(3) and a dimensionless parameter Ω. universe, cosmologists find that Ω m Moreover, the matter is composed of two

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1 Zero-point energy of vauum flutuation as a andidate for dark energy versus a new onjeture of antigravity based on the modified Einstein field equation in general relativity Guang-jiong Ni Department of Physis, Fudan University, Shanghai, 00433, China Department of Physis, Portland State University, Portland, OR707, U.S.A. address: pdx008@pdx.edu In order to larify why the zero-point energy assoiated with the vauum flutuations annot be a andidate for the dark energy in the universe, a omparison with the Casimir effet is analyzed in some detail. A priniple of epistemology is stressed that it is meaningless to talk about an absolute (isolated) thing. A relative thing an only be observed when it is hanging with respet to other things. Then a new onjeture of antigravity the repulsive fore between matter and antimatter derived from the modified Einstein field equation in general relativity is proposed. This is due to the partile antipartile symmetry based on a new understanding about the essene of speial relativity. Its possible onsequenes in the theory of osmology are disussed briefly, inluding a new explanation for the aelerating universe and gamma-ray bursts. I. INTRODUCTION The past deade has witnessed a remarkable progress of osmology and astrophysis. Espeially, the preise data provided by the Hubble Spae Telesope (HST) key projet, the Wilkinson Mirowave Anisotropy Probe (WMAP) team, the Sloan Digital Sky Survey (SDSS) and the disovery of aeleration of the universe expansion, et. have shaped an observable universe quite quantitatively. The present understanding an be summarized basially as follows (see, e.g. [], []): (a) The Hubble law the linear relation between the redshift z of a galaxy and its distane D from earth H 0 z = D () has been verified up to D ~ 000 Mp with the Hubble onstant: H = 7 km / s.mp () 0 (b) Sine 8, observations on the hundreds Type Ia supernovae with Z 0.5 reveal that the expansion of universe is aelerating rather than deelerating as speulated before. () Defining a ritial mass density of universe as 3H 0 3 ρ = 0 g m (3) 8 π G and a dimensionless parameter Ω m = ρ ρ with ρ being the matter density of universe, osmologists find that Ω m 0.3. Moreover, the matter is omposed of two

2 parts, the visible baryon matter and the invisible dark matter. They are orrelated in the spae with their ratio being about : 6. The dark matter is still at large. (d) The temperature flutuations of osmi mirowave bakground radiation (CMB) (around the average value T =.75 K) measured aurately by WMAP show that the spae-time of universe is flat, whih means O = O + Ω = (4) m Λ in the Friedmann model. Here the parameter Λ Ω Λ = (5) 3H 0 was introdued via a osmologial onstant Λ adding to the Einstein field equation of general relativity (GR) as follows[3]: R µ ν - gµ ν R - Λg µ ν = - 8πGTµ ν. (6) (e) Analyses from WMAP, SDSS and the aelerating universe are foused on a value of Ω Λ 0.7, whih verifies the flatness of spae as shown by Eq. (4) and gives an observational value of Λ : -5 - Λ.3 0 m (7) obs II. DARK ENERGY AS THE ZERO-POINT ENERGY IN VACUUM The term dark energy was oined in 0 to reflet the strange feature of Λ, whih makes a positive ontribution to mass density as whereas it exerts a negative pressure as ρ p?? Λ 8πG = (8) 4 Λ 8πG = () In other words, its equation of state an be written as p = w ρ (0) (the subsript Λ is omitted) with w = -. To explain this strange property, on the right-hand side of Eq. (6), a quantum average of the lassial energy-momentum tensor T in the vauum, i.e., the zero-point energy assoiated with the vauum flutuations (ZPEVF) effetive equation as T = ρ va vag, µ v T = ρ va va R - g R - eff g = - 8πGT is added to T 00. One finds an Λ () where Λ e ff = Λ + 8πGρ va () should be identified with the Λobsshown by Eq. (7). There are two possible interpretations: () Assume the lassial osmologial onstant Λ = 0. Then Λ is totally eff stemming from the ZPEVF: Λ eff = Λobs = 8πGρva (3)

