Astrophysics and Space Science, Volume 330, Number 2 / December 2010, pp DOI: /s

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1 Astrophysis and Spae Siene, Volume 0, Number / Deember 010, pp DOI: /s The model of a flat (Eulidean) expansive homogeneous and isotropi relativisti universe in the light of the general relativity, quantum mehanis, and observations Vladimír Skalský Faulty of Material Sienes and Tehnology of the Slovak University of Tehnology, Trnava, Slovakia; vladimir.skalsky@stuba.sk Abstrat Assuming that the relativisti universe is homogeneous and isotropi, we an unambiguously determine its model and physial properties, whih orrespond with the Einstein general theory of relativity (and with its two speial partial solutions: Einstein speial theory of relativity and Newton gravitation theory), quantum mehanis, and observations, too. Keywords General relativity and gravitation Speial relativity Quantum mehanis Cosmology Observational osmology Theoretial osmology Mathematial and relativisti aspets of osmology 1 Introdution In September 8, 1905 Albert Einstein published the artile Zur Elektrodynamik bewegter Körper (Einstein 1905) whih ontained the speial theory of relativity. Einstein in his speial theory of relativity whih represents the theory of physial homogeneous and isotropi spae and time disovered the essential onnetion of the fourdimensional physial homogeneous and isotropi spae-time. This onnetion is expressed by Lorentz transformation (group) (Lorentz 1904): x vt x' =, (1a) v 1 y' = y, (1b) z' = z, (1) v t x t' =, v 1 (1d) where x, y, z are the spae o-ordinates and t is the time in the inertial system, whih move relative to the observer at the veloity v; x, y, z are the spae o-ordinates and t is the time in the observer s own inertial system.

2 In Einstein generalised the speial theory of relativity on the phenomenon of gravity and elaborated the general theory of relativity, in whih he disovered the essential onnetion of matter, spae and time as the unified physial theory of matter-spae-time. The mathematial and physial fundament of the Einstein theory of general relativity represents the Einstein field equations. On Thursday November 5, 1915 at the meeting of the Royal Prussian Aademy of Sienes in Berlin 1 Einstein presented the artile Die Feldgleihungen der Gravitation (Einstein 1915), whih ontained the final version of Einstein field equations 1 Gim = κ Tim gimt, () where G im is the Einstein or onservative tensor, κ Einstein gravitational onstant [κ = (8π G) / 4 ], T im energy-momentum tensor, g im metri or fundamental tensor, and T salar or trae of energymomentum tensor ( T T i i ). Disovery of the general relativity by Einstein has a large signifiane for the osmology, too. Einstein was well aware of it; therefore he attempted to apply the field equations () to the whole universe. Einstein in the spirit of tradition antiipated that the relativisti universe is homogeneous, isotropi and stati. The field equations (), applied to the whole homogeneous and isotropi relativisti universe do not give a stati solution. Therefore, Einstein tried to modify (generalize) them so that the appliation to the whole homogeneous and isotropi relativisti universe would result in a stati solution. The only possible generalization of the Einstein field equations () whih, when applied to the whole homogeneous and isotropi relativisti universe gives a stati solution and does not violate the priniples of general relativity, is an addition by the supplement, representing a hypothetial energy of the physial vauum. On Thursday February 8, 1917 at the meeting of the Royal Prussian Aademy of Sienes in Berlin Einstein presented his artile Kosmologishe Betrahtungen zur allgemeinen Relativitätstheorie (Einstein 1917a), whih ontained historially first model of the relativisti universe, whih is a solution the theoretially most possible generalised version of Einstein modified field equations 1 λ g µν = Tµν g T, () G µν κ µν where λ is the Einstein osmologial onstant. The Einstein supplementary osmologial member λ g µν in the Einstein modified field equations () an have positive, negative, or zero values, depending on the value of Einstein adjustable osmologial onstant λ, whih an obtain all hypothetially (mathematially) possible values, i.e.: λ > 0, λ < 0, or λ = 0. The Einstein field equations (), or (), represent a non-linear system of ten partial differential equations of the seond order for ten unknown funtions of four variables. For their solution general method does not exist. The Einstein theory of general relativity is logially simple, omplete and unambiguously determined theory, whih annot be modified. The general theory of relativity is either valid or not, another possibility does not exist, tertium non datur. 1 Meetings of the Royal Prussian Aademy of Sienes in Berlin took plae on Thursdays.

