A Valid Finite Bounded Expanding Carmelian Universe Without Dark Matter

Transcription

1 Int J Theo Phys (2013) 52: DOI /s A Valid Finite Bounded Expanding Camelian Univese Without Dak Matte John G. Hatnett Received: 2 May 2013 / Accepted: 18 July 2013 / Published online: 18 August 2013 Spinge Science+Business Media New Yok 2013 Abstact The solution of Einstein s field equations in Cosmological Geneal Relativity (CGR), whee the Galaxy is at o cosmologically nea the cente of a finite yet bounded spheically symmetical isotopic gavitational field, is identical with the unbounded solution. I show that this leads to the conclusion that the Univese may be viewed as a finite expanding white hole. This is not the white hole solution of Einstein s spacetime but has similaities to it. The fact that CGR has been successful in descibing the distance modulus veses edshift data of the high-edshift type Ia supenovae means that the data cannot be used to distinguish between unbounded models and those with finite bounded adii of at least cτ ( ch 1 0 ). Accoding to Camelian theoy, whethe o not the Univese is finite bounded o unbounded it is spatially flat all epochs whee it was matte dominated. As a consequence in a finite bounded univese no gavitational edshift in the light fom distant souce galaxies would be obseved. Keywods Camelian cosmology Finite bounded univese 1 Intoduction This pape poposes a model whee the Galaxy is at the cente of a spheically symmetical finite bounded univese. It contends that fits to the magnitude-edshift data of the high-z type Ia supenovae (SNe Ia) [1, 11, 12], ae also consistent with this model. That is, poviding that the adius of the Univese (a spheically symmetical matte distibution) is at least cτ whee c is the speed of light and τ s (o Gy). [7]Heeτ is the Hubble-Cameli time constant, o the invese of the Hubble constant (H 0 ) evaluated in the limit of zeo gavity and zeo distance. This model is based on the Cosmological Geneal Relativity (CGR) theoy [5] but exploes the motion of paticles in a cental potential. In this case the cental potential is the J.G. Hatnett (B) School of Chemisty and Physics and the Institute fo Photonics and Advanced Sensing, Univesity of Adelaide, Adelaide, South Austalia, Austalia john.hatnett@adelaide.edu.au

2 Int J Theo Phys (2013) 52: esult of the Galaxy being situated at the cente of a spheically symmetical isotopic distibution compising all matte in the Univese. This pape is peceded by Hatnett [8] that foms the basis of the wok I pesent hee. Also Oliveia and Hatnett [7] and then late I [10] pogessed the wok by developing a density function fo highe edshifts. Those papes assumed the unbounded model. The eade should at least be familia with [8, 10] befoe eading this. 2 Methods The metic [2, 3, 5] used by Cameli (in CGR), in a geneally covaiant theoy, extends the numbe of dimensions of the Univese by the addition of a new dimension the adial velocity of the galaxies in the Hubble flow. The Hubble law is assumed as a fundamental axiom fo the Univese and the galaxies ae distibuted accodingly. The undelying mechanism is that the substance of which space is built, the vacuum, is unifomly expanding in all diections and galaxies, as taces, ae fixed to space and theefoe the edshifts of distant fist anked galaxies quantify the speed of the expansion. In detemining the lage scale stuctue of the Univese the usual time dimension is neglected (dt = 0) as obsevations ae taken ove such a shot time peiod compaed to the motion of the galaxies in the expansion. It is like taking a still snap shot of the Univese and theefoe only fou co-odinates x μ = (x 1,x 2,x 3,x 4 ) = (,θ,φ,τv) ae used thee of space and one of velocity. The paamete τ H 1 0, the Hubble-Cameli constant, is a univesal constant fo all obseves. Hee the CGR theoy is consideed using a Riemannian fou-dimensional pesentation of gavitation in which the coodinates ae those of Hubble, i.e. distance and velocity. This esults in a phase space equation whee the obsevables ae edshift and distance. The latte may be detemined fom the high-edshift type Ia supenova obsevations [7, 8, 10]. 