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Dissipative Model of the Universe.

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A N N A L E N D E R PHYSIK
7. Folge. Band 46.1989. Heft 7, S. 481-560
Dissipative Model of the Universe1)
1’. X O L N ~
and
K .M. M ~ S Z ~ R O S
Institute of Physics, Budapest University of Technology,
Biidapcst, Hungary
A b s t r a c t . I n the present, study a ti-iitl is carried out to generalize the non-relativistic-dynamic
model of cosmology put forward by 0. HECKMANN. The generalization is performed under the framework of the Gyarmati principle of irreversible thermodynamics for anisotropic inhomogeneous and
viscous case. The equation of mot,ion of the dissipative Universe will be a Xavier-Stokes tensor equation leading to a Riccatian differential equation general in one dimension. The equat.ions of the Heckmaim model compatible with standard cosmology can be obtained from this by means of further simplifying assumption.
Das dissipative Nodell des Universurns
I n h a l t s u b e r s i c h t . Vorliegende Studie versuclit das von 0. Heckmann ausgearbeitete niclit
relativistkche-dynamische kosmologische Model1 zu verallgemeinern. Die Verallgemeinerung bezieht sich im Rahmen des Gyarmati-Prinzips der irreversiblen Thermodynamik auf einen anisotropinhomogenen und viskosen Fall. Die Bewegungsgleichung des dissipativen Universums wird eine
Navier-Stokes Tensorgleichung, die in einer Dimension zu einer allgemeinen Riccati-Differentialgleichung fuhrt. Damus sind durch weiterc Vereinfachungshypotliesen die Gleichungen des Heckmann-?llodells zu gewinnen, die mit der Standard Kosmologie kompatibel sind.
1. Introduction
Generalizing the relativistic cosmologies t o imperfect fluids the energy-impulse
tensoi.iii the Einstein equations will have the form T p v= (u p ) . Ut‘U‘‘- pq”” 4T/””.
The first two terms of the right side define the energy-impulse of perfect fluids. The third
term presents the Lorentz invariant dissipative tensor loaded by all dissipative effecth
+
+
A TP* == -qHfi~H”’W,, - X
(
~
@
+ H ~ Y U ~-) Q6 ~ ~
~
~
” ~
a,uy.
w
(Here a i d in what follows &, := a/axl’ and the iiitlices 6, y , p, Y can assume the values
0, I , 2 , 3.)
I n 4TP” q, E 2 0 are consecutively the coefficients of thermal conductivity, shear
U p .U , is the projection tensor, W,,, =
viscosity and mass riscosity, H,,, = -glLY
2
i w , a,,u, fa.”,
. a,UY the shear tensor and Q, = ?lfZT’ 7‘%,.U,.Vct h e heat
x,
+
+
+
flux \ector (7’ is used for the temperature). By means ofAT[‘’ it is possible t o (letermine
the entropy production generated by the fluid’s motion, the vector cluartett of the entropy flux S”‘ etc. [ 11. The relativistic cosmologies, however, suffer from coritrarlictions
(anomalies, inconsistencies etc.) [21. These problems seems t o hr unresolvnble and their
conseqiiencc>sm8y reac~hhepond the scope of cosmology.
l)
Dedicated t o Prof. I. Qyarmati on his Goth birthday
4s.‘
Ann. Physik Leipzig 46 (1989) 7
0. Heckmaim developed in 1941 a dynamic Kewtonian model assuming homogeneous-isotropic, perfect fluid [3]. The equation of motion pertaining t o this model is
analogous to the f tinclamental equation of standard cosmology [I]. The Heckmann model
(HIVI) is free from the above contradictions and from the Olbers-, Clausius-, Seeligerparadoxom, furthermore i t is able to account for Hubble‘s extragalactic red shift.
The thermodynamics of the HM is reversible and is based on Boltzinann statistics.
Kevertheless, the interpretation according to kinetic gas theory is not compatible with
the phenomenological handling of the continuum’s motion in the HM. Moreover, the
collision of galaxies as gas particles, as well as supposing them to be equal, is not in
agreement with observations.
Recent measiirements and observations have shown that the homogeneity and isotropy of the Universe are not ralid. It can he seen that the applicability of the cosmological principle is pashetl further to the large and less investigated dimensions, notwithstanding the large-scale measiireineiits basetl on the Hubble-law, which is compatible
with the cosmological principle [ 41.
New observations motlifj- the Hubhle-law itself. The velocity-field of galaxies is
anisotropic and inhomogeneous. The relocity-field may be divided in monopole, dipole,
quadropole . . . etc. The monopole term corresponds t o the Hubble-law. The dipole
term correspontles to the translation of Milky Way. The others multipole terms correspond to the anisotropic-inhomogeneous velocity-field of galaxies [5].
May be if the precision of the large-scale measurements increases, the anisotropicinhomogeneous character of the velocity-field and mass distribution will also increase.
That is why reasonable devise which is a, new clynamical cosmological model uses a n
aiiisotropic-inhomogeneons continuum.
The following paradigm (letermines the conception of the dissipative model of the
Universe (DML‘):
1. the Mach principle, 2 . the feasibility of continuum physical (phenomenological)
treatment, 3. the fact that the Gniverse is a dissipative system, 4. the Gyarmati-principle of irreversible thermodynamics, 5. the preferment of new cosmological measurements, 6. the possibility t o esplain red shift in accordance with Hubble, 7. the possibility t o accept the isotiopic SIC black body radiation as background radiation, 8. the
possibility t o interpret the measured (70-30 or 80-20%) H-He proportion by element-genesis extended to the entire metagalaxy, 9. the Occam razor: the maximum
simplicity of the model, no new physical lans are introduced.
2. Creation of the System of Axioms
Using asiomatic treatment the basic concepts (in the present case e.g. iinperfect
fluid, Euler’s-tlescriptioii, Gyarrnati-principle, entropy-representation etc.) need not
he defined, however, in order to (leal with their physical properties it is necessary t o
set out certain postdates for the basic concepts.
