GENERALIZATION OF TAUB’S RELATIVISTIC
RANKTNE- H’UGOMOT EQUATIONS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
Ahmed Shawky EI-Ariny »
1967

 

T'Hibl‘:

  
   
   

  

L I B R A R Y
Michigan 7:309
Universny

This is to certify that the Y

thesis entitled

GENERALIZATION OF TAUB'S RELATIVISTIC
RANKINE—HUGONIOT EQUATIONS

presented bg

Ahmed Shawky El-Ariny

has been accepted towards fulfillment
of the requirements for

___E.b_._Il.___degree in_Mechanical
Engineering

 

fZC/t KW¢L‘7/%%9q \

Major lfirefessor

 

[hm February 23; 1967

0—169

 

 

 

 

 

 

ABSTRACT

GENERALIZATION OF TAUB'S RELATIVISTIC
RANKINE-HUGONIOT EQUATIONS

by Ahmed S. El—Ariny

The present work refers to the relativistic hydro—
dynamics in the presence of the gravitational field. The
velocity of the propagation of signals is assumed to be a
variable in accordance with the proposition by Einstein
(1907), FOk (1955) and others. The present approach is a
generalization of Taub's work in vacuo. The fluid is con-
sidered to be a collection of particles in random motion
and under the influence of a gravitational field. Only
ideal fluid is taken into account. In this case transport
physical aspects like viscosity, heat conductivity, etc.,
are to be disregarded. The flow governing equations
(continuity, momentum and energy) are based on the Kinetic
theory of gases by means of Boltzmann equation. The space-
time based on a variable velocity of propagation of signals
is necessarily Riemannian one. Due to the difficulties
that arise in establishing solutions of flow problems in
the Riemannian space—time, an approximation is suggested
by introducing a piece-wise constant velocity of signals.
This enables us to obtain, in the Euclidean space-time,

-solutions which are valid only at a point. The entire

Ahmed S. El-Ariny

formalism of Taub is transferred to the Euclidean space-
time where the velocity of the propagation of signals is
less than that in vacuo. It is shown that actually the
Taub procedure is transferable to the present case with
small modification involving constant parameters. To
demonstrate the discrepancy between the present approach
and the possible one in the Riemannian space—time, in
Chapter III are some equations which show clearly the
simplification which must be applied to reduce the problem
to the Euclidean space-time. To illustrate the theory a

numerical example is calculated.

GENERALIZATION OF TAUB'S RELATIVISTIC

RANKINE—HUGONIOT EQUATIONS

By

Ahmed Shawky El—Ariny

A THESIS

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

DOCTOF OF rHiLOSOEHY

Department of Mechanical Engineering

1967

set/”773’

(y//&/’ o/

 

To my wife
Malak,

with deep appreciation

ii

 

 

ACKNOWLEDGMENT

The author wishes to express his sincere appreci-
ation to Professor M. Z. v. Krzywoblocki for his guidance,
encouragement and constructive criticism throughout the
course of this work. Appreciation also goes to Pro-
fessors A. M. Dhanak, D. W. Hall and J. A. Strelzoff for
serving on his guidance committee. Special recognition
goes to Mr. F. Lecureux for his help and cooperation in
preparing the computer programming part of this work.
Grateful acknowledgment is made to the U. A. R. government
for their financial assistance. He also wishes to thank
the Mechanical Engineering Department of Michigan.State
University for their financial support during a portion

of this research.

iii

 

 

TABLE OF CONTENTS

DEDICATION
ACKNOWLEDGMENT.
LIST OF TABLES.
LIST OF FIGURES
NOMENCLATURE

INTRODUCTION

Chapter

I. FUNDAMENTALS OF THE APPLIED RELATIVISTIC
MODEL . . . . . . .

1.1. Second Einstein Model of the
Special Relativity
1.2. Fok' s Model . . . .
1.3. The Metric Tensor of the Four-
Dimensional Space-Time.
1.4. Fundamental Mathematical Oper-
ations .
1.5. Relativistic Mechanics of a
Particle in the Four-
Dimensional Space-Time.
(i) Velocity
(ii) Lagrangian and Momentum
(iii) Force
(iv) Energy.
6. Local Orthogonal Coordinates.
1.7. Second Form of the Four-
Dimensional~Space-Time.
1.8. Transformation of Coordinates

iv

Page

ii
iii
vi
vii

ix

.1274:

O\U1

10.
10-
11
14
15

18
21

 

 

 

Chapter

II. RELATIVISTIC FLUID DYNAMICS

2.
2

2.

[\JMMNNNN

|-—io o o o

l.
.2.

OKO GDN O\\J'T Jl‘w

Fundamental Aspects.
The Hydrodynamical Equations.
(i) Boltzmann Equation.
(ii) The Summational Invariants
(iii) Law of Conservation of Mass.
(iv) Laws of Conservation of
Energy and Momentum
Specific Internal Energy
The Fundamental Inequality
Case of an Ideal Gas
One-Dimensional Motion.
Progressive Waves
Rankine-Hugoniot Equations
The Shock Velocity
Concluding Remarks

III. DERIVATION OF THE HYDRODYNAMICAL EQUATIONS
IN THE RIEMANNIAN SPACE-TIME. .

WWWW WWWWLU

\OCDNQ UIJEUUI'UH

Introduction

Boltzmann Equation

The Summational Invariants

Law of Conservation of Mass

Laws of Conservation of Energy and.
Momentum . .

Specific Internal Energy

Case of an Ideal Gas

One—Dimensional Motion.

Concluding Remarks

IV. APPLICATION

A.
A.
A.

REFERENCES

l.
2.
3.

Gravitational Potential
Gas Model .
Shock Model

.u

1

4.... .. .1 .n . . i.‘ .ru.‘.lbd\tl| Iii-1|... .
Jliiufiég 1.... it . o .
n q. . ~ .
. , . .

.

 

LIST OF TABLES

Table Page
1. Comparison Between Flow Quantities in the
Work [18] and Those of the Present Work . A9,
2. The Calculations of the Quantities c_2x(n>
and c-110<n) at the Points-R(n). . . . 75
3. Solution of Eqs. (2.6.17) and (2.6.18) for
e at the Particular Instant t = to= O.A

sec. for Different Values of the ’
Parameter I n. 0-1 and the Correspond-
ing Approximate Solution for e . . . . 81

A. Solution of Eqs. (2.6.17) and (2.6.18) for
u at the Particular Instant t = t0 =-O.A-
for Different Values of the Parameter
I<n) 0-1 and the Corresponding Approxi—
mate Solution for u . . . . . . . 82

vi

 

LIST OF FIGURES

Figure

1. The Dimensionless Gravitational Potential
c-Zx and the Dimensionless Velocity
C'1I(c-1IO) as Functions of the Y—
coordinate

2. The Dimensionless Sound Velocity "a" as a
Function of the Normalized Density 90*.

3. The Quantity ¢ as Function of the Normalized
Density po . . . . . . . . . . .

A. The Dimensionless Sound Velocity "a" as a
Function of the Quantity ¢.

5. The Dimensionless Velocity u(Y,to) vs. Y'at
Time t = t0 = O.A for Different Values of
the Parameter c-lIO and the Corresponding
Approximate Solution. in the Case of the
Variable Signal Velocity IO . . . .

6. The Quantity ¢(Y to)vs.Yat Time t = to = 0.14
for Different Values of the Parameter
c-1IO and the Corresponding Approximate
Solution in the Case of the Variable
Signal Velocity IO .

7. Approximate Solution of Eqs. (2.6.17) and
(2.6.18) for ¢ = ¢(Y,O.A) Using the
Diagonal Numerical Values in Table 3

8. Approximate Solution of Eqs. (2.6.17) and
(2.6.18) for u = u(Y,O.A) Using the
Diagonal Numerical Values in Table A

9. The Pressure Ratio g = p+p_-i vs. the
Dimensionless Velocity c“1 l for Differ-
ent Values of the Parameter c-lI When

, o
C‘2p_p?_—l = O.lO-and y = 1.6lA

vii

Page

76

83

8A

85

86

87

88

89

92

 

Figure

10. The Density Ratio n = p$p9:1 vs. the Di-
mensionless Velocity c-lul for Differ-
ent Values of the Parameter c-llO When
c‘2p_p<3-1 = 0.10 and y = 1.6lA

11. The Dimensionless Velocity c‘lfil on the
Right Side of the Shock vs. the Dimen—
sionless Velocity c'lul on the Left
Side of the Shock for Different Values
of the Parameter C‘llo When c-2p_p2‘1
= 0.10 and y = 1.61A

12. The Dimensionless Velocity c-~Ei vs. c'lui
for Different Values of the Parameter
C- I 0 O 0 O O 0 O O O 0
0

viii

Page

93

9A

95

 

 

 

 

 

 

 

 

NOMENCLATURE

velocity of sound

covariant form of the metric tensor of the
Riemannian space—time—{x}

covariant form of the metric tensor of the
Riemannian space—time-{x}

covariant form of the metric tensor of the
Riemannian space-time-{X}

constant coefficients evaluated at point
"0", (p. 16)

velocity of light signal in vacuo.
operators (p. 39)

the distribution function in the Riemannian
space—time-{x}

the distribution function in the Riemannian
space-time-{y}

an arbitrary function of e (p. A1)

four-force vector in the space-time-{x}

four-force vector in the space-time—{f}

four-force vector in the space-time-{X}

four-force vector in the space—time-{y}

four—force vector in the space-time-{y}

four—force vector in the space-time—{Y}
components of the forces and the work done by
them obtained by differentiation with re—

spect to the time t

ix

 

228 3|

3|

modified four-force vector

components of the external force per unit
mass (p. 26)

covariant form of the metric tensor of the
Euclidean space-time-{y}

covariant form of the metric tensor of the
Euclidean space-time-{y}

covariant-form of the metric tensor of the
Euclidean space-time-{Y}

velocity of propagation of signals in the
presence of the gravitational field

velocity of propagation of signals I evaluated
at a particular point "0"

constant of integration (p. 78)

indices running over the range 1,2,3
Langrangian

pound force

pound mass

rest mass of a particle in vacuo

mass of a particle in space—time-{y}
mass of a particle in space—time—{y}
molecular weight of the gas

mass of a particle in the space-time-{x}
mass flux across the shock (p. A3)

mass of a particle in the space-time—{f}
an integer = l,2,3,...(p. 7A)

number of particles per unit volume (n=ffd3£)

as measured with respect to fixed coordinates
- {y} (or {x})

 

’U

Q Q

"UA "UQ ’UI *6

.0

a8

T*“B

number density of the particles as measured
by an observer moving with the velocity
uJ with respect to the fixed coordinates
{y} (or {x})

hydrostatic pressure (p. 3A)

four-momentum vector in the space—time-{y}

four-momentum vector in the space-time-{y}

four—momentum vector in the space-time-{x}

four-momentum veCtor in the space-time-{f}

magnitude of the particle spatial velocity
whose components = q

the particle velocity components measured with

respect to the x-coordinates

magnitude of the particle spatial velocity
whose components = qJ

the particle velocity components measured with

respect to the x-coordinates
heat input into the system from the outside
degree Rankine
universal gas constant

local relativistic analogs of the Riemann
functions

dd = element of arc length in the four—dimen—
sional space-time

entropy as measured by an observer at rest
with respect to the fluid

contravariant form of the energy-momentum
tensor

contravariant form of the energy-momentum

tensor including the effect of the external
forces

xi

 

 

dimensionless velocity in one dimensional
flow in the 1Euclidean space- -time referred
to IO (u = u lIO

dimensionless four-velocity vector of a flow,
(ud = nO-lUd)

mean value of the spatial components of the
particle velocity VJ

mass—current vector in the space-time—{x}
mass-current vector in the space-time-{y}

magnitude of the particle spatial velocity
whose components = VJ

the particle velocity components measured with

respect to the.y-coordinates

magnitude of the partigle spatial velocity
whose components = vJ

the particle velocity components measured with

respect to the y—coordinates

the fourevelocity vector in the Riemannian
space-time-{X}

the four-velocity vector in the Riemannian
space-time-{Y}

dimensionless velocity in one dimensional flow

referred to I in the space-time-{x}
(w = wlI‘l)

mean value of the spatial components of the
physical velocity qJ of the gas particles

dimensionless four— —velocity vectorl of a flow
in the space— —time- {X} (Wu = no"1 Ua(x))

coordinates of the Riemannian space-time—{x},
x = t

coordinates of the Riemannian space-time—{f},
x = ct

coordinates Xof the Riemannian space—time-{X},
X3 = c"1

xii

 

coordinates of the Euclidean space-time—{y},
y=t-

coordinates of the Euclidean space-time-{y},
y = ct

coordinates of t e Euclidean space-time—{Y},
YJ =-c‘1yJ, Y .=~y

distance from the celestial body (p. 79)

coordinates of the Euclidean space in vacuo
(corresponding to Taub's work)

dimensionless sound velocity referred to IO
integers running over the range l,2,3,A

a function of the shock parameters (p. AA)
specific heat ratio

the characteristics of the governing equations
of the one-dimensional flow in (y,IOt)-plane

operators (p. 71)
Internal energy of the fluid per unit mass

real_energy~of the particle in x- or y-coordinates
(x— or y—coordinates)

total energy of rhe particle in x— or y-
coordinates (x- or y-coordinates)

four-velocity vector in the space—time-{x}
four-velocity vector in the space-time-{i}

—l
O 03 )

density ratio (n = p+

temperature

[dp = (l + c c )—%fd3c] (pp. 28 and 58)

xiii

 

 

pressure ratio (a = p+ p_-l)

four—velocity vector in the space-time-{y}
four-velocity vector in the space—time-{y}

a second order tensor whose covariant derivative
represents the external forces

density of the gas as measured with respect to
a fixed coordinate system, p = nmO

density of the gas as measured by an observer
moving with the mean velocity with respect
to a fixed coordinate system p0 = nomO

normalized density

d1 = arc length in the four—dimension space-
time
d = e (xJ’ t,cj) a transport quantity in the

{x}-space time

c = e (yd, t, 53) a transport quantity in the-
{y}—space time

d¢ = o‘po-ldpo
gravitational potential

summational invariant functions

xiv

 

 

 

 

INTRODUCTION

Relativistic theory of fluid dynamics with the refer-
ence velocity of propagation of-signals in vacuo has been
developed by Taub [l8] and Synge [15]. The former based
his work on the Kinetic theory by means of the relativistic
Boltzmann transport equation, while the latter based his
formulations on Maxwell's Boltzmann statistics. Additional
aspects of the microscOpic approach with the reference
velocity of propagation of signals in vacuo have been in—
vestigated by various authors: Goto [8], Vlasov [20] and
others.