2.1 Phase Space Equation The line element in CGR, [6] ds 2 = τ 2 dv 2 e ξ d 2 R 2( dθ 2 + sin 2 θdφ 2), (1) epesents a spheically symmetical isotopic univese, that is not necessaily homogeneous. It is fundamental to the theoy that ds = 0. In the case of Cosmological Special Relativity (see Chap. 2 of [5]), which is vey useful pedagogically, we can wite the line element as, ds 2 = τ 2 dv 2 d 2, (2) ignoing θ and φ co-odinates fo the moment. By equating ds = 0 it follows fom (2) that τdv = d assuming the positive sign fo an expanding univese. This is then the Hubble law in the small v limit. Hence, in geneal, this theoy equies that ds = 0. Using spheical coodinates (, θ, φ) and the isotopy condition dθ = dφ = 0 in(1) then d epesents the adial co-odinate distance to the souce and it follows fom (1)that, whee ξ is a function of v and alone. This esults in, τ 2 dv 2 e ξ d 2 = 0, (3) d dv = τe ξ/2, (4)

3 4362 Int J Theo Phys (2013) 52: whee the positive sign has been chosen fo an expanding univese. 2.2 Solution in cental potential Cameli found a solution to his field equations, modified fom Einstein s, (see [8]and[2, 5, 6]) which is of the fom, e ξ = R f() (5) with R = 1, which must be positive. Fom the field equations and (5) we get a diffeential equation, f + f = κτ2 ρ eff, (6) whee f() is function of and satisfies the condition f()+ 1 > 0. The pime is the deivative with espect to.heeκ = 8πG/c 2 τ 2 and ρ eff = ρ ρ c whee ρ is the aveaged matte density of the Univese and ρ c = 3/8πGτ 2 is the citical density. The solution of (6), f(), is the sum of the solution (2GM/c 2 ) to the homogeneous equation and a paticula solution ( κ 3 τ 2 ρ eff 2 ) to the inhomogeneous equation. In [5] Cameli discaded the homogeneous solution saying it was not elevant to the Univese, but the solution of a paticle at the oigin of coodinates, o in othe wods, in a cental potential. Now suppose we model the Univese as a ball of dust of adius with us, the obseve, at the cente of that ball. In this case the gavitational potential witten in spheical coodinates that satisfies Poisson s equation in the Newtonian appoximation is, Φ() = GM (7) fo the vacuum solution, but inside an isotopic matte distibution, ( 4πρ Φ() = G 0 2 d + 4πρ d ) = 2 3 Gπρ2 2Gπρ 2, (8) whee it is assumed the matte density ρ is unifom thoughout the Univese. At the oigin ( = 0) Φ(0) = 2Gπρ m 2,wheeρ = ρ m the matte density at the pesent epoch. In geneal ρ depends on epoch. Because we ae consideing no time development ρ is only a function of edshift z and ρ m can be consideed constant. Fom (8) it is clea to see that by consideing a finite distibution of matte of adial extent, it has the effect of adding a constant to f()that is consistent with the constant 2Gπρ 2 in (8), whee f()is now identified with 4Φ/c 2. Equation (5) is essentially Cameli s equation (A.19), the solution to his Eq. (A.17) fom p. 122 of [5]. Moe geneally (5) can be witten as, e ξ = R f() K, (9)

4 Int J Theo Phys (2013) 52: whee K is a constant. This is the most geneal fom of the solution of Cameli s equation (A.17). So by substituting (9) into Cameli s (A.18), (A.21) becomes instead, 1 ( 2ṘṘ RR f ) + 1 (Ṙ2 f + K ) = κτ 2 ρ R 2 eff. (10) Theefoe (9) is also a valid solution of the Einstein field equations (A.12) (A.18) [5]in this model. Making the assignment R = in (10) yields a moe geneal vesion of (6), that is, The solution of (11) is then, f + f K = κτ 2 ρ eff. (11) f()= 1 3 κτ2 ρ eff 2 + K. (12) Fom a compaison with (8) it would seem that the constant K takes the fom K = 8πGρ eff (0) 2 /c 2. It is independent of and descibes a non-zeo gavitational potential