The axiom system of thp model to be introduced seems to be a classical system of
axioms bedded into a paradigm, hecause maximum “completeness”, inclusion of all
“obviously true” elementary facts, optimum characterization of “relevant reality” is
aimed a t [(;I.
The presenting axiomatically, the (libcussion becomes purely mathematical antl also
the statements are formnlatecl on this basis.
In the first step it is necessarjr to yieltl tlefinition for the concept of Universe, creating also a bridge between the paradigm antl the sFstem of axioms.
P. MOLNAR, M. M$SZ~ROS,
Dissipative Model of the Universe
483
P o s t u l a t e 0 (Principium)
There exists only a single Universe and this is identical with the physical universe
as a system unifying the megascopic-submicroscopic levels.
The Principium excludesall interactions withother material systems (isolated system),
however, does not exclude self inter-action [ 71.
The Universe defined in this way can be treated as continuous medium and point
system too. The classical mechanical description of point-system-Universe has been
given by Treder [ 81. Confining oiirselves t o continuum-physical analysis, the most simple
dni
system in continuum-physics appears t o be the dV = - macroscopic unit of volume.
e
The “macroscopic” attribute points out that therearesufficient number of microparticles
in such a unit of volume in order that all macroscopic quantities like @-massdensity,
p-pressure etc. should have here definite values (cellular equilibrium) and these should
change continuously a t the borders of the volume units. The particles can leave the dV
volume units exclusively in consequence of their statistical motion.
The model t o be defined below aims t o describe the inegascopic behaviour of the
system belonging t o the system of the Postulate 0.
P o s t u l a t e I (D-continuum)
The continuum of the model t o be designed is determined by the D = D(D,, D,,D,,
U,) imaging, where the following mutual unequivocal correlation are generated : d V
macroscopic volume units and megascopic volume units with edges of lo8- lo9 light
year lengths correspond t o D,, microparticles and galaxy groups refer t o D,,statistical
movements and peculiar movements correspond t o D,,the d V megascopic volume units
constructed in this way and the points of a Euclidean space of variable metrics are
related by D,.
The Universe “smeared out”, averaged for the dl’ cells, can be assumed merely
using the D, above.
P o s t u l a t e I1 (Axiom of permanency and continuity)
(i) Permanency: Any deformation can be represented exclusively by single valued
and almost everywhere infinite times continuously differentiable transformation. For
the Jacobi determinant
of this transformation 0 < J < co . A material quantity found in a finite and positive
d B volume can not be deformed t o zero or infinite dV‘ volume: dF“ = J d V .
(ii) Continuity in time : Any transformation expressing motion can be differentiated
according t o time continuously infinite times.
I n classic space theories Postulate I1 has basic importance [Y].
If the description is confined t o Eulerian ideas, the motion of the continuum can be
determined in respect of coordinate systems fixed in space. Remaining at the study
of the relation between single coordinate systems, for three points with different spacial
coordinates in the continuum D, the volume units should be dV,, dl’,, d V 3 . Investigating
the relative movements of these it becomes evident, that after averaging the local- and
convection-movements each d V volume unit is at rest in respect of the environment.
Coordinates of dV, relative t o dl’, are r and relative t o d V , are r‘.
Using the identities
r
= r ( x , , x,,
’
I
f
x3), r’ SE r’(xl, x,, x 3 ) , a
= a(ul,a,,
u,), r , r
,a E R3,
(1)
484
Ann. Physik Leipzig 46 (1989) 7
the relations of the three coordinates expressed in components will be
xi = ai
+ xi.
Differentiating ( Z ) , the velocity of dV, is obtained as
dai
I
vi = dt + vi.
(3)
Due to ( 3 ) one can write
The dV-s, the individual coordinate systems, are static in respect of their environment,
thus
vi(r, t ) G 0, if
r = 0,
vi(r’, t ) = 0, if r’ = 0,
(5)
oi1 1 ( r ” ,t ) = 0, if r” = 0 ,
etc.
I
Inserting the second equation of (5) into (4)
is obtained.
Applying the algebraic equation r’ = r - u and (6), one can write
vi(r - a, t ) = vi(r,t ) - vi(a, t ) .
(7)
Recent measurements and observations have shown that in the Universe there are
contiguous objects of (3- 7) x los or lo9 light year extension [4]. This fact is expressed
by the following statement.
P o s t u l a t e 111(Inhomogeneous D-continuum)
The e(r,t ) , p ( r , t ) and vi(r,t ) dynamical parameters of the D-continuum violate the
spatial homogeneity
e ( r ,4 =t. e’(r’,t ) , ~ ( r4 ,?= ( r ’ ,t ) and 4 r , t ) =I=
ui(r’, t ) , if r $. r ’ . (8)
1.e. the mathematical form of the functions
p ( . )and vz(.)is not identical with
those of e’(.),p’(.)and vl(-)(the prime distinguishing these functions cannot be omitted).
(8) means that the e mass density, p pressure and vi velocity are not everywhere the
same.
@(a),
Theorem 1
For inhomogeneous case relation (7) can not be satisfied in space coordinates by any
class of linear functions.
Proof 1 L e m m a (i): If g is linear and g ( s & y) = f ( x )ff(y), then g f f . Proof (i)
Provided g is linear and g f f the equality g ( x IJ, y) = g(z) & g(y) is gained. According
t o the lemma g(s & y) = f(s)ff ( y ) , from which for all s, y f ( s ) 5 f(y) = g(x) & g(y).
From this equality if follows for positive sign, that f ( x ) f ( z )= g(x) g(z), i.e.
Zf(x)= Zg(s). Thus / ( x ) = g(x), what is contradiction. I n this way lemma (i) is proved.
L e m m a (ii): I f f is linear and g(x & y) = f ( x ) & f ( y ) , then g =f. Proof (ii): g(s t y) =
f ( z ) & f(y) = f ( x & y). L e m m a (iii): I f f
g and g(z f y) = f ( z ) f(y), then neigther
f nor g are linear. Proof (iii): By common denial of lemmas (i) and (ii); ( i ’ )If I/
f
then f is not linear, (ii’)If g + fthen f is not linear.
