Is the Cosmic Big Trip Just a Classical Wormhole Artifact?

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International Journal of Astronomy and Astrophysics, 2011, 1, 83-89 doi:10.4236/ijaa.2011.12012 Published Online June 2011 (http://www.scirp.org/journal/ijaa)

Pedro F. González-Díaz Colina de los Chopos, Instituto de Fsica Fundamental, Consejo Superior de Investigaciones Cientficas, Madrid, Spain E-mail: [email protected] Received March 15, 2011; revised April 21, 2011; accepted April 29, 2011

Abstract

Starting with the original sub-quantum dark energy model, the current accelerating phase of the evolution of the universe is considered by constructing most economical cosmic models that use just general relativity and some dominating quantum effects associated with the probabilistic description of quantum physics. Two of such models are explicitly considered. They support an interpretation of dark energy in terms of the entangled energy of the universe. The model only violates the dominant energy condition quantum mechanically, that is by an amount given by the entanglement energy density, and gives rise to an asymptotically anti-de Sitter wormhole that becomes an asymptotically flat wormhole in the classical limit. It is shown that for most cosmic solutions such a wormhole does not predict any big trip phenomenon when it accretes phantom like energy.

Keywords: Lorentzian Dynamic Wormholes, Entanglement Energy Thermodynamics, Thermal Radiation

1. Introduction inflation for large values of the Ricci curvature and describe an accelerating behavior at the smallest curvatures. For the time being, the concept of dark energy continues Certain of such theories are mathematically equivalent to posing one of the biggest problems of all physics which, the introduction of quintessence and phantom fields, but in spite of many attempts and theories intended to solve all of them suffer from the typical problems associated or at least ameliorate it, has hitherto not found a conclu- with having a non Hilbert-Einstein action and may violate sive outcome. Among such attempts and tentative theo- some solar-system tests. ries, without trying to be at all exhaustive, we may count From the observational standpoint the rapidly accu- [1] what has been dubbed as quintessence, a scalar-field mulating data coming from supernova Ia luminosity dis- theory satisfying a equation of state pw=  , with the tance measurements, quasar statistics determinations or parameter w being bounded in such a way that studies of the fluctuations in the cosmic microwave  11w /3, or its phantom-energy extension for background radiation seem all to imply a value for the which w <1 . Also very popular have been the so- parameter of the equation of state which becomes each called cosmic generalized Chaplygin gas theories, where time closer and closer to w =1 , which corresponds to the equation of state adopts quite a more exotic structure, a typical cosmological constant, with a certain ever or the tachyonic models for dark energy that describe stronger bias toward slightly smaller values. Thus, the suitable generalizations from the quintessential scalar realm where our accelerating universe appears to ap- fields. Besides some rather serious difficulties in trying to proximately lie on is one that can be expressed as a phan- fix the observational data, all of the above theories appear tom-like small perturbation of the de Sitter space. Even to be artificial such as inflaton theories are within the though one could eventually accommodate the above dark inflationary paradigm. Quite fashionable have also be- energy and modified gravity models to account for such come in the last few years some forms of modified grav- an observational scenario, that would ultimately appear ity theories in which one does not include any vacuum rather unnatural. Moreover, none of such models can be field but changes instead the gravitational Lagrangian by shown to simultaneously satisfy the following two re- adding some convenient extra terms that are able to match quirements, 1) exactly predicting what observational data

