Proc. Nati. Acad. Sci. USA Vol. 83, pp. 6245-6249, September 1986 Physics Entropy, matter, and cosmology I. PRIGOGINEt* AND J. GtHtNIAUt tFacultd des Sciences, Universitd Libre de Bruxelles, Campus Plaine, CP-231 1050 Brussels, Belgium; and *Center for Studies in Statistical Mechanics, The University of Texas, Austin, TX 78712 Contributed by I. Prigogine, April 7, 1986

ABSTRACT The role ofirreversible processes correspond- these irreversible processes in the basic evolution equations. ing to creation of matter in general relativity is investigated. It would indeed be paradoxical to consider the creation of The use of Landau-Lifshitz pseudotensors together with con- matter as a reversible fluctuation. formal (Minkowski) coordinates suggests that this creation The solution of this problem also sheds new light on the took place in the early universe at the stage of the variation of meaning of gravitational entropy. If matter is generated the conformal factor. The entropy production in this creation through an irreversible process, there is a definite relation process is calculated. It is shown that these dissipative processes between entropy and matter. Gravitational entropy becomes lead to the possibility of cosmological models that start from the entropy necessary to generate matter.* empty conditions and gradually build up matter and entropy. In traditional thermodynamics, the zero point of entropy is Gravitational entropy takes a simple meaning as associated to the perfect crystal at 0 K. Now, the reference becomes the the entropy that is necessary to produce matter. This leads to space-time structure out of which matter may emerge. an extension of the third law of thermodynamics, as now the Inherent in this conception is the view that we may ascribe zero point of entropy becomes the space-time structure out of to the space-time structures considered in cosmology (and which matter is generated. The theory can be put into a corresponding only to longitudinal modes ofthe gravitational convenient form using a supplementary "C" field in Einstein's field) a zero entropy. In this view, matter appears as an field equations. The role ofthe C field is to express the coupling entropy-carrying "contamination" of space-time. between gravitation and matter leading to irreversible entropy We have been encouraged to follow this line of thought production. because ofthe large value ofthe so-called "specific entropy" S, defined as the ratio ofthe number densities ofphotons and baryons (S - 108-10O). It seems reasonable to consider 1. Introduction baryons as nonequilibrium objects that are either in a meta- stable state or decaying, while photons are "waste products" The role of irreversible processes in cosmology has been that can no longer decay into any other form of matter. Such widely discussed (see, e.g., refs. 1-8). The "cosmological a duality is typical for coupled nonequilibrium processes (14). principle" postulates that the universe is isotropic and It is, therefore, tempting to assume that this duality has been homogeneous. If so, the only irreversible processes that may brought in by an appropriate nonequilibrium process. arise are of a "scalar" nature and involve either bulk To analyze the effect of irreversible processes in cosmol- viscosity or "chemical reactions" including the creation of ogy more closely, we started first with an analysis of energy matter. Bulk viscosity does not seem to play a significant role conservation in terms of "pseudotensors," using a (Minkow- (ref. 9, p. 594). Particle creation based on quantum field skian) conformal metric. We have described our results (13), theory was first considered by Parker (10). However, inde- but we summarize them in Section 2. In short, we have shown pendently of the microscopic description ofparticle creation that an irreversible transfer of "gravitational energy" to in the early universe, it seems important to consider this "matter energy" leads to a redefinition of matter density p problem on a phenomenological level: that is, to compare the and pressure p way in which the effect can be incorporated into Einstein's basic field equations, relating the Ricci tensor to the matter stress tensor. 1= F3p, = F3p, [1.1] Curiously, this problem seems not to have been discussed in the literature. One of the reasons may have been the confidence in the equivalence principle as a guide for the where F is the conformal factor [ds2 - F2(dS)2 MinkowskiJ. formulation of the second law. If accepted, there can be no The experimental density and pressure as related to entropy connection between the space-time structure and the second would be 1 and j (and not p and p). As at the origin of the than that the requirement of co- universe F = 0, this leads to the fascinating possibility of law other provided by cosmological models without singularity. The universe would variance [see Tolman (11)]. However, we know now that the present a kind of phase transition corresponding to the transition from a dynamical description to an irreversible transition ofthe conformal metric into a Minkowskian one (F thermodynamic description corresponds to the transition -* 1). It could start empty and cold and develop its matter and from dynamic groups to semigroups. This includes the entropy as processes coupled to the evolution of space-time. consideration of asymptotic boundary conditions and, there- The universe would present us with the most grandiose fore, application ofthe equivalence principle to this situation example of self-organization due to irreversible processes. appears to be doubtful (12, 13). However, our approach can hardly be considered as If the aim of cosmology is also to provide a mechanism for convincing as we are forced to use a coordinate-dependent the generation of matter from some space-time background, description necessary to apply the pseudotensor concept. this requires the incorporation of the entropy generated by For this reason we present in this paper an alternative way to

The publication costs of this article were defrayed in part by page charge *The relation between gravitational entropy as introduced here and payment. This article must therefore be hereby marked "advertisement" the black hole entropy (8) deserves careful consideration, but will in accordance with 18 U.S.C. §1734 solely to indicate this fact. not be discussed in this paper.