of a finite univese measued at the oigin of coodinates. Thee is some ambiguity howeve as to which density to use in Camelian cosmology since it is not the same as Newtonian theoy. Hee ρ eff is used and evaluated at = 0. In the above Camelian theoy it is initially assumed that the Univese has expanded ove time and at any given epoch it has an aveaged density ρ, and hence ρ eff. The solution of the field equations has been sought on this basis. Howeve because the Cameli metic is solved in an instant of time (on a cosmological scale) any time dependence is neglected. In fact, the geneal time dependent solution has not yet been found. But since we obseve the expanding Univese with the coodinates of Hubble at each epoch (o edshift z) weseethe Univese with a diffeent density ρ(z) and an effective density ρ eff (z). Cameli aived at his solution with the constant density assumption. I have made the implicit assumption that the solution is also valid if we allow the density to vay as a function of edshift, as is expected with expansion. Now it follows fom (4), (9) and(12) that, d dv = τ 1 + ( 1 Ω c 2 τ 2 ) 2, (13) whee Ω = ρ/ρ c. This compaes with the solution when the cental potential is neglected (i.e. 0). In fact, the esult is identical as we would expect in a univese whee the Hubble law is univesally tue. The condition on (9)isthat1+ f() K>0, which means that, ( ) 1 Ω 1 + f() K = > 0. (14) c 2 τ 2 Quite obviously (14) will be valid whee Ω<1. But emembe that Ω hee is eally dependent on epoch (o edshift, z). In this simplified analysis a constant density assumption has been made to highlight the physics in a low density univese, but the moe igoous solution of d/dz in (4) is sought by a change of vaiable to z and then it can only be solved numeically. See Sect. 4 of Hatnett [10]. Fom that analysis a valid solution exists fo 0 Ω<2.

5 4364 Int J Theo Phys (2013) 52: Howeve, povided that z<1 Cameli s constant density assumption is easonably good. Theefoe (13) may be integated exactly and yields the same esult as Cameli, cτ = sinh( v c 1 Ω). (15) 1 Ω Since obsevations in the distant cosmos ae always in tems of edshift, z, we wite (15) as a function of edshift whee is expessed in units of cτ and v/c = ((1 + z) 2 1)/((1 + z) 2 + 1) fom the elativistic Dopple fomula. The latte is appopiate since this is a velocity dimension. As mentioned above, Ω is eally edshift dependent and the moe igoous solution of (13) was found valid fo any edshift [10]. Fo this analysis it is not necessay. The main point of this analysis is that whethe o not one uses a finite bounded o unbounded univese Cameli s theoy gives the same esult as above in (15). What is impotant to note is that egadless of the geomety of the Univese, povided it is spheically symmetical and isotopic on the lage scale, (15) is identical to that which we would get whee the Univese has a unique cente, but with one diffeence, which is exploed in the following section. Fo an isotopic univese without a unique cente, one can have an abitay numbe of centes. Howeve if we ae cuently in a univese whee the Galaxy is at the cente of the local isotopy distibution this means the univese we see must be vey lage and we ae cuently limited fom seeing into an adjacent egion with a diffeent isotopy cente. 2.3 Gavitational Redshift In Hatnett [8] the geomety in the model is the usual unbounded type, as found in an infinite univese, fo example. In a finite bounded univese, an additional effect may esult fom the photons being eceived fom the distant souces. The gavitational edshift (z gav ) esulting fom the Galaxy sitting at the unique cente of a finite spheically symmetical matte distibution must be consideed. In this case we need to conside the diffeence in gavitational potential between the points of emission and eception of a photon. In Hatnett [10] it was detemined in a matte dominated Camelian