Nontrivial Pfaffian forms in cosmology

Acta Phudca Hu.4arica O0 (I-4), pp. 31t1-##7 (1989)

NONTRIVIAL PFAFFIAN FORMS IN COSMOLOGY*

B. LUK �93

C�91 Remmrch l~~titute lor Phyeics 15s Budapest, H=~ary

and

G. PA�93

Konkolg Observatorg of the Hunoarian Academy of ,qcience~ 15~5 Bud~pest, Hua~fll

(Received 20 November 1988)

The compatibility of possible continuous cosmologic particle creation with thermodynamics is studied. It is found that with the usual K -- 2 Pfafflan (dQ = TdS) is compatible only in very special cases. K _~ 3 Pfaf¡ can easily be reconciled with continuous creation. Since then the full therrnodynarnic state space is accessible by quasistatic adiabatic processes, such systems show local rather than global irreversibiliW. This property may prevent Heat Death even with indeflnitely old model Universes.

1. I n t r o d u c t i o n

Thermodynamics offers a very economic way of describing physical systems.

Ir condenses the huge set of rnicroscopic variables into a few rnacroscopic ones [1].

This may quite be sufficient for a given study, and then one can avoid the compli-

cated and often even hopeless task of detailed microscopic analysis. Thermodynamic

picture is often used in cosmology, too, and here it also bridges over the substantial

differences in scales in the following way. Using available microscopic informa-

tion, symmetry principles and common sense considerations, ¡ a m�91 fomula t ion is made for the local description of matter, .then this mat ter governs the geometry on megascopic scales according to General Relativity. The first seri- ous success of this approach was the standard hot Universe [2] thermodynamically based on experience from particle physics that st the observed photon/proton ratio n r / n p -,~ 109 photons and light ultrarelativistic particles dominate both energy density and pressure from a hot plasma state upwards in temperature; then • ~ pc2/3. Formulating this as

p ~ (N~r2/90~3c3)T 4, (1.1)

*Dedicated to Prof. I. Gyarmati on his 60th birthday

Acta Physica Hungarim 66, 1989 AkadŸ Kiad£ Budapest

322 B.LUK�93 and G.PA�93

where N is the number of helicity states for light p�91 one can extrapolate back in time, until a past singul�91 approx. 15 billion ye�91 ago; for the present the predictions ave conforto to observations, including e.g. the overall decelerating expansion, time scales and He/H ratio.

Nevertheless, in the last decade it has been cle�91 recognised that the behaviour of matter cannot have been so simple during the whole past. Theoretica[ constructions suggest at least two important phase transitions at high energies: qu�91 con¡ and GUT spontaneous symmetry breaking. From observational viewpoint, some numbers �91 too l�91 in the Universe: e.g. the total b�91 number N within the cosmologic horizon is approx. 10 Ts. Combining this with the photon/b�91 ratio, S ,~ I0 s7 [3], S being about the number of photons. These huge dimensionless numbers are simply unaesthetic initial conditions in the stan- davd model, instead of simply accepting them it would be rather desirable to get explanations how they have been growing from some reasonable initial value {of the order of 1} to the present ones. For entropy one may always believe in primordial vehement nonequilibrium processes, for which natural candidates �91 the phase transitions [3]; for p�91 number the situation is more complicated, e.g. simulta- neous b�91 nonconservation and CP-violation �91 needed 14]. We do not go into the details of horizon, flatness, etc. problems here (cf. review �91 as e.g. [4]}; generally the suggested mechanism may work ir we believe in fortunate fine tunings or extreme values of unknown p�91 It is quite possible that inclusion of the new microscopic theories may solve the above mentioned problems but one should not tell that observations prove that microscopy.

A similar statement is true for the macroscopic, thermodynamic description. The thermodynamics of the QCD and GUT continua is known as f�91 as the microscopy itself is (cf. e.g. [5] and [6]), and practically all phase transition scen�91 suggested in microscopic language have been translated into thermodynamic one [7-10[. The thermodynamic behaviour is ~h�91 enough, still fine tuning or well planned initial conditions are needed [ii[. One is sometimes haunted by the feeling that sornething is fundamentally wrong in the description of e�91 stages. This ma)' be, of course, either General Relativity or thermodynamics, but the ¡ possibi]ity is not too probable. Namely, without new, axbitr�91 p�91 higher curvature terms neglected in the Einstein equation, can be expected to have effects only on Planck scale, speci¡ energy ,~ I019 GeV or density 1093 g/cm 3 [12], so they �91 really negligible even at the GUT phase transition.

The situation is not so cle�91 with the validity of the used thermodynamic formalista. We do not consider here the obvious question ir the thermodynamic equations ave correct for the particular kind of matter; they are formed from the best available microscopy. Rather we ave interested in general problems. Several aspects have already been investigated and checked:

1) The criteria for s'electing the relevant extensives are well known [1] and can be checked for the e�91 Universe [13].