6245 Downloaded by guest on September 24, 2021 6246 Physics: Prigogine and G&h6niau Proc. Natl. Acad. Sci. USA 83 (1986) describe our results through a modification of Einstein's rical pseudotensor that depends only on g,,, and their first equation: derivatives. Then

= /A = G A V A = [1.2] TNT (-g)T,". [2.6] ILV,RM --g-RKTV2 Because of Bianchi's identity, entropy conservation follows However, as pointed out by Landau and Lifshitz (15), the use from Eq. 1.2. To introduce entropy production we have to of Eq. 2.6 requires supplementary conditions such as the replace Eq. 1.2 by asymptotic validity of the Minkowski description. These conditions are satisfied in the case of an open universe using a conformal metric GAP + KCgI, = KTAV [1.3] ds2 = where Ta is an effective stress tensor (involving fi, p in place F2rj.L dedev. [2.7] of p, p), and Cns is a supplementary quantity closely related to entropy production. We can then write F = F(x), where x is the dimensionless This equation is formally identical to Hoyle's equation (see kinematic time x2 = r,., xJLxY, xS = KeS; F is a well-defined ref. 9) introduced to describe the steady state theory. This is function of the cosmological time (16). For example, for a not an accident. Hoyle's (as well as Bondi's and Gold's) universe filled by radiation or highly relativistic particles (p motivation was the elimination ofthe big bang and, therefore, = p13), we have of the fundamental time dissymmetry of the universe, through the introduction of continuous particle and energy F= 1 -x-2. [2.8] creation. But the creation ofmatter is an irreversible process. It is an irony inherent in the steady-state theory that to avoid The origin ofthe universe in kinematic time corresponds to the idea of a history of the universe it had to introduce x = 1; we then have F = 0. However, with increasing values irreversible processes that represent the universe as being in ofx, F tends rapidly to its Minkowski value 1 (see Appendix a nonequilibrium steady state. If).t We now calculate the energy gain as expressed in Eq. In Section 3 we discuss the generalized field equation (Eq. 2.5. This calculation is summarized in Appendix I. Assuming 1.3). We then conclude with a few qualitative remarks in accordance with the remarks presented in Section I that concerning the relation between matter and entropy as seen this "heat" is entirely dissipated and leads to an irreversible from our point of view. increase of entropy, we now obtain, instead of Eq. 2.2,

2. Entropy Production in Conformal Coordinates kTd(gR3) = d(pF3R3) + pF3dR3 = pR3dF3. [2.9] Let us consider an isotropic homogeneous universe filled When F is constant, we recover Eq. 2.2, but in general the with a fluid whose stress tensor is thermodynamic evolution is no longer adiabatic. The in- crease of the conformal factor F leads to a dissipation of TAY = UILUv(P + P) - Pg"v. [2.1] energy. Of course, thermodynamic theory cannot specify the mechanism of entropy production. It may be related to the We then obtain for the time change of the entropy presence of unstable particles that then decay into baryons and photons. kTd(sR3) = d(pR3) + pdR3, [2.2] We see that the density p and the pressure p are now replaced by the new quantities (Eq. 1.1). As the relation where R is the radius of the universe as it appears in the between the rates R and F is simply (16), Robertson-Walker metric. The vanishing of Eq. 2.2 exptress- es that the evolution is adiabatic and, therefore, that enitropy F = Rix, [2.10] is conserved. To include the idea of entropy producti4on, it seems natural to consider conservation laws in a Minkc)wski the use of fi and 3 leads to the fascinating possibility of a frame. This can be done with the help of pseudotensors T, cosmology without matter singularity as the energy pF3R3 in which satisfy the conservation equation "volume" R3 tends to zero for x -- 1. There is a clear analogy with the steady-state model. d AV= 0. [2.3] However, here the particle creation refers only to the early part of the history of the universe (for which F # 1). In Appendix MI, we discuss the time scales involved. Such pseudotensors can be split into a matter part a nd a There are qualitative differences when a flat universe (k = gravitational part 0) is considered. In the case of the flat universe F is not a constant, whatever period is considered. Therefore, the V45 = ijAV + V [2.4] asymptotic boundary conditions required for the use of the Landau-Lifshitz tensor are not satisfied. If, in spite of this, We then have we apply Eq. 2.9 that will mean that particle creation is continuing forever and will ultimately lead to a universe of infinite mass in volume R3. The closed universe presents new uvda TmA = [2.5] -U'aATgR"L [2.5] features near the recollapse, as then the entropy production will become negative. In this sense, we now have an exchange of energy between matter and gravitation. Many pseudotensors have been tThis is in contrast with the description of