3 Aording to quantum field theory (QFT), a kind of partile with mass m and spin j makes a ontribution to the density of ZPEVF being [4, 5] ( ħ = = ): va 3 d k j = ( ) ( j + ) k + m 3 (π ) ρ (4) For the ase of j = 0, m 0, the theoretial estimation reads: ma theor k ρ va k dk ( m ).4 0 ( V) P Ge 4π 0 6π (5) Here, in order to ontrol the ultraviolet divergene of integral, a ut-off ħ ħ kmax mp = =. 0 GeV (the Plank energy) is introdued. G On the other hand, ombination of (3) and (7) yields: obs ρ va 3 0 ( GeV) (6) Then we see a ridiulous ratio being theor ρva 0 obs > 0 (7) ρva () Alternatively, notiing that the theoretial value of ρ va an be negative as shown in (4) with j = ½, one may assume that the lassial Λ ( 0) and quantum obs ρ in () are anelled eah other so deliately that only a tiny value of va Λ is left eff as shown by (7). This fine tuning mehanism is alled as the anthropi priniple we live where we an live. III. DARK ENERGY VERSUS CASIMIR EFFECT The biggest mystery of dark energy lies in its strange property of negative pressure whih implies some antigravity and leads to aelerating expansion of universe. More generally, the negative pressure is haraterized by an equation of state (0) with the parameter w < -/ [6]. The model of osmologial onstant (with w = -) means the dark energy is uniform in spae and onstant in time. Alternatively, the quintessene model with w> - means that the dark energy ould be desribed by some dynamial salar field whih may vary with spae and deay with time. Other models with w < - might be the sign of really exoti physis or some modifiation of GR. However, up to now, the data analyses still favor the simplest ase of w = -. On the other hand, a number of physiists doubt that there is something wrong in the onept of dark energy (see, e.g. [7]). Moreover, some new theories are put forward, e.g. [8], trying to substitute the dark energy. In this paper we wish to argue that the ZPEVF annot be a andidate for dark energy. It was said that some experiment like the Casimir effet an be performed in laboratories, providing some evidene of ZPEVF to explain the dark energy. We don t think so. Let us have a look at the Casimir effet. As is well known, the Casimir fore F is the gradient of Casimir energy F - E d π ħl = d E : = (8) 3

4 And E is the differene between ZPEVF of eletromagneti field in the presene of two (square) metal plates (with side length L and distane d between them) and that in the absene of them [, 0]: E = ħ ω (preseneof plates) - (absene of plates), ħω k α k, α k, α k, α d k // n π n π = ħ L k // k// dn k // () (π ) n= d 0 d Note that: (a) The wave vetors parallel to plates, k //, are the same for two ases (presene and absene of plates); (b) The wave numbers vertial to plates, k, are disrete (with n being disrete integer numbers ) in the presene of plates whereas ontinuous (with n varying ontinuously) in the absene of plates; () For eah fixed wave number, there are two modes of transversal polarization of field exept for the ase of k = 0 (n = 0) where only one mode (vertial to plates) exists. (d) The expression () is finite as shown by (8) and verified by experiments, see, e.g., []. Eq.() an be alulated either by introduing some utoff funtion (e.g.[]) or by resorting to a rigorous mathematial theorem the Plana summation fo rmula as disussed in [] (see also [0]). The reason why the theoretial result is free from any ambiguity is beause E, being a differene, has neither ultraviolet divergene (when k ) nor infrared diverge ne (when k 0). In fat, the tiny ontribution to E is just oming from the differene of a sum and an integral with eah of them being finite at k 0 region. By ontrast, aiming at an explanation for dark energy, the expression for the density of ZPEVF, Eq.(4), was written down in an ambiguous manner from the beginning. It is divergent at k limit. So, as a next step, an arbitrary utoff was hosen to be the Plank energy (~. 0 GeV). Third, atually, no one knows exatly how many varieties of different fields should be added? Fourth, no one really understands why this positive ρ would lead to negative va pressure of dark energy? Fifth, no one an onretely explain how this negative (but isotropi) pressure exerts an outward repulsive fore on a supernova to render it aelerating? Last, but not least, why the theoretial estimation of ρ va differs from that of observation so absurd as shown by (7)? Physis is an aurate siene as shown by the Casimir effet. The whole history of physis has been proving that a orret theory should be able to aount for the experiments quantitatively. One there is ertain but small disrepany between them, it must imply something of less importane being ignored in the theory. If the disrepany turns to be large in quantity or even out of ontrol, it means that something of importane must be wrong. Now we should regard Eq.(7) as a serious warning that we have been wrong in a fundamental way. 4