2} One may expect troubles with the so called nonequilibrium thermodynamics when "cellul�91 n equilibrium {14] breaks down. This may happen for two fundamental reasons. First spatial gradients ma), be too high; then one should correct

Acta Phgdca Hunosrica 06, 1989

NONTRIVIAL PFAFFIAN FORMS IN COSMOLOGY 323

the entropy with gradient or current terms arriving at Gy�91 wave equation [15]. But in the e�91 Universe gradients of intensives are expected to have im- mediately been destroyed by the radiation. Second the time derivatives ma), be too high, preventing the development of ne�91 momentum distributions |independently of currents) [16]; then the entropy should be corrected with terms describing deviations [17]. Such calculations do exist for the e�91 Universe [18,19], with the result that the new effects �91 important only ir the GUT scah par�91 is inconveniently high from paxtich physics' viewpoint.

3) One may have doubts about cle�91 sep�91 of micro-, macro- and megascopic scales, necess�91 for the general approach. But these scales merge only at Planck density, 16 orders of magnitude above chaxacteristic GUT density.

4) There may be �91191 in the operative de¡ of adiabatic isolation, needed for defining the internal energy I20]. However, iN cosmologll, this de¡ can be made unique exploiting the total spatial symmetry [20].

So the most obvious objections �91 at least p�91 answered and still one may be dissatis¡

The present paper mentions a further possible "anomaly", practically not discussed in the literature so fax, which ma), have occurred in the thermodynamic behaviour of the matter of the very e�91 Universe and ma)" have influenced its evohtion. We do not claim its existence, we merely state that it is not ruled out by any known evidence, and call attention to the fact that it seems to be connected with a favourite problem of last century's thinking about the Universe: the Heat Death.

2. Heat death , cont inuous creat ion, and arrow of t ime

Before the advent of General Relativity in 1916 it was highly problematic to get a clear picture about the long range evolution of matter in the Universe. Conservation laws suggested infinite age, while the Second Law of thermodynamics would have led to total corrosion in such a Universe even before the present time. This ultimate corrosion was named Heat Death. V�91 mechanisms were imagined to avoid it. One such was the refusal of Second Law on megascopic scales. E.g. Poincar› expressed doubts about the valiclity of the coacept of heat on such scales [21]. However, these overstretched conceptual constructions were not too convincing mad are now unnecessary with a past singul�91 The present thermodynamic state of matter is quite conforto with its calculated finite age. In moclels containing the past singularity, Heat Death belongs to in¡ future and so to eschatology. In addition, even the closed Universe of General Relativity is nota closed systeIn of thermodynamics, since its total vohme V and energy E are not constant and so variational principles do not automatically lead to maximal entropy.

However, one ma), not be sure that the problem is completely eliminated forever. The problem ma), return in unorthodox but possible cosmologies containing either infinite pre-Planck past lincluding oscillating models) or any kind of steady

Acta Ph~~ica Hunr ~~, I~89


state with continuous creation. (For the advantage and possibility of both ideas cf. e.g. [22]).

Infinite pre-Planck past is profitab]e iŸ one wants to avoid past singularity; although the physics at or above Planck density is unknown, semiclassical approxi- mations to ~Qu�91 Gravity" can suggest static stage for the Universe there [22] of exponential expansion [23]. Then the Universe is inde¡ old. Again, peffect cosmologic principle would require 10 symmetries (including timelike ones) [24], which iR again static or exponential expansion. But then, because of the time-like symmetry, the Universe shou]d be in steady state, e.g. of constant particle density. Ir this density is not O (exact particle-antiparticle symmetry) then the total particle number cannot be conserved, a continuous creation must balance the expansion. It is highly probable that now our observab]e cosmic neighbourhood is not in steady state, but this does not ru]e out a primordial approximate steady state or steady state outside the present horizon.

Continuous creation might a]so explain the present baryon number vŸ evolution, �91 in fact, Dirac's large number hypothesis, based on the order of magnitude acoincidence ~

N ~. (e 2 /Grnp~e) ~ (2.1)

led to a set of models with continuous creation [25]. In such models a particle number generaUy considered constant in closed macroscopic (and therefore thermodynamic) systems grows on megascopic time and distance scales, and this, as shown in Section 6, contradicts familiar thermodynamics.

There is another problem related to Heat Death (and possibly to perfect cosmologic principle). It is Arrow of Time. Ir demonstrates a huge conceptual difference between General Relativity and, say, thermodynamics. In relativity the space-time is a network of events existing in four dimensions. Time reversal is aleo possible, nothing seems to prefer future to pas to r rice versa. This world picture charly contradicts macroscopic experience in two fundamental ways. The first is a very clear human feeling about the existence of a present, i.e. a set of preferred space-like hypersurfaces. (For a very clear formulation of this cf. [26]). In the Universe this set is automatically selected by the symmetries. However, even then there exists the second problem, the orientatedness of elementary steps in one direction of the map. It must be imposed as extra condition on the space-time map, and ir is Arrow of Time. Its origin is a matter of continuous discussions (cf. e.g. [27]); for any case, in thermodynamics its manifestation is Second Law, analogous to the local causality principle of relativity that every motion remains in the light cone and dt ldr > O.