Attention is called to the meaning of the word "vacuo”
used in the above literature as it was demonstrated by
Pauli [1A] that in perfectly empty space no gravitational
field exists. According to Einstein's article in 1907 [6],
and the discussion presented by Fok [7], the effect of the
gravitational field upon the form of the metric of the
space—time may be taken into account. Thus, the constant
reference velocity of propagation of signals in vacuo is
to be replaced by a variable one in the presence of the
gravitational field.

It must be emphasized very strongly that the symbols

0 and I do not necessarily denote and refer to the velocity

 

of Ilight. Actually, they may refer to the velocity of
anyfl other signal. It may be optical, electromagnetic
wavwes, etc. Already Pauli [1A] mentioned about other
posssible signals. It is well known that the velocity of
eleactromagnetic waves depends very strongly upon environ-
merrtal conditions.

In the present work, we introduce a generalization
to Taub's work under the influence of the gravitational
field. Thus, the present approach refers to a space-time
with a reference velocity of signals which is considered
to be a function of the spatial coordinates. Following the
analysis of Synge and Schild [17], local rectangular co-
ordinates are used in the Euclidean space-time to provide
simplified solutions that are valid only at a point.
Therefore, by introducing a piece—wise constant reference
velocity of signals, an approximate solution in the Rie-
mannian space—time is obtained from solutions involving
constant parameters in the Euclidean space-time.

In Chapter I, we demonstrate the basic principle
upon which the configuration of the space—time in question
is based. The relativistic quantities, being used in the
following chapters are developed in accordance with the
chosen forms of the metric of the space-time. In addition,
several transformations of coordinates are established.

The modification of Taub's work in accordance with the
present theory is presented in Chapter II. The fundamental

governing equations of a flow in the Riemannian space-time

 

 

are derived in Chapter III by making use of the Boltz—
mann equation. It is shown that the Riemannian space-
time, due to its curvature, results in the appearance of
additional terms-in the differential equations governing
a flow. These additional terms present the main differ-
ence between the formalism in the Riemannian space-time
and its correspondence in the Euclidean one. However, it
is shown also that the entire formulations obtained in
Chapter III are reducible to their correspondence in
Chapter II upon replacing the variable reference velocity
of signals by a constant one evaluated at a particular
point, i.e., by a piece-wise constant reference velocity.
A numerical solution of an initial value problem in
one dimensional flow and the calculations of some shock
parameters are presented in Chapter IV as an illustrative
example demonstrating the influence of the gravitational

field.

 

 

CHAPTER I

FUNDAMENTALS OF THE APPLIED

RELATIVISTIC MODEL

1.1. Second Einstein Model of the
Special Relativity

 

 

In 1905 Einstein proposed his first model of the
special relativity and in 1907 his second one in which
he took into account the effect of the gravitational
field on the velocity of the propagation of signals.
Einstein [6]1 derived the formula for the metric of the

space-time using the expression

(dA)2 = -(dx2 + dy2 + dzi) + [c(1 + xc'2)]2 dtz, (1.1.1)

where,
X = gravitational potential; x, y, 2 being the
space coordinates,
t = the time coordinate, and c is the velocity of

signals in vacuo.

1.2. Fok's Model

 

Fok [7] proposes the following two metrics of the

space-time in the presence of the gravitational field

 

1Numbers in square brackets refer to the bibli-
ography of standard works.

   

(dAOZ = - (dx2 + dy2 + dzz) + (c2 — 2x)dt2, (1.2.1)
01”
(d4)2 = - (1 + 2xc-2)(dx2 + dy2 + dzz) + (c2 - 2x)dt2.
(1.2.2)

The gravitational potential, x, appearing in the above
metrics, as considered by Fok, should be small. The first

metric, (1.2.1), has a special interest in the present

 

 

work.
1.3. The Metric Tensor of the
Four-dimensional Space-Time
Following the models of the special relativity dis—
cussed above, let us introduce the function I.= I (E3),

J = 1,2,3, defined as
12 = e2 — 2x. (1.3.1)

The function I represents the maximum velocity of the
propagation of signals in the presence of the external
fields of action.

The metric of the four—dimensional {x}-space—time

is chosen in the form
*2 2 _A.2
- c I (dx )

_ _° k
_ 2 = _
(dA) ajkdx dX

.— _O_
aopdx dxp, 3 (1.3.2)

 

 

where

for O a! p . (1-3-3)

xJ being the spatial coordinates, and

it = ct.

Throughout the present work, we use Latin suffixes for the
range 1, 2, 3, and Greek suffixes for the range 1, 2, 3, A,

unless otherwise stated. The contravariant components of

the metric tensor, 5°“, in £09 dfodxp are

-— — — —- — —o
all= a22 = 8.33 =-l, aAA = _ 021 2 a p 0

for c: 7 p . (1.3.A)

From (1.3.2) we obtain

it. “m—2---J-k—J.g
dJI—[I(l _ q I )1 3 q, _ ajkq q a q dt 0 (1.3.5)

1.A. Fundamental Mathematical Operations

 

Assume a metric in {x}-space-time:
(d))2 = aodedep , (1.4.1).

with a metric tensor, aoo’ whose components are functions
of the coordinates, x0. The absolute derivative, denoted
by the symbol 5%; (A being a parameter), of a first order

contravariant tensor, To, is defined by

 

 

6T0 _ ch u dx

where, gig, denotes the ordinary derivative of T0 with

respect to ,A, whereas, {3v}, stands for the Christoffel
symbol of the second kind which is defined by

3a 3a

3a
0 _ l 00 av on _ uv = on
{LIV} " 2'8. ( 3X“ + axv 3X0 ) a [IMHO], (10,403)

where [uv,p] = the Christoffel symbol of the first kind.

The absolute derivative of the corresponding covariant

tensor, To, is_

6T dT
c

.=__g__ u dx

The absolute derivatives of the second order contravariant

and covariant tensors, Top and Top, are given respectively

by
STUD _ dTOp o udev p uodxv .
a» - 53. 4-{uv} T d7 -+{ } T d7 , (1.u.5)
6T dT V V
__oo-__op_u d_x.-_u 9.;
5A ‘ dA {0v} Tumdl {0v} TuodA ' (1.“.6)

The covariant derivatives of the tensors To, To’ TOp and

Top, with respect to the coordinate XV are defined re-

spectively by, (notice the proper notation):

 

 

 

o _ o o u A: = o = c
T |v - T,v + {“V} T , - T,xv T,v , (1.4.7)
3X
T | = T — {“ } T ; (1.4.8)
o v o,v 0v u
op _ co 0 up - o no ,
T Iv - T ,v + {w} T + {W} T , (1.4.9)
_ _ u _ u .
Toplv — Top,v {0v} Tup {pv} Tno . (1.4.10)

When the met

ric of the space—time {x} assumes the form

(1.3.2), i.e.,

the computat

Christoffel

J ‘1. i

s‘.
t
3

'{flt}

_ _ _ 2
dXJka - c 2I 2(dx“)

,2 ’
-(dA) ajk

Eopdf°d%“,i = I(§J) , (1.4.11)

ion of the non—vanishing components of the

symbols gives

.— L+ -
:a kJ (c 21 Iz)’k’{“3} = —(c I )(c I )

—212

—[1n(c >21 (1.4.12)

,3

Hence, carrying out the computations of the components

of the absolute derivative (1.4.2) using (1.4.12) we obtain

_ 1 —k3 "2 2 .
— —— + 5 a (c I ),k , (1.4.13)

 

 

gg” = ggq + % [1n(c-212)],j Tj gg
+ % [1n(c-212)],j T” g . (1.4.14)
Similarly,
2;? = 2:9 - 5 [1n(c 2I2):l,J T4 gé, , (1.4.15)
27:” = 2;“ _ % §kj(c—212),kTJ g3; (1.4.16)

The covariant derivatives in (1.4.7) to (1.4.9), with the

contraction v = 0, become
TOI = T° + £'[1n(c—2I2)] TJ - (1 4 17)
o ’0 2 ,j ’ ' ‘
kc _ k0 1 -2 2 kj
T |O -.T ,0 + 2 [1n(c I )],jT
+ l §k3(c‘212) .TL+L+ 5 (1.4.18)
2 ? a J
40 = 40 i '2 2 43
T |O T ,0 + 2 [1n(c I )],jT . (1.4.19)

1.5. Relativistic Mechanics of a Particle
in the Four—Dimensional»Space-Time

The metric, (1.3.2), of the four-dimensional space-

time {X} is used primarily in this section.

 

 

 

10

(i) Velocity
The contravariant four-velocity vector is defined

by

= —— . (1.5.1)

Using (1.3.5) and (1.5.1) the contravariant and covariant

components of the four-felocity vector are

53 = EJEI<1 - 521—2)t]-1 ,

—h —- — — 1/ _.
c = q“[1(1 — q2I 2)2] 1 ; (1.5.2)
— _ - -0
CO ' ago; 9 (105.3)
where, a” = g? = c.

The magnitude of the four-velocity vector is given by

50050;? = —1 . (1.5.4)

Since the absolute derivative of the metric tensor vanishes,

the absolute derivative of (1.5.4) with respect to,4 gives

a?
O _
57 - O . (1.55)

(ii) Lagrangian and Momentum

 

Let us introduce the Lagrangian function in the form

x = g—mocza 3°30 . (1.5.6)

 

 

 

 

11

The four-momentum vector is defined by

:2: = m 0250 ED . (1.5.7)

 

F =
C

O 0

Substituting (1.3.3) and (1.5.2) into (1.5.7) we obtain

E.
J

_ _ _ t _ _
cmo[c 1I(l — qZI 2)2] lqj ;

F, = -mO[c‘11(1 - 521-2)%]-112 . (1.5.8)

Let us define the inertial relativistic mass by

21—1

M = mo[c‘11(1 — 521‘ )3] . (1.5.9)

Hence, the components of the four-momentum vector take

forms
F3 = cMaj , F, = - M12 , (1.5.10)
$3 = aJOFO = moczf - chJ , F” - 5”“ F0 = 5”“Ft = Mc2
(1.5.11)

(iii) Force

The contravariant four-force vector is defined by

O

F(§) =

a?“
33 (1.5.12)

Using (1.4.13), with TJ replaced by F3, the first three

contravariant components of the four-force vector are

 

 

. —j .
J_ = dP 1 —Jk 2 2 —tdx
F(x) dA'+ 2 a (C I ),kP cu
= [QEJ-+ i 53k(c'212) FgJQE (1 5 13)
dt 2 ,k cw ° ° °

Substituting (1.3.5) and (1.5.11) for %% and F0 in (1.5.13)

we obtain

Fgg) - [c'11(1 - q I-2)2]-1[ddt(qu) + 5a aJkM(12)

.k]

[c‘11(1 - EZI'2)%1'IF%§) , (1.5.14)
where we define the physical spatial force by

F(x) = 5%(M50) + % aim/1112),k = §%(MEJ) . (1.5.15)

The first three covariant components of the four-force

vector are

— _ — 9 _ -1 —2 —2 g -1— —
FJ.(X) - ajoF(X) — [c I(l - q I )] FJ.(X) ,
(1.5.16)
where
_ _ _ _°_ 5 _
FJ(X) = aJOF(X) = 55( qj) . (1.5.17)

