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'Cosmic Horizon' THE MILNE UNIVERSE AND THE MELIA ‘COSMIC HORIZON’. By Richard Blaber. We revisit the Milne cosmological model and examine Melia’s argument for a ‘cosmic horizon’ in relation to that model, before dismissing both in favour of the standard CDM view. Imagine a universe which is conformally ‘flat’ and a Ricci-flat manifold on a large scale, i.e., where C = 0 and R(vacuum solution of the Einstein field equations of the General Theory of Relativity, GTR), k4 = 0 in 4-space and [1] k3= –1 in 3-space . The 4-space curvature parameter and Ricci curvature tensor can be equal iff = 0[2],and the GTR does not apply on the large scale, but only on the local scale[3]. = 0 implies, in turn, that the mass density ∞ = 0, which, given that there is mass in this universe, implies that it is a finite mass confined to a finite portion of an infinite volume of 3-space at any given time period, t = age of the universe, tu. The nature of the confinement has changed; at the earliest epoch, tu = 0, the mass was concentrated in a volume of [1] Here k3 is the 3-space curvature parameter, which is dimensionless and three-valued: –1, when the total mass density parameter, < 1, and space-time is ‘hyperbolic’; 0, when = 1, and space-time is ‘flat’ or Euclidean; and +1, when > 1, and space-time is closed or Riemannian. (Observation indicates that, in our universe, at the present epoch [signified by subscript ‘0’], 0 = 1, k3 = 0.) For more discussion of k4, see below. C is the Weyl conformal tensor (see: Penrose, R [2010], Cycles of Time. An Extraordinary New View of the Universe, London: Vintage Books, pp.130-5, 224 (he refers to C as C); Weyl, H [1918], ‘Reine Infinitesimalgeometrie [Pure Infinitesimal Geometry],’ Mathematische Zeitschrift st 2(3-4):384-411, 1 September 1918, DOI: 10.1007/BF01199420); R is the Ricci curvature tensor, see: Ricci-Curbastro, MMG & Levi-Civita, T (1900), ‘Méthodes de Calcul Différentielle Absolu et leurs Applications [Methods of Absolute Differential Calculus and their Applications],’ Mathematische Annalen 54(1-2):125-201, 1st March 1900, DOI: 10.1007/BF01454201. [2]I.e., the cosmological constant, which appears in the full tensor form of the Einstein 4 equation: R½Rg = T – g, where = 8G/c , g = the metric tensor (see Ricci- Curbastro & Levi-Civita, op.cit.), R the Ricci scalar (scalar curvature), and T the stress- energy-momentum tensor; see: Schoen, R (1984), ‘Conformal Deformation of a Riemannian Metric to Constant Scalar Curvature,’ Journal of Differential Geometry 20:479-95, http://www.math.jhu.edu/~js/Math646/schoen.yamabe.pdf; Kaya, A & Tarman, M (2011), ‘Stress-Energy Tensor of Adiabatic Vacuum in Friedmann-Robertson-Walker Spacetimes,’ Journal of Cosmology and Astroparticle Physics, 2011 Volume, April Issue, DOI: 10.1088/1475-7516/2011/04/040, http://arxiv.org/pdf/1104.5562.pdf. When T = = 0, R = R= 0. If ≠ 0, R = 0, R = –g. [3]The Special Theory of Relativity (STR) would apply at all length scales. 1 space V = 0, a space-time singularity[4], so that then = ∞[5]. As soon as the mass was released from this prison, however, it was liberated into an infinite Euclidean 3-space, with V = ∞, so its density at infinity promptly dropped to ∞ = 0. Its local density at different times since t = 0 has been non-zero positive and finite; in other words, 0 < L ≪ ∞. (However, L → 0 as t → ∞.) Given that it is open, at any given moment t < ∞, the 3-space of this cosmos has negative curvature, and so k3 = –1, as we have seen. The universe we have just described was first adumbrated by EA Milne in 1933[6]. In imaginary Cartesian coordinates, this has the metric[7]: ( ) ( ) (Eq.1a) which is derivable from the Minkowski[8] metric in imaginary Cartesian coordinates, but Milne space only occupies a quarter of Minkowski space[9]: [4]This was not a curvature singularity, but a Newtonian gravitational singularity, produced by ∇2 = 4G (Poisson’s equation for gravity), where is the gravitational potential in J kg-1 and = –GM/r, where r is distance. When = ∞, r = 0, = –∞. [5]We assume that space-time is continuous and infinitely divisible, not discrete. Thus space- time is not quantised in this universe, and, generally, it is classical rather than quantum mechanical (e.g., there is no Casimir or vacuum energy, see Kaya & Tarman, op.cit.). [6]See: Milne, EA (1933), ‘World-Structure and the Expansion of the Universe,’ Zeitschrift für Astrophysik 6:1-95, BibCode: 1933ZA......6....1M, http://articles.adsabs.harvard.edu/cgi- bin/nph- iarticle_query?db_key=AST&bibcode=1933ZA......6....1M&letter=0&classic=YES&defaultp rint=YES&whole_paper=YES&page=1&epage=1&send=Send+PDF&filetype=.pdf.. Another feature to note of this universe is that the ratio of the pressure, p (in Pa) to the energy density in J m-3, w = p/c2 = 0/0 = 0. [7]See: Charmousis, C, ‘Introduction to Anti de