An idea is that somehow the global cosmologic evolution singles out the preferred local directions [27]. Anyway, Second Law exists arid acts; ir organises the evohtions hito reasonabh ordered sets of events and in usual thermodynamics gen- erates irreversibility for any local pieces of matter.

In this paper we show that a continuous creation is compatible with thermodynamics without any probhm if the Pfaflian form in the theory is not the usual


K = 2, but K > 3 type [1] (for details see Sections 6 and 7). However, then local organising activity of Second Law will not generate a "global irreversibi]ity" for the history of matter: the evohtion stages of matter cannot be put into a time-oriented sequence.

3. The d y n a m i c s o f t h e un ive r se

In this Section we give the most fundamental formulae for the dynamics of a general relativistic Univexse. First the geometry. A Universe solution must possess fuU spatial symmetry, i.e. 6 Killing vectors acting on a space-like hypersufface [24]. Then the symmetry group is one of 3 possibilities: SO(4) (k = § closed spherical), E(3) (k = 0, open fiar) or SO(3,1) (k = -1 , open hyperbolic). The line element is

ds 2 = dt 2 - R 2 (ti {dx 2 + f2 (xi (dO 2 + sin 9 0d~2)} , (3.1)

where

sin x f o r k = + l ,

f(x) = z for k = 0, (3.2)

s h z for k = -1 .

[24]. Becanse of the symmetries the energy-momentum tensor has the simple form

T ~~ = pu 'u k + P(g{~ + u~uk), (3.3)

where pis the energy density, while P i s a dynamical pressure. Due to the contracted Bianchi identity [28] the energy-momentum tensor is divergence-free:

T~';r = 0, (3.4)

where the semicolon stands for covariant derivative. In such a symmetric situation the Einstein equation has only two nontrivial components, and Eq. (3.4) is an integrability condition for the system. Finally one arrives at 2 equations

R2 = - k + (8~I3)GpR 2, (3.5a)

+ 3(R/R)(p + ?) = o. (3.5b)

So a balance equation is obtained for the energy density pureIy from General Re]- ativity. This definite]y does not happen for the particle number density; one may take a balance equation

+ 3 ( R I R ) , = ~, (3.6)

but then the source term v i s still axbitrary and should be taken from experiments. For further comparison note that the velocity u ~ of the cosmologic flow has no spati�91 component and therefore

ur;r = 3 R / R . (3.7)

Acta Phj/sica Huncavica 66, j98g


4. Pfaman form of the increment of thermal ener$y

In thermodynamics the form of the increment of thermal energy dQ possesses a central role, because this quantity carries the "non-mechanical" changes {the symbol d stands for infinitesimal changes which are in general not total derivatives, while d means infinitesimal change of a function of the variables of the state space). Following [1] one can start from the decomposition of the change of internal energy

d E = d W +dQ,. (4.1)

Here dW is the nonthermal change; some generalized forces cause generalized defor- mations on the system. This decomposition is not necessarily unique, as mentioned above; it depends on the de¡ of adiabatic isolation, and the problem is discussed in [20]. Having de¡ dW one can proceed. In our case

ttQ = d E + pdV - t~dN, (4.2)

where the coei¡ p and q depend only on the densities

p = E / V , (4.3a)

rt = N/V, (4.3b)

and can be measured in quasistationary processes. Although these coei¡ ma), be various, the form ofdrQ is restricted into

a few classes. Namely, consider the general form

dQ=~-~~ x, (X k) dX ~, (4.4) i=1

where the functions xl ate analytic in the state space. Then, by introducing new variables one can always arrive at one of the series of canonical forros

dQ = d Q ( X K ) ; K = 1,

= ~ ( x K) dS(XK); K = 2,

= dZ{X K} + ~(X~)dS(XK); K = 3, (4.5)

etc., for K _< n [i]. In the case of one chemical component the maximum is K = 3. These forros ate called canonical Pfai¡ forms and the actual class is intimately connected to the type of thermodynamic irreversibility. In order to see this, impose a condition of Second Law type

d'Q > 0 (4.6)

Acta P¡162 H~~. ir 06, lgSg


for closed systems, and take classes K = 1 and 2. For K = 1 there is a function Q(X K) which must not decrease. Therefore by observing the extensives X i in two states of matter one may see which one is the latter stage of evohtion (unless the special case Qz = Q2 happens by chance). Similarly, for K = 2, the function S must have a de¡ direction of evolution. K = i is of little interest in thermodynamics, while K = 2 is the standard thermodyn�91 [1].

In these first two classes the points of the thermodynamic state space are �91 in�91 from at least one 8uitably chosen neighbouring point in quasistationary adisbatic processes, i.e. by cm'ves going through equilibrium st�91 with dQ = 0 [1]. Now, for K ~ 3 the state space does not cont�91 ~ points. This me�91 th�91 any point pair can be linked by means of a drQ = 0 curve. Then the condition d~ >_ 0 cannot prevent us from re�91 �91 st�91 from �91 other in spite of the preferred direction of elementary irreversible processes locally representing Arrow of Time via Second Law. In such a system, therefore, irreversibility is local but not global.

