The Number of Information Bits Related to the Minimum Quantum and Gravitational Masses in a Vacuum Dominated Universe

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1 Wilfrid Laurier University Scholars Laurier Physics and Couter Science Faculty Publications Physics and Couter Science 01 The uber of Inforation Bits Related to the Miniu Quantu and Gravitational Masses in a Vacuu Doinated Universe Ioannis Haranas Wilfrid Laurier University, iharanas@wlu.ca Ioannis Gkigkitzis East Carolina University, gkigkitzisi@ecu.edu Follow this and additional works at: htt://scholars.wlu.ca/hys_faculty Part of the Matheatics Coons, and the Physics Coons Recoended Citation Haranas, I., Gkigkitzis, I. The uber of Inforation Bits Related to the Miniu Quantu and Gravitational Masses in a Vacuu Doinated Universe. Astrohys. Sace Sci. (01) 46:1 18. DOI: /s This Article is brought to you for free and oen access by the Physics and Couter Science at Scholars Laurier. It has been acceted for inclusion in Physics and Couter Science Faculty Publications by an authorized adinistrator of Scholars Laurier. For ore inforation, lease contact scholarscoons@wlu.ca.

2 The uber of Inforation Bits Related to the Miniu Quantu and Gravitational Masses in a Vacuu Doinated Universe Ioannis Haranas 1 Ioannis Gkigkitzis 1 Deartent of Physics and Astronoy, York University 4700 Keele Street, Toronto, Ontario, MJ 1P, Canada E-ail:yiannis.haranas@gail.co 1 Deartents of Matheatics, East Carolina University 14 Austin Building, East Fifth Street, Greenville C , USA E-ail: gkigkitzisi@ecu.edu Abstract Wesson obtained a liit on antu and avitational ass in the universe by cobining the cosological constant, Planck s constant, the seed of light c, and also the avitational constant G The corresonding asses are.010 kg and.10 kg resectively, and in general can be obtained with the hel of a generic diensional analysis, or fro an analysis where the cosological constant aears in a four diensional sace-tie and as a result of a higher diensional reduction. In this aer our goal is to establish a relation for both antu and avitational ass as function of the inforation nuber bit. For this reason, we first derive an exression for the cosological constant as a function of inforation bit, since both asses deend on it, and then various resulting relations are exlored, in relation to inforation nuber of bits. Fractional inforation bits ily no inforation extraction is ossible. We see, that the order of agnitude of the various araeters as well as their ratios involve the large nuber 10 1, that is roduced naturally fro the fundaental araeters of odern cosology. Finally, we roose that in a colete antu avity theory the idea of inforation the ight have to be included, with the antu bits of inforation (q-bits) as one of its fundaental araeters, resulting thus to a ore colete understanding of the universe, its laws, and its evolution. Key words: Cosological constant, antu ass, avitational ass, inforation bit, fractional inforation bit, large nuber hyothesis. 1. Introduction Observational data fro galaxies and avitational lensing of high z asars and cosic icrowave backound radiation, suggest that the 99% of the universe aterial consists of dark atter (Overduin, and Wesson, 00). Therefore the density of the vacuu contributes a high fraction of the corresonding dark atter, whose energy density is given by: c v (1) 8G In this aer all authors have contributed eally.