univese that the Univese is always spatially flat and that the total mass enegy density Ω t = 1. This is also tue hee. As a consequence g 00 = 1 which is the 00th metic component, the time pat of the 5D metic of coodinates x k = t,,θ,φ,v (k = 0 4). In geneal elativity we would elate it by g 00 = 1 4Φ/c 2 whee 4Φ is the gavitational potential. The facto 4 and minus sign aise fom the Camelian theoy when (12)and(8) ae compaed. So what is g 00 fo the lage scale stuctue of a finite bounded univese in CGR? Consideing a finite bounded univese, fom (12), using Ω = ρ/ρ c, I theefoe wite g 00 as, g 00 () = 1 + (1 Ω t ) 2 + 3(Ω t 1) 2, (16) whee and ae expessed in units of cτ. Equation (16) follows fom g 00 = 1 4Φ/c 2 and (8) wheeφ is taken fom the gavitational potential but with effective density, which in tun involves the total enegy density because we ae now consideing spacetime. Clealy fom Ω t = 1and(16) it follows that g 00 () = 1 egadless of epoch. Thus fom the usual elativistic expession, g 00 (0) 1 + z gav = = 1, (17) g 00 ()

6 Int J Theo Phys (2013) 52: and the gavitational edshift is zeo egadless of epoch. As expected if the emission and eception of a photon both occu in flat space then we d expect no gavitational effects. Since it follows fom (16) thatg 00 () = 1 egadless of epoch, g 00 () is not sensitive to any value of. This also means the above analysis is tue egadless of whethe the univese is finite bounded o unbounded. The obsevations cannot distinguish. In an bounded o unbounded univese of any type no gavitational edshift (due to cosmological oigin) in the light fom distant souce galaxies would be obseved. 3 Finite Bounded White Hole Now if we assume the adial extent of a finite matte distibution at the cuent epoch is equal to the cuent epoch scale adius, we can wite, = 1 1 Ωm, (18) expessed in units of cτ. In such a case, = 1.02 cτ if Ω m = 0.04 and = 1.01 cτ if Ω m = It is impotant to note also that in Cameli s unbounded model (15) descibes the edshift distance elationship but thee is no cental potential. In Hatnett [8] and in Oliveia and Hatnett [7] equation (15) was tested against the SNe Ia data and was found to agee with Ω m = without the inclusion of dak matte [4, 9] o dak enegy. Theefoe the same conclusion must also apply to the finite bounded model suggested hee. In ode to achieve a fit to the data, using eithe the finite bounded o unbounded models, the white hole solution of (6)o(11) must be chosen. The sign of the tems in (12) means that the potential implicit in (12) is a potential hill, not a potential well. Theefoe the solution descibes an expanding white hole with the obseve at the oigin of the coodinates, the unique cente of the Univese. Only philosophically can this solution be ejected [15]. Using the Cameli theoy, the obsevational data cannot distinguish between finite bounded models ( > cτ) and finite ( = 0) o infinite ( = ) unbounded models. The physical meaning is that the solution, developed in this pape, epesents an expanding white hole centeed on the Galaxy. The galaxies in the Univese ae spheically symmetically distibuted aound the Galaxy. The obseved edshifts ae the esult of cosmological expansion alone. Though this has similaities to the classical white solution of Einstein s spacetime it is not the same, since it is the poduct of Cameli s spacevelocity. Moeove if we assume cτ and Ω m = 0.04 then it can be shown [7] that the Schwazschild adius fo the finite Univese, R s Ω m = 0.04 cτ. (19) Theefoe fo a finite univese with cτ it follows that R s 0.04 cτ 200 Mpc. Theefoe an expanding finite bounded univese can be consideed to be a white hole. As it expands the matte enclosed within the Schwazschild adius gets less and less. The gavitational adius of that matte theefoe shinks towads the Eath at the cente. This is simila to the theoetical esult obtained by Smolle and Temple [13] who constucted a new cosmology fom the FRW metic but with a shock wave causing a time evesal white hole. In thei model the total mass behind the shock deceases as the shock wave expands, which is spheically symmetically centeed on the Galaxy. Thei pape states in pat...the entopy condition implies that the shock wave must weaken to the point whee it