3 Einstein drew the attention to this relevant property of the general theory of relativity, in the paper What is the theory of relativity? whih was first published in the London Times in November 8, Einstein wrote in it: The hief attration of the theory lies in its logial ompleteness. If a single one of the onlusions drawn from it proves wrong, it must be given up; to modify it without destroying the whole struture seems to be impossible. (Einstein 1919b). Thirty years later in the artile On the Generalized Theory of Gravitation Einstein wrote: In favour of this theory are, at this point, its logial simpliity and its rigidity. Rigidity means here that theory is either true or false, but not modifiable. (Einstein 1950). The logial simpliity, ompleteness and rigidity of the Einstein general relativity theory are manifested in the Einstein field equations, too. The Einstein field equations () ontain only one adjustable parameter: the Newton gravitation onstant G, whose value is gradually being preised on the basis of observations. The Einstein modified field equations () ontain two adjustable parameters: Besides the Newton gravitational onstant G, ontain even the Einstein osmologial onstant λ, whih an be adjusted, based on observation, or determined on the basis of any physial priniple. The Einstein theory of general relativity at present time is the most verified physial theory. By many years of observations of the binary pulsar PSR , whih was disovered in July, 1974 by Russell A. Hulse and Joseph H. Taylor, Jr., the general relativity is verified with the unertainty (Hawking and Penrose 1996, p. 61; Penrose 1997, p. 6). Aording to Roger Penrose... this auray has apparently been limited merely by the auray of loks on earth. (Hawking and Penrose 1996, p. 61). All the preditions of the general theory of relativity and its speial partial solution: the speial theory of relativity have been onfirmed. Just one of the preditions of the general theory of relativity: predition of the gravitational waves was onfirmed only indiretly. The gravitation field on the Earth and in its near surroundings is relatively weak. The veloities of matter objets on the Earth and in its near surroundings in omparing with boundary veloity of signal propagation are relatively small. Therefore, when determining the matter-spae-time properties of matter objets on the Earth and in its near surroundings in most ases we suffie with the Newton gravitation theory and the lassial mehanis. The differenes of alulating mater-spae-time properties in the region of Earth whih we are making using the Newton gravitation theory or the lassial mehanis and using the Einstein general relativity or speial relativity are relative small, prevailingly irrelevant, or even using ommon measuring instruments non-measurable. For example aording to the general theory of relativity the relativisti mass of matter objets on the surfae of Earth, as a result of the loal gravitational field, is higher about approximately of their own (rest, Newtonian or lassial-mehanial) mass. However, if we are to ahieve results with auray provided by urrent top observational tehnology, the Newton gravitational theory and the lassial mehanis (whih abstrats from relativisti effets) is not suffiient. In these ases we have to take into aount generalrelativisti effets aused by the loal gravitational field, and with moving physial objets we have to take into aount speial-relativisti effets. At present time the speial-relativisti and general-relativisti effets are not only a matter of physial observations and experiments, but they are exploited in some high tehnologies. One of them is for example the Amerian satellite navigation system in ommon ommerial use, best known on the aronym GPS (Global Positioning System). Russell A. Hulse and Joseph H. Taylor, Jr. were for the disovery of a new type of pulsar, a disovery that has opened up new possibilities for the study of gravitation awarded the Nobel Prize in Physis 199.

4 The equations of homogeneous and isotropi relativisti universe dynamis The mathematial-physial fundament of the relativisti osmology is represented by the Friedmann equations of the homogeneous and isotropi relativisti universe dynamis (Friedmann 19, 194), whih using the Robertson-Walker metris (Robertson 195, 196a, b; Walker 196) an be expressed in the following form: 8π Gρa Λa a & = k +, (4a) 8π Gpa a a& + a& = k + Λa, (4b) p = wε, (4) where a is the gauge fator, ρ mass density, k urvature index, Λ osmologial onstant, p pressure, w state equation onstant, and ε energy density. The relativisti osmology is based on the assumption of the homogeneous and isotropi distribution of matter objets in spae. The homogeneity and isotropy of the spae means that we an hoose suh a osmologial time that in eah moment the spae metris is the same in all of its points and in all diretions. (Landau and Lifshitz 1988, p. 458). There exist only three geometri spaes of onstant urvature spae: a) Spherial (Riemannian) geometri spae with onstant positive spae urvature. b) Hyperboli (Lobahevskian) geometri spae with onstant negative spae urvature. ) Flat (Eulidean) geometri spae with onstant zero spae urvature. The FRW equation (4a), (4b) and (4) are an appliation of the Einstein modified field equations () for all three geometrial spaes with onstant urvature spae, i.e. they have solutions with urvature index k = +1, k = 1, and k = 0; with all mathematially possible values of osmologial onstant Λ, i.e. with Λ > 0, Λ < 0 and Λ = 0; and with all mathematially possible values of state equation onstant w, i.e. with w > 0, w < 0 and w = 0. The logial simpliity, ompleteness and rigidity of the Einstein theory of general relativity, ombined with the metris with onstant urvature of spae, gives possibility on the unambiguously theoretial determination of the model and physial properties of the homogeneous and isotropi relativisti universe. It follows from these fats: Neither the Einstein field equations (), nor the Einstein modified field equations (), applied to the whole relativisti universe, do not give a stati solution; therefore, the relativisti universe prinipally annot be stati. The FRW equations (4a), (4b) and (4) with the values of the urvature index k = +1, k = 1, k = 0, the osmologial onstant Λ > 0, Λ < 0, Λ = 0, and the state equation onstant w > 0, w < 0, w = 0, desribe an infinite number of the hypothetial homogeneous and isotropi relativisti universes in a linear approximation, in whih we abstrat from their relativisti properties, but do not abstrat from their expansion veloity. This is the fat whih allows theoretially to identify (selet) unambiguously from an infinite set of mathematially possible Einstein in 1917 on the base of the modified field equations () onstruted a model of the spherial stati homogeneous and isotropi relativisti universe (Einstein 1917a). However, Arthur S. Eddington in the artile On the Instability of Einstein s Spherial World (Eddington 190) showed that not even Einstein modified field equations (), applied to the whole homogeneous and isotropi relativisti universe, do not give stati, but only quasi-stati solution, beause the Einstein model of spherial stati homogeneous and isotropi relativisti universe is extremely unstable, therefore, any small flutuation onverted it into a dynami. 4