5. Cosmology wi th s t a n d a r d t h e r m o d y n a m i c s

In this Section we assume that the particle source u identically vanishes, i.e. the particle number N is constant in comoving vohmes, therefore N can be considered a constant parameter in the traditional Carath› construction. Then the number of changing variables is 2 (V and E) and the dimension of reversible surfaces is 1 -- 2 - 1. So, the existence of irreversibility leads to foliation in the (E, V; N) plane, and the usual thermodynamic formalista can be applied.

The balance equations are (3.5b) and (3.6) (the latter with a 0 right hand side); there is �91 entropy density function s = s(p, n) (by Carath› unique, up to gauge terms given in [29]), and the source density of entropy

+ 8uf;, = 8p/~ + 8.h + sur; , = (p - P)ur; ~, (5.1a)

p = (~ - p~,> - ,~~,,)I~,,.

From the positive semidefiniteness

sgn(p - P) = sg . (~r; r),

whose simplest solution is

(5.1£

(5.2)

P = p - ~(p, . ) , , ' ; . ; ~ >_ o, (5.s)

which is just the familiar case of the volume viscosity [2]. Having chosen particular functions s(p, n) and f(p, n) and some initial conditions the evolution is unique. Namely, Eqs (5.1b), (5.3) give the dynamical pressure, then Eqs (3.5b), (3.6) yield

Acta Phl/sica H=ngavica 66, 1989


the evolution of independent extensive densities, and with p Eq. (3.5a) prescribes the expansion.

In the standard cosmologic model fis taken 0, which is quite good an approx- imation for the present Universe and probably back to 1 s aŸ Big Bang; however, in earlier stages the viscosity may have been important. By means of it one can even produce steady state cosmology as lar as only the dynamics and energy density is concerned [30] (of course n ~ R -3 is decreasing, unless it is 0). As mentioned above, ir is more physical to considera model with approximate steady state fora substantial time in the very hot past. This is possible if the viscosity coel¡ is nearly independent of n and P ~ -p. Such a model (hot inflation) is shown in [8] and [9] able to solve some problems of standard early cosmology.

6. C o n t i n u o u s c r e a t i o n

Now let us turn to cosmologies where v > 0, i.e. the particles which appear to be conserved in laboratory experience are in fact continuously created on a larger cosmic scale. We have seen that such a source term does not cause any problem in the General Relativity formalista. However, the term is generally incompatible with standard thermodynamics.

In order to see the problem let us calculate the entropy production:

+ sur;r = ( p - P)ur;r + s .u =- ( p - P)ur;r - (u/T)v. (6.1)

By CP symmetry sgn(/~)--sgn(n) (changing particles into antiparticles and vice versa the energy does not change), so for any kind of creatioa the last term is negative. Still the total entropy production may be positive, but the two terms seem to be independent, so the positivity would happen by chance. Therefore the continuous creation seems to be a possible mechanism for violating Second Law, which is better to avoid. To eliminate this fundamental problem various possibilities exist, which can be classi¡ into 3 main types:

a) Both Pfaffian forro and Second Law unchanged

al) Maybe the irreversibility described by viscosity is not independent of the particle production, rather the fil-St causes the second or at least guarantees the proper conditions for it. While this is quite possible, it is hard to imagine such a physical mech�91 at least up to now none has been suggested.

a2) Maybe the process lesding to particle production should explicitly appear in T ~k (as viscous terms do); then via Ÿ = 0 a term proportional to v may be expected in the balance equation of p as weU, and so the positivity of the entropy production may reduce to an inequality for a coet¡ j u s t a s in pure viscous case of the previous Section. This is a C-field type resolution of the problem [31]; it is quite possible, but then the particle production is not a fundamental process, rather

Acta Pbl/dr Hm~gar/r 66, 1989


a simple transfer between two parts of the common energy-momentum tensor of the matter. Relativity theory does not suggest any connection between the balances for T ~k and n i, and local physics does not seem to know about such effects.

b) Pfaffiaa Ÿ unchanged, but Seconcl La~o re3"ected

Maybe we should not postulate the positivity of entropy production. Second Law comes from m�91 rather than megascopic experience: ir is questionable if ir should be extrapolated to the whole Universe. Technically everything goes as before; the structure of Eq. (6.1) is unchanged, only we do not postulate the sign of the right hand side. Now the above problem that the continuous creation may generate even a decrease of s is n o t a problem anymore.

Ignoring the sign of entropy production the dynamical equations permit any evolution of s. A limiting case is the "true ~ steady state model in which even e /n is constant. (This is caused, of course, by fortunate cancellations.) In order to see this possibility, observe first that p remains constant ir p + P = 0. Then Eq. (3.5a) gives ah exponential expansion (for k = 0 quite satisfactory for the very early Universe}, so 3R/R is constant, too. Then one can choose a constant u maintaining n =const. via Eq. (3.6). Therefore the variables of s keep constant values, so s is constant with aH its derivatives. P is constant too vŸ F~. (5.3), since f is constaat. The compatibiIity condition is Eq. (6.1), which holds ir there is a single algebraic connection among all the constants appe~ring there; this may impose a condition e.g. on the actual value of n. Then the Heat Death is avoided in the hice last century way by having an explicit term diminishing the entropy (and just compensating the usual m�91 tendency) on megascopic scales.