Sitter Black Holes,’ in Papantonopoulos, E, ed. (2011), From Gravity to Thermal Gauge Theories: The AdS/CFT Correspondence, Part I, Berlin: Springer Verlag, pp.3-26, http://www.physics.ntua.gr/cosmo09/Milos2009/Milos%20Talks%202009/1st%20day/Charm ousis%20Paper.pdf, p.3; here and in Eq.1b, represents proper ‘time’ (in units of imaginary length; not to be confused with the Hubble time, = H-1); and H is the Hubble parameter. ‘AdS/CFT correspondence’, also referred to in the literature as ‘Maldacena duality’ or ‘gauge/gravity duality’ is an abbreviation of ‘Anti-de Sitter space/Conformal Field Theories’, and refers to the fact that the conformal boundary of anti-de Sitter 3-space ( < 0) can be treated as the space-time of a conformal field theory (quantum field theory). See: Maldacena, J (1998), ‘The Large N Limit of Superconformal Field Theories and Supergravity,’ Advances in Theoretical and Mathematical Physics 2:231-52, http://arxiv.org/pdf/hep- th/9711200v3.pdf. [8]Minkowski, H (1909), ‘Raum und Zeit [Space and Time],’ Jahresbericht der Deutschen Mathematiker-Vereinigung 18:75-88, English trans. at: http://en.wikisource.org/wiki/Translation:Space_and_Time, pp.5-9. Strictly speaking, expressing Minkowski space in terms of imaginary or complex dimensions gives it a metric 2 ( ) ( ) [10] (Eq.1b) Instead of considering the local universe, UL (defined below), as a finite portion of an infinite volume of Euclidean 4-space containing a finite quantity of mass- energy, so that k4 = 0, but insufficient for closure of the local 3-space, so that k3 = –1, and that local 3-space is consequently hyperbolic (or ‘Bolyai- Lobachevskian’), let us instead begin by accepting Milne’s basic premise that cosmic observers, wherever they may be located, are equivalent (he calls this ‘The Principle of Extended Relativity’[11]) and adopt an ‘observer-centric’[12] view of the universe, which, in the case of an Earth-based observer, would be a geocentric one. One might, with a high degree of accuracy, describe this as subjective Ptolemaic (or neo-Ptolemaic) cosmology. Next, however, in place of the hyperbolic 3-space, let us substitute a flat (k3 = 0; = 1; = 0) local 3-space, with pressure p = 0 (and thus w = 0). Such a space would constitute a finite hyperplane of a static infinite 4-space co- extensive with ℂ4[13], and thus, if our local volume of 3-space is the only part of this 4-space to contain any mass-energy, ∞ = 0 and k4 = 0. The metric of the 4- space is the Minkowski metric (Eq.1b, see n.8); the metric of the 3-space is for a signature of +, +, +, + (as opposed to –, +, +, + or +, –, –, –), which converts it into a complex Euclidean 4-space, or complex Hilbert 4-space (see: Bierens, HJ [2007], ‘Introduction to Hilbert Spaces,’ http://econ.la.psu.edu/~hbierens/HILBERT.PDF, pp.1-5, 17). [9]See: Peter, P & Uzan, J-P (2009), Primordial Cosmology, Oxford: OUP, p.144. [10]See: Tegmark, M (2005-6), ‘General Relativity. MIT Course 8.033, Fall 2005, last revised November 7 2006,’ http://ocw.mit.edu/courses/physics/8-033-relativity-fall- 2006/readings/gr.pdf, p.2. [11]See: Robertson, HP (1933), ‘On E.A. Milne’s Theory of World Structure,’ Z für Astrophys 7:153-66, BibCode: 1933ZA......7..153R, http://articles.adsabs.harvard.edu/cgi- bin/nph- iarticle_query?db_key=AST&bibcode=1933ZA......7..153R&letter=0&classic=YES&default print=YES&whole_paper=YES&page=153&epage=153&send=Send+PDF&filetype=.pdf, pp.154-65. (It is, incidentally, quite wrong for Robertson to refer to Milne’s observers as ‘privileged’: they are anything but privileged, given their equivalence. Milne merely treats each observer’s viewpoint as if it was the centre of the universe; which, from the observer’s perspective, it is.) [12]‘Aisthetescentric’ will probably not have the elegance to thrive as a word. [13]Co-extensive with ℂ4, the set of all imaginary or complex numbers extended in 4 coordinate axes or dimensions from some origin point (Cartesian coordinate system), as opposed to ℝ4, the set of all real numbers extended over four dimensions. This is because we are including time, but measuring time (ict) using imaginary length. 3 [14] complex Hilbert 3-space . The deceleration parameter, q0, at the present epoch, is then given by: (Eq.2) We find that R = c/H = c = RH, the Hubble radius. Furthermore, the Schwarzschild radius of the mass of the local universe, ML, is given by: (Eq.3)[15] If the above prevails, then our universe constitutes the interior of a large black [16] hole , and the distance RS = RH an event horizon, as defined by Rindler (1956)[17]: ‘An event horizon, for a given fundamental observer A, is a (hyper-) surface in space-time which divides all events into two non-empty classes: those that have been, are, or will be observable by A, and those that are forever outside A’s possible powers of observation.
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