3 where is the cosological constant, c is the seed of light and G is the avitational constant. Today s data indicate that which akes that the distance to the horizon aroxiately is g/c - 5 and therefore (Krauss and Starkan, 1999), RH (Krauss and Starkan, 1999). It is now coon knowledge that the cosological constant of general relativity is a araeter derived out of higher-diensional theories in a four diensional reduction, in an effort to unify avity with article hysics (Weston, 1999). Soe of these theories include 10D suersyetry, 11D sueravity and 6D string theories. The aforeentioned theories should resent the natural ound where the antu of action should naturally aear. In articular, theories like the basic extension of 4D Einstein theory and the low-energy liit of higher-d theories constitute the odern incarnation of (non-coact) 5D Kaluza-Klein theory. This has been intensively studied recently, under the naes induced-atter theory (Wesson, 199) and (Mashhoon, et al., 1998), and ebrane theory (Randall, Sundru, 1998) and (Arkani-Haed, 1998). Both theories redict the existence of a fifth force, which ight be the force via with articles can interact (You, 000). Therefore and as a result it ight be ossible to detect assive, and assless or hoton like articles as well as hotons articles in the sacetie. given by: Following Wesson (004) two different ass scales can be fored. First, a antu ass scale c kg, () where and is the cosological constant, is Planck s constant, and c is the seed of light. This is the scale for the iniu antu ass in the universe. Siilarly, the avitational ass scale is given according to the eation (Wesson, 004): c kg, () G where G is the constant of universal avitation. Our goal is to introduce the idea of inforation as a new otential araeter in any of today s natural henoena. In articular, this contribution, investigates Eqs () and () given in Wesson s original aer and their relation to the nuber of inforation bits, via a relation that relates the cosological constant to the nuber of inforation bits. ext, we further investigate various resulting exressions and their relation to the nuber of inforation bits. As an exale, Planck s length P is exressed in ters of the antu article s Coton wavelength and the nuber of inforation bits. ext, the nuber of inforation bits involved in both article asses as defined by Wesson is derived. Finally, using the Margolus-Levitin theore, we calculate the

4 nuber of oerations erfored by the two articles at tie t eal to the age of the universe as a function of the nuber of inforation bits.. Quantu and Gravitational Masses and the uber of Inforation Bits It is now a suggestion that antu echanics is non-local, but at the sae tie a fundaental echanis for that is not known. However, the holoahic rincile indicates a ossible non-locality echanis in any vacuu-doinated Friedann universe. To be ore recise, a holoahic non-local antu echanical descrition can be ossible for a finite aount of inforation in a closed vacuudoinated universe. Today s theories, assue that the universe began by a antu fluctuation fro nothing, underwent inflation and becae so large that it is locally alost flat, and that after the inflationary era the vacuu energy density of the universe is constant. This is the case of the existence a non-zero cosological constant. More inforation on such a universe arising a in antu cosological way is resented in Mongan (001). When such closed universe began it already contained all the inforation that it will ever contain. If nothing exists outside of the closed hysical universe, that would ily that no inforation can coe into the universe fro elsewhere either (Mongan 001). In the case of a de-sitter etric we solve for the avitational radius of such universe we have that: 1 r 0, (4) and the de-sitter horizon can be easily found to be: r H. (5) 1, One of the ost enigatic features of de Sitter sace is its entroy. That de Sitter sace has finite entroy ay be exected, based on the aearance of a horizon in ure de Sitter. This horizon is, however, alitatively different fro a black hole horizon. The osition of the horizon is observer deendent, and because of this it is not entirely clear which concets about black holes carry over to de Sitter sace. In fact, the de Sitter cosological horizon looks in any ways like the Rindler horizon in Minkowksi sace. Let us now roceed with the derivation of how the cosological constant deends on the nuber of inforation bits. We start with the entroy of such black hole that can be written as according Bekenstein (197) to be: k S M AH 4 (6) This is the Bekenstein-Hawking area-entroy law, which says that the entroy S is associated with an event horizon, and where k B is the Boltzann constant, is the horizon area A H divided by 4, where

5 G is Planck s length (Bekenstein, 197). This is a acroscoic forula and it should be viewed c in the sae light as the classical acroscoic therodynaic forulae. It describes how roerties of event horizons in general relativity change as their araeters are varied. Therefore the entroy of such a black is given by (Bousso and DeWolfe, 00): SU H n D k B kb D 1 D kb 1 kb A H 4 P 4 n P 4 P where D =4, is the diensionality of the de-sitter sace, n reresents oints on n-1 diensional shere and is the gaa function of the indicated arguent. Solving for the cosological constant we obtain: km, (8) SU H Taking into account that the nuber of inforation bits relates to the entroy S in the following way is given (Lioyd, 000): S k ln, (9) B where k B is the S is the Boltzann constant. Substituting in Eq. (8) we obtain that: ln ln ax. (10) (7) where 1/ c G (Haranas, 00) the axiu cosological constant, the nuber of ax / inforation bits, is the Planck length. This is the cosological constant as a function of the inforation bits associated with the cosological horizon. Our derived exression aees with the eation given in Mongan (007). ext substituting Eq. (10) in Eq. () and rearranging we obtain that and eations which give the deendence of Wesson asses on the nuber of inforation bits, we obtain the following exressions: c ln 4.50, (11) c solving for the Planck length we can exress this iortant constant of hysics as a function of the inforation bit in the following way: Co P 4.50 c, (1) ln