Similarly, using (1.4.16), with F4 replacing Ti, the

covariant fourth component of the four-force vector is

 

 

dFt . ._q .—

— _ i 1 -Jk -2 2 — dx 1 2 -2 -2 2 —,dx

Ft(x) — d_ - 2 a (c I ),kPJdd — 2 c I (c I ),J ”dd
(1.5.18)

Substituting (1.3.3), (1.5.1), (1.5.7) into (1.5.18) we

get
d?“
"‘ _ __ _ A; ‘2 2 2.—.'k.—.'L+
F4(X) - d0 2(0 I ),k(moc 4 c )
dfit
1 -2 2 2Tk7g _
+ §(C I ),k(mOC I; I, ) - 'dT . (1.5.19)
Hence,
qu 824
- _ __ = __ 22 = _ 5L - 2 QB
F“(X) ‘ d» dt d) [dt(MI )JdA
_ L -
= [1(1 — 2I 2)2] 1Tux) , (1.5.20)
where we define
F (f) = —£L(M12) (1 5 21)
L, dt . . .
The contravariant fourth component of the four—force
vector is given by
4_ —40 - ~44 — 2 -2 -
F(X) = a FO(X) = a Fu(X) = -c I F4(X)
_ _ L _ _ _
= [1(1 - qZI 2)21 1021 2§%(M12) , (1.5.22)

or, we may write it in the form

 

 

14

2 -2<i

4 4
F65) = [1(1 — 521‘2);51“1F“(x),fi(35) = c 1 a-E-(MIZ) . (1.5.23)

(iv) Energy
From (1.5.5), (1.5.7) and (1.5.12) we obtain

2° FO(X) = 0 . (1.5.24)

Substituting (1.5.2), (1.5.16) and (1.5.20) into (1.5.24)

and Simplifying we get

_J'__ _. =
q FJ(X)

ad'E‘M-IZ) . (1.5.25)

Eq. (1.5.25) suggests that we may define the real energy

of a particle by

2* = M12 . (1.5.26)
From (1.5.10) and (1.5.26) we have-

2, = -E¥ . (1.5.27)

i.e., the fourth covariant component of the four-momentum
vector is equal to the negative value of the real energy
of the particle.

Similarly, from (1.5.11), (1.5126) and (1.5.27), we.

_*
may define the total relative energy at by

e = F” = —c IIZFn = c I 5* = Me2 . (1.5.28)

which represents the maximum energy the particle may possess.

 

 

15-

1.6. Local Orthogonal Coordinates
“d

{x

Consider a Riemannian space-time (f) ,Y“}, i” 3 ct,

with the metric

_(dA)2 = Eopd§°d§p . (1.6.1)
The terms that contain dxl are, in general,

a11(d;1)2 + 2§12d5€1d§2 + 2§13dfldf2 + 251ndf1di—u , (1.6.2)

where we assume that gap is symmetric. Similar expressions
hold for dfz, d§3 and di“. The right hand side of (1.6.1)
can be reduced to a sum of four squares, each having as a
coefficient, the corresponding diagonal component of the
metric tensor gap. In that respect, we follow [17] and

write (1.6.1) in the form

-(dJ)2 = a11(‘¥1)2 + 522(W2)2

+ 533(43)2 + 544(1”)2 , (1.6.3)
where, for example, W1 takes the form
4’1 = dil + 5125114652 + 51351718?
+ 51,5518? , (1.6.4)

and where we assume all # O.

 

 

 

16
Hence, we may write

0'

T = b0

—'D _
de , 3(00) 5‘ O . (1.6.5)

If we consider a given point, 0, with coordinates a0 we

may write

5° = (b°p)o(§p — 6°) , (1.6.6)

where the subscript, 0, denotes evaluation at-a particular
point 0. Hence, from (1.6.3), (1.6.4), (1.6.5) and (1.6.6)

we obtain at "O":

-(dfi)2 = §11(d§1)2 + E22(d§2)2 + E33(d§3)2 + E44(d§”)2 :
(1.6.7)
where E11, E11, E33 and g,, are evaluated at the point "0".

In the present work, the components of the metric tensor,

300’ of (1.6.1) are.:

511=azz=a33=l,auu=-CI,Eop=OfOPO#D

(1.6.8)

Using (1.6.4), (1.6.5), (1.6.6), (1.6.7) and (1.6.8) we

 

obtain:
T3 = 855 = d? , w“ = d§“ = d?“ = cdt ; (1.6.9)
’ — ‘ _ — _ -2 2 — _
$11 = 822 = $33 - 1 a 844 - -C I , gO — O for 0 # p .
O 0
(1.6.10)
1

Indices between brackets do not follow the summation
rule.

 

 

 

 

17

Hence, (1.6.7) becomes-

-(dA)2 = 'g'opdyOd'y’p = d§3d§k — Iozdtz . (1.6.11)

83k
In the local {§}-space-time coordinates with the metric
(1.6.11), the four-velocity vector is defined by

—A —A

{A = 91 = dt

3' 3 (1.6.12)

as

“6‘3 = VJEIOM - VZIO'QYHI‘l .
E“ = [c’llo(l - 5210-2)%]-1 , (1.6.13)
with
dt _ —2 -2 % -1—J _ d“j —2 _ —. -J-k
53'- [IO(1 — V I0 ) J ,v - 3%.,v — ngv v

(1.6.14)

If, in Section 1.5, 1 is replaced by IO, 65 by 63, keeping
in mind that all derivatives of 10 with respect to the

space and time coordinates vanish, we obtain the corre—
sponding quantities in the local {§}-space-time coordinates.
The components of the four-momentum vector in {§}-space-

time take the form [see (1.5.10) and 1.5.11)]:

P3 = any , F, =-n"i'IO2 ; P = cnv‘J , F1+ = ficz , (1.6.15)

 

 

 

18

where the relativistic,mass, a, in the {§}-space-time is

defined by

a = mo[c-110(1 - 3210-2)%J-1 . (1.6.16)

The components of the four-force vector in the {§}-space-

time become

Fj(§) = [6‘110(1 — FZIO‘Z) 1'1F5(§), FJ(§) = §%(573), (1.6.17)

l
I_‘I
H
A
'_J
I
<;
H
N
v

F4(§) 3-1F4(Y) : F4(37) = - §?(mI02)

(1.6.18)

The energy equation in the f —Space-time, (1.5.25), has

its correspondence in the y -space—time in the form

(7363(5) = 5%(5102) . (1.6.19)

1.7. Second Form of the Four-
dimensional Space-time

 

 

Here we define the "world point" as a point at a
certain time with three coordinates x1, x2, x3, having
the dimensions of length, and a fourth coordinate x9,
having the dimensions of time t, i.e., x” =.t. The metric

of the space-time (1.3.2) is replaced by

j k
ajkdx'dx

-(dT)2 — c-212(dx”)2 = aopdxodxp 3 (1.7.1)

c-ldA and has the dimensions of time.

where dr

19

Four-velocity

 

 

 

c“ -[c‘11(1 — q I )2] (1.7.3)
Four-momentum
PJ = moczcJ = cZMq.j , Pu = -MIZ; P‘j - moczzg'j - 02Mqj
P” - Me2 = a: , M = mo[c-11(1 - qZI'2)%]“1 (1.7.4)
Four—force
6P 6; _ _ L _ _
FO(X) = 3e° = moczg?o , Fj(x> = [c 1m - qZI 2>21 1F360,
— _ 6 _ d 1 2 ,
FJ(x) — czfich) - CZEa-(qu) + —2-M(I ),J.J , (1.7.5)
F~<x> = [6'11(1 - q21‘2)%1‘1Ft(x) ,
Ft(x) = - 3%(4’112) . (1.7.6)
Energy
31M ) - 341412) = 93* F3(x) = c_212F_ (:2)
q j X ' dt dt ’ qj ’
wa) — CZI-Zd(MI2) (1.7.7)

 

 

 

20-

The following relation (to be used-below) is obtained-

from (1.4.13) and (1.7.5):

5 J

J“!

09

t

=0

—2

-1—J _
mO F (x) —

J _
Q_ + % m 1Makj

t o (C

—22

I

),k

(1.7.8)

Similarly, the metric (1.6.11), in the {y}-space-time, takes

 

 

the form.
—(dT)2 = d 0d p = g ddeyk — 0—21 2(dyL‘)2 dyl+ =‘dt
(1.7.9)
where;
-2 -2 2
$11 = 822 = 833 = C a gun = *0 Io 9 800 =‘0 for 0 ¥ 0
(1.7.10)
The four-velocity
A - EEK j _ jE .1I (1 2I —2)%]—1 j _ 9X:
5 ' d1 ’ a ' V c o ’ V O , V - dt ,
(1.7.11)
_ _ y -
a“ = [c l10(1 — VZIO 2)2] 1 (1.7.12)
The four—momentum
2 _ 2 _ 2 _ *
p j = moo g3 — c mvJ , p” — —mIO — -e (y) , (1 7.13)
* —1 2 2 b__1
p4 = me2 = et(y) , m = mofc 10(1 - V 10- )2] (1.7.14)

 

 

21

The four-force

 

dp d6

 

110(3)) = 2170 = mOCZE-O, 113(3)) = [c-llo(l .. v210'2)"]‘1FJ(y),
(1.7.15)
83(y) = czg%(mvj> , Fu(y) = [c'110(1 — v210'2)*J'1F.(y) ,
(1.7.16)
_ _ d 2
F4(Y) - - 35(m10 ) . (1.7.17)
The energy
* o
VJ§J(y) = §%(m102) = g% (y) , VJFJ(y) = c-ZIOZF”(Y);
— _ 2 -2d. 2
F“(y) c 10 a€(MIO ) . (1.7.18)

1.8. Transformation of Coordinates

 

A simple form of the metric, (1.7.1), can be obtained

by introducing the transformation of coordinates

X3 = c’li , X” = x‘+ = t . (1.8.1)

Hence, the metric (1.7.1) becomes

dTZ = —AJkdXJka + c5212(dx8)2 = —AodeOpr , (1.8.2)
where,
Ajk = 1 for j=k , A1. = -c‘212, A0p = o, for 0 ¢ 9

(1.8.3)

 

 

 

22

The components of the velocity vector in the {X}-

space-time are

- J
VJ(X) = 3% = :30‘1,V“(X) = g?“ = c1’1(1 + v2(X))15
-1 2...
= cl (1 + c c )2, Vj = ch . (1.8.“)
where,
v2(X) = AJkVJ(X)Vk(X) , Aopv°(x)vp(x) = - 1 . (1.8.5)

The spatial force

[moczvj(x)1 = g} (moczcjc'l) c'1F3(x); (1.8.6)

FJ(X) = g%

FJ(X) = c‘lfiJ(x) , F (X) = cF (x) . (1.8.7)

3 J

Similarly, in the local orthogonal coordinates we

apply the transformation
Y'j = c—ly‘j , Y1+ = yL+ = t . (1.8.8)

The metric (1.7.9) has its correspondence in the

form
—(dT)2 = GodeOde = GJKdYJdYk _ 0’2102(dy“)2 , (1.8.9)
where
ij — 1 for j=k , G11 = -c’2102,cOp = 0 for 0 ¢ 9

 

 

 

23

The components of the velocity vector are

3 .
VJ(Y) = g? = ch’l , V”(Y) = gg” = clo'1[1 + v2(Y)]36
_ -1 2 -2 8 - . I
- 010 (1 + a c ) , VJ(Y) - cEJ , (1.8.11)
- J k 0 p _
v2(Y) ~ ijv (Y)V (Y) , GOpV (Y)V (Y) — -1 . (1.8.12)

The spatial force

113(1) = c'lew) , 83(1)

c‘l'fij(y),Fj(Y-) = cF (y)

J

(1.8.13)

 

 

CHAPTER-II
RELATIVISTIC FLUID DYNAMICS

2.1. Fundamental Aspects

In classical relativistic theories of fluid dynamics,
the fluid is characterized by its internal energy-per unit
mass, 5, measured by an observer at rest with respect to
the element of the fluid as a function of the pressure,
p, and the rest density, po.

In the present modified relativistic theory of fluid
dynamics, additional aspects are taken.into account due to
the presence of a gravitational field. We assume a-cer-
tain domain filled out by.a fluid considered as a collection
of particles with rest mass m0. The system in question
possesses certain amount of Kenetic energy, potential

energy and is subject to the work of the external force

fields.

2.2. The Hydrodynamical Equations

 

The fundamentals of the relativistic fluid in a flat
space with a reference velocity of propagation of signals
in vacuo were derived in [18].1 In this work we derive

the generalized formalism corresponding to that-in [18])

 

fi—

1Numbers in square brackets refer to the bibliography
of standard works.