Copyright © 2011 SciRes. IJAA 84 P. F. GONZÁLEZ-DÍAZ point out in a natural way, and 2) an economic principle verse because, besides being regular along its entire evo- according to which one should not include unnecessary lution, it does not show the violent instabilities driven by ingredients such as mysterious cosmic fluids or fields nor a non-canonical scalar-field kinetic term as by construc- modifications of the very well tested background theories tion the model does not have a negative kinetic term nor it such as general relativity. The use of scalar fields in classically violates the dominant energy condition which quintessence or k-essence scenarios is not with standing guarantees the stability of the theory, contrary to what the quite similar to including an inflaton in inflationary theo- customary phantom models do. Another cosmic model ries for the early universe [2]. Even though, owing to the was also obtained which describes an initially accelerat- success of the inflationary paradigm which actually ing universe with equation of state parameter always shares its main characteristics with those of the present greater than –1, that eventually becomes decelerating for universal acceleration, many could take this similarity to a while, to finally contract down to a vanishing size as- be a reason enough to justify the presence of a scalar field ymptotically at infinity. The latter model seems to be less also pervading the current universe, it could well be that a adjustable to current observational data although we are cosmic Occam’s Razor principle would turn out to be not completely sure as this is a toy model. over and above the nice coincidence between predictions In the context of the dark energy models new phenom- of usual models for inflation and what has been found in ena can be predicted which appear to be extremely sur- cosmic observations such as the measurement of back- prising. Thus, if the parameter of the equation of state w ground anisotropies. After all, the medieval opinion that is less than –1, then space-time wormholes should take the simplest explanation must be the correct explanation place as a rather natural ingredient of the cosmic theory has proved to be extremely fruitful so far and, on the because in this case the dominant energy condition (DEC) other hand, the paradigm of inflation by itself still raises is violated. Under these circumstances the existing phan- some deep criticisms. Occam’s Razor is also against the tom energy would be accreted by the wormholes in such a idea of modifying gravity by adding to the relativistic way that the size of their throat quickly increases to even- Lagrangian some convenient extra terms. tually diverge even before the occurrence of the big rip Besides general relativity, quantum theory is the other building block which can never be ignored while con- singularity. The cosmic big trip [7-9] is a process thought structing a predicting model for any physical system. Al- to take eventually place in the far future by which natu- though it is true that a quantum behavior must in general rally existing wormholes accrete phantom energy in such be expected to manifest for small-size systems, cosmol- a way that they will end up engulfing the universe itself. ogy is providing us with situations where the opposite The characteristics of this process have been many times really holds. In fact, fashionable phantom models for the reviewed and its main difficulties discussed, including: current universe are all characterized by an energy density the static character of the used metrics, the asymptotic which increases with time, making in this way the curva- flatness of the wormhole space-times, the intrinsic quan- ture larger as the size of the universe becomes greater. In tum instability of the Cauchy (chronology) horizons and such models quantum effects should be expected to more the violation of the holographic bounds on entropy and clearly manifest at the latest times where the universe information. In this way, some criticisms have been becomes largest. Thus, it appears that quantum theory raised against that process, mainly by Faraoni and Israel should necessarily be another ingredient in our task to [10], Faraoni [11] and others [12]. However, all of such build up an economical theory of current cosmology criticisms have been refuted [13] at the end of the day and without contravening the Occam’s Razor philosophy. therefore, in spite of the rather Dante’s infernal aspect of A cosmological model satisfying all the above re- the phenomenon, the cosmic big trip still stands up as one quirements has been recently advanced [3-6]. It was in of the potential ingredients of current cosmology and fact constructed using just a gravitational Hilbert-instein strongly menaces the future of the universe. In the present action without any extra terms and taking into account the paper we shall consider the problem of dark energy ac- probabilistic quantum effects on the trajectories of the particles but not the dynamical properties of any cosmic cretion in the realm of the above mentioned benigner field such as quintessence or k-essence. The resulting phantom like model. It will be seen that the effect of the most interesting cosmic model describes an accelerating quantum DEC violating term is to covert an asymptoti- universe with an expansion rate that goes beyond that of cally flat (e.g. Morris-Thorne) wormhole space-time into the de Sitter universe into the phantom regime where the its asymptotically anti-de Sitter counter part, which does tracked parameter of the universal state equation becomes not give rise to any big trip phenomenon, but leads to a slightly less than –1, and the future is free from any sin- wormhole throat divergence at exactly the same time at gularity. Such a model, although still a toy one, will thus which the big rip singularity takes place, in close accor- describe what may be dubbed a benigner phantom uni- dance with the expected co-moving evolution [14].