In general, however, in Case b) a governing principle of thermodynamics is lost. Irreversibility is no more a law but only a tendency in human environment where energy and matter ate mu•h" more concentrated than their cosmic average. A basic principle would then radically ditfer on medium and large scales.

c) PfaOian forro changed, Sec~nd Lato preserred

Suggestion b) was that an extrapolation from macroscopy to megascopy might have been unjustified, especially that of Second Law. But maybe the unjustified extrapolation happened not to the existence of irreversibilities but to the structure of the Pfaflian forro, and this alternative will be discussed in the next Section.

7. M o d i f l e d P f n m a n

Let us take the position that we believe in 1. v > 0 for any substantial cosmologic reason; 2. dQ _> 0 for themodynamic reasons; 3. the absence of accidental cancellations in fundamental formulae.

Acta Phj/#ica Hun~arias 66, 1989


Then we are again confronted with the problem of the previous Section, so something is still wrong. Maybe we have an incorrect extrapolation of the notion of quasistatic (or reversible} processes. So, let us start from the beginning.

Consider a system with adiabatic isolation, in which we want to perform Joule-type experiments, but during this the only particle component is continuously created. Then which are the reversible processes here? What is the functional connection among E, V and N during such a process?

The traditional answer was

E = E(V, N = const.) (7.1)

but clearly to another question. Now we are confronted with an alternative. The first possibility is

E = E(V, N) for arbitrary changes of V and N. (7.2)

This is equivalent with the existence of oae function constant in reversible processes:

S = S(E, V, N) = const. (7.3)

Hence, vŸ Carath› construction,

dQ = TdS >_ 0 (7.4)

(ir irreversibflity is postulated). This leads to the case discussed at the beginning of Section 4, because then any change of N ma), be reversible, which clearly needs compensating mechanisms. Thi~ seemed improbable there and note that we have no evidence for the reversibility structure (7.2). In fact, nobody actually performed Joule experiments during substantial baryon number changes.

The second possibility is to observe that a Joule experiment performed at coastant N can prove only the form

= ~(v, E, N)aS(V, E, N)+Z(V, E, lV)dN. (7.s)

Now, there remains the question ir this form is reducible to dQ = a*dS* or not. The standard literature [1] gives the following answer. Consider purely quasistatic adiabatic processes. Then there are two and only two possibilities. Either any point of the manifold is inaccessible by such processes from at last one point of its small open neighbourhood (Case 1) or any point is ay from any neighbouring one (Case 2). The distinction should obviously be done according to fundamental experience inaccessible for us. Now, Case 1 leads to (7.4} [1], not discussed further. Case 2 results in the next simplest canonical Pfaf¡ [1] of K = 3;

dQ = dZ" (V, E, N ) + t i * (V, E, N} dS* (li, E, N). (7.6)

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Hence one may return to a forro (7.5), with dZ instead of dN, which is not canonical, but more trsnsp�91 This forrn may be interpreted in such a way that dQ is gener- ated by 2 disjoint kinds of processes: e.g. "entropy ~ S changes ir "usual ~ transport processes happen, while Z does ir the creation procesa differs from that preferred by some cosmologic principles. As f�91 as our knowledge about the behaviour of p�91 creating systems reaches, such a Pfafli�91 forro is not at all impossible. Then, since al] points �91 accessible even by quasistatic adiabatic processes [1], any final state can be reached from any initial one.

Still one may postulate Arrow of Time in eltmentary processes by an unequal- ity which is Second Law. But here we meet an alternative. Either we require only dQ ~_ 0, or dS ~ 0 and simultaneously dZ ~ O. The existence of this alternative me�91 that now entropy increase (if S is still entropy) is not synonimous to irreversible evolution. (Note the analogous conjecture of [32] for economy, where the relev�91 form seems to be rather (7.5), not (7.4) [33]).

To decide the proper form of the Second Law one should use obeervational evidences rather scarce about the creation processes in the Universe. However, note that d~ >_ 0 is the weaker form; by choosing that forro two interesting direct possibi¡ exist to eliminate the Heath Death. (The signs of a and ~ �91 stiU unknown; here for definiteness' sake we postulate them positive as usual for the temperature.) Namely, one could believe e.g. in an evolution with d~ = 0. Then Arrow of Time does not single ont �91 of the possible directions of evolution, so no ultimate state needs to be approached. This case can be realised in two ways. First, the entropy may increase, dS > 0, dZ < 0, but, second, dS < 0, dZ > 0 is ;Liso possible; then the quantity S precipitately identified with the full measure of thermodynamic evolution can even decrease mimicking a rejuvenation of the system (compensated by the aging of Z) in a Universe. Both cases �91 quantitative reali#ations of the rather obscure last century objections that entropy increase was not proven for the whole Universe.

8. Modela

There remains the question how to select the actual class of Pfaf¡ for the matter filling the (e�91 Universe and what �91 the quantitative consequences po- tentiMly testable by observations.