5 solutions of the FRW equations of the linearized model of expansive homogeneous and isotropi relativisti universe, desribing the expansive homogeneous and isotropi relativisti universe in the first (linear) approximation. Aording to the speial theory of relativity, the matter objets an expand at veloity v in the interval (0, ), therefore, the dynami homogeneous and isotropi relativisti universe whih expands in finite distanes at veloities v, prinipally annot be infinite. The finite dynami homogeneous and isotropi relativisti universe is (must be) losed in spae-time manner, therefore, in priniple, it annot be ontratile. In the expansive homogenous and isotropi relativisti universe the energy density of matter objets derease, the energy density of the hypothetial physial vauum energy, determined by the osmologial onstant λ, does not hange. Therefore, the law of energy-momentum onservation is valid in it only when the λ = 0. All models of hypothetial spherial (Riemannian) expansive homogeneous and isotropi relativisti universes, whih are the solution of the FRW equations (4a), (4b) and (4) with k = +1 and Λ = 0, have the total dimensionless density of matter objets Ω tot > 1. From the Shwarzshild solution of the Einstein s field equations (Shwarzshild 1916) follow unambiguously that the hypothetial expansive homogeneous and isotropi relativisti universes with the total dimensionless density of matter objets Ω tot > 1 would in the initial period of its expansive evolution have to expand at veloities v >. However, aording to the speial theory of relativity, the matter objets in priniple annot expand at the hyper-veloities. It means that an expansive homogeneous and isotropi relativisti universe in priniple annot have total dimensionless density Ω tot > 1, i.e. in the first (linear) model approximation, it annot have onstant positive urvature of spae, determined by the urvature index k = +1. The volumes of spaes of the hypothetial hyperboli (Lobahevskian) relativisti universes with the negative urvature spae are determined by the divergent integral, therefore, they are an infinite (Friedmann 194). It means although that they have total dimensionless density Ω tot < 1, at a finite distane from observers they would expand at veloities v >. But that aording to the speial theory of relativity in priniple it is not possible. It means that an expansive homogeneous and isotropi relativisti universe in the first (linear) approximation in priniple annot have a onstant negative spae urvature, determined by the urvature index k = 1. In the model of the expansive homogeneous and isotropi relativisti universe with the onstant zero spae urvature the Eulid geometry is valid. For the Eulidean sphere is valid the known relation: 4 V = π r, (5) where V is the volume, and r radius. For the mass m of the Eulidean homogeneous matter sphere the relation: 4 m = π r ρ (6) is valid. Therefore, using the relation (6) and the relation: a : = r, (7) the relation for the mass of a flat (Eulidean) expansive homogeneous and isotropi relativisti universe in the first (linear, Newtonian or lassial-mehanial) approximation an be determined: 5

6 4 m = π a ρ, (8) where ρ is the ritial mass density. The FRW equations (4a), (4b) and (4) fulfil the restritive ondition, determined by the relations (8), only with k = 0, Λ = 0 and w = 1/ (Skalský 004). It means that the flat (Eulidean) expansive homogeneous and isotropi relativisti universe (ERU) model determined by the FRW equations (4a), (4b) and (4) with k = 0, Λ = 0, and w = 1/ (Skalský 1991) is the only one model of the expansive homogeneous and isotropi relativisti universe with the flat (Eulidean) geometry (Skalský 004). The model of a flat (Eulidean) expansive non-deelerative non-aelerative homogeneous and isotropi relativisti universe Using the FRW equations (4a) and (4b) with k = 0, Λ = 0, and total zero energy state equation (Skalský 1991) 1 p = ε, (9) we an determine the fundamental matter-spae-time parameters of the ERU model, i.e. the universe model, whih desribes observed expansive homogeneous and isotropi relativisti quantum-mehanial universe in the linear approximation, in whih we abstrat from its relativisti and quantum-mehanial properties (Skalský 1991): m = a = t, (10) G G where t is the (osmologial) time (age of universe). Aording to the relations (10) the fundamental parameters of ERU model, i.e. the mass (of matter objets) m, gauge fator (radius) a, and (osmologial) time t, grow linearly. From the relations (10) result these inreases of the fundamental parameters of ERU model: inrease of universe mass 6 1 m = a = kg m, (11) G 5 1 m = t = kg s, (1) G inrease of gauge fator G 7 1 a = m = m kg, (1) 8 1 a = t = m s, (14) inrease of osmologial time G 6 1 t = m = s kg, (15) t = a = s m. (16) 6

7 In the relations (10) eah from three fundamental mater-spae-time parameters of the ERU model i.e. the mass of universe m, the gauge fator a, and the osmologial time t is unambiguously bounded linearly with other two fundamental parameters. Through FRW equation (4a), (4b) and (4) with k = 0, Λ = 0 and w = 1/, eah from fundamental parameters of the ERU model m, a, and t, is unambiguously linearly bounded and with parameters ρ, p and ε. Therefore, if in the ERU model we determine the relation of any next derived parameter with an arbitrary from mentioned parameters m, a, t, ρ, p and ε, at the same time are unambiguously determined and its relations with all other fundamental and derived parameters of the ERU model. It makes possible a simple introdution of further derived parameters of the ERU model, and gives possibility to larify its properties. The parameters of the ERU model, whih are determined by the relations (9) and (10), an be extended by next derived parameters: about the energy (of matter objets) E, determined by the Einstein relation E = m and by the Hubble parameter H, determined by the relation (45). For better transpareny, the parameters of the ERU model a, t, H, m, E, ρ, ε, and p, are presented in all possible relations and variations (Skalský 004): 4 4 Gm GE a = t = = = = =, 4 a =, (17a) H 8π Gρ 8π Gε 8π G p a 1 Gm GE t = = = = = =, t =, (17b) 5 H 8π Gρ 8π Gε 8π G p H 5 1 8π Gρ 8π Gε p = = = = = =, H =, (17) a t Gm GE 8π G a t E m = = = = = =, m =, (17d) G G GH π G ρ π G ε π G p a t E = = = = m = =, E =, (17e) G G GH π G ρ π G ε π G p 6 8 ρ H ε p = = = = = = =, 8π Ga 8π Gt 8π G π G m π G E (17f) ε H = = = = = = ρ = p, 8π Ga 8π Gt 8π G π G m π G E (17g) H ρ 1 p = = = = = = = ε. (17h) 8π Ga 8π Gt 8π G π G m π G E In the relations (17a)-(17h) we an see, that all (fundamental and derived) parameters of the ERU model are unambiguously linearly bounded eah to other, inlude the relation for the pressure p and the energy density (of matter objets) ε, representing the total zero energy state equation, whih is determined by the relation (9), and presented among the relations (17h), too. In the osmologial literature instead of the (osmologial) time (age of universe) t sometimes is used and the dimensionless onform time η, defined by the relation: 7

