an
additive constant of the motion in all frames of refer-
ence, Eqn (50) must be used rather than Eqn (48).

Along with agreeing with the Ott and Arzelies
transformations Kibble also pointed out that the Planck—

Einstein formulation implies that work can be done on a

22

system when neither the volume or the pressure changes.

This is seen by writing the first and second laws,

dQ = dE - dW, (51.1)
and

as >, dQ/T, (51.2)
and defining two new variables (T1 and Q1) as
T = Tlg(B), d0 = dng(B)

where T and dQ are the Ott temperature and heat and g(B)

v/c. These new vari-

is any function of the velocity, 8

ables transform as

Yon/g(B). (52)

T1 = ITO/9(8); do1

The two laws of thermodynamics, Eqn (51.1) and Eqn (51.2),
will be satisfied in terms of the two new variables if we

redefine the work done on the system as

dwl = dW + dQl[g(B)-l]. (53)

Upon comparing Eqn (8.2) and Eqn (8.3) with Eqn (52), it

is seen that the two new variables correspond to the

Planck—Einstein variables when g(B) = y2. But, from Eqn

(53) one sees that this means that dWl can be non-zero

when neither the pressure or the volume are changed

(dW = 0).

23

At.this same time A. Bbr521, using a microconical
distribution and statistical mechanics methods, argued for
the Ott transformation laws for heat and temperature. In
his development he assumed that the form of the distribu-
tion function was the same for the moving observer as for
the rest observer.

In an obvious response to this article and those
by Ott, Arzeliés, and Gamba, R. Pathria6 used statistical
mechanics arguments to support the Planck-Einstein trans-
formation laws. Also, in the same vein, R. Penney22 pub-
lished a rebuttal in favor of this formulation and against
the Ott formulation.

This flurry of papers was climaxed by a paper by

23

F. Rohrlich which reviewed and compared the two formula—

tions. His conclusion was:

"There is no way to choose between the conventional
(Planck—Einstein) and the manifestly covariant (Ott)
formulation of relativistic thermodynamics in terms
of logical arguments. Both descriptions are consis-
tent with special relativity and classical thermo-
dynamics."

D. Landsberg

 

However, Rohrlich's paper did not end the debate
concerning the transformation laws of thermodynamic vari—
ables. In 1966 P. T. Landsberg24 began publishing a
series of articles in which yet another Special relativ-
istic formulation of thermodynamics was presented. To

avoid problems about how a moving observer would make

24

thermodynamic measurements (temperature, entropy, or in-
ternal energy, for example), he proposed a formulation in
which these measurements need only be done in the rest
frame of the system. In other words, he proposed a formu-
lation of relativistic thermodynamics in which all thermo-
dynamic variables were Lorentz invariant. He, like B¢rs
and Pathria, used statistical mechanical arguments.

His development is as follows: Let IIo be the
maximum entropy distribution function in the rest frame.

This is
ln[no(VoINolEoIPo)] = —ln(Qo) - aloNo
‘ a2oEo ‘ GaoPo (54)

where V0 is the volume, No the number of particles in the
system, E0 the total energy of the system, Po the total
momentum of the system, Q is a partition function and the
0's are Lagrangian multipliers.

He assumed that in an inertial frame traveling at
a velocity X in the +x direction with respect to the rest

frame, the same form of Eqn (54) would be true. Thus,

ln[H(V,N,E,P)] = -1n(Q) - N - azE - P. (54.1)

0‘1 0‘3

Now, if the variables of Eqn (54) and Eqn (54.1) refer to
the same state of the system one can use the special rela-

tivity transformations of N (N = No), E, and P to write

Eqn (54.1) in terms of the corresponding rest frame

25

quantities. Since the entropy is defined such that it is
a Lorentz invariant in this formulation, the two proba-
bilities given by Eqn (54) and Eqn (54.1) must be the
same. This means that the Lagrange multipliers are re-

lated as follows:

V0.30];

c2 (55)

0‘1 = 0‘107 0‘2 = y[a20 +

a3 = y[a30 + vazo].

Also, the partition function must be an invariant:

Q = Q0 (56)

From the way the Lagrange multipliers a2 and d3

transform, one sees that they form a four-vector

a“ = [0012, Q3]. (57)

From the definition of the entropy,

S0 = ~k2No'Eo’PonolnHO, one obtains
So = -k{1n(Qo) + alofio + 01201210 + 0.30130} (58)
where the bars denote averages. In the rest frame of the

system Po = 0.

To identify the Lagrange multipliers Landsberg
assumed that the entropy depended not on E and P separate—
ly, but instead on some combination of them that is called

the internal energy (U). Hence, from Eqn (58)

26

l 380

_( _
k ano

) (59.1)

VoIUo'

1 BS
OL = _ o _ _
20 k(3E Vo INo 1P0
° (59.2)

1 3 U:
3V0 ol 0 3E0 at or o
(59.3)

a 138

_ O '—
3° k(3P )V..N..E.