5 The reason why the evaluation for Casimir effet is orret whereas that for dark energy, Eq.(4), is wrong lies in the following lue: In the later ase we were talking about an absolute energy of the osmos spae without a omparison between two onfigurations (as that in the Casimir effet). Thanks to researhes on the dark energy over deade, we now have a very strong argument supporting the priniple of relativity in physis (also in epistemology, see [3]) No hange, no information. Any information (e.g., the energy, the momentum or the position of a partile, et) is reated right at the ourrene of a hanging proess, i.e., at the measurement performed by the observer on the objet. In fat, numerous experiments, espeially that aiming at larifying the Einstein- Podolsky-Rosen paradox, all lead to the same onlusion that the onrete form of orrelation in a quantum state exhibits itself as a diret onsequene of measurement right at the destrution of original entanglement of state there is no information existing before and after the measurement (for detail, see [0]). Maybe many physiists still not believe in this priniple mentioned above. Let us think about a simplest example: A partile with mass m is resting in the laboratory (S frame). I an say that it has no kineti energy as long as I am staying in the S frame too. However, one I jump onto a train (S frame) moving with a veloity v with respet to S, I will say that the partile has a kineti energy E = mv. The question is: Neither I nor anybody had done any work on the partile. Where its energy is oming from? This simple example shows that it is meaningless to talk about the absolute energy of an isolated objet. Only when two partiles are olliding eah other, an the energy (and momentum) transfer be unambiguously alulated either in S or in S frame. The energy (momentum) onservation law is meaningful exatly only right at the ourrene of ollision, not before or after. Moreover, in fat, the definitions of energy and momentum are preisely meaningful only when they are onserved. There is another problem in the theory of ZPEVF as dark energy the theory of GR had been mixed with the QFT. We will disuss it after we further larify the lassial nature of GR in the next setion. IV. ANTIMATTER AND A NEW MODEL FOR UNIVERSE EXPANSION We wish to propose an alternative explanation for the aelerating universe, i.e., antigravity without invoking the dark energy. In our opinion, we should start from srath. After studying the theory of speial relativity (SR) for deades, we have been learning the essene of SR being nothing but the equal existene of partile and antipartile [4, 5, 0, 3]. Many physiists thought that the so-alled CP violation in weak interations might trigger (in an early epoh of universe expansion) the asymmetry of partiles and antipartiles the former surpass the latter in number up 8 to a fator like 0-0. We don t think so. The CP inversion is equivalent to the soalled time reversal, but the latter is essentially not a time reversal (whih is merely a misnomer) but a motion reversal (see [6], [0]). Hene the tiny CP 5

6 violation has nothing to do with the symmetry between partile and antipartile. Rather, we believe that in the entire universe, the number of partiles is equal to that of antipartiles. Next, thanks again to SR ombining with quantum mehanis (QM) and evolving to relativisti QM (RQM), we notie that there is a mass inversion ( m - m ) symmetry in RQM to reflet the partile antipartile symmetry [7]. This symmetry, together with an interesting disussion of negative mass paradox raised in [8], leads to a natural generalization of Newton s gravitation law [7]: m m F = ±G (0) r Here the minus (plus) sign means the attrative (repulsive) fore between matters or antimatters (matter and antimatter). Note that the mass of matter or antimatter is always positive. Eq.(0) an be dedued from the Einstein field equation in GR by modifying its right-hand side as [7]: eff T T = T -T () where T ( ) is the energy-momentum tensor of matter (antimatter) at the spae T point x µ. This modifiation makes Einstein equation invariant under the transformation m - m. In ref [7], some further onjetures are proposed as follows: (a) The big bang reated equal quantities of partiles and antipartiles. The repulsive fore between them triggered the inflationary expansion. But antipartiles had a head start and flew away faster than partiles did. The latter lagged behind to some extent and our galaxies gradually evolved out of them. On the other hand, distant stellar objets might be evolved from antipartiles and some of them may be just those observed supernovae undergoing aeleration aused by the repulsive fore exerted by inner galaxies omposed mainly of matters. (b) The flatness of our universe is ritially depending on two fators: first, the inflationary expansion; seond, the nearly equal densities of matter and antimatter in a large part of universe after the inflationary expansion. (See Appendix and Fig.). () In the intermediate region where matter and antimatter are overlapping, there may be some probability of ollisions between matter and antimatter. Beause of the long range repulsive fore between them, these ollisions may be of grazing type and so the annihilation proess ours at short distane would have some speial harate r. Maybe it ould explain the mehanism of gamma-ray- bursts (GRBs), the latter are distributed at remote distane on a osmologial sale and they tend to have a roughly onstant explosive energy (espeially for GRBs with typial lifetime T ~ 0s. Most likely, the gamma rays are emitted within two small-angle jets). (d) Furthermore, it is observed that the star formation rate is rising steadily from remote distane with redshift z ~ to z > 6 []. We guess this phenomenon is a refletion that the domination of antimatters is inreasing steadily there. Sine the oexistene of matter and antimatter would suppress the flutuations inside the mixture of them and thus suppress the formation of stars, we guess there might be 6