+
+
+
+
+
P. MOLNAIL,M. & I ~ s z ~ RDissipative
o~,
Model of the Universe
48.5
C o n s e q u e n c e 1: I n the case of inhomogeneous spatial distribution of the dynamical
parameters vi(r, t ) can not be written in the homogeneous-linear form
wi(r,
2
CiZ(t)%,
1
t) =
(9)
used by Heckmann. (9) can be applied only for homogeneous anisotropic D-continuum.
Namely, in this case the form of ( 7 ) will be vi(r - a , t ) = q ( r ,t ) - wi(a,t ) and thus can
be forniulated according to (9).
In (9) and in what follows the indices i, j , k , I , m , . . . can take the values 1, 2, 3 and
the summation is carried out always from 1 to 3.
Recent measurements and observations have revealed anisotropy in the red shift
around the Local group [5]. Thus the following seems to be reasonable:
P o s t u l a t e TV (Anisotropic D-continuum)
Relation (7) is satisfied in the inhomogeneous D-continuum by the anisotropic velocity distribution
vdr, t ) :=
2
C i d T , t)fdr, t ) ,
1
(10)
generalized from (9) and the distribution of the dynamic parameters e(r,t ) , p ( r , t ) is
anisotropic, too.
Utilizing the v ( r ,t ) = A ( t ). r form of the v = A . v homogeneous linear function
(homogeneous linear form), it is easy to confirm that (10) is not homogeneous and linear
in the space coordinates (the form is not homogeneous linear in the changes of space).
It follows from Postulates I11 and IV that the cosmological principle is not valid
and thus there are no equivalent coordinate systems.
However, (10) seems t o be too general for the investigations.
P o s t u l a t e V (Velocity space)
The relocity space of the D-continuum is determined - during the motion - by
v d r , t ) :=
cc d r , t ) q ,
(11)
1
which is gained from (10) reducing it to f l ( r ,t ) = zL, where dim{cil(r, t ) } = 8 - l .
D e f i n i t i o n I (Metrics of the D-continuum)
The square of the arc element of events dF, and dB, are defined by
Z2(r,t ) :=
2
( 12J- c d r , t)q dt)2,
a
derived from (11). Here $ c t Z ( r t, ) dt is the second order metric tensor of R3.
It is easy t o prove that the normalized space (R3,
Z(r,t ) )satisfies the basic properties
of metric spaces with the arclength (norm) E(r, t ) :=
c%Z(r,t)zl d t y ] l / z . This
[7( F
is trivial for r = r ( x l , x2, z3)and for t it becomes evident using the definite integral
p
1,
(.) dt.
I n case of infinitesimsl distances the elementary arc will be
dZ(r, t ) =
[
(T $ ciz(r, t )
d X l dt)2]1’2.
Definition 1 and (12) represent an Euclidean space with variable metrics.
(12)
48G
Ann. Physik Leipzig 46 (1989) 7
P o s t u l a t e VI (Covariance of the elementary arc in the continuum D)
The (12) elementary arc is invariant against that x -+ x ‘ group of coordinate transformations for which x and x’ are idle in respect of their own local environment in D.
(11) is also a non-homogeneous linear form in the space variable.
The following statement provides for the control of the physical system supposed
in Postulate 0.
P o s t iilat e VII (D-coiitinuuin with irreversible thcrmodyaamics)
I n the model to be defined the behaviour of the continuum D is determined by the
Euler-Lagrange equations
which belong to the 6
dp dV = 6
J’ ($
-
Y ) d B = 0 integral principle according
V
V
t o Gyarmati [ 101 written in the entropy representation of irreversible thermodynamics.
Together with the integral principle of Gyarmati, the conditions of its derivation
were postulated, too, e.g. the principal theorems of thermodynamics, the Gibbs-relation,
the equalities of entropy- and masvbalance etc . The &variations occur according t o
the velocities vi.
Postulate VII represents phenomenological thermodynamical description and this
is compatible with the treatment according t o continuum physics.
Postulate VII is in agreement with Postulate 0, because it is alwayspossible to couple
a n L integral to isolated systems. The first variant of this is an extreme value invariant
in respect of group transformations (Noether’s theorems I and I1 [ll]).For example
in case of discussion of Universe as a point system Treder used the Hamiltonian-principle
[81.
The flow velocities vi(r, t ) belonging t o the impulse are intensive quantities, because
they extinguish each-other when the various dV volume elements interact. This means
that in case of homogeneous isotropic D-continuum their volume is the same in any
dV.
P o s t u l a t e VIII (The Lagrange density function of the D-continuum)
The Lagrange density function of the D-continuum has the form
= : QS - Y.
(14)
In (1 3 ) s = s(r, t ) is the specific entropy, Y = Y ( r ,t ) is the dissipation potential, Fi
=
Fi(r, t ) is the i-th component of the mass forces and oii = uj<(r,t ) is the “ i j ” component
of the stress tensor (Newtonian fluid).
Consequence 2: Introducing (14) into (13)) executing the required differentiations,
omitting the summation for index “i” and applying Fi(r, t ) f F i
487
P. M O L X ~ R31.
, M~SZ~RO
Dissipative
S,
Model of the Universe
is obtained. This is the Euler-Lagrange equation, or the Cauch,~equation of motion,
n hich is the continuum niechanical tlifferentiatetl forin of the tl’Alemhert principle.
The geodetic equation (15)expresses locally the coiiserrat ion of inipulse. The Cauchy
equation of motion is the most general non-relatiristic equation of motion in the mechanics of continuums, because it is valid for any continuum (perfect, suffering from friction
and turbulent etc.), any stress tensor (Newtonian fluid: gt7= g , ~stc.), ally field of
force (electromagnetic, gravitational etc.) and any velocity field. Deriving (15) fJ*Olll
( 1 3) and (14) means the incorporation of the Cauchy equation of motion into the franiework of thermodynamics.
Therefore (14) is the most general tliermorlynamical 1,agmnge-tlensity function of
continuum physics.