(3U
3Po

4) (59.3)

= E<gv °)VO 'fi°( V0,.fio ’EO.
For the internal energy in the moving frame, he

proposed the total energy (E) in

_ 1
u s Ell - (93)212 - B, (60)

E
where B is an invariant energy depending at most on N.
Also, sinceIL,= 0, from the transformation law for momen-
tum one sees that % = cP/E. Thus, one can write the in-
ternal energy as

U=E/y-B.

Using the fact that “so is the spatial component
and “20 the timelike component of a four-vector, along
with the definition for the internal energy (U) and Eqn

(59.2) and (59.3), one finds that

_ 1, (61)
_ k

010)
C: U)

[Cd d

20’ _30] )V’N— Y[Cl -X]°

S) — is Lorentz

This is a true four— —vector equation if (— V N

8U

27

invariant. Thus, since the absolute temperature is

usually defined in terms of this partial derivative as

= as
— (fi)v,fi' <62)

HIP

one sees that it is also Lorentz invariant. He also
argued that since the entropy was a Lorentz invariant the
heat (Q) must also be Lorentz invariant.

Thus, in Landsberg's formulation the thermodynamic
quantities are invariant, but the form of the first law of

thermodynamics is changed. It is written as,

TdS = dU + YpdV - udN (63)
instead of in the usual form,

TdS = dU + pdV - udN (64)

The y factor appears in the pdV term because to the moving
observer p = po and dV = y-ldvo. So, for the pdV term to
be Lorentz invariant the y factor is needed to cancel the
y-1 that the moving observer uses in the volume element.

The second law still has the usual form
ds 3 dQ/T. (65)

However, in a later paper25 Landsberg decided to

redefine his heat, so that it is defined as in the Planck-

28

Einstein formulation. This meant that he had to rewrite

the second law, Eqn (65) as,
ds 3_YdQ/T, (66)

since do = Y-lon in the Planck-Einstein formulation and
the temperature and entropy are still to be Lorentz in-
variant. Thus, the y factor in Eqn (66) enters into the

equation for the same reason that it entered Eqn (63).

E. van Kampen

 

In 1968 another formulation was proposed by N. van

Kampen26. In his formulation, the first law is

dQu = dUu - KUdT + dAVu, (67)
where dQu is called the thermal energy-momentum transfer,
is the four-velocity of the system and

U = v U°, where v

H U U

U° the internal energy of the rest system, KU is the Min—
kowski force acting on the system, I is the proper time
of the system, and dAVu is the mechanical energy and
momentum transferred from the system. He then defined a

"heat supply," dQ°, as
dQ° = VudQu. (68)

In other words, the heat supply is the component of the
thermal-energy momentum transfer parallel to the four-

velocity. This is a true scalar.

29

By using Eqn (68) in the second law, one can de-

fine a scalar invariant temperature T such that,
0
ds = dQ /T. (69)

He also suggested that one could define a four-
vector temperature BU = vp/T. Using this it is possible

to rewrite Eqn (69) as
ds = BudQ . (70)

However, if one compares this equation with the
covariant statement of the second law in the Ott formula-
tion, Tuds = dQu, one sees that the only difference be-
tween van Kampen's formulation and Ott's is the definition

of the temperature four-vector. Ott's definition is

T“l = T°v“. (46)
and van Kampen's is

B“ = v“/T°. (71)
Hence,

T“ = T°28“. (72)

F. Historical Resume

 

In View of all the arguments that have been raised
by proponents of different formulations of relativistic
thermodynamics since Ott's work one can only wonder why
the work of Eckart and Leaf was ignored. That their work

was ignored both at the time it was done and then, more

30

than ten years later, during the debates precipitated by
Ott's work is puzzling. The explanation that the neglect
is due to incomplete surveys of the literature may seem
improbable because of the number of different people that
have participated in the debates. But in fact, since each
proponent's concern has been only to support one formula-
tion rather than another, extensive surveys of the litera—
ture were probably not made. Another reason is that
Eckart's and Leaf's work is not readily interpreted in
terms of the familiar thermodynamic variables (Q,T,P,V)
because it involves nonintuitive entropy flow tensors and
the nonintuitive temperature 6.

Of the four formulations of relativistic thermody-
namics (Planck-Einstein, Eckart-Leaf, Ott — vanKampen, and
Landsberg), only Ott's and Eckart's are manifestly covari-
ant. That the Planck-Einstein formulation is not mani-
festly covariant is not surprising, because it was
developed before the meaning of covariance was completely
understood.

The noncovariance of the Landsberg formulation ap—
pears to be the result of his concern about how an observer
would measure the temperature of a system that is moving
at a uniform velocity with respect to himself. He says27
that the only way this problem can be solved is by re-
quiring that the temperature be invariant. This, along

with his acceptance of the Planck definition of heat,

31

guarantees the lack of covariance. His justification28

for the lack of covariance in his formulation is that he
thinks that the condition of manifest covariance is too
restrictive. His definition of covariance is the same as
that used in the Planck-Einstein formulation: The laws of
physics must be of the same form in all inertial frames of
reference.

