We propose a classical mechanism for the cosmic expansion during the
radiation-dominated era. This mechanism assumes that the Universe is a
two-component gas. The first component is a gas of ultra-relativistic "normal"
particles described by an equation of state of an ideal quantum gas of massless
particles. The second component consist of "unusual" charged particles (namely,
either with ultra-high charge or with ultra-high mass) that provide the
important mechanism of expansion due to their interaction with the "normal"
component of the gas. This interaction is described by the
Reissner--Nordstr\"om metric purely geometrically -- the ``unusual'' particles
are modeled as zero-dimensional naked singularities inside spheres of
gravitational repulsion. The radius of a repulsive sphere is inversely
proportional to the energy of an incoming particle or the temperature. The
expansion mechanism is based on the inflating of the "unusual" particles (of
charge $Q$) with the drop of the temperature -- this drives apart all neutral
particles and particles of specific charge $q/m$ such that ${sign}(Q) q/m \ge -
1$. The Reissner--Nordstr\"om expansion naturally ends at recombination. We
discuss the range of model parameters within which the proposed expansion
mechanism is consistent with the restrictions regarding quantum effects.Comment: 9 pages, LaTe

Boyer

Brisudova

Carter

Cohen

de Rujula

Ellis

Emil M. Prodanov

Frolov

Gaida

Gondolo

Guth

Hawking

Kerr

Lucchin

Misner

Newman

Nordstöm

Peebles

Preskill

Reissner

Robertson

Rossen I. Ivanov

V.G. Gueorguiev

Walker

English

arXiv

Prodanov, Emil

Ivanov, Rossen

Gueorguiev, Vesselin

Irish Universities

Reissner–Nordstrom Expansion

We propose a classical mechanism for the cosmic expansion during the radiation-dominated era, assuming the Universe as a two-component gas. The first component is the ultra-relativistic “standard” fraction described by an equation of state of an ideal quantum gas of massless particles. The second component consist of superheavy charged particles and their interaction with the “standard” fraction drives the expansion. This interaction is described by the Reissner–Nordstr¨om metric purely geometrically — the superheavy charged particles are modeled as zero-dimensional naked singularities which exhibit gravitational repulsion. The radius of a repulsive sphere, surrounding a naked singularity of charge Q, is inversely proportional to the energy of an incoming particle or the temperature. The expansion mechanism is based on the “growing” of the repulsive spheres of the superheavy particles with the drop of the temperature — this drives apart all neutral particles and particles of specific charge q/m such that sign(Q)q/m ≥ −1. The Reissner–Nordstr¨om expansion mechanism naturally ends at Recombination. We model the Universe during the Reissner–Nordstr¨om expansion as a van der Waals gas and determine the equation of state

GUEORGUIEV, V.G.