I n the following section of the system of axionis the tlefinite foim of the physical
quantities present is the geodetic equation - among them the equation of niotion (15) will be given.
C o n s e q u e n c e 3 : (i) From Postulate 11 follows the existence [9] of the local balance
equation
a,o + 2 2t(p5i) = 0 pertaining t o (13). Using (11) and p = ? ( r ,t ) the
7
ct
i
equation
of mass conservation mill be
which is one of the secondary conditions belonging t o Postulate TIJ. (ii) The total rest
(visible) inass of any arbitrarily chosen tloinain T in the Universe is entirely continuous
in the case of M = M ( V ) :
lpfT(t) =
j”p(r,t ) d
~
.
T
Differentiating this in respect of time and introducing @ ( r , t ) from (It;), clue t o Postulate 0
ciixi 2$
,o
zi E L C , l
9 2 C i i ] dV :=
~,,I,,,SE(t)
=
s
V
~
-
[
~
+ c
E J
~
~
+
i~
..- 0
~
(17)
is obtained for the mass hahiice of the Universe.
Applying the forin
of the dv,/dt substantial acceleration of the total hytlrotlynamic time differential written
in components and the relation
liTi(r, t ) =
(191
-&@(r, t ) ,
F(r t )
rewritten in components from -= F ( r , t ) = -grad @ ( r ,t ) for (15)
112
is obtainetl. 1ntroc.luci~igthe form (11) of
clevicling by ,o we get the equation
vi
a i d tt,(r, t ) =
2
c b i ( r , t)rlinto
1
(20), then
q
Ann. Physik Leipzig 46 (1089) 7
488
Differentiating (21) in respect of .ri and slimming up for ‘‘i”
(22)
resalts.
It has heen establislietl 1-,-meastireinents that the Hubble constant changes in rlirection antl \ \ i t h distance, thus in a given instant it can be iieighter constant [<>I.According to other measurements the Metagalasj- rotates antl this reveals anisotropy with
four or two poles [la]. Dne t o this the folloniiig assumption seems t o be reasonable:
P o s t rilat e I S (Separation of cglaccording t o r and t )
The secoiitl ouler tensor c z l ( r ,t) IS R qiiasi homogeneous linear function of the “8”-th
component of the Hubhle 1-ector H ( t ) (“reciprocal life time vector”)
c d r , t ) :=
s6I/\(r).
s
(23)
lJA(t)>
in which r , [ ( r ,t ) can he called “Nubble tensor”, irhile bzls(r)can be given the name
“Rubin-Ford teiisor”. b7[$(r)
lins 110 dimension ant1 tlim{Hs(t)) = 8-l.
Separation (23) is a iestriction of mathematical character in respect of structure (form)
of description nnrl does not interfere with physical content.
Tt can be seen that rneaburemeiits [ 5 ] antl [ l a ] are compatible with (23) because in
tlieee measurements tiine relations do not occur. In isotropic caseH,(t) = H,(t) = H 3 ( t )==
H ( t ) is the Hubhle c o n h i i t [:$I.
Introciitcing ( 2 3 ) into (22) ailti separating 2’
S
6,,,H, -’,1 .Y
12 .ri 2,b,,, fj, t a,Y,r
2X hLA
. b , ~ , H , f ft
, 2 .4,~
a h , fI,H7
t,s,r,k,l
7.8
i
2 J bmXi 8 h , H8H, +
P
i
- ~ . l b ~&btlr
zs
HSfir
1 , v J
(24)
(2a)
P. M O L K ~ R&I.
, M~SZ~RO
Dissipative
S,
Model of the Universe
489
(24) can be written as
(27)
Direction of the further generalizations is given by the statement below:
P o s t u l a t e S (D-continuum dominated by viscous matter)
In case of interaction between the dV volume units of the D-continuum, the impulsive
current will be given by
+
u J r , t ) := - p ( r , t ) 6ij &(r, t ) , &Jr, t ) = 0, if i = j .
(28)
In (28) the symmetric S17(r,t ) is usually termed viscosity or (internal) friction tensor.
Sgj(r, t ) is the conductive part of the impulsive current and this is not related t o the
mass flux of the moving D-continuum.
P o s t u l a t e X I (Solid, liquid and gaseous state of D-continuum)
The continuum D is solid or liquid or gaseous regarding internal friction (viscosity)
and the order of magnitude relations between the coefficients of viscosity in each state
of matter are
v,('l
t)>
vu(r9 t )
v&-> t ) > q d 7 - 7 t ) > 102 To(',
t).
(29)
The 77-8 express density of conductive im pulse flux due t o unit change of velocity
and dim{v} = g em-1 s-1.
If one of qs,ill, vu is known, the other two can be determined. From (29) it is clear,
that the intensity of cohesive forces in the different states is highly different, therefore
the interpretation of internal friction should be carried out separately for each state.
Solid (amorphous-solid) D-continuum : Deformation of the continuum, inut ual tlisplac*t,ment of layers requires work against the cohesive forces. The viscosity is in this case
the shear force acting oposite deformation of the continuum D, against the relative
tlisplaceinent of layers, representing the cohesive forces operating between the d V
volume units. Liquid D-continuum: I n this continuum i t might occur that a dV volume
unit becomes disrupted from its neighbours and in this way free space, void is creatctl
among the d JT volnme units. Displacement of the volume units (displacement of adjacent
layers) takes place by transfer of dV-s into the vacancies created nearby. Probability of
this process is small compared t o the previous one. Gaseous D-continuum: The dl;
volume units of the gas layer, flowing with higher speed intercalate the layers flowing
with lower speed and vice versa. I n consequence the velocity of the faster layers is
decreased due t o steady collisions of dV-s, while the velocity of the slower layers increases ant! this interaction is reflected by the internal friction. Collision of the d?'-.s seems
t o be still less probable than the previous process.
Since the components of the Sij(r,t ) friction tensor vanish for resting liquids, these
components can depend merely on the velocity state of the liquid. According t o the
basic principle of kinematics the velocity of matter in a dV volume unit can be devidetl
into translational, rotational and deformational velocities. S i j ( r ,t ) can depend only on
the tleforinational velocities because in the event of translation and rotation the dT'-s are
not displaced relative t o each other.