Figure 2, page 32, gives a short resume of the
transformation laws and form of the first two laws of
thermodynamics for the four proposed formulations of

relativistic thermodynamics.

32

 

 

 

 

 

PROPONENTS do T U LAWS OF
THERMODYNAMICS
PLANCK- -1 -1 dQ = dU + dW
EINSTEIN y dQ° T° y(U°+82PV°)
Tds = dQ
OTT dQ = dU + dW
Yon YTO YUO
TdS = do
LANDSBERG 1 do = dU + deV
y‘ldo° T° U°[1-(EB)2]2
V TdS = ydo
ECKART de° T° U° *See Note

 

 

 

 

 

*Because the form of the first and second laws of
thermodynamics is not the usual form in Eckart's formula—

tion, they are not included in this figure.

Figure 2.--A review of the proposed transformations.

II. THE PROBLEM

My problem is to determine which, if any, of the
formulations of relativistic thermodynamics proposed by
Planck-Einstein, Eckart, Ott, and Landsberg is the correct
formulation.

The usual approach of comparing predictions made
by a theory to the results of experiments cannot be made
here because there is no experimental data on relativistic
thermodynamic systems. It must be remembered that I am
considering systems which are non-relativistic in their
rest frames and any relativistic affects are due only to
the center of mass of the system of interest having a rela—
tivistic velocity with respect to the observer. This
means that the results of experiments on relativistic
plasmas are of no use because the plasma has a zero center
of mass velocity with respect to the observer. Thus, any
relativistic effects in a plasma are due to a sizable
fraction of the number of particles in the system having
relativistic velocities and are not due to the center of
mass of the system having a relativistic velocity with re—
spect to the observer. The lack of experimental informa-
tion means that we must examine the different relativistic

thermodynamic formulations in terms of certain abstract

33

34

properties, to be discussed below, that theories are

usually assumed to satisfy.

A. Consistency

 

The first, and probably most important consider-
ation is whether or not a theory is consistent. Consis-
tency is not easily shown rigorously; however, a necessary
condition is that all predictions for a given situation
are consistent with each other. It is possible to think
of consistency in two ways. First, a theory must be self-
consistent; that is, no predictions made by the theory
may contradict other predictions made by the theory or any
hypothesis upon which the theory is based. Second, be—
cause the domains of physical theories overlap, there must
be no contradictions between predictions made by the theory
of interest and other, established theories when applica-
tion is made to physical situations in the region where
the theories overlap. Since this second concern is with
the consistency of physics, rather than with self-consis-
tency of a single theory, I call this latter consistency a
physical-consistency.

In accordance with the preceding considerations,
an attempt was made to test the physical-consistency of
the formulations of relativistic thermodynamics proposed
by Planck-Einstein, Eckart, Ott, and Landsberg. This at—

tempt involved relating mechanical and thermodynamical

35

quantities. This procedure allows the comparison of the
Lorentz transformation laws of thermodynamic quantities,
in the aforementioned formulations, with those of mechan—
ical quantities, for which the Lorentz transformation

laws are well established and confirmed. However, me—
chanics deals with individual particles and thermodynam—
ics deals with the bulk properties of a system composed of
a large number of individual particles. This gap is
bridged by statistical mechanics. But, because of the
apparent failure, as seen in papers by B¢rle, Pathria6,
and LandsbergZ4, of statistical mechanics to discern the
differences between the different formulations of relativ—
istic thermodynamics, I looked for a model of a physical
process that would allow the direct comparison of mechan-
ical and thermodynamic quantities. In this comparison the
model is used to develop an equation that relates specific
mechanical and thermodynamical quantities.

The role of the model, that is, to give an equation
relating specific mechanical and thermodynamic quantities,
requires that the model satisfy the following conditions.
First, since statistical mechanics is the bridge between
mechanics and thermodynamics, the model must be based on
statistical mechanical considerations. Second, the equa-
tion developed from the model can not include the ratios
of thermodynamic quantities that have the same Lorentz

transformation laws. This is because the transformation

36

coefficients of the thermodynamic quantities, if any,
would cancel, thereby yielding no information about the
physical-consistency of the relativistic thermodynamics
formulation being considered. It was found that the
usual model for the process of evaporation satisfies the
two requirements.

This model is satisfactory because, even though
the process of evaporation is complex, the equilibrium
point is determined only by quantities associated with the
system itself. Therefore, it can be argued that the equil-
ibrium point must be independent of the frame of reference
from which it is observed. In any event, since the posi-
tion of equilibrium determines the entropy, the invariance
of the equilibrium point follows from the invariance of
the entropy, which is asserted in all of the relativistic
formulations of thermodynamics.

