A Numerical Discussion about Willem De Sitter Cosmological Model

The Dutch astronomer Willem de Sitter (1872-1934) gave important contributions to the rise of relativistic Cosmology. The debate from 1916 to 1918 between de Sitter and Albert Einstein (1879-1955) is a fundamental chapter in the history of the scientific view of the universe. In fact, during such a debate both Einstein and de Sitter formulated their own mathematical expressions for the metric of the universe as a whole.

In the fall of 1916 Einstein debated with de Sitter on the problem of suit- able boundary conditions at infinity. According to the Principle of Relativity, Einstein tried to obtain values for the gµν‘s at infinity that was invariant for all transformations. He avoided this difficulty by replacing such boundary conditions with the condition of closure, introducing a ―finite and yet unbounded universe‖.[4] Einstein proposed a spherical model of the universe, in which the matter was uniformly and homogenously distributed. This static solution had line element: R was the radius of curvature of the three-space (x1, x2, x3), that was every- where orthogonal to the time dimension x4. This model fully achieved the relativity of inertia. There was not any independent property of space which claimed to the origin of inertia, so the latter was entirely produced by masses in the universe. The condition of spatial closure ensured that both the gravitational potential and the hypothetical average density of ponderable matter remained constant in space.

constant in time. He inserted the so-called cosmological term, namely the fundamental tensor gµν multiplied by −λ, a universal but unknown constant: In this way field equations expressed the observational evidence of the static equilibrium of the universe. The new constant λ, the radius of the universe R, and the mean density of ―world matter‖ ρ were strictly connected: In Einstein model the metric of the universe could be given as a solution of relativistic field equations with the cosmological term. Both the general covariance and the laws of conservation of momentum and energy were still satisfied.

Right after Einstein model appeared, de Sitter proposed his own solution of field equations. The Dutch astronomer admired Einstein conception of the universe ―as a contradiction-free chain of reasoning‖,[7] and gave a different solution also maintaining the λ-term. However de Sitter preferred the original relativistic theory of gravitation, ―without the undeterminable λ, which is just philosophically and not physically desirable‖.[8]

De Sitter approached the cosmological problem in a different way. It was mainly Paul Ehrenfest (1830-1933) who suggested him a mathematical conception of inertia[9], which led de Sitter to propose a finite and ―empty‖ universe. De Sitter proposed a distinction between the ―world matter‖ and the ―ordinary matter‖. The former was hypothetically distributed through space with density ρ0. The latter corresponded to observable objects as planets and stars, i.e. to locally condensed world matter with density ρ1. By this assumption, de Sitter pointed out that ―inertia is produced by the whole of world matter, and gravitation by its local deviations from homogeneity‖.[13] Neglecting all pressures and internal forces, and supposing all matter to be at rest, the energy-momentum tensor became: According to de Sitter, the three-dimensional finite world proposed by the Feinstein seed the ―material relativity requirement‖,[14] or equivalently the ―material postulate of relativity of inertia‖.[15] The Dutch astronomer pointed out that the relativistic field equations were ―the fundamental ones‖[17]: the postulate that at infinity all gµν‘s were invariant for all transformations was more important than the ―Machian‖ postulate of inertia introduced by Einstein. In fact in Einstein model, for the hypothetical value R = ∞, the whole of gµν‘s degenerated to This set of values was invariant for all transformations for which, at infinity, t0 = t. In other words, in Einstein cylindrical world it was possible to find systems of reference in which the gµν‘s only depended on the space-variables, and not on the ―time‖. However the ―time‖ of such a systems had ―a separate position‖,[18] because it was ―the same always and everywhere‖.[19] For such a reason, according to de Sitter, the time coordinate in Einstein model was nothing else than an absolute time, and there the world matter took ―the place of the absolute space in Newton‘s theory, or of the inertial system‖.[20] De Sitter proposed that the potentials should have degenerate at infinity to the values: According to him, ―if at infinity all gµν‘s were zero, then we could truly say that the whole of inertia, as well as gravitation, is thus produced. This is the reasoning which has led to the postulate that at infinity all gµν‘s shall be zero‖.[21] De Sitter called this requirement the ―mathematical relativity condition‖,[22] or the ―mathematical postulate of relativity of inertia‖.[23] In fact, such a condition corresponded to the possibility that ―the world as a whole can perform random motions without us (within the world) being able to observe it‖[24]: ―the postulate of the invariance of the gµν ‘s at infinity - de Sitter stated - has no real physical meaning. It is purely mathematical‖[25]