2H

 

 

 

25

but using a piece—wise constant velocity of the propa-
gation of signals. The concept of the variable velocity
of propagation of signals leads to the necessity of deal-
ing with Riemannian spaces. As_discussed in Chapter I,
due to insurmountable difficulties in dealing with
Riemannian spaces, we, from the very beginning, intro-
duce an approximation in the form of Euclidean space.
Therefore,we introduce the local orthogonal coordinates,
{y}, (1.7.9). We begin with the hydrodynamical equations
described in terms of a rectangular system of coordinates,

yo, fixed in the space-time {y}. The particle random

J

velocity components v are measured with respect to {y}.

The spatial components of the relativistic velocity vector

are given in the {y}—space-time by (1.7.11), i.e.,
23 = vj[c—1I0(l - v2I0-2)%]—1 , (2.2.1)
from which we obtain
VJ = 0“log“1 + 0-252)_%’C_252 = Sjkfijgk
vjv , (2.2.2)
where, gjk = c"2 for j=k and gjk = O for j # k.

(i) Boltzmann Equation

 

Let us introduce the distribution function f(yJ,t,£j)

in the orthogonal phase—space with coordinates yJ and

velocities £3.

 

26
As shown in-[18] the Boltzmann equation for f is

Df a 3% + v333 + 9333. = A f , (2.2.3)

ay'j 3&3 e

or, substituting (2.2.2) into (2.2.3) we get

"9

3

a 33 + 33333 = A f , (2.2.4)

Df a J 6
8y 3&3

+ c-lloaj(l + 520-2)’%

d

where, JJ= mO-lc-2F3(y) = the external force per unit) mass;
Fj(y) is given by (1.7.16), whereas Aef = the time rate of
change in f due to encounters between the particles.

We define the mean value of a function G by

HI

n<G> = fod3g; n = ffd3a, <G> mean value of G,

d3: = daltza3 . (2.2.5)

Multiplying (2.2.4) by any transport quantity @(yj,t,§j),
and integrating over the entire volume of the (81,52,53)-

space we get

_ .. -l’ .1
f¢Dfd3g 2 f¢[%% + c 11053(1 + c 252) 2§_u +~ 5132133g
a.) J
. y 35
= f®Aefd3€ . (2.2.6)

Integrating (2.2.6) by parts, with the usual
assumptions that products of the form (fo) tend to zero
as gJ tends to i a , and that }J is independent of EJ,

after some algebraic rearrangements, we obtain

 

 

27

_ ° _ -t
f¢Dfd35 E 3%(n<¢>) + c 1I 3L.[n<¢£J(l + c 2&2) 9]

0 J

3y
- . - —L
‘n[<%%> “0 IIOEJU + c 252) ’33—» + 7331.4
ayJ 3:3

= f¢Aefd3€ . (2.2.7)

(ii) The summational invariants

 

Let us associate with (2.2.6) the form

f¢Dfd3g a nA¢ = f¢Aefd€g , ¢ ‘ mean value of 2.

(2.2.8)

There is a certain class of functions, W, characterized
by some conservation properties during encounters in the
sense that the sum of these properties for all the particles
involved in an encounter undergoes no change by the encounter.
Hence, the variation A? = 0, (see [2] and [9]). Such func-
tions are called symmational invariants.

For a gas we may have five summational invariants
corresponding to the physical conservation laws with

w°,o = O,l,2,3,4, inserted for o ix1(2,2,8);

v = m , wj = mogJ , w” = E , (2.2.9)

where E = total energy of a particle, denoted below by

W” in (2.2.11).

 

28

Relation (2.2.8) for such functions takes the form:

fWODfd3£ a n1?“ 0 , (o = 0,1,2,3,u) . (2.2.10)

The condition AFC = O expresses the conservation of
mass during the encounter, AVJ = O expresses the principle
of conservation of momentum, while A?” = O expresses that
of the conservation of energy.

In analogy to the classical relativistic theory,

[7], W” is assumed to be given by [see (l.7.l4)]:

* -1 2 -2 1’ -1
q = = u = 2 = 2 _ 2
w et(y) p mc c mo[c Io(l V I0 ) J
(2.2.11)

Inserting (2.2.2) into (2.2.11), we have
- _ y
W” = chOcIO 1(l + 52c 2)2 . (2.2.12)

(iii) Law of Conservationfiof Mass

In this section we operate interchangebly in both
{yj} and {Y3} coordinates which differ only by the factor
c.

Let us, first, introduce the mass current vector

defined by

1
-2

U0‘ = fVadu , dp = (1 + 22c“2) fd3£ , (2.2.13)

where V“ is given by (1.8.11).

 

 

29

Substituting 2 = 70 = m0 = constant into (2.2.7),

using the transformation (1.8.8), simplifying and re-

arranging we obtain in the {Y}-space-time:
m Ual = o . (2.2.14)

Eq. (2.2.14) expresses the law of conservation of mass.

Let us introduce the notation of the average velocity

-1 ... 2 1/

85 = n-lfvjfd3g = n fc 11053(1 + £20— )‘2fd3g . (2.2.15)

_ 3 _._.
Defining u2 = z uJuJ, and making use of (2.2.5), (2.2.13)
i=1
and (2.2.15), then simplifying and rearranging, we obtain

’2)»2 = (-n'2U°‘Ua)lé . (2.2.16)

_2
(l - u IO
Let the number density as measured by an observer

moving with velocity 33 with respect to the fixed coordinates

(YJ), taking into account the relativistic aspects, be de—

fined by

02

n = n2(l — 5210-2) = -Ua

o = o
Ua, and p n mo. (2.2.17)

Furthermore, we define a dimensionless velocity,

U“. by

(2.2.18)
Hence, from (2.2.17) and (2.2.18) we have

u u = -1 . (2.2.19)

 

 

 

30

The law of conservation of mass, (2.2.14), expressed

in terms of u“ and go then takes the form

(oOu“)la = 0 . (2.2.20)

(iv) Laws of Conservation of
Energy and Momentum

 

 

If we use (1 — 5210-2)-g as a fundamental factor in-
2 _ _
stead of (1 - V I0 2) k, a modified force per unit rest

mass, corresponding to (1.7.15) may be introduced in the

form
"*J(Y) = m ‘61 “1(1 - 621 '2)‘*fij(y) (2 2 21)
s o o o ‘ ° ' '
From (1.7.18), we have
vJFfl(Y) = c“ZIOZF“(Y) . (2.2.22)

Substituting, vJ = c-llogj(1 + €20-2)-% into (2.2.22)

and taking average we get

1

._.' ..l 2 -2 _/
FJ(Y)fc :Oaj(1 + t c ) 2fdat

c‘ZIOZIF”(Y)fd3g, (2.2.23)
or, by making use of (2.2.15), we obtain

nF'J(Y)EJ = 6'2102n<F”(Y)> . (2.2.24)

..1

_ -1
Multiplying both sides of (2.2.24) by n 1mO (cIO .)

2 —2

' _ -L
. (1 — u IO ) 2, using (2.2.21), we get

.7*Jfi- = C—2I 21*4

J o . (2.2.25)

31
where we define

7 = m CI (1 — 321 —2)-%<F4(Y)> . (2.2.26)

Remodelling (2.2.23), we get

fFj(Y)den a FU(Y)IV an c‘lioffi“(v)fdgg

J

c‘lion<fi”(Y)> . (2.2.27)

From (1.8.11) and (2.2.5) we obtain:

1 —2

_ _ _2 2 _ _ _ L
c 1Ion<FL+(Y)> = c IO n<Fq>n 1010 1(1 + 52c )2

.. ..1/ —,
. (1 + 52c 2) 2fd3§ = -<F”>fv.du . (2.2.28)

Inserting (2.2.28) into (2.2.27), making use of

(2.2.13) and rearranging we have

FJ(Y)UJ + <F”>U. = 0 . (2.2.29)

Using (2.2.21), (2.2.26) and (2.2.18) into (2.2.29),

after some algebraic rearrangements we obtain

37 u = O . (2.2.30)

Eq. (2.2.27) may also be rewritten in the following

form

—j = —1 _-2 —2l§ _—2
IF deu c Ion(1 u IO ) (1 u 10

 

 

32

or, using (2.2.17) and (2.2.26);

_2 2 "361+
JFJV as = c I nomo .

j 0 (2.2.32)

Substituting ©(gk) = 90, o a 1, 2, 3, 4, [see (2.2.9)
and (2.2.13], into (2.2.7), introducing the transformation of-
coordinates (1.8.8), using (2.2.21), (2.2.26) and (2.2.32),
simplifying and rearranging we obtain the equations of con—
servation of mementum and energy

TO‘BIB = 603*“, (2.2.33)

Where we define the energy momentum tensor by

To"8 = mocszaVBdu . (2.2.34)

The right hand side of (2.2.33) represents the external
forces and the work done by them on the fluid.
*
In order to bring the forces, pOTE-a, to the same form

as in the left hand side of (2.2.33), let us assume the exist-
8

ence of a second order tensor Ha such that
.f>;*“ = HQBIB . (2.2 35)
The form of 110‘8 is chosen below.
Hence, an energy—momentum tensor, T*a8, can be
introduced in the form
T*88 a T88 - n88 . (2.2.36)

As a consequence of (2.2.35) and (2.2.36), the
equations of conservation of momentum and energy, (2.2.33),

Take simple form

 

 

 

33

95
T o‘BIB = 0 . (2.2.37)

2.3. Specific Internal Energy

 

According to the manipulations and the discussion
presented in the work [18], we define the internal energy

of the fluid, 5, per unit mass in the {Y}—coordinates by:

2 0—2
m (0 ) Ta

0 0803 = Ta uauB = 60(62 + e) , (2.3.1)

B B

where we used (2.2.17) and (2.2.18) to obtain the second

invariant quantity in (2.3.1). The tensor TaB is the co—

variant form of the energy-momentum tensor, Tag, defined

by (2.2.34).

2.4. The Fundamental Inequality

 

The internal energy, a, per unit mass of the fluid
defined by (2.3.1) undergoes certain restrictions when it
is considered as a function of the pressure and the rest
density. The restriction imposed on 8 appears in the
form of an inequality derived by [18] in {Y}-space—time

coordinates:

E: >// ippO-l + 02{[l +191-(C—2po

2 0‘1)2]% — 1} . (2.4.1)

The inequality (2.4.1) holds also in the {y}-space-time
coordinates.

As stated in [18], the significance of the inequality
(2.4.1) for a flow, is that it imposes a restriction on the

types of functions, €(p,po), furnished by the relativistic

 

 

 

34

kinetic theory of gases. This contradicts the macroscopic

viewpoint which allows e to be any function of p and p0.

2.5. Case of an Ideal Gas

As in [18], we assume in {Y}-space-time coordinates:

T“8 = poc2[1 + c-2(e + ppO-l)]uauB + meB . (2.5.1)

where p is the hydrostatic pressure.
Let us choose the tensor, Has, given by (2.2.35), in

the form

08

H = xGaB . (2.5.2)

As proposed in Chapter I, [see (1.3.1)], the function
I depends on the gravitational potential. Since I = ID =

constant in the {Y}-space-time, it follows that:

x = x0 = constant , (2.5.3)

where x0 is evaluated at point 0.
Inserting (2.5.1) and (2.5.2) into (2.2.36), using

(2.5.3), we have:

*a8 -2 _1 -
T = TGB _ HOB = DOCZEl + C (8 + poo )juau8_

+ (p _ xom.“B . (2.5.4)

The equations governing the motion of the fluid are

[see (2.2.20) and (2.2.37)]:

 

 

35

(ooua)|a = 0 ; (2.5.5)

T*“B|B = 0. . (2.5.6).

Inserting (2.5.4) into (2.5.6), taking into account
(2.5.3) and (2.5.5), simplifying and rearranging we ob-

tain:

ooczuBEHUGJIB + p,BG°‘B = 0 , (2.5.7)

where we define

u = 1 + c‘2(e + ppO'l) . (2.5.8)

Multiplying (2.5.7) by (-ua), using (2.2.19), (2.5.5)

and simplifying we obtain

DO[€,BUB + p<p°‘1) us] = 0 , (2.5.9)

,8
or, we write (2.5.9) in the form:

0’1) u8,d(9°-1) = (po‘l) uB

dc + pd(p ’8

= O , de = 5,8

(2.5.10)

Eq. (2.5.10) expresses the first law of thermo-
dynamics with dQ = O, (dQ being the elementary heat input
into the system from the outside). Accordingly, we may
introduce the notion of the absolute temperature a, and
the specific entropy S, as measured by an observer at rest

with respect to the fluid, such that:

 

 

36

8

663 = dQ = 0 , as = S,8u8,dQ = Q, u . (2.5.11)

8
Combining (2.5.10) and (2.5.11) we have
0—1
de + pd(p ) = @dS . (2.5.12)

2.6. One-Dimensional Motion

 

A11 quantities in one-dimensional motion are assumed
to be functions of the local orthogonal coordinates, Y1
and Y” = t, which are introducedin Section 1.6.