Mechanical quantities are brought into the calcu-
lation by interpreting the canonical distribution function
as giving the probability of finding a particle with a
given kinetic energy (5), rather than making the usual in-
terpretation of it giving the number of particles in the
system with a given kinetic energy. This is done by
dividing the distribution function by N, the number of
particles in the system. Because all observers agree on
the number of particles in the system, N, this renormaliza—

tion of the distribution function does not affect any

37

predictions made in terms of the function. The change in
entropy of the system resulting from the evaporation of a
particle from the liquid to the vapor state of the system is
given in terms of the change in kinetic energy of the par-
ticle and the temperature of the system. Hence, since the
Lorentz transformation law for energy is agreed upon, the
physical-consistency of the Lorentz transformation law for
the temperature in the formulations of relativistic thermo-
dynamics proposed by Planck-Einstein, Eckart, Ott, and
Landsberg can be examined.

In the calculations to follow, the evaporation

29 is followed. Because of

the invariance of the distribution function30, it will be

equation as given by Rodebush

necessary only to carry out the necessary calculations in
the system rest frame. The distribution function in that

frame is
P(el) = K exp{—el/kT}, (73)

where P(el) is the probability of finding a particle with
the kinetic energy 51 arising from velocity components
normal to the liquid surface. As the particle leaves the
liquid surface it loses the energy As. The remaining

kinetic energy of the particle is 62, where

81 - A6. (74)

Because of equilibrium between the liquid and vapor phases,

38

the same distribution function holds for both of the
phases. Hence, the probability of finding the particle

with a kinetic energy 82 in the vapor phase is
P(ez) = K exp{-62/kT}. (75)

This means that the change in entropy (AS) associated with

the evaporation of this particle is

AS = k 1n P(€2) - k 1n P(e (76)

1).

Upon substituting Eqn (73), (74) and (75) into Eqn (76),
we get that

AS = AE/T. (77)

Due to the invariance of the distribution function an ob-
server moving uniformly with respect to the liquid-vapor
system would conclude that in his frame of reference

(primed) the change in entropy is
AS' = A€'/T'. (78)

Since the equilibrium point is Lorentz invariant,
the entropy: and hence entropy changes, are invariant, and

we can equate Eqn (77) and (78). This yields,

Ae/T = Ae'/T',
or

A€'/A€ = T'/T. (79)

39

Because of the way we use the distribution function
for individual particles, As and As' denote mechanical

energies. This means that

 

Ae'/A€ = 1/ /rl - vz/c2 = y, (80)

where y.is the velocity of the moving observer with re-
spect to the system rest frame observer. Thus, examination

of Eqn (79) and (80) shows that

T'/T Y (81)

must be true for each of the four formulations of relativ-
istic thermodynamics being considered. Figure 3 shows the
Lorentz transformation laws of the entropy and the temper-
ature for the formulations of Planck-Einstein, Eckart,
Ott, and Landsberg. We see that even though all four for—
mulations have the Lorentz invariance of entropy which was
used to equate Eqn (77) and (78), only one of them, the
Ott formulation, has the value of Y for the ratio of the
temperatures as seen by the moving and rest frame system
observers respectively.

Consequently, because there can be no contradic-
tion between different theories, I must conclude that the
contradictions between the Planck-Einstein, Eckart, and
Landsberg temperature transformation laws and that for
mechanical energy, when the model for the process of

evaporation is used to relate mechanical and thermodynamic

40

 

 

 

 

 

 

 

 

 

 

Formulation S'/S T'/T
Planck-Einstein 1 Y—1
Eckart 1 1
Ott 1 Y
Landsberg l 1

Figure 3.—-Ratio of entropy and temperature transforma—

tions for the formulations of relativistic
thermodynamics being considered.

41

quantities, eliminate them from consideration as being the
correct formulation of relativistic thermodynamics. This
leaves only Ott's formulation as a contender for being the

correct formulation of relativistic thermodynamics.

B. Covariance

 

Even though the Planck-Einstein and Landsberg
formulations are not manifestly covariant, the fact that
the Eckart formulation is not physically consistent with
the invariance requirement of a specific model including
the transformation of mechanical energy shows that de-
velOping a manifestly covariant theory does not protect it
from contradictions in application. Hence, the signifi-
cance of manifest covariance must not be related to
physical-consistency as a physical theory.

If a theory is cast in a manifestly covariant form
we know that the theory is automatically in proper rela-
tivistic form, as discussed earlier. However, it is pos-
sible to formulate a theory in a non-manifestly covariant
form and still have it be relativistically correct. An
example of this is Maxwell's Equations as usually presented
at the undergraduate level. But, because of the intimate
relation between manifest covariance and the space—time
geometry required by the relativity postulates all theo—
ries, if they are relativistically correct, must be able

to cast in a manifestly covariant form. This means that

42

the significance of manifest covariance, in the evaluation
of a theory, is that with respect to its being relativis-
tically correct, but is not a guarantee of its yielding
physically consistent results.

Since it was discussed in the first chapter, I will
not discuss the covariance of Ott's formulation. However,
it will be remembered that it was stated that his formula-
tion is manifestly covariant. Thus, we know that it is

relativistically correct.