8 dt η = ±. (18) a ( t) Therefore, any hosen parameters of the ERU model, expressed in the dimensionless onform time η, determined by the relation (18), are shown in the Table 1. Table 1 Parameters of the expansive homogenous and isotropi relativisti universe model with the total 1 zero energy state equation p = ε (0 < η < ) Curvature index k Gauge fator a Cosmologial time t Hubble parameter H Energy density ε Dimensionless density Ω 0 t e η = t 0 t e η = 0 a e η = 1 t t 0 e 8π Gt η 0 1 = 8π G t 1 Note: Aording to Skalský (1991) 4 The observed and model properties of the universe Based on the observations at present time we reliably know that the observed universe at smaller osmologial distanes is non-homogeneous and anisotropi, strutured into a hierarhial gravitationally bound rotating systems (HGRSs) with superritial mass density and only one resultant entre of gravity. HGRSs form (in ase of negleting the smaller systems): the galaxies, lusters of galaxies and super lusters. From these fats, it results unambiguously, that in smaller osmi distanes (i.e. in the range of the largest HGRSs), the universe has the superritial mass density. Beause only under this ondition an exist the HGRSs, in whih the gravitational interation of matter objets is ompensated by their inertial rotational motion. In larger osmi distanes (than are dimensions of the largest HGRSs), observed universe annot have superritial mass density. Beause if it had superritial mass density, HGRSs would have to exist with larger dimensions than super lusters have, i.e. they would have to exist super-super lusters, super-super-super lusters... et. What would we with the present level of observational tehniques undoubtedly observe. In larger osmi distanes (than the dimensions of the largest HGRSs), observed universe is expansive, homogeneous and isotropi. At present time with relatively high auray we know some of the physial and model parameters of the observed universe. For example: In 009 G. Hinshaw et al. published the artile Five-year Wilkinson Mirowave Anisotropy Probe (WMAP) observations: Data proessing, sky maps, and basi results (Hinshaw et al. 009) with the osmologial parameters derived from the WMAP measurements, and with the osmologial parameters, derived from the WMAP data ombined with the distane measurements from the Type Ia Supernovae (SN) and the Baryon Aousti Osillations (BAO). Some of them you an see in Table. 8

9 Table Seleted Cosmologial Parameters Desription Symbol WMAP-only WMAP+BAO+SN Seleted Parameters for Standard ΛCDM Model Age of universe t ± 0.1 Gyr 1.7 ± 0.1 Gyr Hubble onstant H km s Mp ± 1. km s Redshift of deoupling z * ± ± 0. 7 Age of deoupling t * yr Mp yr 1 Total density Seleted Parameter for Extended Models Ω tot Note: Aording to Hinshaw et al. (009) The observed expansive homogeneous and isotropi relativisti quantum-mehanial universe represents a maximum atual whole of physial reality, whih from a maro-physial point of view has the relativisti properties and from a miro-physial point of view has the quantum-mehanial properties. The relativisti and quantum-mehanial properties are omplementary. The quantummehanial objets (partiles) generate the relativisti maro-world and vie versa, the partiles an exist only in the relativisti maro-world. The observed universe from the relativisti point of view represents the relativisti matterspae-time (or the matter-spaetime), in whih the matter objets determine the properties (geometry) of the spae-time, and the spae-time has influene on the relativisti properties and movement of matter objets. Therefore, to omplete observed physial properties (and with it unambiguously bounded model properties), of observed expansive homogeneous and isotropi relativisti quantum-mehanial universe we need to know: a) total mass (energy) of the universe, or: b) spaetime properties of the universe. Using the total energy of the universe, or using the spaetime properties of the universe, we an determine the model and the physial matter-spaetime properties of the observed universe. The FRW equations (4a), (4b) and (4) desribe the models of homogeneous and isotropi relativisti universe in the first (linear) approximation. In the observed expansive homogenous and isotropi relativisti quantum-mehanial universe in the first (linear) approximation the Newtonian relations are valid. In the ERU model the Eulid geometry is valid and the same geometry is valid in the Newton theory of general gravitation. The ERU model and the Newton gravitation theory desribe the physial maro-world in the linear approximation (whih abstrats from the relativisti and quantum-mehanial properties). It means that the ERU model is a speial partial solution of the Newton gravitational theory. 9

10 We an onvine ourselves about it: In the Newton gravitational theory esape veloity Gm v es =. (19) r If in the relation (19) for v es we put the veloity, we reeive the relation for Shwarzshild ritial (gravitational) radius Gm r =. (0) If in the relation (0) instead r we put the gauge fator a, we reeive: Gm a =, given among the relations (17a). From the relations (17a) and (0) unambiguously results: The ERU model is also a speial partial solution of the first non-trivial spherial symmetrial exterior (vauum) solution of the Einstein field equations, found by Karl Shwarzshild (1916). Using the relation (6) we an rewrite the relation (0) into the form: r a = =. 8π Gρ (1) If in the relation (1) instead r we put the gauge fator a we get the relation:, 8π Gρ given among the relations (17a). From the relation (1) it results: ρ =. () 8π Gr If in the relation () instead r we put the gauge fator a, we get the relation: ρ = 8π Ga, given among the relations (17f). Aording to the Einstein general relativity, for the total mass m tot of an arbitrary Eulidean homogeneous matter sphere with the radius r is valid the relation: m tot 4 p = π r ρ +. () In the expansive homogeneous and isotropi relativisti universe the positive energy of the matter objets is exatly ompensated by their negative gravitational energy. It means that:... the total energy of the universe is exatly zero. (Hawking 1988, p. 19). Therefore, for the total energy E tot and the total mass m tot of the expansive homogeneous and isotropi relativisti universe are valid the relations: E tot = m = 0. tot (4) 10