the universe in terms of defined in the literature, but the choice favored by LaLndau cosmic time where the open universe retains forever its non- and Lifshitz (15) has been adopted, as it leads to a synimet- Euclidean form. Downloaded by guest on September 24, 2021 Physics: Prigogine and Gdhdniau Proc. Natl. Acad. Sci. USA 83 (1986) 6247 3. Generalization of Einstein's Field Equations energy-momentum tensor), while the entropy density S remains constant. The results summarized in Section 2 leave us with a rather Clearly we come here to the steady state model introduced unsatisfactory situation as the redefined density fi and pres- by Bondi, Gold, and Hoyle (ref. 9, p. 459 and p. 616). If, as sure 6 differ from the quantities that appear in the stress is often assumed (17-19), the first stage of the universe can tensor (Eq. 2.1) and, therefore, in Einstein's equations (Eq. be described by a de Sitter universe, the actual reason may 1.2). For this reason it appears natural to introduce a new be either the role of the cosmological constant that leads to "effective" stress tensor T,,, defined in terms of fi and j, Eq. 3.7 or the existence of particle creation. In the second case, there is no need for negative pressure. T,, = uu,,(fi + P) - jg0, [3.1] The close relation between the inflationary model based on together with a new tensor C,,, which, for reasons of Eq. 3.4 and the steady state model has been emphasized by has the form Hoyle (9) and Narlikar (20). The basic difference from our symmetry, perspective is that the inflationary universe corresponds to an Cr = -au~uv + [3.2] adiabatic entropy conserving evolution (with negative pres- bgj, sure!) while the matter creation model includes irreversible where a and b are two scalars characterizing the entropy processes. change. Obviously, (a) Incoherent Fluid. Here we can apply the arguments of Section 2. We take a = p + p - (p + p), b =j-p. [3.3] j5 =p = 0. [3.11] We see that dissipation expressed in terms ofa and b changes Then using Eq. 3.3, the definitions of density and pressure. The energy/matter conservation now becomes, in covariant form b=0, a= p- p. [3.12] aENu~+ a-dE uS = a,(a - b)VguA As we may take T as a constant in Eq. 3.6, we have (see Eq. + baAV u , [3.4] 1.1) which has to be compared with the covariant form ofEq. 2.2. kTif = = F3p, [3.13] kTa sVguA = a, pV-ug + pdAVus. [3.5] and Eq. 3.6 reduces to Similarly, we now introduce the second law of thermody- kTaMVu = p-u ,,F3, [3.14] namics (see Eq. 2.9): corresponding to a positive entropy production for F increas- kTaMAVsulA = d,,puS + pd/suA ing with time. = O,(a - b)VguM + gu/4 - 0. [3.6] (Ni) Black-Body Radiation or Highly Relativistic Particles. ba,,V We may assume that the equation of state remains To emphasize that we use here fi and p (and not p and p), we have introduced the notations T and S. We also use P = p/3 [3.15] Boltzmann's constant k for dimensional reasons. Let us consider a few examples. as the produced particles are immediately converted into (i) de Sitter Universe. The first example we consider radiation. We also accept the usual thermodynamic relations corresponds to the de Sitter universe. While it is somewhat outside the approach considered in Section 2, it illustrates p = AP4 and kti = 4p3/3 [3.16] nicely some features of our approach. As is well known (9, 11), we have then together with Eq. 1.1. We obtain then the entropy production 4 p+p = 0, p = constant. [3.7] Oa = pVu/La F3 or3gN/'--g The constancy ofp, in spite ofthe expansion ofthe universe, or means that matter is generated. The mechanism that is producing matter is the existence ofa negative pressure. This k~d(iR3) = 4pR3dF3. [3.17] matter creation is occurring reversibly without any entropy production. The effective entropy in "volume" R3 Using Eq. 3.3 we can present alternative pictures. If we request S =R3 = 4 ! A"14(pR4)314F914 3 k P= P, [3.8] is growing as F914 from zero at x = 1. The effective energy in volume R3 then a=b=bi++p, [3.9] E = 1R3 = pR4F3/x = and has a maximum at x where is an arbitrary constant. Eq. 3.6 then leads to is=-v3zero at x 1, increases quickly, kT&=a= i+p. [3.10] (iv) Mixture of Incoherent Matter and Radiation. This example is studied in Appendix III. The total energy has a In an expanding universe O,,N\guW> 0, the total entropy in maximum at a finite value of the kinematic time x. The volume R3 increases (as a = b is positive for a usual existence and the position of this maximum is closely Downloaded by guest on September 24, 2021 6248 Physics: Prigogine and Gdhdniau Proc. Nad. Acad. Sci. USA 83 (1986) correlated to the order of magnitude of the specific entropy would mean that "time precedes existence." Indeed either S (see Section 1). time, as irreversibility, is-as was stated by Einstein-an Other examples as well as estimates of the maximum illusion; or irreversibility has to appear at all levels of temperature are deferred to a separate paper. physical description including cosmology.