The &(r, t ) deformation velocity is usually [ 131 tlefiiietf as
490
Ann. Physik Leipzig 4G (1989) 7
Confinjng the treatment t o infinitesimal linear tleformations we have
P o s t u l a t e XI1 (The form of viscosity tensor)
Components of Sij(r,t ) are quasi homogeneous linear functions of the &(r, t )
deformation velocities
is
qt,Zm(r,
t ) is the viscosity coefficient tensor [13], the scalar invariant spur of which
of 7’ are ?lihlm =
2 qzzzz
( r ,t ) = : q(r,t ) = qy(r,t ) + qz(r, t ) + qJr, t ) . Symmetries
L
qlmak
= “/k&lm= qahml.
The internal friction (mass viscosity) is a kinetic energy dissipation process coupled
with heat prodiiction antl happens t o be expressed by therinotlynamically irrevrrsible
impulse transfer pointing from the high velocity places t o the low velocity ones. By
means of (28), (30), (31) antl using the symmetry properties of 9Izllrn, the dissipation
potential of !P defined in (14) will be
Taking into accoiint ( l l ) ,(19) antl (2:3)
(33)
is obtained for the specific entropy defined in (14).
div Js = g 2 0 will become with (33)
The entropy production [LO]
+
e 2 (2b i l , b ~ z , I ~ s H+
s x ~ 2@
i,j
z,s
) + div J , = o 2 0 ,
dv?
(34)
which is the other secondary condition belonging t o Postulate VII. IVithin the framework
of the theory, the following relation will he valid for the friction force f i
fi
-
1
ay/ =+ 0 (local form)
dVi
d J’ Y / d V
=
-
2 J’rj dvj CEV
i
or
(global form)
(35)
P. N O L N ~M.
R ,M I ~ S Z ~ R ODissipative
S,
Model of the Universe
4!)1
Inserting first (11) into (30), then the resulting relation into (31) and applying
(28)
is obtained for the second term on the right side of equation (27).
Defining a n Pi(r, t ) function from the first terms of the right sides of (27) ant1 (36)
Fj(r, t ) := -z~?#J
+
C (aie-1) ajp6ij
j
+C
j
Q-1 aiajpbi,
(37)
results.
(In what followx the function F i ( r , t ) represents invariably the right side of (37).)
The second term of the right side of (36) can be written in the form
and then the definition of Gi8is unequivocal. Inserting the relations (37) and (38) into
(27) the tensor equation
results, which determines the behaviour in time of the continuiiin D a t any point r .
(39) yields local description and therefore boundary conditions are not needed. Due t o
this fact application of equation (39) to the entire Universe seeins to be meaningless.
The fundamental system of equations of the continuum D consists of three equations:
(i) the continuity equation (IG), (ii) the equation of entropy production (34), (iii) the
equation of motion (39).
492
Ann. Physik Leipzig 46 (1989) 7
The initial condition belonging to (39) is
H,(t,)
= El:.
(40)
Comment 1
Setting other axioms instead of 0 antl I among the postulates 0, I-XI1 and interpreting via new assumptions the vector H ( t ) given by (23), the system of equations
(16, 34, 39) might characterize any continuum with irreversible thermodynamics.
Using (23)
vi(r,t ) =
ZI:cil(r, t)xz = C [biza(r). H,(t)I. xz
(41)
1,s
is gained for the velocities (ll),which can be devided into symmetric antl antisymmetric
parts
(42)
The first term on the right side of (42) is a component of the deformation tensor
1
E ( r , t ) := - [ C ( r , t )
C ( r ,t ) ] . while the second term leads to the E ~ mi
,
components
2
of the tensor products E @ r , co @ r between the angular velocity pseudo tensor
1
w ( r ,t ) := - [ C ( r , t ) - k(r,t ) ]and the r position vector. The measurements of Jodrell
2
Bank [la] which render possible the four pole anisotropy and the measurements [9]
supporting (23) are coupled within the framework of the theory by (42) and o j ( r , t).
+
From (23) and (42) there results for the second order tensors
where
(2)
E(r,
(2)
t ) and ~ ( rt ) ,that
49 3
P. M O L N ~ RM.
, MEszb~tos,Dissipative Model of the Universe
P o s t u l a t e XI11 (Meaning of the H(t) Hubble vector)
The s-th component of the Hubble vector H(t) has the following relation with the
s-th component of the cosmic vector scalefactor R(t)
Rs(t)represents the dimensionless measurement standard giving the time variation of
the geometric scale of space in direction “s”.
C o n s e q u e n c e 4 : Due t o (44) the tensor equation (39) is a Navier-Stokes equation.
Applying (44) the form of (39) will be
. .
BierRs Rr i,s
i,s,r
Rs Rr
R
2 Gi, -2
+ cF i
i,s
R,
i
(45)
and the relevant initial condition can be written as
Rs(to)= R: and Rs(to)= R:.
(46)
Using (44) it can be seen that in homogeneous isotropic case (41) will be wa(r,t ) =
!@xi,
4 4
what is in agreement with the statement of the Heckmann model for vi(r, t ) .
Exploiting (13), (23), (41) and (44)
is obtained for the arc length (12), what would correspond in the Heckmann model ([ln 1 R(t)I
~ 1 2 4 ~ ) I~n ’(17)
~.
carrying out consequent computation - to l(r, t ) =
+
R,(t) 2 1, meaning the anisotropic expansion of the. sphere with unit radius. Due t o
term 6) of the Paradigm the extragalactic red shift of Hubble can be given by
What was said above would describe the behaviour of the continuum D in case
of any field provided i t has a potential.
P o s t u l a t e XIV (D-continuum with Kewtonian gravitation)
I n the D-continuum the potential @ ( r t, ) is determined by the Newtonian gravitational law, i.e. summing the first terms of (37)’s right side for ,‘i” one will obtain because
of (39) and (44)
where G is the gravitational constant. (48) excludes the selfinteraction of the system.