11 For the total mass of the expansive homogeneous and isotropi relativisti universe in the linear approximation m tot with the non-zero values of the gauge fator a and the mass density ρ an be valid: 4 p m tot = π a ρ + = 0 (5) only on the ondition (Skalský 00, 004): + p ρ 0. = (6) For the mass density ρ and the energy density ε is valid the relation: ε = ρ, (7) therefore, the relation (6) we an using the relation (7) rewrite into the form: ε + p = 0. (8) If in the relation (8) we express the value of pressure p, we reeive: the total zero energy state equation 1 p = ε, whih is shown above as the relation (9) and among the relations (17h). From the above mentioned unambiguously results: The ERU model is only one non-formal model of the expansive homogeneous and isotropi relativisti quantum-mehanial universe in the linear approximation with the total zero and loal non-zero mass (energy). The ERU model, determined by the FRW equations (4a), (4b) and (4) with k = 0, Λ = 0 and w = 1/, is the only model of the expansive homogeneous and isotropi relativisti universe in the linear approximation with non-zero mass density ρ in whih the total energy E tot is unhanged. It means that: The ERU model is the only non-formal model of the expansive homogeneous and isotropi relativisti universe in the linear approximation in whih the law of energy onservation is valid. In 197, Edward P. Tryon in the journal Nature published an artile Is the Universe a Vauum Flutuation? in whih he postulates the hypothesis aording to whih the observed relativisti quantum-mehanial universe is a vauum flutuation (Tryon 197). The Tryon hypothesis is based on a ombination of quantum-mehanial properties of the physial vauum and mathematial-physial properties of the expansive homogeneous and isotropi relativisti universe with the total zero energy. The ERU model is the only one non-formal model of the expansive homogeneous and isotropi relativisti quantum-mehanial universe in the linear approximation, whih has a total mass (energy) equal to zero. It means that: The ERU model is the only model of the universe, whih in the linear approximation desribes the expansive homogeneous and isotropi relativisti quantum-mehanial universe, whih may be regarded as a vauum flutuation (Skalský 00). The expansive homogeneous and isotropi relativisti quantum-mehanial universe with the total zero mass (energy) annot have any other aeleration than zero. The expansion of the homogeneous and isotropi relativisti quantum-mehanial universe in the linear model approximation onforms to the Newton general gravity law. The negative aeleration (i.e. deeleration), of the matter objets a on the surfae of a Eulidean homogeneous matter sphere is determined by the relation: Gm a =. (9) r 11

12 If in the relations (9) we substitute the mass of the Eulidean homogeneous matter sphere m by the total mass m tot = 0 and the radius r by the gauge fator a, we get: Gm tot a = = 0. (0) a The relation (0) an be expressed using the relation (5), too. If in the relation (0) instead m tot = 0 we put the relation (5), we obtain: 4 p a = π Ga ρ + = 0. (1) The relations (0) and (1) mathematially and physially express that what we already knew thanks to a simple, trivial onsideration: The ERU with the total energy E tot = m tot = 0 throughout the whole expansive evolution expands at a onstant veloity. The aeleration a in the relation (9) an be zero only under ondition that the quantity whih we put instead of the mass m is zero. It means that: The ERU model is the only model of expansive homogeneous and isotropi relativisti quantum-mehanial universe with non-zero gauge fator a, non-zero mass density ρ and aeleration a = 0. In the expansive homogeneous and isotropi relativisti universe with total zero-energy gravitational interation of matter objets is ompensated by their expansion, determined by the pressure p in the relations (17h), i.e. the matter objets in larger distanes (than are the dimensions of the largest HGRSs), are moving away from eah other by a onstant veloity. Therefore: In the expansive homogeneous and isotropi relativisti universe with the total energy E tot = 0 gravitational interation of matter objets does not our, it affets only their speialrelativisti properties that are a result of their relative uniform retilinear motion. The same onlusion we reah also by identifying the spaetime properties of the universe. Aording to the observations, the universe expands in finite distanes by finite veloities (Hubble 199). Aording to the Einstein speial theory of relativity (Einstein 1905) physial objets may expand at veloities v in the interval (0, ). Therefore: A homogeneous and isotropi relativisti universe in priniple annot expand at veloity v >. An expansive homogeneous and isotropi relativisti universe in whih the physial objets in finite distanes expand at veloities v is finite in the spae-time manner. From the fat that the observed finite expansive homogeneous and isotropi relativisti quantum-mehanial universe from the relativisti point of view represents mater-spae-time, unambiguously result: The finite expansive homogeneous and isotropi relativisti universe is losed in spae-time (Einstein 1919a). A bakward extrapolation of evolution of the expansive homogeneous and isotropi relativisti universe from the relativisti point of view leads to a geometrial point ( beginning limit osmologial singularity). From the bakward extrapolation of the universe expansion it unambiguously results: The finite expansive homogeneous and isotropi relativisti universe an be losed in spaetime only in one possible way: by the initial limit osmologial singularity. The finite expansive homogeneous and isotropi relativisti universe an be limit-singularly losed in spae-time only if during the whole expansive evolution in the maximum (limit) distane from eah observer expands at the maximum (limit) veloity of signal propagation, i.e. only on the assumption if the gravitational properties of matter objets in it are exatly ompensated by their expansion and due to their relative movements only their speialrelativisti properties are manifested. These properties result also from the fat that observed expansive relativisti universe in larger distanes (than the dimensions of the largest HGRSs) is homogeneous and isotropi. 1