Arrow@TUDublin

REISSNER–NORDSTROM¨ EXPANSION

Technological University Dublin ARROW@TU Dublin Articles School of Mathematics 2007-01-01 Reissner–Nordstrom Expansion Emil Prodanov Technological University Dublin, emil.prodanov@tudublin.ie Rossen Ivanov Technological University Dublin, rossen.ivanov@tudublin.ie Vesselin Gueorguiev University of California, San Francisco Follow this and additional works at: https://arrow.tudublin.ie/scschmatart  Part of the Cosmology, Relativity, and Gravity Commons, Other Astrophysics and Astronomy Commons, and the Physical Processes Commons Recommended Citation Prodanov, E., Ivanov, R. & Gueorguiev, V.G. (2007) REISSNER–NORDSTROM¨ EXPANSION, in H.V. Klapdor-Kleingrothaus and G.F. Lewis, (eds.) "Dark Matter in Astroparticle and Particle Physics - Proceedings of the 6th International Heidelberg Conference", World Scientific, 2007. This Article is brought to you for free and open access by the School of Mathematics at ARROW@TU Dublin. It has been accepted for inclusion in Articles by an authorized administrator of ARROW@TU Dublin. For more information, please contact yvonne.desmond@tudublin.ie, arrow.admin@tudublin.ie, brian.widdis@tudublin.ie. This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License Technological University DublinARROW@TU DublinConference papers School of Mathematics2007REISSNER–NORDSTROM¨ EXPANSIONEmil ProdanovTechnological University Dublin, emil.prodanov@dit.ieRossen IvanovTechnological University Dublin, Rossen.Ivanov@dit.ieV.G. GUEORGUIEVUniversity of California, San FranciscoFollow this and additional works at: https://arrow.dit.ie/scschmatconPart of the Astrophysics and Astronomy CommonsThis Conference Paper is brought to you for free and open access by theSchool of Mathematics at ARROW@TU Dublin. It has been accepted forinclusion in Conference papers by an authorized administrator ofARROW@TU Dublin. For more information, please contactyvonne.desmond@dit.ie, arrow.admin@dit.ie, brian.widdis@dit.ie.Recommended CitationProdanov, E., Ivanov, R. & Gueorguiev, V.G. (2007) REISSNER–NORDSTROM¨ EXPANSION, in H.V. Klapdor-Kleingrothaus andG.F. Lewis, (eds.) "Dark Matter in Astroparticle and Particle Physics - Proceedings of the 6th International Heidelberg Conference",World Scientific, 2007.March 4, 2008 3:37 WSPC - Proceedings Trim Size: 9in x 6in 31˙Prodanov304REISSNER–NORDSTRO¨M EXPANSIONEMIL M. PRODANOV∗, ROSSEN I. IVANOV†School of Mathematical Sciences, Dublin Institute of Technology,Kevin Street, Dublin 8, IrelandE-mails: ∗Emil.Prodanov@dit.ie; †Rossen.Ivanov@dit.ieand V.G. GUEORGUIEVUniversity of California, Merced, USAE-mail: vesselin@mailaps.orgWe propose a classical mechanism for the cosmic expansion during theradiation-dominated era, assuming the Universe as a two-component gas. Thefirst component is the ultra-relativistic “standard” fraction described by anequation of state of an ideal quantum gas of massless particles. The secondcomponent consist of superheavy charged particles and their interaction withthe “standard” fraction drives the expansion. This interaction is described bythe Reissner–Nordstro¨m metric purely geometrically — the superheavy chargedparticles are modeled as zero-dimensional naked singularities which exhibitgravitational repulsion. The radius of a repulsive sphere, surrounding a nakedsingularity of charge Q, is inversely proportional to the energy of an incomingparticle or the temperature. The expansion mechanism is based on the “grow-ing” of the repulsive spheres of the superheavy particles with the drop of thetemperature — this drives apart all neutral particles and particles of specificcharge q/m such that sign(Q)q/m ≥ −1. The Reissner–Nordstro¨m expansionmechanism naturally ends at Recombination. We model the Universe duringthe Reissner–Nordstro¨m expansion as a van der Waals gas and determine theequation of state.In 1971, Hawking1 suggested that there may be a very large numberof gravitationally collapsed charged objects of very low masses, formed asa result of fluctuations in the early Universe. A mass of 1014 kg of theseobjects could be accumulated at the centre of a star like the Sun. Hawk-ing treats these objects classically and his arguments for doing so are asfollows:1 gravitational collapse is a classical process and microscopic blackholes can form when their Schwarzschild radius is greater than the PlanckMarch 4, 2008 3:37 WSPC - Proceedings Trim Size: 9in x 6in 31˙Prodanov305length (Gh/c3)−1/2 ∼ 10−35 m (at Planck lengths quantum gravitational ef-fects do not permit purely classical treatment). This allows the existence ofcollapsed objects of masses from 10−8 kg and above and charges up to ±30electron units.1 Additionally, a sufficient concentration of electromagneticradiation causes a gravitational collapse — even though the Schwarzschildradius of the formed black hole is smaller than the photon’s Compton wave-length which is infinite. Therefore, when elementary particles collapse toform a black hole, it is not the rest Compton wavelength hc/mc2 thatis to be considered — one should instead consider the modified Comptonwavelength hc/E, where E ∼ kT >> mc2 is the typical energy of an ultra-relativistic particle that went to form the black hole.1 Microscopic blackholes with Schwarzschild radius greater than the modified Compton wave-length hc/E, can form classically and independently