solution of the metric of the universe as a whole, actually the second relativistic model in modern Cosmology. The Dutch astronomer considered field equations with the λ-term and without matter, i.e. by assuming ρ0 = 0: These equations could be satisfied by the gµν‘s given by the metric: The coordinates (x, y, z, t) could have infinite values, on condition that gµν ‘s were null at infinity. Such a condition was equivalent to the finiteness of the world in natural (proper) measure. In fact the length of any semi-axis in natural measure was: A finite world (i.e. a finite value of Lα) necessary implied gαα = 0 for xα → ∞, and vice versa.[27] De Sitter pointed out that in his model no world matter was necessary, and the insertion of the λ-term satisfied the mathematical postulate of relativity of inertia. In this system there was not any universal time, nor any difference between the ―time‖ and the other coordinates: none of these coordinates had any physical meaning.[28] The cosmological constant determined the value of the curvature radius R: by using an imaginary ―time‖-coordinate ξ4 = ict, the geometry of de Sitter world was that of a 4-dimensional hyper-sphere which could be described in a 5-dimensional Euclidean space: In hyper-spherical coordinates the metric of such a four-dimensional world resulted: where 0 ≤ θ ≤ 2π; 0 ≤ ψ, ζ , ω ≤ π. Equivalently, by replacing the imaginary ―time‖-coordinate ξ4 with a real time-coordinate (ξ4 → iξ4), the geometry of de Sitter world corresponded to a 4-dimensional hyperboloid in a 4+1- dimensional Minkowski space-time: By pseudo-spherical coordinates (with iω = ω), the metric of space-time resulted: where 0 ≤ θ ≤ 2π; 0 ≤ ψ, ζ ≤ π; −∞ < ω0 < +∞. The potentials in the hyper-spherical coordinate system were: Thus the metric proposed by de Sitter, ―If a single test particle - de Sitter wrote to Einstein - existed in the world, that is, there were no sun and stars, etc., it would have inertia‖.[30] Essentially, in the universe proposed by de Sitter a suitable metric was obtained without any ―physical‖ masses.[31] Such forms of matter as stars and nebula were to be regarded as ―test particles‖ in a fixed background metric, which curvature was determined by the cosmological constant.[32]

Einstein acknowledged de Sitter‘s solution to be ―very interesting‖,[35] but ―must have been disappointed‖,[34] and tried to discard this anti-Machian solution: ―I cannot grant - Einstein wrote to de Sitter - your solution any physical possibility‖.[35] In fact, the cosmological term took a fundamental role in de Sitter model in order to involve a sort of spatial (and not material) origin of inertia. ―The gµν field – Einstein replied to de Sitter - should be fully determined by matter, and not be able to exist without the latter ‖.[36] At first Einstein objected that the hyperboloid surface was a singularity. On this surface there was a discontinuity, because the g44 term ―jumped‖ [37] from +∞ to −∞, and gαα ‘s from −∞ to +∞. Such a surface lied in the physically finite, but it was not possible to assume infinite values for the potentials, because of the supposed static nature of the universe and the small velocities measured on stars.[38] Moreover, the four-dimensional continuum proposed by de Sitter did not have the property that all its points were equivalent.[39] In De Sitter replied that the hyper-surface involved a finite natural spatial dis- tance and an infinite natural temporal distance. Thus the discontinuity was only apparent, and this problem was ―not interesting‖.[40] Also the supposed preferred point was later shown to be a geometrical consequence of that choice of coordinates, and not a true physical aspect.[41] ―My four-dimensional world - de Sitter remarked to Einstein - also has the λ-term, but no world matter‖.[42]

In order better to compare his own model with Einstein solution, de Sitter proposed another expression of the metric.[43] By using spherical polar coordinates, he represented the hyperboloid universe (system B) as the Einstein universe (system A), i.e. as 3-dimensional hyper-spheres embedded in a 4-dimensional Euclidean space: De Sitter acknowledged Einstein remark to be correct, but gave a different interpretation. According to the Dutch astronomer, such a remark involved a philosophical, and not a physical requirement.[52] In fact, the ―equator‖ at r = π R was at a finite distance in space, but was physically inaccessible.[53] The velocity of a material particle became zero for r = π R. Thus a material particle which was on the polar line on the origin could have no velocity, nor energy. ―All these results - de Sitter stated - sound very strange and paradoxical. They are, of course, all due to the fact that g44 becomes zero for r = π R. We can say that on the polar line the four-dimensional time-space is reduced to the three-dimensional space: there is no time, and consequently no motion‖.[54] The time needed by a ray of light, or by a material particle, to travel by any point to the equator was infinite. Thus the singularity at r = π R could never affect any physical experiment.[55]

―At the present time - de Sitter wrote in 1920 - the choice between the systems A and B is purely a matter of taste. There is no physical criterion as yet available to decide between them‖.[59] However de Sitter noticed that these systems differed in their physical consequences. In fact, in de Sitter world a particle at rest would not have remained at rest unless it was at the origin. This mass test would have escaped far away because of the presence of the cosmological constant. Thus the de Sitter system to all appearances was static, and required a positive radial velocity for distant objects. This effect of recession was many nebula. The interest in de Sitter effect survived until 1930, when truly non-static theoretical models of the universe were proposed to explain the red-shift problem and the astronomical evidences of a cosmic recession .[60]

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Assistant Professor of Mathematics, Pt. C. L. S. Govt. College, Sector -14, Karnal, Haryana rkdubey2004@yahoo.co.in