13 The expansive relativisti universe an be homogeneous and isotropi only on the assumption that during the whole expansive evolution it expands at the maximum possible veloity of signal propagation. Therefore, the maximum (boundary, limit) veloity of signal propagation is the only veloity, whih is not dependent on the veloity of its soure, and therefore, nor on the veloity and loation of the observer. By the fat that in the larger distanes of the expansive homogeneous and isotropi relativisti universe with the total zero energy are manifested only speial-relativisti properties of matter objets all its other physial and model properties are given. The observers in the expansive homogeneous and isotropi relativisti universe due to the Lorentz time dilation, determined by the relation (1d) are ontemporaries of all osmologial times, inluding a limit beginning of the expansive evolution of the relativisti universe. It means that: The expansion veloity of the relativisti universe is determined by the veloity at whih evolution of relativisti universe expansion began, beause as a result of the Lorentz time dilation is idential with it. Therefore: The relativisti quantum-mehanial universe throughout the whole expansive evolution may expand at only one possible veloity: boundary (maximum, limit) veloity of signal propagation. A hypothetial universe whih would expand at a veloity v <, would be non-homogeneous and anisotropi, would have only one privileged entre and would not be losed in the spaetime manner. Therefore, an assumption of an expansive homogeneous and isotropi relativisti universe, whih expands at veloity v <, represents ontraditio in adjeto. The observed expansive homogeneous and isotropi relativisti universe in whih the gravitational interation of material objets is ompensated by their expansion is pseudo-flat (pseudo-eulidean), i.e. it has the Minkowski pseudo-eulidean geometry, whih differs from Eulidean geometry in suh a way that it is influened by the speial-relativisti effets equally straightforward expanding inertial matter objets. The pseudo-flat (pseudo-eulidean) expansive homogeneous and isotropi speial-relativisti universe in the linear approximation is a flat (Eulidean). In the model of the expansive homogeneous and isotropi relativisti quantum-mehanial universe (in whih we abstrat from the relativisti and quantum-mehanial effets, i.e. in the model universe whih desribes the observed universe in linear, i.e. non-relativisti approximation), the Eulidean geometry is valid (i.e. de fato linearized Minkowski pseudo- Eulidean geometry, in whih we abstrat from the speial-relativisti effets), and Galilean transformation: x' = x vt, y' = y, z' = z, t' = t, () i.e. de fato linearized Lorentz transformation, determined by the relations (1). For the gauge fator a and the osmologial time t of the homogeneous and isotropi relativisti universe with the total zero and loal non-zero mass (energy), whih expands at a onstant maximum possible (limit) veloity, is valid the relation (Skalský 199, 1989): a = t, () whih is shown among the relations (17a), too. From the relations (7), (19) and () it results that matter objets in the ERU model in any distane r a expand at an esape veloity (Skalský 004) r v es =. (4) a From the relation (4) it results that the model of the ERU in the distane of gauge fator a expands at the esape veloity v es =, in the distane r = a/ expands at esape veloity v es = /, in the distane r = a/ expands at esape veloity v es = /... et. 1

14 That is indeed the ase, we an be persuaded by a simple alulation: From the relations (17a), (17d) and (17f) or from the relations (8), (6), (7) and (8) it results that the expansive non-deelerative non-aelerative homogeneous and isotropi relativisti universe with the total energy E tot = 0 at ertain osmologial time t, for example at osmologial time t =15 Gyr, (5) x in the first (linear) approximation will have: gauge fator a x mass = t x mx = πρ 6 x m, tx 4 5 mx = π ax ρ x = kg, (7) G mass density mx 7 ρ x = = kg m. (8) 8π Gt 4π a x x The expansive homogeneous and isotropi relativisti universe with the total zero energy in the osmologial time, determined by the relation (5), in the linear approximation will have mass density ρ x, determined by the relations (8), mass m x, determined by the relations (7), and in the distane of gauge fator a x, determined by the relations (6), will aording to the relation (19) expand at the esape veloity v es = m s 1 =. The sphere with the radius r = a x / = m, with mass density ρ x, determined by the relations (8), have aording to the relation (8) the mass m = kg, and aording to the relation (19) in the distane r = a x / expand at the esape veloity v es = m s 1 = /... et. As mentioned above, the bakward extrapolation of the evolution of expansive homogeneous and isotropi relativisti quantum-mehanial universe leads to the initial limit osmologial singularity. Therefore, in the initial period its expansive evolution the relation () must be onsistent also with the onditions arising from the Plank quantum hypothesis and from the Heisenberg unertainty priniple. In the years in the Royal Prussian Aademy of Sienes in Berlin, Max K. E. L. Plank presented with five sequels of his artile Über irreversible Strahlungsvorgänge. On Thursday June 1, 1899 he presented the fifth and final sequel of the named artile (Plank 1899). Plank in it, using four onstants: the Newton gravitational onstant G, onstant veloity of light in vauum, Plank quantum onstant h and Boltzmann onstant k B, determined the fundamental physial units of mass, temperature, length and time that are now named after him. The Plank mass m P, Plank temperature T P, Plank length l P and Plank time t P at present time are presented with the following values: m P h = = G 8 kg, (6) (9) T P = 5 h G k B = K, (40) 14