2. Asymptotically Flat Space-Time Holes H t =0 , (5) c 4πG One of the most interesting implications from the so- called benigner phantom and dark energy models respec- After which the universe entered a contracting phase which would be maintained until t =  . If H  4πG , tively corresponding to the solutions a and a is 0 the interpretation [6] that the potential energy density is then the present model would no longer be valid. It could nothing but the entanglement energy density of the uni- be at first sight thought that the universe might now be in 2 verse, that is VSQ   , where VSQ is the so-called the phase tt< a of solution aa =exp00 HtGt 2π  , sub-quantum potential density [3-6]. In what follows we but current constraints on w [1] seem to preclude that it can shall use such an implication in all the calculations of the be greater than –1. Besides thermodynamical considera- vacuum energy accretion that we will consider. tions, perhaps another argument against solution a [3-6] Thus, if we place a Schwarzschild black hole with ini- be the fact that for this kind of solution, while the accretion tial mass M 0 in the universe described by the suggested of the sub-quantum energy onto a Morris-Thorne worm- model for a , the mechanism advanced by Babichev, hole leads to a progressive decrease of the wormhole size 2 Dokuchaev and Eroshenko [15] would imply that the according to the law bb=/10  π Dt , the size of a black hole will accrete this sub-quantum phantom energy black hole of initial mass M 0 will progressively increase so that it would progressively lose mass down to finally with sub-quantum energy accretion so that MM=/0 2 2 vanish at t =  , according to the equation t1 π DM 0  . In this way, at a time tD*0=1/π  M M the black hole would blow up. For a supermassive black M =0 , (1) hole at a galactic center one would then expect that by the 1 π2 DMt 0 present time the black hole had grown up so big that its With D a constant. If we place a Morris-Thorne astronomical effects would be probably observable. How- ever, one might also expect that we still are evolving on a wormhole with initial throat radius b0 instead, the corresponding accretion mechanism [7-9] leads now to a too early time for making such observable effects observ- progressive increase of the wormhole size governed by able. All the above results have been obtained in the case b 0 that the entanglement energy density associated with the b =2 , (2) 1 π Dbt 0 sub-quantum potential would dominate over any other with D another constant, bringing us to consider the type of energy. More realistic models where contribu- existence of a big trip process [7-9] by which, relative to tions from dark and observable matters are taken into an asymptotic observer at r =  , the wormhole will account as well will be considered elsewhere. quickly grows up to engulf the universe itself, blowing up at a finite time in the future given by 3. Violation of Classical DEC

1  The quantum violation of the dominant energy condition t =2 . (3) π Db 0 that was entailed by the benigner models described in Ref. In this case, on times tt>  the wormhole converts into [6] has not any classical counterpart and therefore is an Einstein-Rosen bridge which decays into a black hole physically allowable. We shall investigate in what fol- plus a white hole that will in this case progressively lose lows the sense in which that violation would permit the mass to vanish at t =  [7-9]. This result holds both for formation of Lorentzian wormholes and the nature of a static wormhole metric and when the throat radius is such wormholes. Choosing the simplest mixed en- allowed to be time-dependent [7-9]. ergy-momentum tensor components and the ansatz that correspond to a static, spherically-symmetric wormhole We shall now briefly consider solution a . If 1/2 space-time with vanishing shift function, d=ds22 t CH00== >>4π G , then this solution corre-  222 2 sponds to an initial period of accelerating expansion with erdd r 2 (where d2 is the metric on the unit an equation-of-state parameter w greater, though very two-sphere), we can obtain a wormhole space-time solu- close to –1. This situation would stand until a time tion from the corresponding Einstein equations containing the extra sub-quantum or entangled energy density and HG 4π  0 pressure plus the stress-energy tensor components that ta =, (4) 4πG correspond to a Morris-Thorne wormhole [16], that is which corresponds to w = 1/ 3 . After ta the universe 2  18πG 9r0 would keep expanding but now in a decelerating way ee24 1=   until a time r rG3 8π r

2 On the other hand (2), let us consider a three-geometry 18 πG 3r0 2ep1= 4  at t = const. respecting spherical symmetry and r <  . rG3 8π r  For, it suffice choosing the slices  =,0 , so that 2 2 18 πG 3r 2dr 22  0 d=s r d. (12) ep=,4 2 23r r 22 8πGr 10   r r 2 P where [3-6] We wish now visualize the slice (12) as removed from 2  6πGH 1 H  (6) metric (8) and embedded in three-dimensional Euclidean space which can be described by cylindrically symmetric 3H 2 pGH=4π  122  1= wt, (7) coordinates rz,,   2H 22222 d=ddszrr d. (13) With H =4 G  and H =4GH , sup- 0 The embedding surface is describable by a function plemented by the condition  p =  , and a Lagran- zzr= , Thus, gian density from which we can derive the result that the   2 auxiliary scalar field  [3-6] satisfies  =0 and 22dz 22 2 d=1sr dr d. (14)  =1, to finally obtain  dr dr 2 d=dst22  r 22 d, (8) Metric (14) will be the same as metric (12) if we iden- r 2 2 10 22 r tify the coordinates r, of the embedding space with r 2 P those of the wormhole and we require the function zr() 2 to satisfy where  P plays the role of a negative cosmological 1/ 2 2 constant, with r0 the radius of the spherical wormhole dzr throat and  the Planck length. Note that if   p =1. (15) P dr rr224   was positive then no cosmic wormhole could be obtained, 0 P such as it happens for the de Sitter space. Now, if we want the wormhole to be connectible to a To such a rr -asymptotically anti-de Sitter space-time asymptotically anti-de Sitter space-time, the embedding we shall continue just denoting as an asymptotically surface should flare outward at or near the throat [16]. anti-de Sitter Morris-Thorne wormhole for the sake of This can be mathematically expressed by the condition simplicity. Metric (8) still shows an apparent event hori- d/d>022rz , which is in fact satisfied as it can be zon at checked from Equation (15), that is 22 2 224 2 14 P r0 1 d r rr0   P rh =2 , (9) =. (16) 22242 2 P dz rr0   P Which is the quantum-mechanical equivalent to the Finally (3) the considered wormhole can be shown to classical event horizon at rr= of the usual Morris- 0 be traversable as a two-sphere surrounding one of its Thorne wormhole. mouths where space-time is nearly anti-de Sitter can be Metric (8) is by itself nevertheless an actual asymp- regarded as an outer trapped surface to an observer totically anti-de Sitter wormhole because, (1) if that met- looking through the wormhole from the other mouth. ric is written as It can be readily checked that for   p >0 there is 22222 d=dst d  r d,2 (10) no metric like (8) which can show properties (1) - (3). then the new parameter [16] However, even though the asymptotically anti-de Sit- ter wormhole can be converted into a time machine with r rrd1  == line element r 2224 0 2  2 rr0  P r P 2222222 d=1sgFt d  d     rr0 sin  d, 22rr2224 r  22 r 1 (17) ln PPP0 (11) 22 (where we have assumed the right mouth to move with 14  Pr0  relative velocity v with respect to the left one, so induc- goes from  (when r =  ) to zero (at rr= ) and ing time travel to be allowed, g =d/d 2 vt is the ac- 0 1/2 finally to  (when r =  again), such as it is ex- celeration of the moving mouth,  =1 v2 , and pected for a wormhole with a throat at r= r0 which is F() is form factor vanishing on the left half (   0 ) of traversable and can be converted into a time machine. the wormhole and increases monotonously from 0 to 1 as