4. Concluding Remarks APPENDIX I In the literature on cosmology it is often suggested (18, 21) In case Eq. 2.6 one easily finds the identity that the creation of the universe may have been a "free lunch." It appears indeed likely that no deviation from the =a0_pF7u' + pF3OlF4u/ -pF4u'AaA conservation of total energy (field plus matter) is involved, F-1ul~aTA but the price had to be paid in entropy increase. and, thanks to Eq. 3.5, In this way there appears a simple relation between the matter energy that is created and the entropy that has to be ascribed to it (see the examples investigated in Section 3). In F-lu 8 TA' = F4( -p)ud- special relativity, there appears an "equivalence principle" between energy (E) and mass (m), while there appears here thus, an equivalence principle between entropy and matter, a,,pF3F4uM + pF30aF4u'0 = F4pu~dF. [A.1] Ts- p [4.1] We give in Eq. 2.9 and Eq. 3.6 a thermodynamical interpre- tation to Eq. A.1, through the introduction of the effective with a numerical coefficient depending on the model (inco- energy density and pressure (Eq. 1.1). herent matter . . ) considered. While the production of matter is as a whole an entropy APPENDIX II increasing process, the share of entropy carried by various types of particles (baryons, leptons, and photons) may be Time Scales Involved. In case Eq. 2.8 the cosmological time quite different. We hope to come back to this problem in a t is related to x through separate publication; however, let us note that the reference state of zero entropy is here the non-Euclidean vacuum state t = c-1K-'P x (x + x-1 - 2) [A.21 and not as in conventional thermodynamics the crystal at zero temperature. We can, therefore, conceive that some where (-K2) is the scaling constant in the usual form of the particles ifproduced alone would have a negative entropy (or Robertson-Walker metric, and (p is an integration constant a positive information) in contrast to the vacuum state. This (see ref. 16). is in line with the current statement that particles can no Einstein's equations together with Eq. 2.8 lead to longer be considered as "elementary." The analogy with biomolecules (including DNA) comes Kp= 12(Kcp-)2(l -x-2)-4x-4. quite naturally to mind. For such molecules, entropy content is smaller than that of the molecules out of which they are We the constants built, and their production is coupled to the production of can, therefore, eliminate integration by other entropy generating molecules. forming the product Our approach combines features of the standard big bang model with features of the steady state theory. Our main B tVp/2V3 = (1 + x-1)-2x-1. emphasis is, of course, on the important role of irreversible processes. Ifindeed the total energy ofmatter and field in the Thus universe is close to zero, the characteristic feature of the universe would reside mainly in its entropy. To obtain the X-1 = (1 - 2B - V /4B)/2B.- supplementary C tensor in Eq. 1.3, the traditional approach is in the coupling of the gravitational field with some mass fields (10). However, the irreversibility inherent in the C For small B, x-1 B. The present value of B is 8 x 10-4, tensor can only be achieved if the time symmetry is already and this is the present value of x-1, too. Thus, from Eq. A.2 broken in these fields. Therefore, the possibility of a semigroup description of fields involving breaking of sym- c-'K-lfo ==8 x 10-13 sec. [A.3] metry is a prerequisite condition. As briefly outlined in Section 2, there is a drastic difference Atx = V5l between the three traditional possibilities (k = -1, 0, and + 1) included in the Friedman-Lemaltre cosmology when dissi- t - 5 x 1013 sec. pation is included. We may conceive