15 l P h = = m P hg = m, lp hg 44 tp = = = s. 5 (4) The value of the Plank mass m P, determined by the relation (9), from maro-physial point of view is very small (approximately two hundred thousandth of gram). The Plank temperature T P, determined by the relation (40), is aording to the Plank quantum hypothesis theoretially the maximum possible temperature, therefore, from point of view of its effet is giganti (maximum possible). The mass m manifests itself inertially and gravitationally. The temperature T manifests itself repulsively (by pulling or negative pressure). From omparison of the Plank mass m P with the Plank temperature T P taking into aount quantization of the mass-spae-time of the universe it results that the expansive evolution of universe began at the maximum possible veloity. This dedutive onlusion onfirmed and speifies the values of Plank length l P, determined by the relation (41), and Plank time t P, determined by the relation (4), from whih result: l P = t P. (4) If in the relation (4) instead Plank length l P we put a Plankian gauge fator a P, defined by the relation: a P := l P, we obtain: a P = t P, (44) whih is the speial partial solution of the relation (). Therefore, from the relations (9)-(44) it results unambiguously that aording to the Plank quantum hypothesis the universe its expansive evolution begin at only one possible veloity: at the boundary veloity of signal propagation. In 197 Werner Heisenberg in the artile Über den anshaulihen Inhalt der quantentheoretishen Kinematik und Mehanik (Heisenberg 197) postulated an unertainty priniple, aording to whih is not possible with unlimited preision determine simultaneously both the position and the momentum of any partile. From the Heisenberg unertainty priniple (relations) it results that the partile annot remain on ertain plae beause it would have an exat position and exat (i.e. zero) momentum but it must permanently flutuate. The observations onfirmed that if we minimise the spae in whih the partile an flutuate (i.e. if we speify its position), then its flutuations are aelerated, and in result of the trembling motions its unertainty of momentum grows. The universe at the beginning of it expansive evolution had minimum size parameters, therefore, the partiles in it flutuate at the maximum possible veloities. The result of these flutuations was the maximum possible negative pressure, whih ompensated their mutual gravitational interation and was one of the auses of the maximum possible veloity of the inrease of matter-spae-time of the universe. It means that even aording to the Heisenberg unertainty priniple (one of the fundamental priniples of the quantum mehanis), an expansive evolution of the universe began his expansive evolution at the only possible veloity: at the maximum possible (limit) veloity of signal propagation. These dedutive onlusions are onfirmed by the observations, too: In 199 Edwin P. Hubble disovered the expansion of the universe (Hubble 199). Hubble on the basis of astronomial observations of the nebulae (galaxies) found: a roughly linear relation between veloities and distanes among nebulae (Hubble 199, p. 17). (41) 15

16 At present time this relation is known as the Hubble law and is written in this form: v = HR, (45) where v is the veloity of a osmi distant objet, R is its distane, and H Hubble onstant (oeffiient, parameter). From the relations () and (45) it results the relation for the Hubble parameter H and the osmologial time (age of universe) t (Skalský 1991): v 1 H = = = =, (46) R a t t shown among the relations (17), too. Aording to the WMAP measurements (Hinshaw et al. 009): (present) age of universe t = Gyr, (47) 0 ± and (present value of) Hubble onstant H 0 = 71.9 km s Mp = km s Mp. (48).7 From the relations (46) and (47) result: H 0 = = 71.4 km s Mp = km s Mp. (49) t The value of H 0, determined by the relation (49), is in the frame of measurement unertainty of the value of H 0, determined by the relation (48). Aording to the WMAP+BAO+SN measurements (Hinshaw et al. 009): t 0 = 1.7 ± 0.1 Gyr, (50) and H 0 = 70.5 ± 1. km s Mp = km s Mp. (51) From the relations (46) and (50) result: H 0 = = 71.7 km s Mp = km s Mp. (5) t The maximum value of H 0, determined by the relation (5), differs from the maximum value of H 0, determined by the relation (51), by the value km s 1 Mp 1. From the omparison of the relations (48) and (49) and the relations (51) and (5) result, that aording to the WMAP and the WMAP+BAO+SN observations, determined by the relations (47), (48), (50) and (51), the observed universe in the frame measurement unertainty expands at the boundary veloity of signal propagation. Aording to the WMAP measurements (Hinshaw et al. 009): age of deoupling t * = yr, (5) 5841 and redshift of deoupling z * = ± 0.95 = (54) From the relations (54) and (77) result veloity of deoupling v * : v v v, (55) z = * z =

17 where v z = = = , and v z = = = Aording to the WMAP+BAO+SN measurements (Hinshaw et al. 009): + 16 t * = yr, (56) 167 and z = ± 0.7 = (57) * From the relations (57) and (77) result: v v v, (58) z = * z = where v z = = = , and v z = = = From the relation (55), or (58), taking into aount the age of the universe, determined by the relation (47), or (50), and the age of deoupling, determined by the relation (5), or (56), it results that the WMAP and the WMAP+BAO+SN observations onfirmed that the observed universe in the frame measurement unertainty expands at the veloity. The expansive non-deelerative non-aelerative homogeneous and isotropi relativisti universe with the total zero energy whih during the whole expansive evolution expands at the esape veloity v es =, has the ritial mass (energy) density, i.e.: total (dimensionless) density (of the universe) Ω = 1. tot (59) Aording to the WMAP measurements (Hinshaw et al. 009): Ωtot = = (60) The value of Ω tot, determined by the relation (59), differs from the minimum value of Ω tot, determined by the relation (60), by the value Aording to the WMAP+BAO+SN measurements (Hinshaw et al. 009): Ωtot = = 0,9989 1,0110. (61) The value of Ω tot, determined by the relation (59), is in the frame measurement unertainty of the value of Ω tot, determined by the relation (61). 5 The model and physial properties of the expansive homogeneous and isotropi relativisti universe The ERU model, determined by the FRW equations (4a), (4b) and (4) with k = 0, Λ = 0 and w = 1/, desribes the expansive homogeneous and isotropi relativisti quantum-mehanial universe in the linear approximation (in whih we abstrat from its relativisti and quantummehanial properties). The ERU model is flat (Eulidean), however, the real expansive homogeneous and isotropi relativisti universe is a pseudo-flat (pseudo-eulidean), i.e. it has the Minkowski pseudo- 17