C. Compatability with Related Theories

 

Even though it has been shown that three of the
four formulations of relativistic thermodynamics are in-
consistent, the fourth formulation, Ott's, must still be
examined to see if it is compatible with the related
fields of relativistic fluid dynamics and statistical
mechanics. This compatibility is very important because
the non-relativistic formulations of thermodynamics, fluid
dynamics, and statistical mechanics are closely related.
In fact, because of the different hypotheses upon which
fluid dynamics and statistical mechanics are developed, it
is possible to think of thermodynamics as being the bridge
that relates the two. It would be expected that the cor-
rect formulation of relativistic thermodynamics also played
this role with respect to relativistic fluid dynamics and

statistical mechanics.

43

1. Statistical Mechanics

 

In order to show the relationship between Ott's
formulation and relativistic statistic mechanics I must
give a rough outline of the way relativistical statistical
mechanics is developed.

An early formal attempt to develop a relativis-
tically correct form of statistical mechanics was carried
out in 1951 by Peter Bergmann3l. It is fascinating to
note that he states that his reason for developing a rel-
ativistical statistical mechanics was to remove relativis-
tic thermodynamics from the phenomenological level and
place it on the underlying statistical concepts. His ap-
proach is to include the time as one of the canonical var-
iables, with the negative of the Hamiltonian being its
canonical conjugate. Since the time is involved a system
is now represented by its trajectory in the expanded phase
space and not by a point, as in non-relativistic statis—
tical mechanics. However, because the negative of the
Hamiltonian is one of the coordinates of what he calls
"expanded phase space" the system is required to travel
on a hypersurface of the space. This means that an en-
semble has to be described in terms of the density of a
field of system trajectories on a hypersurface. In spite
of the complexities of the problem, Bergmann managed to

develop a distribution function of the form

44

_ '1 _
with

. dg' .... ngf, (83)

Z = f(6,H)J exp{-ZiBiAl

where (8,H) is the Poisson bracket of the expanded phase
space, 6 is an arbitrary parameter 6(qi,pi,t,-H), J is the
Jacobian of the equations leading from the canonical coor-
dinates to the generalized parameters (51,8, and H), Ai
are the additive constants of the motion for the specified
system and Bi are a set of parameters determined by the
specified values of the ensemble averages of the additive
constants of the motion.

If we specify that there be no angular momentum
in the system, but only linear momentum, the only additive
constant of the motion for the system is the magnitude of

its four-momentum (PU). This means that the distribution

function has the form

U = z‘1 exp{-BUPU}. (84)

We can give physical meaning to BU by comparing Eqn (84)
to the non-relativistical distribution function for a sys—

tem with a non-zero linear momentum (P). 'This function is

u = 2'1 exp{E1,—r-(E - 12)}. (85)

Upon comparing Eqn (84) and (85) we see that we can iden-

tify the four-vector B11 as

45
u _ _l_ )1

This is, except for Boltzman's constant, van Kampen's four-
vector temperature, Eqn (71), which was earlier shown to
be simply related to Ott's four-vector temperature.

It is interesting to note that Bergmann completely
ignored temperature in his paper, but did show that the
entropy was an invariant quantity and that heat carried
linear momentum. This is contrary to Planck's definition
of heat; however, he made no mention of this fact.

In 1963, W. Israel32 published a paper in which he
develops a relativistic kinetic theory of an ideal gas.

He uses a four-vector approach; however rather than re-
lating the four-vector directly to the temperature, he de-

fines a scalar temperature
u l
T = l/k(B BU)2' (87)

His reasoning behind this is not clear to me.

Starting in 1967 R. Hakim33

began publishing what
has turned out to be virtually a book in which he has de-
veloped a relativistic statistical mechanics that is much
more complex than Bergmann's. In this development he is

concerned with plasmas. He starts with the Lorentz-Dirac
equations of motion for a charged particle in a system of

N identical particles. He then defines a phase space of

lZN-dimensions in which he defines his "Gibbs" ensembles.

46

His development from then on is strictly a Green's func-
tion approach with its associated perturbation expansions
and "bubble" diagrams. However, it is interesting to note
that even though he suggests that, because certain quan—
tities are arbitrarily defined, more than one temperature
be defined, one of these is a four-vector as in Bergmann's
work. This temperature is what he calls the kinetic tem-
perature, and is based on Lagrange multipliers of the four—
momentum when the system is in equilibrium. However, the
other temperature, which he calls the equilibrium tempera—
ture, is defined through the Juttner-Synge distribution
function and is neither a scalar or a vector, and is the
same as Planck's temperature. This defining of two tem—
peratures is probably due to the fact that he has problems
with defining what is meant by equilibrium, since the con—
cept of temperature only has a unique meaning when the
system is in equilibrium in non-relativistic statistical
mechanics. The equilibrium problem arises from the inclu-
sion of the electromagnetic fields in the equation of mo-
tion. Hakim also states, without proof or explanation,
that information theory requires that the entropy be a
true scalar. Then he states that the Lorentz transforma-
tion laws are uniquely determined for all thermodynamic
variables when the law for the temperature transformation

is combined with the scalar property of the entropy.

