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. 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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. 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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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