scholarly journals Effect of the Cubic Torus Topology on Cosmological Perturbations

Universe ◽  
2021 ◽  
Vol 7 (12) ◽  
pp. 469
Author(s):  
Maxim Eingorn ◽  
Ezgi Canay ◽  
Jacob M. Metcalf ◽  
Maksym Brilenkov ◽  
Alexander Zhuk
Keyword(s):  
Gravitational Potential ◽  
Yukawa Potential ◽  
Cosmological Perturbations ◽  
The Third ◽  
Original Mass ◽  
Present Universe ◽  
Delta Functions ◽  
Yukawa Potentials ◽  
Topology Of The Universe ◽  
The Universe

We study the effect of the cubic torus topology of the Universe on scalar cosmological perturbations which define the gravitational potential. We obtain three alternative forms of the solution for both the gravitational potential produced by point-like masses, and the corresponding force. The first solution includes the expansion of delta-functions into Fourier series, exploiting periodic boundary conditions. The second one is composed of summed solutions of the Helmholtz equation for the original mass and its images. Each of these summed solutions is the Yukawa potential. In the third formula, we express the Yukawa potentials via Ewald sums. We show that for the present Universe, both the bare summation of Yukawa potentials and the Yukawa-Ewald sums require smaller numbers of terms to yield the numerical values of the potential and the force up to desired accuracy. Nevertheless, the Yukawa formula is yet preferable owing to its much simpler structure.

scholarly journals Gravitational Interaction in the Chimney Lattice Universe

Universe ◽  
2021 ◽  
Vol 7 (4) ◽  
pp. 101
Author(s):  
Maxim Eingorn ◽  
Andrew McLaughlin ◽  
Ezgi Canay ◽  
Maksym Brilenkov ◽  
Alexander Zhuk
Keyword(s):  
Helmholtz Equation ◽  
Gravitational Potential ◽  
Fourier Expansion ◽  
Gravitational Interaction ◽  
Alternative Solution ◽  
The Third ◽  
Present Universe ◽  
Delta Functions ◽  
Yukawa Potentials ◽  
The Universe

We investigate the influence of the chimney topology T×T×R of the Universe on the gravitational potential and force that are generated by point-like massive bodies. We obtain three distinct expressions for the solutions. One follows from Fourier expansion of delta functions into series using periodicity in two toroidal dimensions. The second one is the summation of solutions of the Helmholtz equation, for a source mass and its infinitely many images, which are in the form of Yukawa potentials. The third alternative solution for the potential is formulated via the Ewald sums method applied to Yukawa-type potentials. We show that, for the present Universe, the formulas involving plain summation of Yukawa potentials are preferable for computational purposes, as they require a smaller number of terms in the series to reach adequate precision.

RESTRICTED PROBLEM OF THREE BODIES WITH NEWTONIAN + YUKAWA POTENTIAL

2004 ◽  
Vol 13 (05) ◽  
pp. 783-806 ◽  
Author(s):  
FERNANDO KOKUBUN
Keyword(s):  
Gravitational Potential ◽  
Restricted Problem ◽  
Yukawa Potential ◽  
Coupling Constant ◽  
Third Body ◽  
Length Scales ◽  
The Third ◽  
Newtonian Case ◽  
Coupling Parameters ◽  
Yukawa Term

Trajectories of the third body in the Restricted Problem of Three Bodies including a Yukawa term to the Newtonian gravitational potential are analyzed. It is shown that this modified gravitational potential changes some important aspects of the Restricted Problem of Three Bodies. Depending of coupling constant α, motions obtained in the pure Newtonian case are qualitatively different when Yukawa term is included. Depending of coupling parameters α, the nature of dynamics change from regular to chaotic (α<0) or from chaotic to regular (α>0) and in both cases using the same length scales λ.

On the matter of proving Euclidean fifth postulate and the origin of non-Euclidean geometries

2019 ◽  
Vol 950 (8) ◽  
pp. 2-11
Author(s):  
S.A. Tolchelnikova ◽  
K.N. Naumov
Keyword(s):  
Celestial Mechanics ◽  
Natural Philosophy ◽  
Relativity Theory ◽  
Theory Of Relativity ◽  
The Third ◽  
Stellar Astronomy ◽  
Spherical Surfaces ◽  
Mathematical System ◽  
Solar System Bodies ◽  
The Universe

The Euclidean geometry was developed as a mathematical system due to generalizing thousands years of measurements on the plane and spherical surfaces. The development of celestial mechanics and stellar astronomy confirmed its validity as mathematical principles of natural philosophy, in particular for studying the Solar System bodies’ and Galaxy stars motions. In the non-Euclidean geometries by Lobachevsky and Riemann, the third axiom of modern geometry manuals is substituted. We show that the third axiom of these manuals is a corollary of the Fifth Euclidean postulate. The idea of spherical, Riemannian space of the Universe and local curvatures of space, depending on body mass, was inculcated into celestial mechanics, astronomy and geodesy along with the theory of relativity. The mathematical apparatus of the relativity theory was created from immeasurable quantities