47

Even though there are obvious problems in relating
Ott's formulation, with its four-vector temperature and
heat, to Hakim's formulation of relativistic statistical
mechanics, Ott's formulation is compatible with Bergmann's
and Israel's work in relativistic statistical mechanics

where there are no problems with equilibrium.

2. Fluid Dynamics

 

The relation between Ott's formulation of relativ-
istic thermodynamics and relativistic fluid dynamics at
first seems to be very poor. This is because the usual
formulations35 of relativistic fluid dynamics use a scalar
formulation of thermodynamics similar to Eckart's work and
not a four-vector formulation as in Ott's. However, in

1966 L. Schmid36

started publishing a series of papers in
which he uses variational techniques to develop a formula—
tion of relativistic fluid dynamics with which Ott's for—
mulation of relativistic thermodynamics is compatible.

His approach is to start with the four—vector

Euler equation for adiabatic flow of an ideal gas (this

is a gas with no viscosity or thermal conduction),

IQ:

(m*v“) = a“ P, (88)

Q.)
UIH

T

where m* = m(l + h/cz), with m being the particle rest
mass and h the enthalpy per unit mass, n the number of

particles per unit volume in the rest frame, and P the

48

pressure. He then uses the following thermodynamic

identity:

l a p = m8 h - mTa s (89)
n U U H

where T is the scalar temperature and s the entropy per
unit mass, to eliminate the particle density n from Eqn

(88). This procedure yields

IQ.-

(m*vu) = mauh - mTBus

D.)

T
(90)

3“(m*c2) — mTaus.

The temperature T is a scalar here because it refers to
the local rest frame of the fluid. Schmid then eliminates

the temperature by defining a scalar function, 6,
—— = vuaue 2 T (91)

The function 6 is commonly called the temperature integral.

Applying 6, Eqn (91), and the adiabatic constraint,
SE = o, (92)
to Eqn (90) yields

vu{[3“(m*vy) - 3Y(m*v“)] - m[(8“s)(aY6) +

- (Bys)(8u6)]} = o. (93)

Because the terms in the braces on the left side of Eqn

(93) are orthogonal to the four-velocity, it is possible

49

to relate these terms to a tensor qu in the following

way:
[3“(m*vY) —aY(m*v“)1 — m(a“s)(aYe)-(3Ys)(a“e)1 +
+ 2m*w“Y = 0, (94)

where wUY is antisymmetric and completely unspecified ex-
cept that,

v why = - wuyv = 0. (93
M Y

Schmid calls MUY the intrinsic vorticity because when h,

 

s and 8 are constants,
(BUY = __:_ (BUVY _ BYVU), (96)

which is the relativistic vorticity. Thus he regards wUY
as the contribution to the total fluid vorticity that is
not produced by thermal effects.

The second term in Eqn (94) can be rewritten as

(aud (8Y8)-(8Y9 (8“8)=a“(saYe) - 8Y(sa“8),
allowing us to rewrite Eqn (94) in the form,

2mm”Y = —[8“(m*vY—msaYe)—8Y(m*v“-msa“e)1. (97)

This equation says that the tensor 2m"‘(1)“Y is expressible
as the curl of a four—vector [m*vY-msaY6] plus the four—
gradient of a scalar function C. Calling this four—vector

b11 we have,

50

2mm“Y = a“bY - 8Yb“, (98)

where
-bu = m*vU - msaue + auc. (99)

Schmid then develops an argument for allowing the four-
vector bU to be written in terms of two unspecified con—
stants of the fluid motion which he represents by M and

Q. The resulting equation between M, T and bU is,
b“ = Ma“0, (100)
where because M and T are constants of the motion,

= ¢ = 0. (101)

o
3
Q

Q.)
v-1
Q.)
r-]

Upon substituting Eqn (100) into Eqn (99) one gets,
m*v“ = —a“c + msa“0 — MBUT, (102)

which is the formal integration of Euler's equation of
motion, Eqn (90).

Schmid then develops a variational formulation for
the flow of a single ideal gas. This results in the fol-

lowing statements,

adev4 = 0, (103)

where dV is a four-volume element, and,

4

51

_ 2 u l
L — -nm(l + U/c )c(v v“)2 +
u l u
+P[1 - (v vu)2/c] - nv 80C +

+nmsvu3u0 - nMv“aU¢, (104)

where U is the internal energy per unit mass. In Eqn
(104), P, C, 0, M, 9 and s are undetermined multipliers.
The Euler-Lagrange equation that results from the varia-

tion of, as an example, n is

3)) [fig—”571 - g% = 0. (105)

Upon varying 9 and 5 one gets,
86: §§-= 0. (106)
s: g§-= %E{m(1 + U/c2)]nc2 = (gg)n 2 T. (107)

Thus, the Lagrange multiplier G is just the thermal in—
tegral defined in the development of the formal integra-
tion of the Euler equation of motion. However, Eqn (106)
says that T is the Lagrange multiplier associated with the
adiabatic condition (g? = 0).