Eq. (2.2.15) can be remodelled by making use of

(2.2.13), (2.2.17) and (2.2.18) as follows

_' _ .. — ...} _ _
uJ = n'lfc 11023(1 + c 252) zfdgg = n IIOIVJdu==n 1IOUJ

'11 05 = 1 (1 — u—ZI -z)%uj , (2.6.1)

(1’1 0 O O O O

or, rearranging we get
_- _ _ _ _t _ _-_
= uJI 1(1 - u2I 2) 2 , u2 = 0. uJuk . (2.6.2)

0 O

_1 -1
Let us denote the dimensionless velocity u IO by

the symbol u, then Eq. (2.6.2.), for j = 1, becomes

u(1 — u2)-% . (2.6.3)

1.11

_L
Substituting, ul u(1 — uz) 2, u2 a u3 a 0, into

(2.2.19) and rearranging we get

u” = cIO-l(1 — u2)'15 . (2.6.4)

 

 

 

37

Substituting (2.6.3) and (2.6.4) into (2.5.5) and
(2.5.6), using (2.5.4), and rearranging we obtain in the

(y,t)—plane:

[Io'lp°(1 — uW’fiH+ + [6°u(1 - u2)"/21,1 = o 5 (2.6.5)

_1 _ _
[IO czp°uU(l - uz) 11,. + [ooczuu2(l — uz) 1

+ (p - x0)],1 = O . (2.6.6)

Carrying out the differentiations in (2.6.5) and
(2.6.6), simplifying and rearranging we obtain in the

(y,t)—plane:

—1 o-1 o

(l — u2)(IO o p .t + uoo'l

—1
p°,y) + IO uu,t + u,y = 0 ;

(2.6.7)
-1 —1 -1 _1
u(1 — u2)(IO u u,t + Up u,y) + Io u,t + uu,y
+ (1 — u2)2po_lc-2u—lp,y = O , (2.6.8)
where
-2 _1
u = 1 + c (e + poo ) . (2.6.9)
Differentiating (2.6.9) we get
czdn = de + pd(po_l) + po‘ldp . (2.6.10)

 

 

 

38

Substituting (2.5.10) into (2.6.10) we get

czdu = pO-ldp . (2.6.11)

Let us introduce the auxilliary quantity:

a2 = pop £10 , i.e., dzpo 1dp0 = u du . (2.6.12)
do

Multiplying (2.6.11) by e'zn‘l, using (2.6.12), we

get

_ _ _1 _
@209 ldpo = C 2n p0 1dp . (2.6.13)

Similarly, we introduce the auxilliary function,

¢, defined by
d¢ = apO-ldpO . (2.6.14)

Hence, we may write (2.6.12) and (2.6.13) in the

forms:
u—ldu = dd¢ 3 (2.6.15)

C-Zpo-lu-ldp = ad¢ . (2.6.16)

Substituting (2.6.15) and (2.6.16) into (2.6.7) and

(2.6.8) and simplifying we obtain:

2 ‘1 _1 =
(l - u )(IO ¢,t + u¢,y) + u(1O uu,t + u,y) O , (2.6.17)

6(1 - u2)(IO_lu¢ t + ¢ ) + I u + uu = 0 . (2.6.18)
,

 

 

39

Hence, Eqs. (2.6.17) and (2.6.18) constitute a
system of partial differential equations for the unknown
variables 6 and u.

Addition and subtraction of (2.6.17) and (2.6.18)

yields, respectively,

(1 - u2)D+¢ + D+u = 0 ; (2.6.19)
(1 - u2)D_¢ — D_u = 0 , (2.6.20)
where

D = (1 + du)I 'IJL + (a + u)JL (2 6 21)

+ 0 at ay ’ ° ‘
D = (1 - du)I "1—3— — (a — u)1 (2 6 22)
— 0 3t 3y ° ' ° -

Let us introduce the identity
2 _1 _1 g

(l — u ) Diu = Diln[(1 + u)(l - u) ] . (2.6.23)

Hence, substituting (2.6.23) into (2.6.19) and

(2.6.20) we obtain respectively

U
...3
ll
O
0

(2.6.24)
D s = O . (2.6.25)

where,

r = e + ln[(1 + u)(1 — (1)412, s = 0 — 1n[(1 + u)(1 — u)'11’2

(2.6.26)

40

As stated in [18], the functions r and s are the
local relativistic analogs of the Riemann functions which
occur in the classical theory of propagation of one-
dimensional waves of finite amplitude.

The characteristic curves of (2.6.24) and (2.6.25)

along which r and s are constants, are respectively, given

by
(§%)I = (a + u)(1 + au)-1IO , (2.6.27)
d _ -1
(5%)11— —(e — u)(l - au) 10 . (2.6.28)

Let us define

a = aIO . (2.6.29)

It is shown below that the quantity, a, represents
the velocity of sound, whereas a is its dimensionless form

referred to lo.

2.7. Progressive Waves

 

According to the definition of [18], a disturbance
is said to propagate as a progressive wave if either r or
s is constant.

If we assume that:

s = to = constant; (2.7.1)

 

 

41
we then obtain from (2.6.26)

u = tanh(¢ - to) . (2.7.2)

Inserting (2.7.2) into (2.6.24), carrying out the

differentiation and simplifying we get:

_1 _
1O ¢.t + 1‘(¢)¢’y — 0 . (2.7.3)

where
m.) = (a + u)(1 + au)-1 , (2.7.4)

whereas a is a function ofj¢ determined by (2.6.14).

The general solution of (2.7.3) is of the form:

f(¢) = y - r(¢)10t . (2.7.5)

where f(¢) is an arbitrary function.

It follows from (2.7.5) that ¢ is constant along the
straight lines of slope P(¢) in the (y,IOt)-plane. Hence
a represents the dimensionless velocity of propagation of
a sound wave referred to IO; the form of a is given below.
According to the classical theory, the internal energy, a,

for a perfect gas can be written in the form (see [4]).
_1 -1
e = (y - 1) Doc . (2.7.6)

Differentiating (2.7.6), using (2.5.10), and

simplifying we obtain:

 

 

 

QRb = 7900-1 . (2.7.7)
do

Substituting (2.7.7) into (2.6.13), making use of

(2.6.9), we get after some.rearrangements:

_2 _ —2 — —1 —1
92 = C 7900 1[l + 0 7(7 - l) 1poo 1 . (2.7.8).
If we consider a medium of high temperature, for
which c-2ppo-1 is large compared to one, Eq. (2.7.8) then

approximately becomes
2
a —+ (Y "' l) o (2.709)

Hence, for y>2, sound waves propagate with velocity
greater than the maximum velocity of propagation, i.e.,
IO. This contradiction implies that-the equation for 5,
(2.7.6) for y>2 is not-a possible one.

A physically possible flow, for which (2.7.5) is a
solution, exists only if the curves ¢ = const. do not
intersect in the (y,lot)-plane ([4] and [12]). If this
condition is not satisfied, one-dimensional motion will
suffer a discontinuity in the form of-shock waves accord-

ing to the classical theory.

2.8. Rankine-Hugoniot Equations

 

The relativistic Rankine-Hugoniot equations were
derived by [18] in the flat space-time with the reference
velocity of propagation of signals "c" in vacuo. Similar

equations, having identical forms as those of [18], are

 

   

 

43

obtained in the {Y}-space-time coordinates with a refer-
ence velocity "IO". We assume that both 16 and x0 re-
main constant at their corresponding values at a point
"0". Only the flow variables p°,u8, p and e are subject
to Jump discontinuities across the shock. We choose our
coordinate system in such a way that the discontinuity is
at rest and is perpendicular to the Yl-axis of the {Y}-
space—time. We put down the relativistic Rankine—Hugoniot
equations without derivation as obtained by [18] in one-

dimensional flow:
(mass): po+u+(l — n+2)”;5 = po_u_(l.- u_2)J5 = M ;
(2.8.1)

(from momentum):

M = c‘lup+ — p_)(u_oo_l- u+oo+-1)_11% ; (2.8.2)

(energy):

...1 -1
M2c2(u+2 — u_2) = NIP-(9+ - p_)(u+oo+ + 4-93 )

(2.8.3)

In the above formulations, we assume that the fluid
moves from right to left across the fixed shock. Quantities
on the right side of the shock are denoted by the subscript
(—) whereas those on the left side are denoted by the sub-

script (+).

I

 

44

2.9. The Shock Velocity

Following [18], we introduce the quantities:

-1 o 0—1

—1 -2 -1
5 = p+p_ , 0 = 9+9_ 8 = Y+(Y+ — l) c p 9°

3

(2.9.1)

Rewriting (2.6.9) in terms of quantities (2.9.1)

making use of (2.7.6) we have:
_1 —1 —1
u+ = 1 + Bén , u_ = l + Y_Y+ (Y+ - l)(Y_ -l) 8

(2.9.2)

As stated in [18], and hence B may be functions

Y+

0—1
of p+p+

However, they are assumed to be slowly vary-
ing functions and for the purposes of the discussion be-
low, it is sufficient to consider y+ to be a constant.

Hence, the second of (2.9.2) becomes (with y+ = y_):

u = 1 + B . (2.9.3)

From the inequality (2.4.1) and the fact that

e > 0, it follows that:
5/3 2.7+ > 1 . .(2.9.4)

Substituting (2.9.1) and the first of (2.9.2) into

(2.8.3) we obtain after some algebraic rearrangements:

 

45

8(2 + 7. - 1)826'2 + [(7+ + l)§ + (7+ - 1)]86‘l

- {[B(¥+ - 1)(£ - 1) + u_Y+1u_ — 7+} = 0
(2.9.5)

Eq. (2.9.5) is a quadratic form for the quantity Bn-l

Consequently, if_we solve for the positive value of Br)-1

we have:
88-1 = {R - [(7+ + 1)£ + (7+ — l)]}{2t[a + (7+ — 1)l}'1 .
(2.9.6)
where
R = ((y+ - l)2(€ — 1)2 + 45(2 + 7+ - 1)
. [7+u_2 + Bu_(Y+ - l)(€ - 1)]12 . (2 9.7)

After some manipulations, the author of [18] obtained

the following inequality:

9 - u 0—1;u_(€ + 7+ - l)_1[(€ - l)(2 - 7+)

— +

.(._ - 7+)(y_ — 1)‘11 . (2.9.8)

Substituting (2.8.1) into (2.8.2), using (2.9.1)

and rearranging we get:

 

 

46

u_<1 - u_2)“% = [(y+ - l)B(£ - l)1%[7+(u_ - n+6”)?15

(2.9.9)

As mentioned in [18], u_ is less than one whenever
the right hand side of the inequality (2.9.8) is positive.
According to the convention presented in Section 2.8, the
gas moves from right to left across a fixed shock. The
velocity of the gas on the right side of the shock is de-
noted by Ei, whereas that on the left side is denoted by

ui. The shock is considered to be stationary with respect
to a suitably chosen coordinates {Y}. Let us assume now
that the fluid on the right side of the shock is at rest
and the.shock moves across.the medium. In order to find
the shock velocity, let us superimpose.the velocity of the
magnitude 6i upon the entire system in the direction
opposite to the moving fluid. The gas will be at rest

in the moving new system {Y*}, and the shock will move
with the velocity 5: from the left to the right.

The transformation of coordinates {Y*}+{Y} is of

the Lorentzian type:

y*1 (Yl _ c'lfiit)[1 _ (5:)210-21-5 ’ y*2 = Y2, Yna = y3

t9!-

_... — — —l’ —
(t - cIO 2 :Yl)[l - (ui)21O 2] 2 , ui = u I

 

 

47

which leaves (d7)2 invariant in the four—dimensional space—

time, i.e.,

-(dT)2 (811)2 + (dY2)2 + (dY3)2 - c

2 2 2
IO (dt)

-2

(dY*1)2 + (dY*2)2 + (dY*3)2 — c I02(dt*)2

(2.9.11)

In our main problem of the association between
Riemannian and Euclidean spaces we solve the problem of
shock in rectangular coordinates related piece-wise to
the curvilinear coordinates. Hence, the velocity u: is
considered to be momentarily constant. This implies that
the above transformations (2.9.10) is valid momentarily in
a piece-wise sense. In conclusion, the velocity of the
shock relative to the gas into which it is traveling is
less than the signal velocity 10. The remaining reason—

ings of the discussion that follows in the work [18] are

valid in the present approach.

2.10. Concluding Remarks

 

A passage from-the present work in the {Y}-space-
time, with reference velocity 10, to that of [18] in the
{Y’}-space-time, with reference velocity c, can be made

through the transformation of coordinates,

Y’J = YJ , Y’“ = c‘lioy“ . (2.10.1)

 

 

48

As a consequence of the coordinate transformation
(2.10.1), the relation between quantities in the above
reference frames are presented in Table 1. It follows
from this table that the flow variables are independent
of the above coordinate transformation. This is due to
the fact that the fundamental factors (1 — u’2)15 and
(l - u2)1/2 are equa14and that the distribution function
f(Y,t,§j) is an invariant under (2.10.1) (see [8]).