7 some anti orrelation between the star formation rate and the spatial distribution of GRBs. Fig. shows a shemati diagram of universe expansion. It is a separating proess of matter from antimatter with (the osmologial) time. Hene their overlapping region dereases gradually and its two boundaries an be estimated via the measurements on the farthest and nearest GRBs (loated at G fand G ) n respetively. Our Earth is loated near the enter of matter region beause: (a). The distribution of GRBs is isotropi in all diretions. (b). The CMB has a bipolar anisotropy showing the veloity of Earth being only V = 365 km/s with respet to the osmos rest frame (CRF), whose origin should be the point surrounded by CMB 5 isotropially as shown by two symmetri soures on the surfae at t A = 3.8 x 0 y. Indeed, if resorting to the Hubble law, Eq.(), and substituting the value of V = 365 km/s into the z = V/, a distane of Earth from the origin of CRF an be estimated to be 5 Mp whih aounts only 0.% of the length sale of matter region ( estimated to be 000 Mp at z =0.5). (). Luky enough, this loation of Earth ensures a safer environment for human being s evolution. Other plaes in the universe would be suffered from more dangerous radiations emitted by nearer GRBs in the past billion years. V. SUMMARY AND DISCUSSION (a) The essene of SR or that of the priniple of relativity has two linked aspets, one in physis and one in epistemology [3]. The former is used to establish Eqs. (0) and () while the latter forms the main rux of our argument of why the ZPEVF annot be the dark energy. (b) The physial essene of SR is nothing but the symmetry of partile and antipartile, whih an be stated in one of two equivalent invariane, the invariane under the newly defined spae-time inversion (x -x, t -t) at QM level [0] or that under the mass inversion (m -m). The latter an be used either at QM level or at lassial level. Being a lassial field theory and treating only the gravitation among matters, the GR should be and an be modified to inorporate antimatter so that a new approah of big bang theory of osmology might be established from srath, treating matter and antimatter on an equal footing. However, even the modifiation shown in () has been made, it learly refuses the addition of a quantum average of ZPEVF (even if it is onsidered ambiguously) to the lassial energy momentum tensor beause the ontribution from matter will anel that (if onsidered separately) from antimatter ( beause both of them should have the same sign), a onlusion just in ontradition with the predition of Eq.(4) by QFT (where the ZPEVF of partile and antipartile is onsidered as a whole). So we see that the disussion of () through (7) is merely onfusion in onept to ombine a lassial theory with quantum theory inorretly. We still have no orret QFT for gravitation yet. () Every law in physis is a disovery about the objetive nature, also an innovation of human being as the subjet. An isolated thing is absolute or abstrat in the sense of being devoid of any ognition. We an only onstrut a meaningful theory based on (at least) two opposite things. Usually, in an experiment, one of them 7