Schmid then examined reversible heat transfer be-
tween two ideal gases. The Lagrangian densities for the
two fluids were of the form of Eqn (104). He then shows

that the Lagrangian density for the total system (L )

total

is of the form,

._ I _ :
Ltotal — L + L + O(¢ 0 ), (108)

 

52

where primed quantities represent one fluid and unprimed
quantities represent the other fluid. Due to the form of
the Lagrangian density, Eqn (108), only the Euler-Lagrange
equations for 0, 0' and 0 will differ from the equations
for the isolated fluids. The results of these three var-

iations are,

30: 3u(nmsvu) = o, (109)
89': 3U(n'm's'vp ) = -o, (110)
80: 8 = 6'. (111)

The addition of Eqn (108) and (109) yields,
3U[nmsvu + n'm's'v“ ] = 0, (112)

which reiterates the requirement that the heat flow be
reversible because it, Eqn (112), states that the total
entrOpy flux is conserved. Thus, the meaning of O is that
if it is positive, it is the time rate of entropy increase
per unit volume in the unprimed gas and the time rate of
decrease of entropy per unit volume in the primed gas.
Thus, it is seen that the role temperature usually plays
in reversible heat transfer in non-relativistic thermody—
namics, fluid dynamics, or statistical mechanics is in

Schmid's formulation played by the thermal integral, 0.

 

This means that for reversible heat transfer to occur be-

tween two systems in thermal contact, it is not the

53

temperatures of the two systems that must be equal but

their thermal integrals (0 and 0') must be equal. Thus,

 

'
since vu # vu , the scalar temperatures,

_d6_u
T 37'— v BUB, (113)
and
d0' 0' .
T dT' v u (114)

are not necessarily equal.
From here Schmid goes on to show that the Euler

equations for each of the two gases are of the form,

d(m*vu) = 3U

 

caue
n d? P + ’ (115)
and,
*I u.
n' d‘md,Y > = sup. — 0399-, (116)

where P and P' are the partial pressures of each of the
gases. Upon comparison of Eqn (115) and (116) with those
for the nonrelativistic case, it is seen that the four-
vector BUG plays the role of a four-vector temperature T“.

In fact, Schmid defines a four-vector temperature,
Tu = ca“8, (117)

which is just Ott's four-vector temperature.
Thus, in spite of the fact that the usual formula-
tions of relativistic fluid dynamics use a scalar thermo—

dynamics, Schmid has developed an equivalent formulation

54

which uses the four-vector relativistic thermodynamics

formulated by Ott.

 

III. CONCLUSION

Since there is no experimental data on relativis—
tic thermodynamic systems, the usual approach in the
evaluation of a proposed theory, of comparing theoretical
prediction with experimental data, can not now be under—
taken. Hence, in order to evaluate the formulations of
relativistic thermodynamics proposed by Planck—Einstein,
Eckart, Ott, and Landsberg other criteria were selected.
These were that the correct formulation of relativistic
thermodynamics must be physically-consistent with rela-
tivistic mechanics and that it had to be compatible with
the relativistic formulations of statistical mechanics and
fluid dynamics. This latter compatibility requirement was
used because of the compatibility between thermodynamics,
fluid dynamics, and statistical mechanics in the non-rela—
tivistic region. With respect to the consistency require-
ment, it was found that the Planck-Einstein, Eckart, and
Landsberg formulations were not physically-consistent with
respect to the Lorentz transformation law for mechanical
energy when application is made to the model of evapora—
tion, which is assumed to be physically correct.

Hence, only the Ott formulation remained as a pos—

sible candidate for being the correct formulation of

55

56

relativistic thermodynamics. It was shown that there are
formulations of relativistic statistical mechanics and
fluid dynamics with which Ott's formulation is compatible.
I must therefore conclude that Ott's formulation of rela—
tivistic thermodynamics is correct with respect to the
criteria used to evaluate the proposed formulations of
relativistic_thermodynamics.

At the annual joint meetings of the American
Physical Society and the American Association of Physics
Teachers in February, 1969 there was a symposium on rela-
tivistic fluid dynamics. In an invited paper on relativ—
istic shock waves, P.A. Koch predicted that experimental-
ists would have relativistic shock waves in the next five
years. When these are observed it will be possible to
evaluate the proposed formulations of fluid dynamics in
the way all theories must in the final analysis be tested,
by comparing theoretical predictions with experimental
data. At that time it will be possible to compare predic-

tions made by Ott's formulation with experimental data.

REFERENCES

10.
ll.

.12.

13.

14.

REFERENCES

M. Planck, Sitzber. Preuss. Akad. Wiss. Berlin,
(1907), 542-570.

 

 

M. Planck, Ann. d.Phys.,2§J (1908),1-34.

M. Planck, Vehr. d. Deutsch, Phys. Gesellu 8,
(1906), 136-141. Reprinted in Max Planck, —
Physikalische Abhandhungen und Vortrage, V. 2.,

pp. 115—120. Friedr. Vieweg and Sohn, Braunschweig,
1958.