The magnitudes (5:1 and 51) of the velocities of the shock
waves, in the above frames of references, relative to the
gas into which they are traveling are governed by the

relation

6:1 = c1 "131 . (2.10.2)

which shows that 611:»61

However, their dimensionless magnitudes, u: and u_
referred to c and 10, are equal. The same argument holds
for the velocities of sound a' and a. Thus, in conclusion,
only 31 and a are affected by introducing IO in place of

C .

 

 

 

 

49

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CHAPTER III

DERIVATION OF THE HYDRODYNAMICAL EQUATIONS

IN THE RIEMANNIAN SPACE-TIME

3.1. Introduction

 

In this chapter, we derive the hydrodynamical
equations in the Riemannian space-time, {x}, with the
reference velocity of signals I = I(x%. We demonstrate
below that these equations reduce to their corresponding
equations in Chapter II, when we set I = IO = constant
and x = X0 = constant.

The hydrodynamical equations are described in terms
of a curvilinear system of coordinates, x0, fixed in the

space-time {x}, (1.7.1). The particle random velocity

J

components q are measured with respect to {x}.

3.2. Boltzmann Equation

 

Let us introduce the distribution function f(xj,t,;j)
in the Riemannian phase-space, with coordinates, xj, and
velocities, cj. As stated in [8] the distribution
function f(xj,t,cj) is an invariant. The variation in

the number of particles during the interval of time dt is
[f(xJ + dx'j,t + dt,c;,J + dg‘j) - f(xJ,t,;J)']d3xd3r.

= Aefd3Xd3§dt , (3.2.1)

51

52

where, d3x = dxldxzdx3, d3; = dcldczdg3; whereas Aef
= the time rate of change in f due to encounters between
the particles.

Expanding the first term on the left hand side of
(3.2.1) in Taylor series around (xJ,t,;j), retaining only
the first order differential terms, dividing all-through

by d3xd3;dt and rearranging we get

4..
I
l
+
I
II
p.
“2

%§ dt 3 a? J e . (3.2.2)
The validity of the Operations of the ordinary
differentiation carried out in (3.2.2) follows from the
fact that the ordinary derivative of a scalar (an in-
variant) is identical with its absolute derivative (see
[171).
Solving (1.7.2) for q'j in terms of cj we get

_ ° - -1 - °
qj = c lI§J(1 + c 2:2) 2 , c 2:2 = ajkcJ k . (3.2.3)
From (1.7.8) we obtain
dcj_ -2 -1 —j 1 k] 2
35 — 0 m0 [F (x) - §Ma (I ),k] . (3.2.4)

Substituting for Qfi = qJ using (3.2.3) and (3.2.4)

dt
into (3.2.2) we obtain

53

- - -1/ - - —
Df s §£-+ c 11:3(1 + c 2:2) 23:. + c 2m 1[Fj(x)
at J 0
8X
- —1— MaJk(I7-) 1111. = A r . (3.2.5)
2 ,k 3:3 e

Similar to (2.2.5), we define the mean value of a

function G by:

n<G> = fod3; , n = ffdgg , <G> E mean value of G.

(3.2.6)
Multiplying (3.2.5) by any transport quantity

9(xj,t,cJ) and integrating over the entire volume of the

(C1, C2, C3)-space we get

- _. 2 _12 _ _ _'
fthd- = 19(33 + c 1123(1 + c 2; ) 233 + c 2m 1[FJ(x)
8t 3X.) 0
_ % a3kM(12) k]3§ }d3; = f¢Aefd3C . (3.2.7)

) BCJ

Integrating by parts making use of (3.2.6) we

obtain
at“ -2 ii -2. ii.
J¢§Ed3g — §€f¢fd3t - fatfd3c — at(n<<1>>) — n<3t>, (3.2.8)

 

 

54

_3_
J’

222.1:

f¢cjc_ll(1 + c—ZCZ) J

d3§ = [fC—II¢CJ(1 + C_2§2)—%fd3§1

8X 3X

- ° _ ...1/
- I{Ji.[c 119:3(1 + c 2:2) 211fd3€
axJ

= (c—II)JLJ[I¢;J(1 + c-2c2)'%fd3;]
8X

_ ° _ -1
+ [Jij(c 1I)]f4>;J(1 + c 2:2) 2fd3c
3X

_ ' - -9
_ fiLj[c 11¢;J(1 + c 2:2) 2de3;
6x

= (C_ll)[n<¢cj(l + c'2:2)'2>1-j

+ (6‘11) J.(n<<z>.J(1 + C_2§2)-2>]

- n<[(c‘11)¢cj(1 + c'222)'81 j> . (3.29)

For j = l, we have

fff{c—2mo_l¢[F1(x) - %allM(12),113£1)dcldtzdc3
a;

1-
= ff{C_2f¢mO[F1(X) - %ailM(I2) i1} dCZdC3
a 1

C =-oo

— f/IfiL {c-2m0-1¢[E1(X) — %a11M(IZ) ]}d3; . (3.2.10)

8:1 J

The same is valid for j = 2,3.

 

 

55

As mentioned before, we assume that products of

the form (f0) tend to zero as CJ tends to 1w. Adding

Eq. (3.2.10) for j 1 and its correspondence for J 8 2,3,

we obtain:

fc'zm ’10[Fj(x) - LaJkM(IZ) .]32 d3;
0 2 ,k BCJ

= - IrJL.(c‘26[m ‘123(x) - %m 'lMaJk(IZ) k1}d3C
BC'J L. O 3
= — f{c_2m 1[FJ(x) lMaJk(IZ) 133

+ oiLJ[c-2m -1FJ(x) - %c-2m “lMaJk(IZ) k]}fd3c

—n<c_2m -1Fj(x)33.> + n<%m ’lMaJk(12) 33 >
o a: a o , BCJ

%c-2mO-1Majk(12),k]>

- n<¢JL [c-Zm -1FJ(x)]> + n<0—8—jE
J o
a; 8:

(3.2.11)

The mass, M, can be expressed in terms of c2 and I

as follows:

_ - - - - L
M = mOcI 1(l - q2I 2) g = mOcI 1(1 + c 2:2)2 . (3.2.12)
Hence, with I = I(x), we have
—2n[%m ’IMaJk(c‘212) k]= éeaifie'ziz) ,iL.[ei‘1
- 1 - ~ _ _t
(l + c 2:2)21 = (c 1I) .§J(1 + c 2:2) 2 . (3.2.13)

,J

   

 

 

 

5"

56

Substituting (3.2.8), (3.2.9), (3.2.11) and (3.2.13)

into (3.2.7) and rearranging we obtain

7 _1 - ->
f¢Dfd3C = 3%(n<4>>) + (c I)[n<¢cj(l + c 2:2) 2>1 3
,
_ _ -1
+ (c 1I) [n<9cJ(1 + c 2:2) 2>1 - {n<33>
:3 at
+ n<[(c'll)¢cj(l + 0—2C2)_%1 j>
,
— — _ — 1 -
+ n<c 2mO 1Fj(x)§$ > — n<(l + c 2:2)1—2 > a3k(c 1I) k
3:3 8:3 ’
-J
+ n<c'2m ‘192—E > - n<¢;j(l + c‘2;2)15>(c-1I) .}
= f¢Aefd3C . (3.2.14)

3.3. The Summational Invariants

 

The summational invariants WC (y), (2.2.9) and
(2.2.12), in the flat space-time, {y}, have their corre—

spondence in the Riemannian space-time, {x}, in the form
0 . .
I (x) = mo , 73(x) = mOcJ .
_ - 1
v“(X) = czmo(cl 1)(l + c 2:2)2 . (3.3.1)

3.4. Law of Conservation of Mass

Substituting 9(cj) = 70 = m0 = constant into (3.2.14)

we obtain

 

 

57

2 2 -1 .
c ) 2>1,J

+

(n<mo>),t (e‘li)[n<mO;J(1 + c-

—‘ . ' .. _1
+ (c ’I) J.[n<moz;‘](l + c 2:2) 2>]

9

- ‘ - -}r
_ {n<m > + n<[(c 11)mO;J(l + C 262) 2193>

o,t

. -2 —1—j
+ n<c. mo F (X)mo,;3>

-~ 1 ~ -
_ n<(l + c 2g2)2aka J(c 1I)

- n<mogj(l + 6-222)'2>(e'11) j} = 0 . (3.4.1)

Simplifying and rearranging we obtain

m {n + c-ll[n<qj(l + c-242)'%>] ,
O ,8 :J
- ° _ _1 _ _
+ (c 1‘1) jn<cJ(l + c 2:2) 2>} = c 2n<F,;j >
3

(3.4.2)

Introducing the transformation of coordinates
(1.8.1), with the usual assumption that the force is
independent of the velocity ;J and with I = I(XJ), Eq.

(3.4.2) takes the form

- . , - , ° —1
m [c 1I(IVidII) + c 11(JVJdu) . + (c 1) .IVJdu] = 0,
0 31+ )J ’J

(3.4.3)

 

58
where,
at = (1 + 0-2;2)—%fd3c . (3.4.4)
Let us define
U“(x) = Iv“(x)du , (3.4.5)

Eq. (3.4.3) can be rewritten in the form, (after

multiplying all through by cI_l):
4 + UJ. + l 1 '2 2 . 3 = 0 . .4.
mO{U’q ’J 2[ n(c 1 )]’JU 1 (3 6)

Comparison of (1.4.17) and the left hand side of
(3.4.6) suggests that U0t can be considered as a contra-
variant four—vector, (the mass current vector), in the

Riemannian {X}-space-time, so that we may write
moual = 0 . (3.4.7)

For operations below, we need to introduce the notion

of the average velocity defined by

-_‘ _ - _ _ _3/
wJ =,n lqufdgz = n lfc 1123(1 + c 222) 2fd3; , (3.4.8)

where we used (3.2.3) in (3.4.8).
Defining w2 = Ajkijk, using (3.2.6), (3.4.5) and
(3.4.8) we have

 

”‘7'

 

59

(1 - wZI—Z) = 1 - Ajkijk

1‘2 = (n-lffd3c)(n—1ffd3c)

2 2

- - - -l’
lfc l125(1 + c t ) 2fd3§]-

- A I-2[n

jk

1/

‘1 2293;]

-[n fc—11;k(l + 0-262)—

_ - _ _ 1 ._ -1
= n 20 2IzUcl 1(l + c 2c2)2(1 + c 2:2) 2fd3q].

- - 1/ - ...1/
. [fCI 1(1 + c 222)2(1 + c 222) Zfdgg]

— Ajkn—2(fc-1deu)(fc—lckdu)

= —n*2[(rvidu)(I-c’zizv”du)

+ Ajk(ijdu)(kadu)] = —n-2[(fV“du)(fVudu)

+ (fVJdu)(fV dp)] = -n-2UaUa . (3.4.9)

J

or,

(1 — 621‘2)% : (—n‘20“U )72 . (3.4.10)

0.

Let the number density as measured by an observer
moving with velocity wfl with respect to the fixed coordi—
nates (X3), taking into account the relativistic aspects,

be defined by

n02(X) = n2(X)(1 - 621-2) = —Ua(X)Ua(X) . (3.4.11)

 

 

 

 

60
o

The corresponding density, p , is.

p = nom . (3.4.12)

Let us, further, define a dimensionless velocity

w = n Ua . (394-13)

WOW = -l . (3511.111)

Similarly, inserting (3.4.12), (3.4.13) into (3.4.7)

we get

(6°w“)|a = 0 . (3.4.15)

Hence, Eqs. (3.4.7) and (3.4.15) are alternative

expressions for the law of conservation of mass.