6 where Co / c is the Coton wavelength related to the iniu antu ass scale article in the universe as it is given in Wesson (004). ext, solving for the inforation bit nuber related to the iniu antu ass we obtain that: Co 0.51 P 0.51 Co ax, (1) therefore we find that the nuber of inforation bits involved in Wesson s iniu antu ass is given by the sare of the ratio of its corresonding Coton wavelength over the Planck length. Siilarly, for the avitational ass in the universe we obtain the following inforation bit nuber relation: c G ln c G solving for the nuber of inforation bits we obtain that: where G R ax c P, (14). (15) R is the avitational radius of the corresonding avitational ass. Eation (9) deonstrates that the nuber of inforation bits involved in the avitational ass is eal to the ratio of half the article s avitational radius over its wavelength Coton wavelength. Fro Eqs. (7) and (9) we have that: R Co. (16) Let us now assue that the antu ass can be eal to the Planck ass. This will resuose the following condition on the cosological constant. To find the condition on labda we eate the Planck ass to the antu ass derived by Wesson (004) and therefore we have: c, (17) G c fro which we obtain that: c ax, (18) G where ax is the value of the cosological constant during the Planck era (Haranas, 00). cosological constant = ax resuoses a tie era slightly earlier that the Planck tie liit, an era A

7 the eations of hysics fail to describe. Siilarly, reesting that results in the sae eation like Eq. (18), which again corresonds to an era earlier that the Planck era. According to the Margolus-Levitin theore, the axiu rate at which logical oeration could be erfored by a hysical syste with energy E is Pl E / (Margolus and Levitin, 1998). Therefore, the axiu nuber of oeration that could have been erfored by the asses of the articles redicted by Wesson in the observable universe when written as a function of the nuber of inforation bits becoe. For the antu ass liit we obtain: n c t u c H 0, (19) where t u = 1/H 0 is the age of the universe, and H 0 = c (Isla, 199) is Hubble s constant. We see that Eq. (19) roves to be indeendent of the nuber of inforation bits. Siilarly, for the avitational ass we obtain the following total nuber of oerations that the avitational ass can erfor in a tie eal to the age of the universe to be: 4 c c c 1 6 c 6 ax n tu G G H G which, in ters of the nuber of inforation bits, the nuber of oerations n 0, (0) n becoes: 6c ln ln. (1) G ext, following Faus (010) and using an exression that connects the cosological constant to the Hubble araeter we can derive an exression of how the Hubble araeter relates to the inforation bit H0 1 nuber. Eating Eq. (10) to 1 c c t c H t 0 (Faus, 010) and (Isla, 199) we obtain that () We obtain an -1/ deendence of the resent value of the cosological constant on the inforation bit nuber. Using the tie deendence of the cosological constant as it is given by Faus (010) we derive the deendence of inforation nuber bit on cosic tie t, and therefore we obtain the following exression: t ln tp t ()

8 5 44 where t = G / c (010) the ass of the ion s is the Planck tie. Siilarly, with reference to Hajduković 1/ H /Gc (Hajduković, 010) we obtain that the ass of the ion deends on the nuber of inforation bit and tie in the following way: t t t, (4) G G P Gc 1/ 5 where t P G/c is the Planck tie, and G is the avitational constant. Therefore we find that the following ion ass deendences on the inforation bit and tie t are / 1/ t. ext, with reference to Santos (010), and using his derived exression for the redicted by hi avitational density given by the eation Santos (010): 6 G av 0, (5) 4 where 4 / can be written as function of inforation bit in the following way 0 av ax G G Gc t (6) And therefore we find that the avitational density as derived by Santos (010) deend on the inforation bit and tie t are 1 t. av. Discussion and uerical Results 6 To evaluate our findings let as use.10 (Wesson, 004) kg and fro that we obtain the following Coton wavelength to be Co , or eivalently Co R uni, and Co therefore we obtain that the ratio of the antu to avitational Coton wavelengths is: Co Co (7) Siilarly, the corresonding nuber of inforation bits related to the antu and avitational ass scales are given by: 107 bits, (8) bits. (9) The corresonding relation between the nubers of bits nuerically becoes:

9 (0) Furtherore, nuerical values for the values of inforation bits involved in various resulting scenarios are calculated. We first calculate the aount of inforation bits involved in today s value of the cosological constant. Using, the density of the vacuu to be g/c - (Krauss and Starkan, 1999) and the distance to the horizon to be aroxiately RH (Krauss and Starkan, 1999) we obtain, that the cosological constant in the resent era is Therefore substituting for we obtain that the nuber of inforation bits involved in the resent value of the cosological constant is: bits. (1) ln Our redicted nuber aees with that given in Funkhouser (008). c 69 Max (Haranas, 00) we obtain that: G Siilarly in the case where 1.40 bits, () ln ax which corresonds to an entroy of aroxiately S 9.8 k B. Eq. () indicates a fractional nuber of inforation bits. Since inforation bits can only take values of zero and one fractional inforation bits would ily no inforation extraction. Therefore taking the ratio of / we find that: () ax Finally, to find the nuber of inforation bits that the de-sitter horizon contains we write Eq. (5) in ters of the nuber of inforation bits via the cosological constant in the following way: r H r ln ln ax H First, in the case where the antu ass ass 1 bits. (4) redicted by Wesson (004) becoes eal to the Planck Pl, the cosological constant, is indeendent of the Planck length and involves a secific nuber of inforation bits that is just a nuerical constant. Furtherore, in the case where the antu ass redicted by the above theories is eal to the Planck ass Pl the cosological constant involves a secific constant nuber of inforation bits that is indeendent of the Planck length and it is eal to: ac

10 G ln c ln P 4. 5 bits. (5) Here again fractional inforation bits corresond to now inforation. Siilarly, for the ratio of avitational to the antu ass as redicted by Wesson (004) we obtain that: c ax G (6) In ters of the nuber of inforation bits, Eq. (6) can be written as: ax ln Pl 0.1, (7) and therefore, the avitational ass scale redicted by Wesson relates to the antu ass scale in the following ways: ax, (8) or eivalently in ters of the inforation bits the two ass scales relate in the following way: ln. (9) Siilarly, the ratio of the ass of the universe to that of antu ass as redicted by Wesson (004) becoes: M uni (40) Finally, the nuerical value of the total nuber of oerations erfored in the age of the universe by a avitational article is: 1 n, (41) Therefore the their ratio is n n (4) The nuber 10 1 aears in an enseble of ure nubers naturally roduced fro fundaental cosological araeters that ight constitute a new-large nuber coincidence siilar to that ostulated by Dirac. These nubers constitute a coelling, new large nuber coincidence roble (Funkhouser, 008). In this aer, we deonstrate that new large nuber coincidence also exist in the relations involving the inforation bits, and nuber of oerations n, in its relation to the fundaental ass scales redicted by Wesson (004), and also their relation to the cosological constant. All these are ossible after a relation relating the cosological constant to the inforation nuber bit is derived.