on competing quan-tum processes. Hawking suggests that these charged collapsed object mayhave velocities in the range 50 – 10000 km/s and would behave in manyrespects like ordinary atomic nuclei.1 When these objects travel throughmatter, they induce ionization and excitation and would produce bubblechamber tracks similar to those of atomic nuclei with the same charge. Thecharged collapsed objects survive annihilation and, at low velocities (lessthan few thousand km/s), they may form electronic or protonic atoms:1the positively charged collapsed objects would capture electrons and thusmimic super-heavy isotopes of known chemical elements, while negativelycharged collapsed objects would capture protons and disguise themselves asthe missing zeroth entry in the Mendeleev table. Such ultra-heavy chargedmassive particles (CHAMPS) were also studied by de Rujula, Glashow andSarid2 and considered as dark matter candidates. Dark Electric Matter Ob-jects (DAEMONS) of masses just above 10−8 kg and charges of around ±10electron units have been studied in the Ioffe Institute and positive resultsin their detection have been reported3 — observations of scintillations inZnS(Ag) which are excited by electrons and nucleons ejected as the relicelementary Planckian daemon captures a nucleus of Zn (or S). The DAMA(DArk MAtter) collaboration also report positive results4 in the detectionof such particles using 100 kg of highly radiopure NaI(Tl) detector.These superheavy charged particles can serve as driving force for the ex-pansion of the Universe during the radiation-dominated epoch in a classicalparticle-scale model, which we recently proposed.5 Along with this type ofparticles, within our model, magnetic monopoles can also play the same rolefor the expansion of the Universe: it has been suggested6 that ultra-heavyMarch 4, 2008 3:37 WSPC - Proceedings Trim Size: 9in x 6in 31˙Prodanov306magnetic monopoles were created so copiously in the early Universe thatthey outweighed everything else in the Universe by a factor of 1012.The classical mechanism of the cosmic expansion relies on the assump-tion that the Universe is a two-component gas. One of the fractions is thatof ultra-relativistic “standard” particles of typical mass m and charge qwith equation of state of an ideal quantum gas of massless particles. Theother component consists of the superheavy charged particles of masses M(of around 10−8 kg and above) and charges Q (of around ±10 electroncharges and above) — exactly as those described earlier.For an elementary particle such as the electron, the charge-to-mass ra-tio is q/m ∼ 1021 (in geometrized units c = 1 = G), while for the su-perheavy charged particles, M <∼Q. In view of this, the general-relativistictreatment of elementary particles or charged collapsed objects of very lowmasses also necessitates consideration from Reissner–Nordstro¨m (or Kerr–Newman) viewpoint — for as long as their charge-to-mass ratio remainsabove unity. We also treat the superheavy charged particles classically (inline with Hawking’s arguments outlined earlier). That is, the superheavycharged particles are modelled as Reissner–Nordstro¨m naked singularitiesand the expansion mechanism is based on their gravito-electric repulsion.Instead of the Schwarzschild radius, the characteristic length that is to beconsidered now and compared to the modified Compton length,1 will bethe radius of the van der Waals-like impenetrable sphere that surroundsa naked singularity (see Cohen et al.7 for a very thorough analysis of theradial motion of test particles in a Reissner–Nordstro¨m field). As shown inProdanov et al.,5 for temperatures below 1031K, the radius of the impene-trable sphere of a superheavy charged particle of mass 10−8 kg and charge±10 electron units is greater than the modified Compton wavelength of thesuperheavy charged particle itself. The “standard” particles of the expand-ing Universe are therefore too far from the superheavy charged particles forquantum interactions to occur between the two fractions.Consider a “standard” particle of specific charge q/m, and a super-heavy charged particle of charge Q, such that sign(Q)q/m ≥ −1, with the“standard” particle approaching the superheavy charged particle from in-finity. The pseudo-Newtonian potential of the field of the naked singularityis given by5 U(r) = −(mM)/r + (qQ)/r + [(m/2)(−M 2 + Q2)]/r2 andthe field is characterized by three regions.5,7 The region between r = 0and r = r0(T ) is impenetrable due to the condition for reality of the ki-netic energy of the incoming test particle . The turning radius r0(T ) isgiven by5,7 r0(T ) = Q(q +m)/(kT ) for very high temperatures. This canMarch 4, 2008 3:37 WSPC - Proceedings Trim Size: 9in x 6in 31˙Prodanov307be thought of as the radius of an “impenetrable” sphere, surrounding thenaked singularity, that grows with the drop of the temperature: the higherthe energy (or the temperature), the deeper an incoming particle will pen-etrate into the gravitationally repulsive field of the naked singularity. Theregion between the turning radius r0(T ) and the critical radius rc ≥ r0(T )is repulsive. At this critical radius, repulsion and attraction interchange:5rc = M(Q2/M2 − 1)/[1 − (qQ)/(mM)] and the region above rc is attrac-tive. As the temperature drops, the superheavy charged particles “grow”(incoming particles have lower and lower