8 is just the apparatus. However, in QM, in order to desribe an abstrat quantum state vetor ψ of a partile onretely, one has to resort to a basis vetor x, t (whih represents a fititious apparatus, see [0, 3]) for obtaining the wave funtion: ψ ( x, t) = x, t ψ () whih is nothing but a probability amplitude of fititious measurement to detet the potential possibility of finding the partile at position x and time t. The wave funtion is really a great innovation to endow the QM a predition power before the real measurement is made. Another example is the Casimir effet. The existene of two plates allows us to ompare the ZPEVF in two onfigurations (the presene and absene of two plates). In this ase, every thing is finite and an be alulated unambiguously. By the way, all ontributions of ZPEVF of massive partiles to Casimir effet an be ignored. (d) The divergene, i.e., the emergene of infinity has two aspets in mathematis: () it is a huge number; () it is unertain. However, we physi ists often emphasized the first one whereas overlooked the seond one. We often introdued some utoff to render the divergene under ontrol like equation (5). Thus we overlooked the unertainty in mathematis implying a warning in physis it implies that we expeted too muh in trying to alulate something beyond our ability, or most likely, that thing doesn t exist at all. In short, we were wrong in some basi way. (e) For example, in QFT, the divergene emerges every where. The situation was put by Dira in the following words: (see [0]) It would seem that we have followed as far as possible the path of logial development of the idea of QM as they are present understood. The diffiulties, being of a profound harater, an be removed only by some dra sti hange in the foundation of the theory, probably a hange as drasti as the passage from Bohr s orbit theory to the present QM. It seems to us that one important neessary hange in basi onept is just the following answer to the famous Einstein s question: Is the moon there when nobody looks? [] The reality should be defined at two levels: Originally, every thing ontains no information at the level of thing-in-itself isolated from other things. Then it turns into the thing-for-us during (not before or after) a ertain measurement by us, refleting a series of observed phenomena. Every thing is infinite whereas our experimental data and theoretial interpretations must be fixed and finite before they an be meaningful. We had expeted too muh in QFT and been bothered by divergenes for deades. Eventually, we got rid of them just beause we began to distinguish what is existent thing whereas what is not. And we began to understand that we may get more only after we expet less. (See [3,, 3].) (f) The partile -antipartile symmetry prevails throughout the entire universe but it is spontaneously broken into two regions. We are living in the partile region and, very likely, near its enter. (g) Nowadays, one of the most important frontiers of physis lies in the fast developing field of osmology and astrophysis. Among the greatest unsolved problems in siene [4], the mystery of dark energy oupies a prominent position. 8

9 We are luky to enounter a divergene like Eq.(4) one again whih unveils a disrepany between theory and observation as in (7) that is unpreedentedly huge. This should be regarded as a new, serious warning from Mother Nature. It s time for us to listen to her. Aknowledgements: I am grateful to S. Q. Chen, P. T. Leung, E. J. Sanhez and Z.Q. Shi for helpful disussions. APPENDIX. ACCELERATING UNIVERSE EXPLAINED BY MODIFIED EINSTEIN FIELD EQUATION If similar to the Friedmann model [3], now the Einstein field equation ((6) without the Λ term) with modifiation () is tentatively solved for the time (the osmologial time oordinate) evolution of a dimensionless osmologial sale fator R(t) in the Robertson- Walker metri. Being a measure of the deviation from the Hubble law, the time derivative of Hubble parameter H( t) = R ( t) R( t) ould be derived approximately as ( =): d H ( t) = H ( t) = 4π G [( ρ ρ ) + ( p - p dt (A.) C Here ρ ( ρ ) and p(p ) are the mass density and pressure of matter (antimatter) in the flat osmos with zero urvature. However, beause antimatter flew away faster than matter, depending on the initial ondition at the end of inflationary expansion era, there should be two o-moving oordinate systems related to them separately. This ompliation renders Eq. (A.) and the following three possibilities only meaningful qualitatively: (a) In the matter dominant region nearer to our galaxy, ρ ~ p ~ 0, Ḣ (t) < 0, the universe expansion is deelerating. (b) In the intermediate region of universe where matter and antimatter are overlapping with nearly equal densities, ρ~ ρ, p ~ p, Ḣ (t) ~ 0, the linear relation of Hubble law remains valid and shows a oasting osmos that is neither aelerating nor deelerating. () In the outer region of universe where the antimatter dominates, ρ ~ p ~ 0, Ḣ (t) >0, the universe expansion began to aelerate as shown by the supernovae (with z > 0.5) observed after 8. Of ourse, a more rigorous treatment is needed. But it seems likely that the osmologial onstant (dark energy) is merely a false appearane of antimatter distributed at the remote distane. REFERENCES [] Partile Data Group, Review of Partile Physis, Phys. Lett. B5(004) -0. [] M. Tegmark et al. Phys. Rev. D 6(0) 004,035. [3] S. Weinberg, Gravitation and Cosmology (John Wiley, 7). [4] P. West, Introdution to Supersymmetry and Supergravity (World Sientifi, Singapore, 0).p.6. )]