11 The involveent of the cosological constant in any of today s cosological relations introduces a direct relation of the cosological araeters to the inforation nuber bit. Certain cosological scenarios involve fractional nuber of inforation bits, which ilies no inforation extraction is ossible. Our result is in aeeent with Faus (011). Few authors have given various exlanations, for exale, Funkhouser (008) has deonstrated ilicit hysical ure nuber relations that result fro the standard cosological odel. Our ain interest is to exress the basic relations in this aer as functions of the nuber of inforation bits. Thus we have found that the iniu antu ass and its corresonding Coton wavelength scale as 1/, where the nuber of inforation bits involved in the iniu antu ass scales as the ratio of the antu Coton wavelength over the Planck length sared. Siilarly, the avitational ass scales as 1/, where the corresonding nuber of inforation bits is eal to the half of the avitational radius of the avitational ass divided by Coton wave length of the avitational article. It ight be iortant that the nuber of inforation bits enters Wesson s definition of two different ass scales through the deendence of the cosological constant that itself deends on the inforation bit. Therefore the avitational inforation bit and also the antu of inforation bit also called the q-bit that is itself a icroscoic syste, such as an ato, or nuclear sin, or hoton (Caurro and Hjørland, 00), ight be the corresonding units at these extree scales of oeration of nature s henoena, where antu avity oerates. It can be of a eat interest to exaine the antu sacetie fluctuations fro an inforation oint of view, and try to lace inforation boundaries on the ultraviolet cut off, but this will be our next aer. Thus an ultiate inforation theory ight find its lace at the heart of this antu avity theory, as well as in all the other theories entioned above. In a universe that exands the nuber of inforation bits reired to define a article increase, and therefore the total aount of inforation also increases. If the cosological constant varies as 1/ t, this would result in an increased nuber of inforation bits, as the entroy of the universe increases. Therefore, at uch later ties in our universe a very large aount of inforation bounded by the light cones would be necessary to describe its evolution. 4. Conclusions We have used the results redicted by Wesson in order to investigate the deendence of the iniu antu and avitational asses in a vacuu doinated Friedann universe on the nuber of inforation bits. The nuber 10 1 aears in an enseble of ure nubers naturally roduced fro fundaental cosological araeters. Using our derived exression of the cosological constant as a function of the inforation bit, we have found that the iniu antu ass involves an -1/ deendence on inforation bits, where the avitational ass scale has an 1/ deendence resectively.

12 Finally, we roose that a colete antu avity theory ight have to include the antu bit of inforation as one of its fundaental araeters for a ore colete descrition of the universe, its laws, and its evolution. Acknowledgeents: The authors would like to thank the editor in chief of Astrohysics and Sace Science journal Dr. Michael Doita, for encouraging us to resubit a second revised version of our aer. References Arkani-Haed,., Diooulous, S., Dvali, G. R. Phys. Lett. B49, 6, Bekenstein, J. D., Black Holes and Entroy, Phys. Rev. D 7,, 197. Bousso, R., DeWolfe, O., and Myers, R. C., Unbounded entroy in saceties with ositive cosological constant, arxiv: he-th/005080v 18 Jun 00. Caurro, R., and Hjørland, B., Annual Review of Inforation Science and Technology Ed. B. Cronin, Vol. 7, Chater 8, , 00. Faus, A. The case for the Universe to be a antu black hole, Astrohys Sace Sci. vol. 5, 11, 010. Faus, A. A., Recent Researches in Artificial Intelligence, Knowledge Engineering and Data Bases, Plenary Lecture WSEAS Aiked 11 Cabridge, 1- February 011. Funkhouser, S., A new large-nuber coincidence and a scaling law for the cosological constant, Proc. R. Soc. A. 464, , 008. Funkhouser, S. A., Fundaental scale of ass for black holes fro the cosological constant, arxiv:hysics/070189, 008 Haranas, I., The Teerature of a Black Hole in a De-Sitter Sace-Tie, Ro. Astron. Journal, Vo. 1 o., 00. Isla, J.,. An introduction to atheatical cosology, Cabridge Univ. Press,. 70, 199. Krauss, L. M., and Starkan, G. D., Life, The Universe, and othing: Life and Death in an Ever Exanding Universe, arxiv: astro-h:99018v1, 1 Feb 1999 Lloyd, S., Coutational caacity of the universe, arxiv: ant-h/ v1 4 Oct 000. Margolus,., and Levitin, L. B., The axiu seed of dynaical evolution, Physics D, 10, , Mashhoon, B., Wesson, P. S., Liu, H., Gen. Rel. Grav. 0, 555, Mongan, T., Gen. Rel. Grav., , 001. Mongan, T. R., Holoahy and non-locality in a closed vacuu-doinated universe, Int. J. of Theor. Phys , 007. Overduin, J. M., Wesson, P.S. Dark Sky, Dark Matter, Institute of Physics, London, 00.

13 Randall, L., Sundru, R. Mod. Phys. Lett. A1, 807, Santos, E., Dark energy as sace-tie curvature induced by antu vacuu fluctuations, arxiv: v [-qc] 9 ov 010. Wesson, P. S., Phys. Lett. B76, 99, 199. Wesson, P. S., Sace, Tie, Matter, World Scientific, Singaore, Wesson, P. S., Is ass antized, arxiv :-qc v1 0 Se 00. You, D., Phys. Rev. D 6, 08400, 000.