3.5. Laws of Conservation
of Energy and Momentum

 

 

As discussed in Chapter II, we introduce the corre-

sponding modified force—vector, [see (2.2.21) and (2.3.26)]:

3"- — _ — - ...}, -— _° _.
9 a = mo ch 1(1 + wzl 2) 2<F“(X)> , <FJ> = Fj . (3.5.1)

Similar manipulations to those presented in Chapter

II lead to the expressions:
8
7 (x)wa = 0 3 (3.5.2)

fF-J.(X)deu = 0-212n0m07*4(X) . (3.5.3)

 

 

 

61

Substituting ¢(Ck) = wk = mock into (3.2.14), we

obtain

1 k -1 , k j —2 2 —g
(nxmoc >),t + (c I)[n<moc c (1 + c C ) >1,j

j k —2 2 —a _ k
I) JImmot : (l + c c ) >1 {n<(moc ),t>

3

+ (c-1

_2 2 -g —2 —j k
> + n<c m F x m >
c ) l j 0 ( )( 0: )’CJ

3

+ n<[c-1Imongk(l + c

, —2 2 t k 13 -1 '2 ‘1 k‘fi J
— n\(1 + c c ) (mo; )’63>a' (c I) + n<c mO mo; F :C >

,J

- n<mogkgj(l + c‘2;2)—%(c-1I) J.} = 0 . (3.5.4)

Simplifying and rearranging, with the assumption

that FJ is independent of :3, we get
-2 2 1 -2 2 -1
[mofck(l + c 4 )2(1 + c C ) 2fd3C] H
3

1/
2fd3€J .
3

-' ,- ° -22-
+ c iILmOfCKCJU + c C ) J

- ' _22_1/
11) j[mOkaCJ(l + c c ) 2fdgc]

3

+ (c

(6" I) j[mortkcju + c_ZCZ)—%fd3€]

+

3

_ _2 2 -1
(c 1I) J.[mofckcjfl + c C ) 2fdac]

- - 1’ - -l’
1) J.aJKntOu + c 2:2);5(1 + c 2:2)2(1 + c 2:2) 2fd3c
.3

—1
+ (c

-2

= c p mo"1Fk(x) , 0 (3.5.5)

ll
:3
S

   

62

Applying the transformation of coordinates, (1.8.1),

with 1 =.1(x3), Eq. (3.5.5) reduces to

C[(C-II)(mOqudeu)’q + (c—II)(mOijdeu),XJ]

i k

+ e'1(6521 )(c-II) JAka fV“V”du + C(C_1I) 3m IV v‘ du
,X o ,X o
= c‘lpmo‘dfik(X) . (3.5.6)
Let us define
Tka = mocszkVadu , (3.5.7)
Eq. (325.6), after we multiply all through by
(cZI—l), becomes
L.
T k + 13k . + l(c‘212) AJkT”“
:4 :J 2 3.1
- ° *
+ §I1n<c 212)] 313k = 603 k(X),
3
- - _ *
61 1mo 1Fk(X) = 00 9 k(X) , (3.5.8)

where we used (3.4.11) and (3.5.1) to obtain the right
side of (3.5.8).

Similarly, if we substitute 9(cJ) = W“ = moc2(cl'l)

-. 1
(1 + c 222)2 = MC2 (3.3.1), into (3 2.14), simplifying,
3

keeping in mind that F'J is independent of 43, and re-

arranging we get

63

[m c2n<(cI-1)(l + C-2C2)%>1
O :4

+ (c-II)[moczn<(cI_1)(l + 6‘2;2)%tj (1 + 6‘222)'2>],J

_ _ _ 1 ° - -1
+ (c 1I) 31mc02n<(cl 1)(1 + c 2:2)2CJ(1 + c 2:2) 2>1

3

 

— mOmO-ln<Fj;j(l + c—2;2)-%>(CI'1)

+ (C-II),kajkmOCZn<(1 + C-2§2)%(CI-l)§g(l + 6'222)'5>

+ moc2n<cI-1(l + c52§2)%; (1 + 0-2;2)_%> (0-11),J = O .
(3.5.9) “

or, rearranging (3.5.9), with I = I(xj), we get

(c-II)[mOczn<(cI-l)(l + 0—2;2)%(cI-1)(l + 6‘2;Z)*(1 + 6‘222)'*>] 4

+ (c_ll)[moczn<(cI-l)(l + c_2§2)%;j(l + c-2;2)-%>]

.J

_- _ _ 1 - _1
+ (c 1I) jfmoczn<(cl 1)(l + c 2t2)2tj(l + c 2:2) 2>1
3

_ _- .- -1
cI 1n<FJ(X)Cj(l + c 2:2) 2

>

_ .. _e1‘ _ ..1
+ (c 11) jimoczn<(cl 1)(l + c 2c‘)2cJ(l + c 2:2) 2>1
3

+

(0711) 31m002n<(01—1)(l + c'2c2)%cj(l + 972:2)'%>1 = O

3

(3.5.10)

 

 

 

64

Applying the transformation {x}+{X}, (1.8.1),

dividing all through by (c-II) and simplifying we get

(m cZIVqquu) + (m cZIViVJdu) .
0 31+ 0 3t]

*
p07 4

+ %[ln(c-212)] (mocsz9deu)

J

c21'2I83(X)V du = 00 7*(X) , (3.5.11)

3

where, we used (3.5.3) to obtain the right hand side of

(3.5.11).

Let us define

17° = mocsz“V°du , (3.5.12)
Eq. (3.5.11) then becomes

1““. + ngj + 511n<c'212)1 jT“j = o°'7*“(x) . (3 5.13)

Comparison of (1.4.18) and (1.4.19) with the left
hand sides of (3.5.8) and (3.5.13), suggests that T“8
can be regarded as a second order contravariant tensor in
the Riemannian {X}-Space-time.

Therefore, (3.5.8) and (3.5.13) are combined into

the tensorial form

18818 = 60J*“ , (3.5.14)

 

 

 

65
where the energy—momentum tensor is defined by

98

T = mocszaVBdu . (3.5.15)

The right hand side of (3.5.14) represents the
external forces and the work done by them on the fluid.
0 ]*0I
In order to bring the forces, p , to the same

form as in the left hand side of (3.5.14), let us assume
8

the existence of a second order tensor Ha such that-
*
60.7 a = nasls . (3.5.16)
The form of flu8 is chosen below.
*
Hence, an energy-momentum tensor, T a8, can be intro-
duced in the form
*
T “B s T88 — mas (3.5.17)

Hence, from-(3.5.14), (3.5.16), (3.5.17) we get

*
T o‘Bls = 0 . (3.5.18)

3.6. Specific Internal Energy
Similar discussion as that presented in Section 2.3
holds here. The internal energy of the fluid, a, per

unit mass, similarly, is defined by:

TanawB = 90(02 + e) . (3.6.1)

 

 

 

 

66

3.7. Case of An Ideal Gas

 

Following [18], we assume:

8

Ta .0002[l.+ c-2(e + ppO—l)]wawa + pAO‘8 . (3.7.1)

where p the hydrostatic pressure.

98

Let us choose the tensor, H , given by (3.5.16),

in the form

08

H = anB

(3.7.2)
Hence, inserting (3.7.1), (3.7.2) into (3.5.17) we
get

T = 606211 + e-2(. + p.0'1)1 wawB + (p - X)A“B

(3.7.3)'

The equations governing the motion of the fluid are

[see (3.4.15) and (3.5.18)]:

I
O
V

(3.7.4)

(powa)|a ‘

I
O

1
T o“3|... - (3.7.5)
Substituting (3.7.3) into (3.7.5), taking into
account (3.5.16), (3.7.2) and (3.7.4), simplifying and

rearranging we get:

*
AOIB=DO; d

-2 .-
8 U: l + c (8 + p00 1)
3 3

ooczw8(uw°‘)lB + p

(3.7.6)

 

 

 

67

Multiplying (3.7.6) by (-wa), using (3.4.14),
(3.5.2) and (3.7.4), and simplifying we get:

po[.’BwB +p(po_1),BwB]= 0 , (3.7.7)

or, we write (3.7.7) in the form:

-1 - _
de + pd(pO ) = O , d6 = e 8W8 d(pO 1) = (00 1) BWB .
3 3 3

(3.7.8)

Eq. (3.7.8) expresses the first law of thermodynamics with
dQ =10, where dQ = the elementary heat input-into the
system from the outside.

If we introduce the notion of-the absolute tempera-
ture 0 and the specific entropy S, as measured by an ob-

server at rest with respect to the fluid, such that

ads = dQ = 0 , d8 = s BWB’ dQ = QBwB , (3.7.9)

Eq. (3.7.8) combined with (3.7.9) then becomes

d8 + pd(p°'1) = eds . (3.7.10)

3.8. One-Dimensional Motion

 

All quantities in one—dimensional motion are con-
sidered to be functions of the coordinates Xl and-XL+ = t,
also we assume W2 = w3 = 0. Eq. (3.4.8) can be remodelled

by making use of (3.4.5), (3.4.11), (3.4.13) as fOllOWS:

 

68

773 = n-lfc-llg‘ju + c-2§2)-;§fd3§ = n-lfIVJdu

= n-lIUj = (n- no)nO-1IU'j = (l — w21_2)%wjl
(3.8.1)
or, rearranging we get
wj = t31‘1(1 _ 221'2)‘3, t2 = A. tflwk . (3.8.2)

Jk

Let us denote the dimensionless velocity fill-1 by

the symbol w, then Eq. (3.8.2) for j = 1, becomes

 

w(l - w2)-% (3.8.3)

W1

Substituting w1 = w(l — w2)-;5,w2 5 w3 s 0 into
(3.4.14), and rearranging we get
4 ‘1 2 —%
w = CI (1 - w ) . (3.8.4)

'Writing (3.7.4) in full, making use of (1.4.17) we get

+ %[1n(c'212)1 oowl = o . (3.8.5)

(0 O
(0 W1) + (o W”)
1 9 91

3 9+

Substituting (3.8.3) and (3.8.4) into (3.8.5),

simplifying and rearranging we get in the (x,t)-plane:

(1 — w2)(I—1pO—1po,t + w pO-lpo,x) + I_lw w t
3

+ w’x = — %w(l _ w2)(1n12),x . (3.8.6)

 

 

 

69

Similarly, writing (3.7.5) in full making use of
(1.4.17), (3.5.16): (3.7.2), (3.7.3), (3.8.3), (3.8.4),
simplifying and rearranging we obtain in the (x,t)-
plane:

1 —1 —1 -1

w l — w2 I— + w + I w + ww
( )( u “,t u u,X) ,t ,x

 

+ (l - wz)2;)O"1C:_211-1p,x = - 3 (l - WZ)(lnIZ)’X
+ (l - wz)2c>o_l<:_‘2II-1>(’X . (3.8.7)
where
I = 1 + 6-27. + poo-1) . (3.8.8)
Differentiating (3.8.8) we get
02du = de + pd(pO-1) + pO—ldp . (3.8.9)
Inserting (3.7.9) into (3.8.9) we get
chu = 60’1dp . (3.8.10)

Let us introduce the auxiliary function, 9, defined by

u du = a p do (3.8.11)
From (3.8.10) and (3.8.11) we obtain
C-2u_lpo_1dp = dpo—ldpo . (3.8.12)

 

 

70

Substituting (3.8.12) into (3.8.7) and rearranging we

get
a2(1 — w2)(wI-1pO-1p?t+ pO-lp?x) + I—lw,t + ww,x
= (1 - w2)160‘1c'21‘1(1 - Wm,X - 2(1912>,x3
(3.8.13)
Furthermore, let us define the auxiliary function 6 by
d¢ = de-ldpo , (3.8.14)

Eqs. (3.8.6) and (3.8.13) in terms of 6 become respectively

2 -1 -1
(l - w )(I ¢,t + W¢,x) + u(I ww,t + w’x)

= - %QW(1 — w2)[1n(12)1’X , (3.8.15)

(1 - w2)(w1_1¢ t + ¢ X) + I-lw + ww
3 3

,t ’x

= _ 2 -2 0-1 -1 _ 2 _ l 2
(1 w )[c o u (l w )x,X 2(lnI ),x]

(3.8.16)
Adding and subtracting (3.8.16) we obtain respectively

I 12=- 2; 2
(1 — w2)8+¢ + 6+N . (l — w )[2(1 + dw)(1nI ),x

—2 0—1

- c 0 .‘1(1 — W2)x,x1 ; (3.8.17)

 

 

 

 

71

(1 — w2)6_¢ — 6_w = (1 — w2)[%(l — aw)(ln12) X

— c'zpo’lu'lu - w2)x’X] . (3.8.18)
where,
5+ = (1 + awn-13%- + (a +-w)-§§ ,
6_ = (1 - aW)1‘1§% - (a — w)§% . (3.8.19)

Let us introduce the identity
.- . _ L
(1 - w2) léiw = 611n[(l + w)(l - w) 112 . (3.8.20)

Substituting (3.8.20) into (3.8.17) and (3.8.18) we

obtain respectively

6+r = pO-lc-Zu—l(1 — w2)x’X - %(l + aw)(ln12),X;
(3.8.21)
8's = %(1 - onw)(1nI2),X - pO-lc-zu—1(l — W2)X,x°
(3.8.22)
where
r = ¢ + ln[(l + w)%(l _ w)_%] ; (3.8.23)
8 = ¢ _ ln[(l + w)%(l _ w)-%] . (3.8.2u)

 

 

 

 

72

3.9. Concluding Remarks
The one—dimensional hydrodynamical equations
(3.8.15) and (3.8.16) or their modified forms (3.8.21)
and-(3.8.22) in the Riemannian space—time can be reduced
to (2.6.17), (2.6.18), (2.6.24), (2.6.25) in the flat

space-time if we set I = IO = constant, and X = X0 -

constant.

All derivatives of I.and x then vanish and the
right hand sides of the above equations (3.8.15), (3.8.16),

(3.8.21) and (3.8.22) are equal to zero.