For the ¡ qu~stion the answer is twofold: cosmologic �91 laboratory. First, cosmologic considerations �91 either compatible with the standard thermodynamics or not. Here we list three different possibilities for incompatibility. Ir, for �91 strong reason of global cosmologic principles, one is convinced about continuous creation of Dirac ot Hoyle type [22, 25] but without �91 C-¡ then, as it was demonstrated in Section ti, it is hardly compatible with K = 2. Ir one prefers a Superuniverse without evolution [34], then obviously static csses �91 excluded by Einstein equ�91 tion �91 the only possibility is to accept the perfect cosmologic principle a steady state with exponential exp�91 �91 continuous creation of p�91 (Then the observed Universe is nota representative sample of the whole, but rather a Super

Acta Phgmica Hu~arir $6, 1989


Duper Cluster.) A steady state needs unevolving mat ter for which a fully accessible state sp�91 is, indeed, very convenient, li', finally, one wants to make a pre-P|anck qu�91 Universe to avoid past singu]arity, then the matter is inde¡ old but not infinitely evolved. This can easi]y be �91 by recurrences to earlier stages in spite of Arrow of Time. Of com'se, a K = 3 Pfaffian has dynamical consequences as wel], testable in principle from observations. Neverthe]ess, these consequences are rather indirect. Namely, q does not occur in the Einstein equation p(p, n) alone thus can mimic systems with K -- 2. So, �91 astronomers' viewpoint, one should investigate the system a s a physicist, i.e. on local, microscopic and macroscopic scales.

Here we make only a short comment aSout microscopy. As lar as Boltzmann's H-theorem works, the Pfai¡ seems to be K = 2. Namely, the negative of the H-function is a quite good entropy density with nonnegative production, which is a function of the parameters of the dist¡ function, so of the extensive densities as well (at least in ce lh lar equilibrium [14]). So then there i8 a nondecreasing entropy, expressing the existence of irreversibility in the familiar way. We do not want to go into the details of the discussions about the validity of the H-theorem; this again a signal that K > 2 needs unfamiliar situations. We wiU return to this problem on macroscopic scales.

Now, in macroscopy a phenomenological local description can be used. Let us imar that somehow we have been able to grasp a piece of mat ter from the very early Universe, and now we want to rneasure the class of the PlatiCan. How to do this?

Assume that the functions p(p, n), i~(p, n) are measured. Then these quantities have to fulfil integrability conditions for K < 2. Case K = 1 is almost trivial (e.g. p = p(n)); for K = 2

p = Ts +pn- p,

T = I/sp,

~�91 = --Sn/Sp (8.1)

amd hence

p . - . ~ . = (p + . ) . . - . p . . (8 .2 )

Ir this Condition does not hold then the class is K > 2. Ir ir holds, still a K > 2 Pfai¡ can be constructed, but it is redund�91 To see this we derive some thermodynamic relations for K = 3 needed anyway. Let us start from the Pfal¡

dE + pdV - ~ d N = dZ + TdS, (8.3)

where T is the ~* of (5.6), using the familiar notation by �91 (Since ir appears in the canonical form of the Pfaffian, it is "almost" unique similarly to K = 2,

Acta Ph~~4 H~~~,~c4 eS, 1#89


cf. [29] for that case, but this question was not investigated in details.) Then, introducing densities p and none arrives at

p = z + T s + p n - p, (8.4)

T = (1 - zp)/sp, ( 8 . 5 )

p = -z~ - T , , . (8.6)

Hence, for the increment density of thermal energy,

dQ/V = (z + Ts) (dV/V) + dz + Tds. (s.7)

Now, let us consider first the case when Cond. (8.2) is ful¡ Then a K = 2 entropy density s* can be constructed vis (8.1), and then

T = T(s ; ) ,

= - 8 / T 8 . , ,

z = (8*/s*p2)(T/T' + s'p). (s.s)

But then

~ l V = ( , ' ( ~ r + d~ ' ) ) /~; , (8.9)

j u s t a s for K = 2, so the quasistatic �91 curves �91 the same, and then accessibility and therefore the K value must be the same as well. So then z is indeed redundant. There remains the case when Cond. (8.2) does not hold. Then K > 2 .

We do not have any experience about such matter , even do not know ir it may exist. On the one hand, for two functions the generic case is when an integrability condition does not hold. On the other, f o r a system of particles in thermal equilib- r ium one can prove K = 2 from the principle that pe rpe tuum mobile of second type be excluded [35,36]. Let us postpone this question until the end of this Section and proceed by accepting the possibility of such mat te r under unfamiliar conditions. Obviously, without any information ir is pointless to construct particular-equations of state. But there is an interesting limiting olas8 in which continuous creation can go without irreversibilities, C-field or violation of Second Law. Let us require:

P = p; dQ - 0 with a rb i t ra r , /v , (8.10)

Acta Phywica Hunr $6, 1989

3 3 4 B.LUK�93 and G.PA�93

(the first condition expresses the absence of viscous irreversibilities and any C-field). VŸ (8.7), hence

v{z.sp - s.z. + 8.} - O. (8.11)

The bracketed term then must vanish, so

= p § ~(8), w(8) isarbitrary, s=s(., p) i sa rb i t r �91 (8.12)

Tha t is, one of the two ~potentials" remains completely free. From Eq. (8.6)

~ - o (8.z3)

for any a and p. This is a natural enough result and one might tell that with p = 0 continuous

creation might go even in the class K = 2. This is true; however, there one could not achieve p = 0 independently of n. In order to see this, let us assume CP symmetry. Since , is negative for antiparticles, the symmet ry leads to an even 8(p, n) in n, and then/~ = - 8 , / 8 p at r~ = 0, but not at generic r�91 In contrast to this, we have seen that for K = 3 q = 0 still permits an arbitrary even function for s, and then w is even as we]l.