struction that the asymptotically anti-de Sitter wormholes could not enter the big trip phenomenon in a multiverse scenario [5,17-19]. Nevertheless, asymptotically anti-de Sitter wormholes can still undergo a swelling process when accreting phantom energy though their throat radius can never exceed the size of the universe where it originally existed. Let us in fact consider the process of dark energy accretion onto one such wormholes. Then, following the procedure put forward elsewhere [7-9], one can finally derive for the variation rate of the throat radius the expression 2 rrQ0 =4 π 

r j 2 –drTrr i ,,,,0 P  r r  1,0 22rp  eqf (19) r 2 P where Q is positive constant, P and  respectively are the pressure and energy density of dark energy,

2 r0 rqf = 0 (20)  P  is the quantum-mechanical extension of the usual asymptotic flatness. It therefore corresponds to the value of Figure 1. Kruskal diagram of the rt,  of the maximally the radial coordinate that makes flat the metric classi- extended space-time of the quantum-mechanical asymp- cally;  is a given function defined in terms of the totically anti-de Sitter Morris Thorne wormhole whose j classical analogous is the classical asymptotically flat Mor- components of the stress-energy tensor Ti [13], and we ris Thorne wormhole. Null geodesics are at 45 degrees have used coordinates so that G==c1. In the present to the vertical. The regions where V and W simultaneously formalism there is not anything like an asymptotically have the same sign are forbidden. flat space-time because of the existence of the entanglement energy density which in turn is the origin of the one moves rightward from the throat to the right mouth), asymptotically anti-de Sitter wormholes. Evaluating then and its metric can be maximally extended to smooth out this expression at rr= qf , that is at values of the radial its apparent singularity at rr= h (Figure 1), coordinate that approach infinity, we get the simpler rate

2 2221 2 2rrQp =4π  . (21) d=sV 4ddWr 4  VW 1 d , 0 qf   42 VW P 02 P Now, for a general equation of state pw=  and a (18) simple quintessence field where [14]

31 w 2 =/at a =13/21C w t t  (with 00 0 0 (22) Vt=exp2   r * , Wt=exp2  r *,  P   P  with C =8π0 /3, one may integrate the above rate where equation to finally have

1 2 2224 22   rr*= ln 2PPPr0 r  2r  1 2     4πQwtt10 2  P rr=exp  , (23) 00i 3  2  11wCt  t  and  P 0   2  2 221 VW 1 rr=4 ), with r0i the initial radius of the wormhole throat. 4VW 0 2  P For da rk energy with w >1 we see that the throat radius slowly decreases, tendin g to a constant radius these wormholes cannot insert their mouths into the as- 2 ymptotically flat regions of larger universes because of rQ0iPexp( 4 /  ) as t . In the case that the obvious topological reasons. It is for this topological ob- quintessence field behaved like phantom energy,