 

 

CHAPTER IV

APPLICATION

n.1, Gravitational Potential

 

Let us assume that there exists a celestial body

with a.magnitude of the gravitational potential at its

surface:
H
x(1) = 8.5(10 )xs . (u.1.1)
where
Xs = gravitational potential of the sun at its
surface,
x = 7.3M(10u) mi2 sec—2 . (H,1.2)

S

The velocity of propagation of light signal in

vacuo is
c = l.86272(105) mi sec-l . (H.l.3)

Using (4.1.1) and (4.l.3) to calculate the dimension-

less quantity

2c-2x(l) = 0.36 . (4.1.8)

73

 

 

 

 

74

Substituting (4.1.H) into (1.3.1) and calculating,

c-llo(l), we obtain:

0-110(1) = 0.8 , (u.1.5)
where 10(1) denotes the signal velocity at the surface of
the body.

(0) n R(l),

The gravitational potential at points R

measured outward from the surface of the body is given by
x(n) = n-lx(l) , (4.1.6)

where, R(l) = radius of the body, n = l,2,3,....
Consequently, the velocity of propagation of signals
(n)

at points R are calculated using the formula [obtained

from (1.3.1) and (M.1.6)]:

10(n) = (1 — 0.36n-l)%c. (4.1.7)

Table 2 shows the calculations of the quantities

c‘4x‘n) and c‘110(n) at the points R<n).
The quantities c_2x and c-llo, considered as functions
of the distance Y from the surface of the body, are shown

in Figure l.

 

 

 

 

75

 

 

 

mmm.0 000.0 050.0 350.0 0N0.0 momeo :mm.0 wmm.0 m00.0 000.0 AQVOHHIO
0H0.0 0m0.0 mmm0.0 mmm0.0 0m0.0 0m0.0 mzoyo 000.0 000.0 00H.0 Acvxmlo
com. 3 gm a Em m Em i. Em s Em m 3m 11 Em m 2.1 m Ex 2%

 

o
. m_caog m m 0 cm x o mm a coda m .o mQOflpmHsQ Mo m 1:.
Acvm p . 3p 8 Acv HH. 8 A20 m- Hp.p as H H gs m mqm<e

 

 

 

 

 

 

76
W 0.20-
1 0 — 0.16
0.954 Ic‘1 0.12-
-1 x
I c 0
O I
N
0.90~ 0.08
x0"2
0.85a 0.0a
0.80 . . ." r . .2 r . 0.0

 

O l 2 3 4 5 6 7 8 9

Y (one unit of Y = R(l))

Figure l.--The dimensionless gravitational potential
c‘2x and the dimensionless velocity c’iIO(0'1I) as
functions of the Y-coordinate.

 

 

77

A.2. Gas Model

 

We assume a hypothetical medium consisting only of
electrons at a very high temperature. The governing
equation of state is assumed to follow that of the per-

fect gas law, i.e.,

poo-l = Qmw'le , (11.2.1)
where
02 = universal constant = 15u5.33-ft 116f mole-lOR-l;
mw'= molecular weight of the gas, mW = (1836)—1 lbm

for electrons; .
0 = temperature of the gas in degrees Rankine;
2
)s

p '= density of the gas (1pmft_3).

p = pressure of the gas (lbfft-

The flow of the electron gas is assumed to be governed

’.

by (2.6.17) and (2.6.18) in the {Y}+space-time with the

following initial conditions at t = O:

u(Y,O) = uOEl + Y(l + Y)'l] , (11.2.2)
¢(Y,O) = §¢O[% - 1(1 + Y>’ll . (u.2.3>

where uO and to are constants whose values are given below.
The origin of the {Y}-coordinate is located at the surface
of the celestial body described in Section A.l.

At t = O and Y-= O, we assume

u(0,0)

II
C
N
O
N
u

(14.2.14)

 

 

 

 

78

O

po(o,0) p = 10‘15 lbm rt‘3 3 (4.2.5)

0

0(0,0) — 1.225(109)OR . (4.2 6)

m
G)
|

Using (4.2.l), (4.2.5) and (4.2.6) we calculate the

following quantity at t O and Y = O:

C—2pop ; = 0.115 . (4.2.7)

From (2.4.1), (2.7.6) and (4.2.7) we find that

YO < 1.61“ . (4&2.8)

As mentioned_in Section 2.9 y is considered to be
a constant. We take y to be equal to 1.614 throughout the
calculations below. 1

Integrating (2.7.7), making use of (4.2.5) and (4.2.7),

we obtain the isentropic relation:

p = K 90) , c_2KO = 1.8676(108) . (u.2.9)

Normalizing po with respect to its value at t = O

and Y = O, i.e., 0:, we write:

)6 _
oo = 000: l . (4.2.10)

From (2.7.8), (4.2.9) and (4.2.10) we obtain a as a
*
function of 00 as shown in Figure 2. Similarly, by inte-

grating (2.6.14), making use of (2.7.8), (4.2.9) and

 

 

   

 

79

*
(4.2.10) with the requirement that-0 = 0 when po = 0

(see [4]), we determine 6 as a function of 00* as shown
in Figure 3. The quantity 00, (4.2.3), is determined
from Figure 3 corresponding to the value 00: = l,
(00 = 0.354466).

From Figures 2 and 3, the quantity 0, to be used
in the calculations below, is determined as a function of
0 as shown in Figure 4.

Figures 5 and 6 represent the numerical solutions
of Eqs. (2.6.17) and (2.6.18) with the initial conditions
(4.2.2) and (4.2.3) for 0 and u for different constant

(0)

parameters IO at a particular instant t =tO for the

range of Y = [0,0.35].
‘ (n)

Using Figure 1 we determine the positions, Y = Y

= R(n) at which the values of the parameter IO = Io<n)

are chosen (see Table 1). In Figures 5 and 6, vertical.

(n).

lines are drawn at each point Y Points of inter—

sections of these vertical lines with the corresponding
curves drawn for the corresponding parameter IO = Io(n)
are determined. Due to the small range of Y for which
diagrams 5 and 6 are drawn, only one point of intersection
(corresponding to the vertical line Y = 0) is shown on
each of these diagrams. However, the numerical values of
¢ and u at the points of intersections can be obtained,
alternatively, using the diagonal numbers of Tables 3 and

4. Curves drawn through these points representing approxi-

mate graphical solutions for 0 and u in the case of a

 

 

 

 

8O

variable reference velocity IO are marked in Figures 5

and 6 as dashed lines for the range of Y = [0,0.35]. The

same curves representing the approximate solutions for

0 and u are drawn in Figures 7 and 8 for the range of

Y = [0:7Jo

 

 

k1 ‘

 

 

81

 

 

 

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82

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83

 

Figure 2.-—The dimensionless sound velocity 0 as a
function of the normalized density 90*.

.O

 

 

 

 

 

 

 

84

 

 

Figure 3.——The quantity 6 as a function of the normal-
ized density 00*.

 

 

 

 

 

 

85

 

 

 

Figure 4.—-The dimensionless sound velocity 0 as
a function of the quantity 6.

 

 

41!“

 

 

86

   

 

 

 

 

0.3.1
0.334
.0.32~
@-
I c - 0. 0
0.31- O 9 5
Ice"; = 0.938
10C 1 1.00
1
0.30 ~
0.29 1 I T l I \ l
0.00 0.05 0.10 0.15 0.20 0. . 0.35

Y (one unit of Y = R(l))

Figure 5.--Solution of eqs. (2.6.17) and (2.6.18) for 0 at the

particular instant t = t0 = 0.4 for different values of the parameter

I c--1 and the corresponding approximate solution for ¢ (shown in dash-
e line) in the range of Y [0,0.325].

 

 

 

O.(

0.

.260

 

87

 

1 c'1 = 1.000
O
Ioc’l = 0.963
« IOc-l = 0.938
I 6‘1 = 0.905
I 6‘1 = 0.800 -—
—J

 

 

 

/ ./ / /
.MHJ-::;// /;//
/9/
)7/
235—%;’
E: 0 J~—~ - r t . . .— .
0.00 0.05 0.10 0.15 0 20~ 0.25 0.30 0.
Y (one unit of Y = R(l))

p
I

J

(V)

8

(2.0.17) and
for different

Figure 6.-»Solution of eqs.
cular instant t tO 0.4

i
C 1

rt
line) in the range of Y =

[0 , 0.35].

(2.6.18)

for u at the

values of the parameter

and the corresponding approximate solution for u (shown in dash~

U-‘

 

LT]

 

 

 

 

88

 

 

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89

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.29 ~

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.230 r r l I I l
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Y (one unit of Y = R(l))

Figure 8.--Approximate solution of eqs. (2.6.17) and (2.6.18) for
u = u(Y,0.4) using the diagonal numerical values in Table 4. The vari—
ation of the curve u as a function of Y at t = 0.4 due to the variations
of the parameter IOC”1 is so small that it cannot be shown clearly on
this diagram.

 

 

 

90

4.3. Shock Model

Let us assume that the shock is moving away from
the celestial body described in Section 4.1. As dis-
cussed in Sections 2.8 and 2.9 a coordinate system {Y}
is introduced such that the shock becomes stationary and
perpendicular to the Yl-axis. As in many practical
problems, we specify the shock parameters 03 and p_ on
the right side of the shock, and we choose either the
pressurep+ on the left side or the pressure ratio a as
an additional parameter describing the strength of the
shock, (see [9]). The remaining shock parameters (pi or
n, u_ and u+) are calculated from (2.8.1), (2.9.6),
(2.9.9), taking into account (2.9.1), (2.9.2) and (2.9.3).

For a chosen constant value of the quantity p_pS-l,
(i.e., the temperature on the right side of the shock is
kept constant), Figures 9, 10 and 11 show the relations
(E, ii), (0,3i) and (0;,Ei) respectively, for different
constant values of the velocity parameter IO. The linear
relation between a; and u, for different values of I0 is
also shown in Figure 12.

In conclusion, Figures 9 and 10 indicate that the
shock parameters E and n increase as the gravitational
potential X increases or IO decreases keeping 31 con-
stant; or, for fixed values of the shock parameters a and

n, the velocity El increases as X decreases or IO in-

creases. Similarly, Figure 11 indicates that the velocity

 

 

 

 

91

ui increases as X decreases or IO increases keeping Hi

= constant.

A critical shock strength line "Ecr." is drawn in.
Figure 11. This line shows that the value of H: on the
left hand side increases as X decreases or IO increases,
whereas 31 on the right hand side of that line increases
as x increases or IO decreases, keeping 51 = constant in

both cases. The latter case has not been investigated in

the present work.

 

 

 

 

 

O

 

 

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I*I“

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95

 

 

 

 

 

1.0
—l—l _ -l
c ui — c uiIO
0.8
106‘1 = 1.00.
-1 _
Ioc — 0.95
0 6 I c-1 = 0 9
o
c—i-H
I:
I 1
0 I c_ - 0.85
0.4 O
I c'1 = 0.80V
o
0.2
0. , I I 1
0 0 0 2 0 4 0.6 0 8 l 0
c—lu

Figure 12.—-c—lfil vs. c_1u¢ for different

values of the parameter c-llo.

 

 

 

 

REFERENCES

Aller, L. H. The Atmosphere of the Sun and Stars.
The Ronald Press Company, second ed., 1963.

Chapman, S. and Cowling, T. G. The Mathematical
Theory of Non-uniform Gases. Cambridge at the
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Collatz, L. The Numerical Treatment of Differential
Equations. Springer-Verlag, Berlin, third ed.,
1960.

Courant, R. and Friedrichs, K. O. Supersonic Flows
and Shock Waves. Interscience Publishers, Inc.,
1948.

 

Eckart, C. Thermodynamics of Irreversible Processes.
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Einstein, A. Ueber das Relativitaetsprinzip und die
aus demselben geZOgenen Folgerungen. Jahrb. d.
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Fok, V. A. The Theory of Space-Time and Gravitation.
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Goto, K. Relativistic Magnetohydrodynamics. Pro-
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Hirschfelder, J. 0., Curtis, C. F. and Bird, R. B.
Molecular Theory of Gases and Liquids. Wiley,
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Krzywoblocki, M. Z. v. On the General Form of the
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96

 

 

 

 

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

 

97

Krzywoblocki, M. Z. v. Special Relativity——A
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Mises, V. R. Mathematical Theory of Compressible
Fluid Flow. Academic Press, Inc., 1958.

Moller, C. The Theory of Relativity. Oxford at
the Clarendon Press, 1952.

Pauli, W. Theory of Relativity. Pergamon Press,
1958.

Synge, J. L. The Relativistic Gas. Interscience
Publishers, Inc., 1957.

Synge, J. L. Relativity: The Special Theory.
North—Holland Publishing Company, Amsterdam, 1955.

Synge, J. L. and Schild, A. Tensor Calculus.
University of Toronto Press, 1949.

Taub, A. H. Relativistic Rankine-Hugoniot Equations.
Phys. Rev., Vol. 74, No. 3, 1948, pp. 328-334.

Tolman, R. C. Relativity, Thermodynamics and
Cosmology. Oxford, The Clarendon Press, 1934.

Vlasov, A. A. Many—Particle Theory and its Appli-
cation to Plasma. Gordon and Breach Science
Publishers, Inc., 1961.