 

 

M. Planck, Ann. d. Phys. 26, (1908), 1-34.

 

A. Einstein, Jahrb. d. Radioakt. u. Elektr. 4,
(1907), 411-462.

 

R. Pathria, Proc. Phys. Soc., 88, (1966), 791-799.

 

H. Minkowski, "Raum und Zeit," reprinted in The
Principle of Relativity, Dover Publications, New
York, pp. 75-91, followed by notes by A. Sommer-
feld, pp. 92-96.

 

A. Sommerfeld, Ann. d. Phys., 32, (1910), 749—776.
Ann. d. Phys., 33, (1911), 649-689.

 

 

A. Eddington, The Mathematical Theory of Relativity.
Cambridge, 1923, p. 34 of Second Edition (1924).

 

C. Eckart, Phys. Rev., 58,( 1940), 269.

 

C. Eckart, Phys. Rev., 58,( 1940), 272.

 

c. Eckart, Phys. Rev., §_8_, (1940), 919.

 

B. Leaf, Phys. Rev., 84, (1951), 345.

 

M. v. Laue, Die Relativitatstheorie, V. 1, p. 171.
Friedr. Vieweg and Sohn, Braunschweig, 1952. The
previous edition, published in 1921, does not con-
tain this footnote. This is the reason for my as-
suming that von Laue is speaking of Eckart's work
in this footnote.

 

57

58

15. H. Ott, Zeits. f. Phys., 175,( 1963), 70.

 

16. C. Mdller, The Theory of Relativity, Oxford Univers-
ity Press, London, 1952, pp. 106-106.

 

17. H. Arzeliés, I1 Nuovo Cimento, 81, N. 3, (1965), 792.

 

18. R. Tolman, Relativity, Thermodynamics and Cosmology,
p. 156, Eq. (69.11), Oxford University Press, London,
1934.

 

19. A. Gamba, Il Nuovo Cimento, 81, N.4, (1965), 1792.

 

20. T. W. B. Kibble, Il Nuovo Cimento, 41B, N.l, (1966),
72.

 

21. A. B¢rs, Proc. Phys. Soc., 88, (1965), 1141.

 

22. R. Penney, Il Nuovo Cimento, 43A, N.4, (1966), 911.

 

 

23. F. Rohrlich, Il Nuovo Cimento, 88,( 1966), 76.

 

24. i) P. T. Landsberg, Proc. Phys. Soc., 88, (1966);

1007;

ii) P. T. Landsberg, Nature, 214, N. 5091, (27 May
1967), 903;

iii) P. T..Landsberg and K. A. Johns, Il Nuovo
Cimento, 888, (1967), 28;

iv) P. T. Landsberg and K. A. Johns, Proc. Roy. Soc.
A, 888, (1968), 477.

 

 

 

25. P. T. Landsberg and K. A. Johns, Il Nuovo Cimento,
52B, (1967), 28.

 

26. N. G. van Kampen, Phys. Rev., 173, (1968), 295.

 

 

I
27. P. T. Landsberg, Proc. Phys, Soc., 88, (1966), 1007. 1
1

28. This argument was given by Landsberg at the con-
ference "A Critical Review of Classical and Relativ—
istic Thermodynamics" in April, 1969.

29. W. H. Rodebush, A Treatise on Physical Chemistry, (
Vol. II, H. Taylor (Editor). D. van Nostrand, New
York, 1925. Pp. 1178-1179.

 

30. P. A. M. Dirac, Proc. Roy. Soc. 106A,(1924), 581.

 

31. P. Bergmann, Phys. Rev., 88, (1951), 1026.

 

32. W. Israel, J. Math. Phys., 8, (1963), 1163.

 

33.

R. Hakim, (A),

59

. Math. Phys.
Math. Phys.
Math. Phys.
Math. Phys.

, (1967), 1315;
, (1967), 1379;
, (1968), 116;

, (1968), 1805.

 

J
(B). J
(C). J
(D), J

 

 

(whdaflm

 

R. Hakim, J. Math, Phys. _9_, (1968), 126.

(A))

(B),

(C),

(D):

 

Landau and Lifshitz, Fluid Mechanics, Chap. 15,
Addison-Wesley, Reading, Mass. (1959).

J. Serrin, Handbuch der Physik, Vol. 8/1, C.
Truesdell, Editor, Springer, Berlin (1959),
125—263.

A. Lichnerowicz, Relativistic Hydrodynamics and
Magnetohydrodynamics, Benjamin, New York,
(1967);

P. Bergmann, Handbuch der Physik, Vol. 4, C.
Truesdell, Editor, Springer, Berlin, 1962, 159—
166.

 

 

 

 

 

L. Schmid, (A), Phys. of Fluids, 9, (1966), 102;

 

(B), I1 Nuovo Cimento,—47B, (1967), l;

(C), I1 Nuovo Cimento, 52B, (1967), 288;

(D), 11 Nuovo Cimento, 52—, (1967), 313;

(E), Goddard Space Flight Center, Green-
belt, Maryland, publication no. x-
641-69-118 (1969).