For the remaining intensives we get

T=-to', (8.14a)

p = tu - 8w'. (8.14b)

From the observed remnants of earlier stages maybe p(p, n) could be deduced, but it is only one part ial differential equation for two functions. T would need the clsrification of its statistical meaning to 5ecome observable, and tha t is rather obscure since the remarks above about the consequences of the H-theorem.

If, however, we still want to proceed farther, Eq. (8.14b) shows that

p = p(p, -) = p(,(p, -)) (8.18)

and from it

~(~) = ,{Wo + / ,'-~p(~') a~', o

(8.16)

Acta Phlwir Hw~u~r ~ , 198r


where Wo is a zero point constant. Eq. (8.1(5) demonstrates that a true ~-law

p -- ('y - 1)p; "y = const. (8.17)

is not possible, because then nothing would depend on n. However, continuous creation was invented to maintain constant density. Let us write

p(8(p, n)) ~ ("/(n) - 1)p. (8.18)

This equation of state for any constant no reduces to a ~/-law. Maybe at the eaxly stages there was ~/ ~ 0 f o r a long time, so p ~ - p and inflation, R ~.. e at [3]; later, with a decrease of n, ir may have switched to ~ ~ 4/3, similar to a radiation- dominated case. With the present absence of knowledge it would be pointless to try to tel] more. Obviously, a cfQ = 0 history is oversimp]ified, not exploiting the full possibilities of K = 3. It would be interesting to include cyclic processes with d~ > 0 sti]l preserving eternal youth. However, for this further investigation, and at least some physical idea would be needed.

We have postponed the question of existence of such systems to this point. Of com'se, it is impossible to give a positive answer. However, ir continuous creation played an important ro]e in the very early universe, then the mat ter must have been in a state violating Cond. (8.2). This may have led to perpetua mobiles of second type in that epoch; we do not have any evidence against their existence in an infinitely long pre-Planck Quantum Gravity history. Ir one wants to exclude the possibility of such processes even in the very remote and completely unfamiliar past, one may do ir, but then continuous creation is excluded as well.

One more very vague remark can be made here. In c]assical physics almcet all interactions follow the Le Chatelier-Brown principle, i.e. have negr feedback, or tendency for equilibration of differincis. (Electric forces at tract opposite chaxges, so neutra]izing.) Some short range ones (as nuclear forces) do not necessarily do this, but they can be regarded sufface forces, so do not violate the thermodynamic formalista. The only volume force with positive feedback is gravity, and one may in. deed find problema with the additivity of energies of subsystems in a self-gravitating system. In General Relativity gravity is not ah interaction but geometry, so the problem is formally absent. However, when the different scales merge, i.e. at Planck energy, gravity (as gQuantum Gravi ty ' ) appears even in the equation of state. So~ a self-amplifying effect may be mixed into the thermodynamic quantities and then anything may happen.

9. C o n r

Because of lack of positive information this Conclusion will be rather short. Since cosmologic models with continuous particle creation ate hopeful for explaining some global data of the Universe, we investigated the compatibility of continuous creation with thermodynamics. The result is that in the generic case continuous

Acta Ph~~ir Hcn~arim ~0, IgSQ

3 3 6 B.LUK�93 snd G.PA�93

creation is incompat ible with the usual cfQ -- Td• the rmodynamics (clase K = 2 for the Pfaf¡ Then one has an al ternative: either the idea of continuous creation is improbable (at least needing very complicated h a n d m a d e mechanism) or the Pfal ª was different in the ma t t e r of the very early Universe. Sinr with the inclusion of Q u a n t u m Gravi ty the behaviour of the ma t t e r can become very unfamiliar (and is stiU completely unknown) , we then chose the second possibility and tried to fill the Universe with K = 3 mat te r . This is possibl i in principle. Then the state space is fully accessible even by means of quasistatic adiabatic processes~ so even requireddQ _> 0 asa version of Second Law, the system does not necess�91 show gioba[ irreversibility in its history. By this way Heat Death can be �91 even in aQuantum Cosmology ~ modela with infinite past, suggested recently. Of course, some statements above �91 rather trivial in thermoclynamics, but not in cosmology.