From Postulate XIV i t can be seen that D is a single component continuum. Based
on (29) the probability relations of three independent processes are
Ann. Pliysik Leipzig 46 (1989) 7
494
1
Instead of (48) one could have introduced e.g. the relation
z
&a,@ = 4 n G ~+ - @
12
as well. I n this case the form of the force law would be
F = -G . W L . M . r-2 exp(--/1} (I r/).). Due t o (48) the time measurement is referred
t o a n efemeris tiinescale [ 141.
The 7-th and 8-th terms of the Paradigm can be realized in a D-continuum with
two components (electromagnetic and gravitational). Then di =
dig and (48)
4n Gp,.
has the form 2 8&@ = (432 eo)-l pem
+
a
+
+
D e f i n i t i o n 2 (DMU)
The dissipative model of the Universe is defined by t h e solution of the initial value
problems (39, 40), or (46, 46), belonging t o Postulates 0, I-XIV.
Fig. 1. illustrates - based on relations (41) and (42) - the velocity, deformation
(viscosity) and rotational relations of the Navier-Stokes tensor equation (39) determining t h e time dependence of H , in any point r .
Fig. 1. Tlic velocity, deformation and rotatiold relations of DMU for arbitrarily chosen coordinate
system and d V volume unit
3. Farther Consequences
Knowing the ei(r, t ) velocity field, the
X i ( r , t ) = grad ei(r,t )
cosmic thermodynamic forces can be calculatetl.
P. MOLX~R,M. MESZ~ROS,
Dissipative Model of the Universe
495
The cosmic current density (flux) conjugated to X i will be
t ) , qL(r,t ) , aiid q,,(r-,t ) viscosity coefficients
Depending on the magnitude of the qS(rr
of the continuurn D,if the mean speed of cosmic flows exceeds a definite GCrit(q)value,
the small disturbances go over into vortexes and in this manner the flow will become
turbulent (see Fig. 2). Treder investigated isotropic turbulences and gave estimation of
relaxation time [15]. I n the continuum D - clue to the large scale of the motions turbulencies exist on a large scale which cause similarly to the mass density e ( r ,t ) and
pressure p ( r , t ) inhomogeneities and anisotropies in temperature T = T ( r , t ) . The temperature distribution of the continuum D reveals similar properties as the velocity clistribution. Due to free convection the anisotropic-inhomogeneous I'(r, t ) tends to even
distribution in the various dV volume units.
Introduction of local temperature is carried out using the substantially derived form
[ 101 of the generalized Gibbs-relation
I n (52) the a, specific values of the f independent extensive A , quantity determine the
s =s
a2, . . ., a,). When rl= T-l, ul = u, and I
', = -T-lI'? are used (52) will be/
come s = P 1 u
-
7l-1
2 r;"ui,
where u is the specific internal energy. Restricting the
i=2
treatment to one component, in respect of heat conduction amorphous solid-approximation
(53)
is gained. I n the thermodynamics of tleforinations the first principal theorem has the
form [13]
du = dq
+2
IIL~
(54)
dt,),
%j
where dq = dq(r, 1 ) is the irreversible specific heat-influx. The substantial balance equation compatible with equation (54) are
eu
+ div J,, = u&2 0
and eq
+ div Jg =
II*
2 0,
(55)
where J,, is the internal energy flux and Jg is the heat flow [lo]. 4. must be determined
by measurements and assumptions. Introducing the form ic = q
2 otl&t of (54)
+ i,j
and (ll),(23), (28), (30), (3 t) as well as the derivative of (33) into (53) we get
Ann. Physik Leipzig 46 (1989) 7
496
X a$irn
+
C XfiHsHnbijS anzb~kn-1 C ITsHnb~msbijn]]
k,S,Ik
8,t&
(56)
by elementary calculations.
Due t o Postulate XIV the temperature (56) results froin the dissipation of the work
done against gravitational cohesive forces and i t is not identic with the temperature of
the measured 3K black body radiation [16]. Taking into account the 7-th and 8-th theses
of the Paradigm, in case of multicomponent - electromagnetic, gravitational - systems,
t h e 3 K temperature of the black botly radiation should be derived from the dissipation
of the work accomplished against electromagnetic cohesive forces. Mathematically i t
is possible that high spatial density weak radiation sources should dominate a general
radiation space without showing up in the radio-astronomy source investigations
carried out yet [17].
The drift-field (deformation) of t h e continuum D can be seen on Fig. 2 - referring
t o the above said facts.
Fig. 2. 1
domains
chosen
Co m m e n t 2
I n the elaborated model the measurements and assumptions (input) are coupled
t o the quantities t o be determined (output) by t h e flow chart in Fig. 3.
Returning to the initial value problem (39, 40) which determines the time dependence
of H , and introducing the symbols
(57)
P. BIOLN~R,
JI. M E S Z ~ R O SDissiptirc
,
JIodel of the Unirersr
497
The Model Established
Measurements and
Assumptions
I
Determinable
Buantit ies
I
r
Fig. 3. Flow clinrt coupling the measurements ,incl nssuniptioiis (inpiit) to the qiiantities which can
be determined (output)
the Sarier-Stokes tensor equation (39) can be w i t t e n in the following forin
2 ASH, + 2
B~JIJ?,- 2 G,Hs = 2 F,.
1,s
S
S
(:JY)
6
Transforming the B,, second order tensor t o diagonal foriii (Bls-+ B,, A , -+ as, G,
F , + f 8 , Ifl-+hi) and omitting the summation for inrles "s"
-+
qs,
+
a&
b8& - gJgs = f ,
(59)
is gained, which is the one tliinensional form of (39). These forin three equations for t h e
three unknown h,(t) functions. Dividing (59) by a, and neglecting the inclices as \tell
as introducing
and reordering in an arbitrary point r the general Riccatiaii differential equation is
obtained
+
+
h =ah2
P(t)7& y ( t ) ,
which can not be solved by means of quadrature in general
7t(t,) = h0 belongs t o (61). Using the equivalence
[le].