10 [5] C. Bek and M.C. Makey, ArXiv:astro-ph/ [6] R.R. Caldwell, Physis World, May, 004, 37. [7] S. Sarkar, Physis World, July, 004, 5. [8] E.W. Kolb, S. Matarrese, A. Notari and A. Riotto, ArXiv: hep-th/ [] C. Itzukson and J-B Zuber, Quantum Field Theory (MGraw-Hill In. 80) p [0] G-j Ni and S-q Chen, Advaned Quantum Mehanis (Press of Fudan University, 000, 003); English Edition was published by the Rinton Press,00. [] A. Lambreht, Physis World, Sept. 00,. [] G-j Ni, M. Zhang and X-w Gong, High Energy Physis and Nulear Physis, 5(8) 65(); English Edition, 5() 8. [3] G-j Ni, Priniple of Relativity in Physis and in Epistemology, aepted by Relativity, Gravitation, Cosmology, ArXiv: physis/ [4] G-j Ni and S-q Chen, Journal of Fudan Univ. 35,35(6) (in Chinese); and in Photon and Poinare Group (Edit: V.V. Dvoeglazov, NOVA Siene Publisher, In. ) 45-6, ArXiv:hep-th/ [5] G-j Ni, Progress of Physis (Nanjing, China),3,484(003) (in English). [6] J. J. Sakurai, Modern Quantum Mehanis (Hohn Wiley & Sons,In. 4). [7] G-j Ni in Relativity, Gravitation, Cosmology (Edit: V.V. Dvoeglazov and A.A. Espinoza Garrido, NOVA Siene Publishers, In.) Chapt.0, 3-36, ArXiv: physis/ [8] C. Will in The New Physis (Edit: P. Davies, Cambridge Univ. Press,8) p.3. [] B.E. Shaefer, The Astrophysial Journal, 583:L67-L70(003). [0] J. J. Sakurai, Advaned Quantum Mehanis, (Addison-Wesley Publishing Company, In. 67) P.5. [] N. D. Mermin, Physis Today, April, 38 (85). [] G-j Ni, S-y Lou, W-f Lu and J-f Yang, Siene in China (Series A) 4, 06 (8); ArXiv:hep-ph/8064. [3] G-j Ni, G-h Yang, R-t Fu and H-b Wang, Int. J. Mod. Phys. A6, 873 (00). [4] J. R. Vaa, The World s 0 Greatest Unsolved Problems, Prentie Hall, 005. [5] J. Hjorth, C. Kouveliotou and S. Woosley, Physis World, Ot. 004, 35. Figure aption for Fig.: 0

11 γ γ Shemati diagram (not in sale) of matter-antimatter universe evolving with the osmologial time t. The origin is the big bang at t = 0. The three-dimensional spae of osmos at t = onst is simplified into a two-dimensional spherial surfae, whose ross-setion with this paper is shown as a irle with its right and left parts being filled by matter and antimatter separately. Their overlapping region is shrinking with time. Two points at the boundaries G and G denote the farthest and nearest GRBs f 0 observed on Earth (E) today ( t E =.37 0 y). They an be tentatively identified with the GRB0003 (at distane D=.3 0 ly) and GRB0303 (D= 0 ly) respetively [5]. So they exploded at time tb =.4 x 0 y and t D =.7 x 0 y respetively. Similarly, the point S denotes a supernova with D ~ 7 x 0 ly loated in the antimatter region and exploded at time t C ~ y. It was aelerating away from us as observed. Points P and P show two soures of CMB emitted at 5 t = y. A n