The situation is snalogous with the causality problem of General Relativ- ity. There �91 space-times, where local conditions (traveUing always forward and within the light cone) do not gu�91 global cansality (i.e. the possibility of going back in time by complicated routes). Such space-times �91 generMly excluded by principles (Cosmic Censorship) on the basis that time travel is never observed and would lead to p�91 Simil�91 K > 2 Pfai¡ may be excluded by requi¡ g[oba[ irreversibility. However, we do not have evidences about the quantum past of the Universe, and the behaviour of the matter there, so i/cosmology needs really continuous creation, it may have.

Ir is more decent, however, to close this paper with the rather neutral (and proven) statement: Heat Death is intimately connected with continuous creation. Ir Arrow of Time exists, then continuous creation is permitted and Heat Death can be avoided ir in the Planck era the class of the Pfai¡ of the increment of thermal energy was l�91 than 2.

A c k n o w l e d g e m e n t s

One of the authors (B.L.) acknowledges that the origina/idea for using K = 3 Pfaman for avoiding Heat Death occurred to hito from a comment of Dr. K. Martin• The present work is intended to be a flrst attempt to follow a suggestion by P.T. Landsberg, given asa footnote in P~f. [1], p. 51.

]~_~erenc eB

I. P.T.Lsndsberg, Thermodynsmics, Interscience, N. Y., 1961. 2. S.Weinberg, Gravitation and Cosmology, Wiley, N. Y., 1972. S. A. Guth, Phys. Rey., D23, 347, 1981. 4. J.D.Bsrrow, Fund. of Cosmic Phys., 8, 83, 1983. 5. J.Kuti, B.LukŸ J.Pol£ and K.Szl~hŸ Phys. Lett. 9aB, 75, 1980. 6. L.Di£ Bettina Keszthelyi, B.LukŸ and G.Pa�91 Astron. Nachr., 306, 213, 1985. 7. B.K�91 and H.Schultz, Z. Phys., G'21,351, 1984.

Act4 P~~:4 H~~~r~a 6~o I~Q


8. L.Di£ Bettina Keszthelyi, B.Luk• and G.Pa• Acta Phys. Pol., B15, 909, 1984. 9. L.Di£ Bettina Kes~thelyi, B.Luk• and G.Pa• Phys. Lett., 157B, 23, 1985.

10. B.K~mpfer, B.Luk• and G.Pa• Phys. Lett., 187B, 17, 1987. 11. B.Luk• Proc. 4th Marcel Grossmann Meeting, ed. R. Rut¡ Elsevier, 1986, p. 1393. 12. B.Luk• Unpublished Theses, 1988 (in Hungarian). 13. B.Luk• and K.Martin• Acta Phys. Slov., 36, 86, 1986. 14. I.Gyarmati, Acta Chem. Hung., 80, 147, 1962. 15. I.Gyarmati, J. Non-Eq. Thermodyn., 2, 233, 1977. 16. J.Ehlers, in Relativity, Astrophysics and Cosmology, ed. W. Israel, D. Reidel, Dordrecht-

Boston, 1973. 17. B.Luk• and K.Martin• Ann. Phys., �91 102, 1988. 18. L.Di£ Bettina Keszthelyi, B.Luk• and G.PmŸ Acta Phys. Hung., 60, 299, 1986. 19. B.LuklŸ K.Martin• and T.Pacher, Astron. Nachr., 307, 171, 1986. 20. B.Luk• K.Martin• and G.Pa• in Gravity Today, ed. Z. Perj› World Scientiflc, Singa-

pore, 1988, p. 247. 21. H.Poincar› Ler sur les hypoth› cosmogoniques, Hermann, Paris, 1911. 22. J.V.Narlikar, The Primeval Universe. Oxford University Press, Oxford, 1988. 23. Ya.B.ZePdovich, Usp. Fiz. Nauk, 133, 479, 1981. 24. H.P. Robertson and T.W.Noonan, Relativity and Cosmology, Saunders, N.Y. 1969. 25. P.A.M.Dirac, Nature, 139, 323, 1937; Proc. Roy. Soc. Lond., A165, 199, 1938; Proc. Roy.

Soc. Lond. A338, 439, 1973. 26. F.Hoyle and G.Hoyle: Fifth Planet, Preface. Penguin, Harmondsworth, 1963. 27. R.Penrose, Proc. Ist Marcel Grossmann Meeting Trieste, North-HoUand, Amsterdam 1977,

p. 173. 28. L.P.Eisenhardt, Riemannian Geometry, Princeton University Press, Princeton, 1950. 29. B.Luk• and K.Martin• Phys. Lett., 114A, 306, 1986. 30. M.Heller, Z.Klimek and L.Sus--ycki, Astroph. Space Sci., 20, 205, 1973. 31. F.Hoyle, Mon. Not. R. Astr. Soc., 108, 372, 1948. 32. Beatrix Pa• K~i--g. Szemle XXXV, 229, 1988 (in Hungarian). 33. B.Luk• Acta Oec. (in print); KFKI-1988-56. 34. I.L.Roy.ental, Big Bang Bounce, Springer, Berlin, 1988. 35. M.Planck, Ann. Phys., 19, 759, 1934. 36. N.Hausen, Z. Phys., 35, 517, 1934.

Acta PhyKc~ Hungarica 66, 1989

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Nontrivial Pfaffian forms in cosmology