(61)
The initial coiidition
one obtains the equation
5 - B(t)& + "Y(t)!f = 0
(63)
from (61). The second older linear differential equation (63) can be written in the SturmLiuville form
1
i Q ( t ) x = 0 , nliere ~ ( t :=
)
(63)
- --2(t)
xy(t)],
4
if the Iliuville-traiisforni at 1011
+
+
'
(65)
is applied. Froni the qualitative theory of second order equations it is nell known that
if beside t, 5 t < 03 and E > 1/4, Q ( t ) 2 k/t,, then the equation (64) is oscillatory.
If to 5 t < 03, furthermore Q ( t ) < J/4 t2, then the equation (64) is non-oscillatory [19J
Since the coefficients a, p(t) and y ( t ) are functions of the space coordinates, the (64)
498
Ann. Physik Leipzig 46 (1989) 7
differential equations belonging to t.n o arbitrarily chosen rl and r2points can be investigated also referring to the Sturm's Comparison Theorem [ 191.
Assuming that the p(r, t ) pressure, Q ( rt,) density and q2yLm(r,
t ) viscosity are time
independent, however, ha\-e anisotropic-inhomogeneous distribution, one can see from
(00) that in any point r , P ( t ) = -1 constant and y ( t ) = y = constant. Thus (64) has
the form
z=o,
(66)
[
what is the incompressible-approximation of the continuum D . This is in agreement
i + ay--p
a
9
1
with the amorphoussolid approviniation. I n this case only stationary material flows can
occur in the Universe. In a given point r the solutions of (66) can be subdivided into
three types according to the fact that P 2 > 4ay or f l 2 = 4ay or P2 < 4ay. Solving (66)
antl using (65) as \\-ell as (62) the solutions of (61) are
1
p.
2
Solutions of the (43, 46) initial value problem are taking into account also (44),
(67, 68, 69) and the principal axis transformation R,+ 9,
It
= - f4ay -
-s,
-_
1
+
(iii) P 2 < 4ay, 9 = c3e 2a sin " ( x t
v).
(72)
Graphs of the solutions (G7, 68, 69) and (70, 71, 72) for some particular cases are
shown in Figs. 4 and 5. The el, c2, c3 antl cp integration constants should be chosen according t o the initial conditions.
I n order t o determine T ( t ) [67, 68, 6!1] must be inserted into [66].
Theorem 2
Let t
> - C1and p ( t ) = const, y ( t ) EE const. Then
for
p2 = 4ay the he = -/3/2a equi-
c2
librium solution of the differential equation (61) is globally uniformly asymptotically
stable in case of t > -cl/c2.
c
P r o o f 2 . (i) Let be c2 > 0 and to := - 2.
From the first equation of (68) it can
c2
be seen that the variation of c1 does not influence the stability because it leads to the
shift of the h(t) function along the time axis. Therefore it is enough to consider the
influence on stability of the changes of c2 value, because the variation of the initial
d u e (or the perturbation) is occiiring ria the change of c2. From the form of h ( t ) i t
P. MOLNAR,If. M~SZAROS,
Dissipative Model of the Universe
can be seen, that independent'ly from the value of c2 ( > O )
means the asymptotic stability of
P [20].
he = - 2n
499
lim h(t) = -
t - to->m
B
-,
2DL
what
I n the case investigated (61) is an
autonomous system, therefore from the asymptotic stability follows its uniformly asymptotic stability [all. Since Ih(to)- he(to)I < 00 also consequently that lim h(t) = he,
t -to -00
Fig. 4. Graphs of the solutions of h ( t ) for a selection of constants as follows:
..
mtq
Fig. 5. Graphs of the solutions R ( t )for a selection of constants as follows:
...
500
Ann. Pliysik Leipzig 46 (1989) 7
the uniformly asymptotic stability of he is global [20]. (ii) For c2 < 0 the proof is analogous to (i).
B
-&
Using he = - 7and (44) RJt) = c3e 2 s will be gained and from ( 4 7 ) one could
201
see that the expansion is linear.
4. DMU and HM
In the case of homogeneous-isotropic perfect fluidum free of pressure we get
J
p ( r , t ) = qijLm(r,
t ) = 0, M ( t ) := 0 .
Abandoning est,ablished thermodynamics, i.e. assuming (15) instead of the Postulates
VII and VIII, the dissipative model of the Universe changes in the limiting case of (73)
t o the cosmologic model of Heckmann. E.g. if (73) is introduced into (20) the
special Riccatian differential equation of the Heckmann model results. This corresponds
in (61) t o the extremal choice
4nG
a(.) = -1, P(r, t ) = 0, y ( r , t ) = - -e(t).
(75)
3
(74) is valid automatically for the principal axis form. Substituting H ( t ) = h ( t) R- l( t)
ant1 p ( t ) == 3M[4nR3(t)]-lfrom the first two equations of (73) into (74),then multiplying
the resultant equation by R ant1 integrating, the equation
is gained, in which E is a n arbitrary integration constant. (76) isanalogous to the fundamental equation of Standard cosmology. Treder showed that the connectioii hetween
Newtonian cosmology iising point system and Heckmann-ansatz using perfect fluiduin
is also given by (73) [ S ] .
Approximating H M by material free Universe ~ ( t=
) 0. Then (74) takes the form
+
Sr H 2 = 0 .
(77)
The solution of (77) is H ( t ) = (t c4)-l.
Using (44) and H ( t ) one will obtain R(t) = c5t c6. From the form of H ( t ) and R(t) it
can be seen that the empty Universe of HM is not static. Since HM can be derived from
DMU in the limiting case (75), the empty Universe of DMU is equivalent t o the empty
Universe of HM.
+
+
Refereriees
[l] WEINBEBQ,
S.: Gravitation and Cosmology. New Pork: John Wiley and Sons, 1972, p. 53-58.
[a] MESZ~ROS,
M.; MOLXAR,
P.: Ann. Physik 45 (1988) 155; Ann. Physik46 (1989) 153; ...;Ann.