Wormholes and large-scale topology of the Universe

1996 ◽  
Vol 111 (12) ◽  
pp. 1433-1438 ◽  
Author(s):  
Zhen-Qiang Tan ◽  
You-Gen Shen
Keyword(s):  
Large Scale ◽  
Topology Of The Universe ◽  
The Universe

ON THE ENERGY OF A BOSE–EINSTEIN CONDENSATE IN AN OPTICAL LATTICE

2007 ◽  
Vol 19 (04) ◽  
pp. 371-384 ◽  
Author(s):  
AMANDINE AFTALION
Keyword(s):  
Optical Lattice ◽  
Critical Case ◽  
Periodic Potential ◽  
Einstein Condensate ◽  
Gamma Convergence ◽  
The Third ◽  
Bose Einstein Condensate ◽  
Delta Functions ◽  
Convergence Type ◽  
Bose Einstein

In this paper, we study the Gross–Pitaevskii energy of a Bose–Einstein condensate in the presence of an optical lattice, modeled by a periodic potential V(x3) in the third direction. We study a simple case where the wells of the potential V correspond to regions where V vanishes, and are separated by small intervals of size δ where V is large. According to the intensity of V, we determine the limiting energy as δ tends to 0. In the critical case, the periodic potential approaches a sum of delta functions and the limiting energy has a contribution due to the value of the wave function between the wells. The proof relies on Gamma convergence type techniques.

The Corrupted “Wheel of Life”: An Essay on the Ouroboros

2020 ◽  
Vol 30 (6) ◽  
pp. 239-282
Keyword(s):  
Jacques Lacan ◽  
Layer By Layer ◽  
Closed Cycle ◽  
Human Spirit ◽  
The Third ◽  
Conceptual Schemes ◽  
Different Dimensions ◽  
Human Desire ◽  
Symbolic Image ◽  
The Universe

The focus of this article is a symbolic image often found in world mythology - a giant snake or a dragon biting its own tail. This image is usually denoted by the Greek word “ouroboros” ( οὐροβόρος ), which means literally “eating its own tail.” This essay is devoted to an interpretation of this symbol, which the author sees as leading to the much broader topic of human unfreedom and the forms that this unfreedom takes. The first section deals with the unique features of Gnosticism which have made it appealing in extremely varied times and situations. Theauthor’s reflections start from understanding the Gnostic worldview as an expression of apprehensiveness about the radical otherworldliness of the human spirit and its alienation from the universe. The second section deals with the symbolism of the ouroboros and its place in Gnostic conceptual schemes as a reference to the closed cycle of nature that enslaves the human spirit. The third section attempts to decipher layer by layer the Gnostic conceptions associated with the ouroboros. Various levels of interpretation are identified: literal, mythological-magical, psychological-ascetic and socio-political. In the fourth section, the author connects Gnostic ideas with Christianity by interpreting St. Paul’s Epistles, particularly his ideas concerning rulers and authorities. The place occupied by the ouroboros in the Christian universe is analyzed. The last section relies on the ideas of René Girard, Jacques Lacan and Alain Badiou to illustrate the manifestations of the ouroboros in different dimensions of human existence, both individual and collective, with special emphasis on human desire and its futile circlings.

Signature of Topology of the Universe

2018 ◽  
Vol 2 (01) ◽  
Author(s):  
Vipin Kumar Sharma
Keyword(s):  
General Relativity ◽  
Energy Density ◽  
Local Geometry ◽  
Microwave Background ◽  
Main Motivation ◽  
And Topology ◽  
The Standard Model ◽  
Topology Of The Universe ◽  
The Universe ◽  
Insight Into

The main motivation to write this article is to relate the cosmology and topology in order to gain some insight into the topological signatures of the Standard model of Universe. The theory of General Relativity as given by Einstein only describes the local geometry of space but not global, hence leaves the possibility to explore the topology of the space (simply- or multi-connected). By expressing the cosmological model in trms of energy density parameters, we attempt to understand the geometry of spacetime. This is followed by a discussion on the possibility to detect the signatures of topology of space imprinted on the Cosmic Microwave Background (CMB).

The Origins of Chinese Mountain Painting: Evidence from Archaeology

2003 ◽  
Author(s):  
JESSICA RAWSON
Keyword(s):  
Third Century ◽  
The Third ◽  
Major Subject ◽  
Mountainous Landscapes ◽  
The Aesthetic ◽  
The Universe

Mountainous landscapes, with massive crags and narrow fissures between rocks, through which water spouts, are among the principal subjects of paintings in China. This chapter addresses the question, why, in the first place, were these subjects chosen? It focuses on developments made during the Qin (221–207 bc) and Han (206 bc–ad 220) dynasties, from the third century bc onwards. It explores the ways in which the conditions prevailing in the Qin and Han periods moulded some aspects of the later Chinese practice. It is argued that the ways in which the Chinese from the Han period onwards viewed the cosmos determined their choice of mountains as a major subject for painted images. The chapter discusses attitudes to the cosmos and the aesthetic consequences of these views. It considers the whole range of ideas about the universe and not simply with depictions or models of mountains as representing one part of the cosmos.

Topology of the Universe

1987 ◽  
pp. 461-475 ◽  
Author(s):  
L. Z. Fang ◽  
H. J. Mo
Keyword(s):  
Topology Of The Universe ◽  
The Universe
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