7 4366 Int J Theo Phys (2013) 52: settles down to an Oppenheime Snyde inteface, (bounding a finite total mass), that eventually emeges fom the white hole event hoizon of an ambient Schwazschild spacetime. This esult then implies that the Galaxy could in fact be somewhee nea to the physical cente of the Univese. Smolle and Temple state [14] that With a shock wave pesent, the Copenican Pinciple is violated in the sense that the eath then has a special position elative to the shock wave. But of couse, in these shock wave efinements of the FRW metic, thee is a spacetime on the othe side of the shock wave, beyond the galaxies, and so the scale of unifomity of the FRW metic, the scale on which the density of the galaxies is unifom, is no longe the lagest length scale [emphasis added]. Thei shock wave efinement of a citically expanding FRW metic leads to a big bang univese of finite total mass. This model pesented hee also has a finite total mass and is a spatially flat univese. It descibes what may be viewed as a finite bounded white hole that stated expanding at some time in the past. 4 Conclusion Since the Cameli theoy has been successfully analyzed with distance modulus data deived by the high-z type Ia supenova teams it must also be consistent with a univese that places the Galaxy cosmologically nea the cente of a spheically symmetical isotopic expanding univese of finite adius. The esult descibes paticles moving in both a cental potential and an acceleating spheically expanding univese without the need fo the inclusion of dak matte. The obsevational data cannot be used to exclude models with finite extensions cτ. Refeences 1. Astie, P., et al.: The supenova legacy suvey: measuement of Ω M, Ω Λ and w fom the fist yea data set. Aston. Astophys. 447, (2006) 2. Beha, S., Cameli, M.: Cosmological elativity: a new theoy of cosmology. Int. J. Theo. Phys. 39, (2000) 3. Cameli, M.: Cosmological geneal elativity. Commun. Theo. Phys. 5, 159 (1996) 4. Cameli, M.: Is galaxy dak matte a popety of spacetime? Int. J. Theo. Phys. 37, (1998) 5. Cameli, M.: Cosmological Special Relativity. Wold Scientific, Singapoe (2002) 6. Cameli, M.: Acceleating univese: theoy vesus expeiment (2002). axiv:asto-ph/ Oliveia, F.J., Hatnett, J.G.: Cameli s cosmology fits data fo an acceleating and deceleating univese without dak matte no dak enegy. Found. Phys. Lett. 19, (2006). axiv:asto-ph/ Hatnett, J.G.: The distance modulus detemined fom Cameli s cosmology fits the acceleating univese data of the high-edshift type Ia supenovae without dak matte. Found. Phys. 36, (2006) 9. Hatnett, J.G.: Spial galaxy otation cuves detemined fom Camelian geneal elativity. Int. J. Theo. Phys. 45, (2006) 10. Hatnett, J.G.: Extending the edshift-distance elation in cosmological geneal elativity to highe edshifts. Found. Phys. 2008(38), (2008) 11. Knop, R.A., et al.: New constaints on Ω M, Ω Λ and w fom an independent set of 11 high-edshift supenovae obseved with the Hubble Space Telescope. Astophys. J. 598, (2003) 12. Riess, A.G., et al.: Type Ia supenovae discoveies at z>1 fom the Hubble Space Telescope: evidence fo past deceleation and constaints on dak enegy evolution. Astophys. J. 607, (2004) 13. Smolle, J., Temple, B.: Shock-wave cosmology inside a black hole. Poc. Natl. Acad. Sci. USA 100, (2003) 14. Smolle, J., Temple, B.: A shock wave efinement of the Fiedmann-Robetson-Walke metic (2003). temple/aticles/temple1234.pdf 15. Gibbs, W.W.: Pofile: Geoge F.R. Ellis: Thinking globally, acting univesally. Sci. Am. 273(4), (1995)