mechanisms that prevent antithermodynamic behavior through appropriate limitations of particle production. We can in this way go from the usual Robertson-Walker We are obviously in front ofa new problematic model. Still description to the conformal description. we consider it to be very encouraging that cosmological models without initial singularity (in energy) can be con- APPENDIX III structed in such a way that at each moment the second law Mixture of Incoherent Matter and Radiation. We take F is satisfied. However, the explicit form ofthe second law may from a paper by Brout et al. (18): an arbitrary linear be different in different stages of the cosmological evolution, combination corresponding to different explicit forms taken by the addi- tional tensor C,,,. If indeed the large scale structure of the F = (p [(1 - + (1 + 8)Fm] [A.4] universe can be traced back to irreversible processes, that 2 8)F1 Downloaded by guest on September 24, 2021 Physics: Prigogine and Gdhdniau Proc. Natl. Acad. Sci. USA 83 (1986) 6249 of 1. Misra, B. (1978) Proc. Nati. Acad. Sci. USA 75, 1627-1631. 2. Misra, B., Prigogine, I. & Courbage, M. (1979) Physica A 98, = ip(l - x-2) and Fm = p(1 - x-')2. 1-26. Fy 3. Misra, B. & Prigogine, I. (1983) in Long Time Predictions in Dynamics, eds. Horton, J. R., Reichl, L. E. & Szebehely, Using Einstein's equation (Eq. A.4) gives A. G. (Wiley, New York), pp. 21-43. 4. Martinez, S. & Tirapegui, E. (1985) Phys. Lett. A 110, 81-83. P = Pm + Py' 5. Linde, A. D. (1984) Rep. Prog. Phys. 47, 925-986. KPm = 6K2C2p(1 + 8)R 3, 6. Brandenberger, R. H. (1985) Rev. Mod. Phyv. 57, 1-60. 7. Page, D. N. (1983) Nature (London) 304, 39-41. KPy = 3K2C2p2(1 - 8)2R-4. 8. Misner, C. W., Thorne, K. S. & Wheeler, J. A. (1973) Gravi- tation (Freeman, San Francisco), pp. 889-891. The material part Em = pmF3R3 ofthe energy in "volume" R3 9. Weinberg, S. (1972) Gravitation and Cosmology Principles is growing as F3. The radiation part has a maximum at Applications of the General Theory ofRelativity (Wiley, New 1 York). -{3(1 + 8) + [9(1 + -8)22 208]1/2}. 10. Parker, L. (1977) Proceedings of the Symposium on Asymp- 2 totic Properties ofSpace-Time (Plenum, New York). The ratio 11. Tolman, R. C. (1950) Relativity, Thermodynamics and Cos- mology (Clarendon, Oxford). Ev/Em = P1/Pm = -(R-'(j - 6)2/(1 + 8) 12. Misra, B. & Prigogine, I. (1983) Lett. Math. Phys. 7, 421-429. 2 13. G6heniau, J. & Prigogine, I. (1986) Found. Phys. 16, in press. 14. Glansdorff, P. & Prigogine, I. (1971) Thermodynamic Theory decreases as R-1. The ratio (constant) of the number densi- of Structure, Stability and Fluctuations (Wiley-Interscience, ties of photons 3.7 py/kTy, (6, p510), and baryons nm - New York). PM/Mc equals 15. Landau, L. D. & Lifshitz, E. M. (1962) The Classical Theory ofField (Pergamon/Addison-Wesley, Oxford), inc2 (18- p. 344. 3.7 ---(p I(1 -8)2 (109 x 6 x )2 16. Tauber, G. E. (1967) J. Math. Phys. 8, 118-123. k RT2 (1+8) 1+8 17. Barrow, J. D. & Tipler, F. J. (1986) The Anthropic Cosmolog- ical Principle (Oxford Univ. Press, Oxford). The order of magnitude (109) is related to the experimental 18. Brout, R., Englert, F. & Gunzig, E. (1977) Ann. Phys. 115, one (for small 8). 78-106. 19. Brout, R., Englert, F., Frere, J. M., Gunzig, E., Nardone, P., We are grateful for discussions with Professors E. Gunzig, M. Truffin, C. & Spindel, Ph. (1980) Nucl. Phys. B 170, 228-264. Henneaux, B. Misra, and L. Shepley. This work was supported in 20. Narlikar, J. V. (1984) J. Astrophys. Astron. 5, 67. part by a grant from the Robert A. Welch Foundation of Houston, 21. Gribben, J. (1986) In Search of the Big Bang (Bantam, New TX. York). Downloaded by guest on September 24, 2021