energies and turn back fartherand farther from the naked singularity). When the temperature gets suf-ficiently low, the radius r0(T ) of the impenetrable sphere of a superheavycharged particle grows to rc (but not beyond rc, as in the region r > rc theincoming probe is attracted and cannot turn back). This means that incom-ing particles have such low energies that they turn back immediately afterthey encounter the gravitational repulsion. Incoming particles of charge qsuch that qQ > Mm do not even experience attraction — the repulsiveregion for such particles extends to infinity (the gravitational attractionwill not be sufficiently strong to overcome the electrical repulsion) and forthem r0(T ) has no upper limit. The interaction between the two fractions ofthe Universe results5 in power law expansion with scale factor a(τ) ∼ √τ ,corresponding to the expansion during the radiation-dominated era.In our picture, the Universe has local Reissner–Nordstro¨m geometry, butglobally, the geometry is that of Robertson–Walker. Namely, we confine ourattention to the local spherical neighbourhood of a single naked singularityand consider the Universe as multiple copies (fluid) of such neighbourhoods.It is plausible to assume that these naked singularities are densely packedspheres that fill the entire Universe. Thus, the volume V of the Universe,at any moment during the Reissner–Nordstro¨m expansion, would be ofthe order of the number N of these particles, times the “volume” of therepulsive sphere of a superheavy charged particle: V ∼ Nr30(T ). Therefore,the number density of the superheavy charged particles is of the order ofr−30 (T ). At Recombination, the free ions and electrons combine to formneutral atoms (q = 0) and this naturally ends the Reissner–Nordstro¨mexpansion mechanism — a neutral “normal” particle will now be too farfrom an “usual” particle to feel the gravitational repulsion (the density ofthe Universe will be sufficiently low).The standard treatment of a Robertson–Walker Universe uses theisotropicity and the homogeneity for modeling the energy-momentumsources as a perfect fluid. This applies for matter known observationallyMarch 4, 2008 3:37 WSPC - Proceedings Trim Size: 9in x 6in 31˙Prodanov308to be very smoothly distributed. On smaller scales, such as stars or evengalaxies, this is a poor description. In our picture we model the Universe asa van der Waals gas and we use small scale density and pressure variables.Such modeling is possible in the light of the deep analogies between thephysical picture behind the Reissner–Nordstro¨m expansion and the classi-cal van der Waals molecular model: atoms are surrounded by imaginaryhard spheres and the molecular interaction is strongly repulsive in closeproximity, mildly attractive at intermediate range, and negligible at longerdistances. The laws of ideal gas must then be corrected to accommodate forsuch interaction: the pressure should increase due to the additional repul-sion and the available volume should decrease as atoms are no longer entitieswith zero own volumes. Therefore, at Recombination, the radius r0(T ) of asuperheavy charged particle will be of the order of Rc = N−1/3trecomb . Werequest that, once r0(T ) becomes equal to Rc, i.e. when the charged parti-cles recombine, then the potential of the interaction between a naked sin-gularity and a particle of charge q, such that qQ > mM , becomes zero. Forparticles of charge q, such that sign(Q)q/m ≥ −1 and also qQ ≤ mM , thepotential of the interaction becomes negligible earlier: when r0(T ) reachesrc.With N superheavy particles, the effective space left for the motion ofthe “standard” fraction consisting of n particles, is reduced by a factorof N times the volume of the repulsive sphere of a superheavy particle.Then the “van der Waals” equation p+ (N 2α)/V 2 = nkT [1 + (Nβ)/V ]/Vhas: α = 2π∫ Rr0(T )U(r) r2 dr = πm(M2 − Q2)[R − r0(T )] + πmM [1 −(qm)/(QM)][R2−r20(T )] and β = 2π∫ r0(T )0 r2dr = 2πr30(T )/3. Here R = rcif sign(Q)q/m ≥ −1 and also qQ ≤ mM , or R = Rc if qQ > mM . Theequation of state is p = ηρ4/3 − α/β2, where η = const. The second termdepends on the temperature via α and β and becomes irrelevant towardsthe end, as α→ 0 when r0(T )→ R.References1. S. Hawking, Mon. Not. R. Astr. Soc. 152, 75–78 (1971).2. A. de Rujula, S.L. Glashow, and U. Sarid, Nucl. Phys. B 333, 173 (1990).3. E.M. Drobyshevski, Detection and Investigation of the Properties of DarkElectric Matter Objects: the First Results and Prospects, astro-ph/0402367.4. R. Bernabei, P. Belli, F. Montecchia, F. Nozzoli, F. Cappella, A. d’Angelo,A. Incicchitti, D. Prosperi, R. Cerulli, C.J. Dai, H.L. He, H.H. Kuang, J.M.Ma, and Z.P. Ye, Recent DAMA Results, in Astroparticle, Particle and SpacePhysics, Detectors and Medical Physics Applications, World Scientific Pub-lishing 158 (2006).March 4, 2008 3:37 WSPC - Proceedings Trim Size: 9in x 6in 31˙Prodanov3095. E.M. Prodanov, R.I. Ivanov, and V.G. Gueorguiev, Reissner–Nordstro¨m Ex-pansion, Astroparticle Physics 27 (150–154) 2007, hep-th/0703005;E.M. Prodanov, R.I. Ivanov, and V.G. Gueorguiev, Equation of State for avan der Waals Universe during Reissner–Nordstro¨m Expansion, forthcoming.6. J. Preskill, Phys. Rev. Lett. 43 (19), 1365 (1979).7. J.M. Cohen and R. Gautreau, Phys. Rev. D 19 (8), 2273–2279 (1979).

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