18 Eulidean geometry, whih differs from Eulidean geometry only in that, that it is influened by the speial-relativisti effets of the expanding inertial matter objets. This fat makes it possible by omparing the linearized (i.e. non-relativisti) properties of the ERU model and the non-linearized (i.e. speial-relativisti) properties of the atual observed expansive homogeneous and isotropi relativisti universe to get a ertain idea about the relationship between them, and about possibilities of using the ERU model in the relativisti osmology. Probably you annot imagine the evolution of the observed four-dimensional pseudo-flat (pseudo-eulidean) expansive non-deelerative non-aelerative homogeneous and isotropi relativisti universe. However, you an imagine it without problems in the linear (Eulidean) model approximation, in whih we abstrat from its speial-relativisti properties. Therefore, we start the omparison of the ERU model with the real pseudo-flat expansive homogeneous and isotropi relativisti universe by this 4-dimensional image: The relativisti universe during its whole expansive evolution expands at a onstant, maximum possible veloity of signal propagation in the distane of the gauge fator a. The maximum veloity of signal propagation is not dependent on the veloity of its soure and hene nor on the veloity and loation of the observer. Therefore, all observers in the relativisti universe are in its entre and in Eulidean approximation (in whih we abstrat from its relativisti properties), it an be imagined as an expanding Eulidean homogenous matter sphere, whose surfae is moving away from them at a onstant veloity. If we separate the time omponent from spatial omponents and if we abstrat from one spatial dimension, the evolution of 4-dimensional expansive homogeneous and isotropi relativisti universe in the Eulidean projetion whih you have just imagined an be presented in -dimensional linear (Eulidean) approximation in the form of a time one, whih we show in Figure 1. Fig. 1 The evolution of 4-dimensional expansive homogeneous and isotropi relativisti universe in the -dimensional Eulidean (non-relativisti) presentation in the osmologial times t 1, t,... t n. 18

19 Fig. Two-dimensional Eulidean (non-relativisti) projetion of the 4-dimensional expansive homogeneous and isotropi relativisti universe in an arbitrary osmologial time t. The ellipses in Figure 1 represent the -dimensional Eulidean (non-relativisti) projetion of the -dimensional pseudo-eulidean spae of expansive homogeneous and isotropi relativisti universe in the linear approximation in the osmologial times t 1, t,... t n. If the -dimensional Eulidean projetion of the expansive homogeneous and isotropi relativisti universe (shown in Figure 1), is redued by another spatial dimension and the times t 1, t,... t n are redued to only one, we get -dimensional linearized spae-time (Eulidean) projetion of the 4-dimensional expansive homogeneous and isotropi relativisti universe in an arbitrary osmologial time t, shown in Figure. In Figure the absissa onneting the point t (in whih is the observer), with point A, and point t with point B, represent the radius of Eulidean sphere r, i.e. the gauge fator a of the ERU model, and the absissa onneting point A with point B represents the diameter of Eulidean sphere d, i.e. 1-dimensional model projetion of the -dimensional spae of the expansive homogeneous and isotropi relativisti universe in the linear (Eulidean) projetion at any osmologial time t. As mentioned earlier, the Minkowski pseudo-eulidean geometry differs from Eulidean geometry only therein, that is influened by speial-relativisti effets of the inertial matter objets, expanding at onstant veloities, therefore, in the plae of the observer (i.e. with zero veloity of matter objets), the Minkowski pseudo-eulidean geometry is idential with the Eulid geometry. These fats allow us to onstrut -dimensional pseudo-eulidean model of 4-dimensional expansive homogeneous and isotropi relativisti universe. We an do it in suh a way that in the 19

20 -dimensional Eulidean spae-time model of ERU (projeted in Figure ), we take into aount the speial-relativisti spae-time effets of expanding inertial matter objets. From the fourth equation of the Lorentz transformation, in this artile shown as the relation (1d), results dimensionless dilated time t t' =. (6) v 1 Dimensionless proportion v/ in the relation (6), representing the veloity of matter objet v, expressed as a fration of the veloity of light at present time prevailingly designated by letter β is known as the dimensionless veloity parameter v β =. (6) The expansive homogeneous and isotropi relativisti universe throughout its whole expansive evolution expands at onstant veloity. For its gauge fator a and the osmologial time t is valid the relation (). Therefore, for the dimensionless distane of the matter objet r, expanding at veloity v is valid the relation: r = vt. (64) From the relations (), (6) and (64) it results the dimensionless proportion r/a, representing a distane of expanding matter objet r, expressed as a fration of the gauge fator a, whih represents the dimensionless distane parameter r R =, (65) a expressing a linearized (non-relativisti) distane of the expanding matter objet. From the relations (), (6), (64) and (65) it results: r vt v R = = = = β. (66) a t The dimensionless inverted value of the root, presented in the relation (6) at present time mainly designed by the letter γ is known as the dimensionless Lorentz fator 1 γ =. (67) 1 β Using the relation (67) we an rewrite the relation (6) into the form: t' = γ t. (68) As a result of the time dilation, determined by the relation (6), or by the relation (68), looking into the distane, in ertain sense, we look into the past. Stritly speaking, we observe the events, whih from observers point of view who are loated on the observed plae are already in the past. The objet expanding at an arbitrary veloity, whih is determined by the dimensionless veloity parameter β, is observed in the orresponding dimensionless 0