Physik 46 (1989) 331. This paper cotains some misprints: the fundamental equation of Standard Cosmology should be K Z ce* = 8 n G u R2(3e2)-1 in the introduction. I n (1)it is cp
and Planck‘s distribution is exp [(hc/kTl)-l]in the first line of pbge 383 there.)
[3] HECKMANN,
0.: Theorien der Kosmologie. Berlin: Springer-Verlag, 1942, chap. 1.
[4] PEEBLES,
D. J. E.: The Large-Scale Structure of the Univeise. Princeton: Princeton University
Press 1990; OORT,H. J.: Ann. 11.Rev. Astron. Astrophys. 91 (1983) 373; GIOVANELLI,
R.; Hay-
+
P. ~ ~ I o L NM.~ ~R I, ~ S Z ~ R
Dissipative
OS,
Model of the Universe
301
M.: Sky and Telescope, 6 (1983); B-ATUS.U~I,
D. J.; BURNS,J. 0.: Astrophys. J. 299 (1985)
5; KOLB,E. IT.;TURNER,
M. S.; LINDLEY,D.; OLIVE,K.; SECKEL,
D.: Inner Space, Outer
Spnce. Chicago: Chicago University Press 1986; DAVIS,31.; PEEBLES,
P. J. E.: ,4nna. Rev.
Astron. Astrophys. 21 (1983) 109.
[51 RUBIK,V. A. e t al.: Astrophys. J. Lett. 183 (1973) 53; LE DENIL~T,
G. e t al.: Aatron. Astrophys.
48 (1973) 219; Nature 256 (1976) 773; KAROJI,H. et al.: Nature 267 (1976) 31; AARONSON,
&I.
rt al.: Astrophys. J. 302 (1986) 536; DRESSLER,
A. e t al.: Bstrophys. J. 313 (1987) L 37.
J. R.: Mathematical Logic. Addison Wesley series in logic, Reading, Mess. 1967.
LCi] SHOEXFIELD,
[i] HOYLE,
F.; XARLIKAR,
J. V.: Action ixt a Distance in Physic and Cosmology. Sail Francisco :
W.H. Freeman and Company 1974, and references.
[8] TREDER,H.-,J.: Elementitre Kosmologie. Berlin: Akademie-VerliLg 1975, Chap. 3.
[9] TRUESDELL,
C.; TOUPIN,R. A. : The Classical Field Theories, Hendbuch der Physik. Berlin :
Springer-Verlag 1960, Band IIIjl. 226.
[lo] GYARMATI,
1.: Non-Equilibrium Thermodynamics. Berlin: Springer-Verlag 1970; -4nn. Physik
31 (19i4) 18.
[ll] KOETHER,
E.: Gottingen Nachrichten 235 (1918).
P. e t al.: Pu’ature, 598 (1982) 451, and references; EDDINGTON,
A. S.: Rotation of the
[l?] BIRCH,
Galaxy. Oxford: Halley Lecture 1930; ROWAN-ROBINSON,
M.: Piature 269 (1976) 97; JAAKOLA,
T. et. al.: Mon. Not. It. Astron. soc. 177 (1976) 191; KAISER,N.; SILK,J.: Nature 324 (1986)
699.
[la] LANDAU,
L. D.; LIFSHITZ,E. &I.:Course of Theoretical Physics, Vol. 7, Theory of Elasticity.
Oxford: Pergamon Press 1975, Transl. by SYKES,J. B. and REID, W. H., chap. V., 5 34.
[11] DIRAC,P. A. M.: Direction in Physics. (ed. by HORA,H. and SHEPANSKI,
J. R.) New York:
John Wiley and Sons 1978, chap. 5, p. 71-92; In the Physicist’s Conception of Nature. (ed. by
RIEHRA,J.). Dordrecht: Reidel 1973; MILNE,E. A.: Proc. Roy. Soc. A 158 (1936); ibid. 1.59
(1937) 1 7 1 ; ibid. 159 (1937) 626; Z. Astropliys. 6 (1933) 1.
[lj] TREDER,
H.-J.: Ann. Physik, 43 (1986) 621.
[I61 PENZIAS,
A. A.; WILSON,R. W.:Ap. J. 158 (1969) 799; Ap. J. 1*2 (1965) 419; PENZIAS,A. A.;
J.; WILSON,R. W.: Ap. J. 157 (1969) 149; HAZARD,
C.; SALPPTER, E. E.: Ap. J .
SCIIRAML,
157 (1969) 1S7.
[li] JAAKOLA,
T. e t al.: Found. Phys. 1 (1975) 267.
[18] MISES,PH.F.-R.: Die Differential und Integralgleichungen dcr Mechanik und Physik, Vol. 1.
Braunschweig: FIiedr. Vieweg und Sohn 1961; KORN,G . 8.;KORN,T. M.: Mathematical Handbook for Scientist and Engineers. New York: McGraw-Hill Book Company 1961, chap. 9.
Sec. 6.; chap. 13. Sec. 6.
[19] LEDERMANN,
W.; VAJDA,S.: Handbook of Applicable Mathematics, Vol. IV.: Analysis. New
Tork: John Wiley and Sons 1982, chap 7. sec. 7, p. 278-286.
[20] KOUCHE,
E.;HABETS,P.; LALOY,M.: Stability Theory by Liapanov’s Direct Method. New York:
Springer-Verlag 1977; BELLMAN,
R. : Stability Theory of Different’ial Equations. Xew Tork :
McGraw-Hill 1963; CESARI,L. : Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations. Berlin: Springer-Verlag 1963; HAHN,W.: Stability of Motion. Berlin:
Springer-Verlag 1967.
[91] YOSHIZAWA,
T.: Stability Problems by Liapunov’s Second Method. Tokyo: Thc Math. SOC.of
Japan 1966, 30.
SESS,
Bei der Redaktion cingegangen am 5. Oktober 1987.
Anschr. d. Verf.: Dr. P. M O L N ~ R
Dr. M. MBSZAROS
Institute of Physics
Budapest University of Technology
H-1521 Budapest
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