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 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 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.

On the total mean curvature of a compact space-like submanifold in Lorentz–Minkowski spacetime

2017 ◽  
Vol 148 (1) ◽  
pp. 199-210 ◽  
Author(s):  
Oscar Palmas ◽  
Francisco J. Palomo ◽  
Alfonso Romero
Keyword(s):  
Euclidean Space ◽  
Compact Space ◽  
Upper Bound ◽  
Mean Curvature ◽  
Light Cone ◽  
Minkowski Spacetime ◽  
Totally Umbilical ◽  
Total Mean Curvature ◽  
Lower Estimation ◽  
Compact Submanifold

By means of several counterexamples, the impossibility to obtain an analogue of the Chen lower estimation for the total mean curvature of any compact submanifold in Euclidean space for the case of compact space-like submanifolds in Lorentz–Minkowski spacetime is shown. However, a lower estimation for the total mean curvature of a four-dimensional compact space-like submanifold that factors through the light cone of six-dimensional Lorentz–Minkowski spacetime is proved by using a technique completely different from Chen's original one. Moreover, the equality characterizes the totally umbilical four-dimensional round spheres in Lorentz–Minkowski spacetime. Finally, three applications are given. Among them, an extrinsic upper bound for the first non-trivial eigenvalue of the Laplacian of the induced metric on a four-dimensional compact space-like submanifold that factors through the light cone is proved.

Harmonic evolutes of B-scrolls with constant mean curvature in Lorentz–Minkowski space

2019 ◽  
Vol 16 (05) ◽  
pp. 1950076 ◽  
Author(s):  
Rafael López ◽  
Željka Milin Šipuš ◽  
Ljiljana Primorac Gajčić ◽  
Ivana Protrka
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Ruled Surfaces ◽  
Null Curve ◽  
Base Curve

In this paper, we study harmonic evolutes of [Formula: see text]-scrolls, that is, of ruled surfaces in Lorentz–Minkowski space having no Euclidean counterparts. Contrary to Euclidean space where harmonic evolutes of surfaces are surfaces again, harmonic evolutes of [Formula: see text]-scrolls turn out to be curves. In particular, we show that the harmonic evolute of a [Formula: see text]-scroll of constant mean curvature together with its base curve forms a null Bertrand pair. This allows us to characterize [Formula: see text]-scrolls of constant mean curvature and reconstruct them from a given null curve which is their harmonic evolute.

Harmonic curvatures of the strip in Minkowski space

2018 ◽  
Vol 11 (04) ◽  
pp. 1850061
Author(s):  
Filiz Ertem Kaya ◽  
Ayşe Yavuz
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Strip Theory ◽  
First Time ◽  
Cylindrical Helix

This study aimed to give definitions and relations between strip theory and harmonic curvatures of the strip in Minkowski space. Previously, the same was done in Euclidean Space (see [F. Ertem Kaya, Y. Yayli and H. H. Hacısalihoglu, A characterization of cylindrical helix strip, Commun. Fac. Sci. Univ. Ank. Ser. A1 59(2) (2010) 37–51]). The present paper gives for the first time a generic characterization of the harmonic curvatures of the strip, helix strip and inclined strip in Minkowski space.

scholarly journals Testing the surrogate zeta-function method

1986 ◽  
Vol 64 (5) ◽  
pp. 633-636 ◽  
Author(s):  
Alan Chodos ◽  
Eric Myers
Keyword(s):  
Euclidean Space ◽  
Zeta Function ◽  
Minkowski Space ◽  
Function Method ◽  
Casimir Energy ◽  
Kaluza Klein

Use of the surrogate zeta-function method was crucial in calculating the Casimir energy in non-Abelian Kaluza–Klein theories. We establish the validity of this method for the case where the background metric is (Euclidean space) × (N sphere). Our techniques do not apply to the case where the background is (Minkowski space) × (N sphere).

scholarly journals A New Angular Measurement in Minkowski 3-Space

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 56 ◽  
Author(s):  
Jinhua Qian ◽  
Xueqian Tian ◽  
Jie Liu ◽  
Young Ho Kim
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Angular Measurement ◽  
Frenet Frame ◽  
Measuring Methods ◽  
Null Curve

In Lorentz–Minkowski space, the angles between any two non-null vectors have been defined in the sense of the angles in Euclidean space. In this work, the angles relating to lightlike vectors are characterized by the Frenet frame of a pseudo null curve and the angles between any two non-null vectors in Minkowski 3-space. Meanwhile, the explicit measuring methods are demonstrated through several examples.

scholarly journals The Dirichlet Problem of the Constant Mean Curvature in Equation in Lorentz-Minkowski Space and in Euclidean Space

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1211 ◽  
Author(s):  
Rafael López
Keyword(s):  
Dirichlet Problem ◽  
Euclidean Space ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Elliptic Equations ◽  
Constant Mean Curvature ◽  
Euclidean Case ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
The Mean

We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the spacelike condition in the Lorentz-Minkowski space allows dropping the hypothesis on the mean convexity, which is required in the Euclidean case.

scholarly journals QCD Factorization and PDFs from Lattice QCD Calculation

2015 ◽  
Vol 37 ◽  
pp. 1560041 ◽  
Author(s):  
Yan-Qing Ma ◽  
Jian-Wei Qiu
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Lattice Qcd ◽  
Correlation Functions ◽  
Parton Distribution ◽  
Distribution Functions ◽  
Matrix Elements ◽  
Parton Distribution Functions ◽  
Qcd Factorization ◽  
The Matrix

In this talk, we review a QCD factorization based approach to extract parton distribution and correlation functions from lattice QCD calculation of single hadron matrix elements of quark-gluon operators. We argue that although the lattice QCD calculations are done in the Euclidean space, the nonperturbative collinear behavior of the matrix elements are the same as that in the Minkowski space, and could be systematically factorized into parton distribution functions with infrared safe matching coefficients. The matching coefficients can be calculated perturbatively by applying the factorization formalism on to asymptotic partonic states.

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