scholarly journals Group Structure and Geometric Interpretation of the Embedded Scator Space

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1504
Author(s):  
Jan L. Cieśliński ◽  
Artur Kobus
Keyword(s):  
Euclidean Space ◽  
Special Relativity ◽  
Group Structure ◽  
Geometric Interpretation ◽  
Commutative Group ◽  
Original Formulation ◽  
Extended Space

The set of scators was introduced by Fernández-Guasti and Zaldívar in the context of special relativity and the deformed Lorentz metric. In this paper, the scator space of dimension 1+n (for n=2 and n=3) is interpreted as an intersection of some quadrics in the pseudo-Euclidean space of dimension 2n with zero signature. The scator product, nondistributive and rather counterintuitive in its original formulation, is represented as a natural commutative product in this extended space. What is more, the set of invertible embedded scators is a commutative group. This group is isomorphic to the group of all symmetries of the embedded scator space, i.e., isometries (in the space of dimension 2n) preserving the scator quadrics.

scholarly journals I: Geometric Interpretation of the Minkowski Metric

2019 ◽  
Author(s):  
Thomas Merz
Keyword(s):  
Euclidean Space ◽  
Special Relativity ◽  
Minkowski Space ◽  
Lie Algebras ◽  
Geometric Interpretation ◽  
Minkowski Metric ◽  
Basis Element ◽  
Direct Mapping

A geometric interpretation of the Minkowski metric and thus of phenomena in special relativity is provided. It is shown that a change of basis in Minkowski space is the equivalent of a change of basis in Euclidean space if one basis element is replaced by its dual element. The methodology of the proof includes infinitesimal changes of basis using the Lie-algebras of the involved groups. As a consequence, a direct mapping between Euclidean and Minkowski space is defined.

scholarly journals I: Geometric Interpretation of the Minkowski Metric

2018 ◽  
Author(s):  
Thomas Merz
Keyword(s):  
Euclidean Space ◽  
Special Relativity ◽  
Minkowski Space ◽  
Lie Algebras ◽  
Geometric Interpretation ◽  
Minkowski Metric ◽  
Basis Element ◽  
Direct Mapping

A geometric interpretation of the Minkowski metric and thus of phenomena in special relativity is provided. It is shown that a change of basis in Minkowski space is the equivalent of a change of basis in Euclidean space if one basis element is replaced by its dual element. The methodology of the proof includes infinitesimal changes of basis using the Lie-algebras of the involved groups. As a consequence, a direct mapping between Euclidean and Minkowski space is defined.

scholarly journals On the Superstability of Lobačevskiǐ’s Functional Equations with Involution

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Jaeyoung Chung ◽  
Bogeun Lee ◽  
Misuk Ha
Keyword(s):  
Functional Equation ◽  
Euclidean Space ◽  
Functional Equations ◽  
Direct Consequence ◽  
Functional Inequality ◽  
Commutative Group ◽  
Bounded Functions

LetGbe a uniquely2-divisible commutative group and letf,g:G→Candσ:G→Gbe an involution. In this paper, generalizing the superstability of Lobačevskiǐ’s functional equation, we considerf(x+σy)/22-g(x)f(y)≤ψ(x)orψ(y)for allx,y∈G, whereψ:G→R+. As a direct consequence, we find a weaker condition for the functionsfsatisfying the Lobačevskiǐ functional inequality to be unbounded, which refines the result of Găvrută and shows the behaviors of bounded functions satisfying the inequality. We also give various examples with explicit involutions on Euclidean space.

Accelerated frames

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Euclidean Space ◽  
Special Relativity ◽  
Reference Frame ◽  
Inertial Frame ◽  
Physical Space ◽  
Minkowski Spacetime ◽  
Curvilinear Coordinates ◽  
Accelerated Frames

This chapter shows how, within the framework of special relativity, Newtonian inertial accelerations turn into mere geometrical quantities. In addition, the chapter states that labeling the points of Minkowski spacetime using curvilinear coordinates rather than Minkowski coordinates is mathematically just as simple as in Euclidean space. However, the interpretation of such a change of coordinates as passage from an inertial frame to an accelerated frame is more subtle. Hence, the chapter studies some examples of this phenomenon. Finally, it addresses the problem of understanding what the curvilinear coordinates actually represent. Or, similarly, it considers the question of how to realize them by a reference frame in actual, ‘relative, apparent, and common’ physical space.