Copyright © 2011 SciRes. IJAA 88 P. F. GONZÁLEZ-DÍAZ w <1 , the radius of the w ormh ole throat exponentially its semiclassical thermal properties, including a negative increased to reach an infini te value just at the big rip sin- temperature and a phantom like radiation, even though gularity time, ttbr =2/310  wC . Therefore, an its thermodynamical formulas all obviously depend on  2 asymptotically anti-de Sitter Lorentzian wormhole does the cosmological constant  P as well. not give rise to the phenomenon of big trip, which re- The results derived in the current paper appears to quires the wormhole size to blow up before reaching the quantum-mechanically prevent the emergence of the big big rip singularity (Figure 2). In the most consistent trip phenomena which is left to just be a cl assical artifact, 2πGt 2  Ht a quantum effect that was expected to take place due to situation where we use the solution aae= 0  0 be instead induced by the bigness reached by both the [3-6] the increasing of the wormhole throat will take wormhole throat size and the universe itself. place in a rather quick fashion but yet it only tends to This work was supported by MEC under research pro- infinity as time does too. In this way, it also follows that ject no. FIS2008-06332. The author also acknowledges neither in this case we find the big trip phenomenon Alberto Rozas-Fernández from IFF and Carmen L. Thus, all situations involving a big trip that can be even- Sigüenza of the Estación Ecológica de Biocosmologa de tually considered would require the presence of worm- Medelln, Spain, for useful discussions and comments. holes which are asymptotically flat, not those that are asymptotically anti-de Sitter. One can finally remark that, quite the opposite to what happens with phantom energy accretion, would be expected to occur in the semiclassical thermal process that leads to the vanishing of an AdS Morris-Thorne quantum wormhole. In fact, by extending the results derived for the case of the spherically symmetric Morris-Thorne wormhole by using the so called 2 + 2 Hayward formalism [20] to the present case, we finally obtain that our quantum-mechanical AdS Morris-Thorne wormhole should be characterized by an entropy given by

22 π 14 P r0 1 SAdSMTw   SAdSMTw == 2 . (24) 4 2 P It can also be derived that these Lorentzian quantum wormholes are able to typically emit, through a semiclassical process, phantom like radiation at a negative temperature which is associated with a positive definite surface gravity  AdSMTw by

 AdSMTw 1 22 TrAdSMTw==1. P 0 (25) 2π 2πr0 Of course, Equations (24) and (25) reduce to their as- Figure 2. Evolution of the radius of an asymptotically ymptotically flat Morris-Thorne values [20] in the limit anti-de Sitter wormhole throat, b0 , induced by accretion of where 0 . phantom energy. At time tt= br , the negative exotic mass Summing up, we have derived a quantum-echanically becomes infinite and then starts steadily decreasing while originated asymptotically anti-de Sitter Lorentzian worm- the universe contracts, always keeping a smaller radius than hole metric w hic h cannot enter any big trip phenomena that of the universe. There is no b ig trip for these wormholes because of 1) the topological obstruction that the nega- but just an identification of the throat with the universe tive cosmological constant would oppose to a smooth when this reaches an infinite size. Thus there could be no insertion of the asymptotic wormhole regions on the uni- disruption of the causal evolution of the whole universe. verses that makes up the multiverse, and 2) a late quan- Because the big rip should be dominated by still unknown tum-mechanically induced swelling of the throat that quantum-gravity effects that presumably prevents the clas- postpones the emergence of the wormhole size diver- sical singularity, one is tempted to conjecture that at the big gence exactly to the moment at which the big rip singu- rip the universe becomes a quantum universe which is un- larity takes place. Such a wormhole shares with its as- distinguishable of the throat of a asymptotically anti- de ymptotically flat classical relative the essential nature of Sitter wormhole.

4. Acknowledgements [10] V. Faraoni and W. Israel, “Dark Energy, Wormholes, and the Big Rip,” Physical Review D, Vol. 71, No. 6, 2005, This work was supported by MEC under research project Article ID 064017. doi:10.1103/PhysRevD.71.064017 no. FIS2008-06332. The author also acknowledges Al- [11] A. V. Yurov, V. A. Yurov, and S. D. Vereshchagin, as- berto Rozas-Fernandez from IFF and Carmen L. tro-ph/0503433. Siguenza of the Estacion Ecologica de Biocosmologia de [12] V. Faraoni, “No ‘Big Trips’ for the Universe,” Physics Medellin, Spain, for useful discussions and comments. Letters B, Vol. 647, No. 5-6, 2007, pp. 309-312. doi:10.1016/j.physletb.2007.02.048

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