Minkowski spacetime

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Euclidean Space ◽  
Special Relativity ◽  
Three Dimensional ◽  
Minkowski Spacetime ◽  
Dimensional Euclidean Space ◽  
Fourth Dimension ◽  
Relativistic Dynamics ◽  
Newtonian Theory ◽  
Space And Time ◽  
Set Of Points

This chapter presents the main features of the Minkowski spacetime, which is the geometrical framework in which the laws of relativistic dynamics are formulated. It is a very simple mathematical extension of three-dimensional Euclidean space. In special relativity, ‘relative, apparent, and common’ (in the words of Newton) space and time are represented by a mathematical set of points called events, which constitute the Minkowski spacetime. This chapter also stresses the interpretation of the fourth dimension, which in special relativity is time. Here, time now loses the ‘universal’ and ‘absolute’ nature that it had in the Newtonian theory.

scholarly journals Unifying the Galilei Relativity and the Special Relativity

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Alexandre Lyra ◽  
Marcelo Carvalho
Keyword(s):  
Euclidean Space ◽  
Special Relativity ◽  
Lorentz Transformation ◽  
Dimensional Euclidean Space ◽  
Low Velocity ◽  
3 Dimensional ◽  
Absolute Time ◽  
Starting Point ◽  
Relativistic Spacetime ◽  
The Absolute

We present two models combining some aspects of the Galilei and the Special relativities that lead to a unification of both relativities. This unification is founded on a reinterpretation of the absolute time of the Galilei relativity that is considered as a quantity in its own and not as mere reinterpretation of the time of the Special relativity in the limit of low velocity. In the first model, the Galilei relativity plays a prominent role in the sense that the basic kinematical laws of Special relativity, for example, the Lorentz transformation and the velocity law, follow from the corresponding Galilei transformations for the position and velocity. This first model also provides a new way of conceiving the nature of relativistic spacetime where the Lorentz transformation is induced by the Galilei transformation through an embedding of 3-dimensional Euclidean space into hyperplanes of 4-dimensional Euclidean space. This idea provides the starting point for the development of a second model that leads to a generalization of the Lorentz transformation, which includes, as particular cases, the standard Lorentz transformation and transformations that apply to the case of superluminal frames.

scholarly journals One Observer Universe: A Geometric Interpretation of Speed of Light and Mass

2020 ◽  
Author(s):  
Ahmed Farag Ali
Keyword(s):  
General Relativity ◽  
Special Relativity ◽  
Geometric Interpretation ◽  
Space Time ◽  
Red Shift ◽  
General Covariance ◽  
Speed Of Light ◽  
The Universe

In this paper, we investigate how Rindler observer measures the universe in the ADM formalism. We compute his measurements in each slice of the space-time in terms of gravitational red-shift which is a property of general covariance. In this way, we found special relativity preferred frames to match with the general relativity Rindler frame in ADM formalism. This may resolve the widely known incompatibility between special relativity and general relativity on how each theory sees the red-shift. We found a geometric interpretation of the speed of light and mass.

scholarly journals Geometric Interpretation of the Minkowski Metric

2019 ◽  
Author(s):  
Thomas Merz
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Lie Algebras ◽  
Minkowski Spacetime ◽  
Geometric Interpretation ◽  
Basis Set ◽  
Measuring Device ◽  
Minkowski Metric ◽  
Basis Element ◽  
Direct Mapping

A novel geometric interpretation of the Minkowski metric is provided, which offers a different and more intuitive approach to phenomena in special relativity. First it is shown that a change of basis in Minkowski space is the equivalent of a change of basis in Euclidean space if a basis element is replaced by its dual element, constituting a mixed basis set. The methodology of the proof includes infinitesimal changes of basis using the Lie-algebras of the involved groups. As a consequence, a direct mapping between Euclidean and Minkowski space is defined. Second, a measuring device called a local, flat observer is defined in Euclidean space and it is shown, that this device uses a mixed basis when measuring distances. Combining these steps, it is concluded that a local, flat observer in a four-dimensional Euclidean spacetime measures a Minkowski spacetime.

scholarly journals The Space-evolution Frame as Alternative to Space-time

2021 ◽  
Author(s):  
Xiaonan Du
Keyword(s):  
Euclidean Space ◽  
Special Relativity ◽  
World Line ◽  
Minkowski Spacetime ◽  
Line Length ◽  
Dimensional Euclidean Space ◽  
Time Space ◽  
Coordinate Rotation ◽  
Self Consistent ◽  
New Perspective

Abstract As a alternative to Minkowski spacetime frame, this paper propose a four dimensional Euclidean space that combine three spacial dimension with evolution instead of time. It is called space-evolution, in which time is considered as world line length and is absolute. The space-evolution frame provide a new perspective for understanding of time, space and special relativity. It is self-consistent without losing compatibility to special relativity, the Lorentz transform and its predictions could be derived geometrically by simple coordinate rotation.

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