scholarly journals Affine Symmetry Tensors in Minkowski Space

2016 ◽  
Vol 13 (3) ◽  
Author(s):  
Isaac Ahern ◽  
Sam Cook
Keyword(s):  
Minkowski Space ◽  
Computational Algorithm ◽  
Minkowski Spacetime ◽  
Lie Derivative ◽  
Killing Tensor ◽  
Spacetime Dimension ◽  
Killing Vectors ◽  
Killing Tensors ◽  
Affine Symmetry

Killing vectors are generators of symmetries in a spacetime. This article defines certain generalizations of Killing vectors, called affine symmetry tensors, or simply affine tensors. While the affine vectors of the Minkowski spacetime are well known, and partial results for valence n = 2 have been discussed, affine tensors of valence n > 2 have never been exhibited. In this article, we discuss a computational algorithm to compute affine tensors in Minkowski spacetime, and discuss the results for affine tensors of valence 2 ≤ n ≤ 7. After comparison with analogous results concerning Killing tensors, we make several conjectures about the spaces of affine tensors in Minkowski spacetime. KEYWORDS: Affine Symmetry Tensors; Affine Vectors; Killing Tensors; Killing Vectors; Minkowski Spacetime; Dimension; Maple CAS; Lie Derivative; Generalized Killing Tensor

scholarly journals Complete set of quasi-conserved quantities for spinning particles around Kerr

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Geoffrey Compère ◽  
Adrien Druart
Keyword(s):  
Linear Order ◽  
Conserved Quantities ◽  
Killing Tensor ◽  
Kerr Geometry ◽  
Spinning Particles ◽  
Kerr Spacetime ◽  
Killing Tensors ◽  
Mixed Symmetry ◽  
Constants Of Motion ◽  
Complete Set

We revisit the conserved quantities of the Mathisson-Papapetrou-Tulczyjew equations describing the motion of spinning particles on a fixed background. Assuming Ricci-flatness and the existence of a Killing-Yano tensor, we demonstrate that besides the two non-trivial quasi-conserved quantities, i.e. conserved at linear order in the spin, found by Rüdiger, non-trivial quasi-conserved quantities are in one-to-one correspondence with non-trivial mixed-symmetry Killing tensors. We prove that no such stationary and axisymmetric mixed-symmetry Killing tensor exists on the Kerr geometry. We discuss the implications for the motion of spinning particles on Kerr spacetime where the quasi-constants of motion are shown not to be in complete involution.

scholarly journals The unitary representations of the Poincar\'e group in any spacetime dimension

2021 ◽  
Author(s):  
Xavier Bekaert ◽  
Nicolas Boulanger
Keyword(s):  
Arbitrary Dimension ◽  
Minkowski Spacetime ◽  
Irreducible Representations ◽  
Theoretical Treatment ◽  
Spacetime Dimension ◽  
Field Equations ◽  
Negative Mass ◽  
Relativistic Field ◽  
Fundamental Particles ◽  
Covariant Field

An extensive group-theoretical treatment of linear relativistic field equations on Minkowski spacetime of arbitrary dimension D\geqslant 3D≥3 is presented. An exhaustive treatment is performed of the two most important classes of unitary irreducible representations of the Poincar'e group, corresponding to massive and massless fundamental particles. Covariant field equations are given for each unitary irreducible representation of the Poincar'e group with non-negative mass-squared.

HIGHER ORDER SYMMETRIES AND THE KOUTRAS ALGORITHM

2002 ◽  
Vol 11 (03) ◽  
pp. 337-351 ◽  
Author(s):  
G. AMERY ◽  
S. D. MAHARAJ
Keyword(s):  
Conformal Factor ◽  
Higher Order ◽  
Conformal Killing ◽  
Killing Vectors ◽  
Killing Tensors ◽  
Conformal Killing Vectors

We investigate the form of Killing tensors, constructed from conformal Killing vectors of a given spacetime (M, g), by utilizing the Koutras algorithm. As an example we find irreducible Killing tensors in Robertson–Walker spacetimes. A number of theorems are given for the existence of Killing tensors in the conformally related spacetime [Formula: see text]. The form of the conformally related Killing tensors are explicitly determined. The conditions on the conformal factor Ω relating the two spacetimes (M, g) and [Formula: see text] are determined for the existence of the tensors. Also we briefly consider the role of recurrent vectors, inheriting conformal vectors and gradient conformal vectors in building Killing tensors.

scholarly journals CONFORMAL STRUCTURES AND TWISTORS IN THE PARAVECTOR MODEL OF SPACETIME

2007 ◽  
Vol 04 (04) ◽  
pp. 547-576 ◽  
Author(s):  
R. DA ROCHA ◽  
J. VAZ
Keyword(s):  
Clifford Algebra ◽  
Clifford Algebras ◽  
Isometry Group ◽  
Twistor Space ◽  
Minkowski Spacetime ◽  
Adjoint Representation ◽  
Inner Product ◽  
Spacetime Dimension ◽  
Conformal Maps ◽  
Spacetime Structure

Some properties of the Clifford algebras [Formula: see text] and [Formula: see text] are presented, and three isomorphisms between the Dirac–Clifford algebra [Formula: see text] and [Formula: see text] are exhibited, in order to construct conformal maps and twistors, using the paravector model of spacetime. The isomorphism between the twistor space inner product isometry group SU(2,2) and the group $pin+(2,4) is also investigated, in the light of a suitable isomorphism between [Formula: see text] and [Formula: see text]. After reviewing the conformal spacetime structure, conformal maps are described in Minkowski spacetime as the twisted adjoint representation of $pin+(2,4), acting on paravectors. Twistors are then presented via the paravector model of Clifford algebras and related to conformal maps in the Clifford algebra over the Lorentzian ℝ4,1 spacetime. We construct twistors in Minkowski spacetime as algebraic spinors associated with the Dirac–Clifford algebra [Formula: see text] using one lower spacetime dimension than standard Clifford algebra formulations, since for this purpose, the Clifford algebra over ℝ4,1 is also used to describe conformal maps, instead of ℝ2,4. Our formalism sheds some new light on the use of the paravector model and generalizations.

SPACE–TIME STRUCTURE IN THE ULTIMATE THEORY

2011 ◽  
Vol 20 (02) ◽  
pp. 253-267 ◽  
Author(s):  
NOBORU NAKANISHI
Keyword(s):  
Quantum Gravity ◽  
Minkowski Space ◽  
Particle Physics ◽  
Space Time ◽  
Time Structure ◽  
Spontaneous Breakdown ◽  
Space Time Structure ◽  
Affine Symmetry ◽  
Poincaré Symmetry

After criticizing the various existing attempts at extending the concept of the Minkowski space, the following problem is considered: If there is the ultimate theory at all, how should the space–time in it be formulated? The principle of "quantum priority" is proposed, and under this principle, it is argued that the ultimate theory should have the affine symmetry in the framework of quantum gravity. It is shown that the Poincaré symmetry of particle physics is realized as a result of spontaneous breakdown of the affine symmetry.

scholarly journals How Does Spacetime “Tell an Electron How to Move”?

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2283
Author(s):  
Garnet Ord
Keyword(s):  
Special Relativity ◽  
Minkowski Space ◽  
Continuum Limit ◽  
Minkowski Spacetime ◽  
Discrete Space ◽  
Kinematics And Dynamics ◽  
Uniform Motion ◽  
Classical Particles

Minkowski spacetime provides a background framework for the kinematics and dynamics of classical particles. How the framework implements the motion of matter is not specified within special relativity. In this paper we specify how Minkowski space can implement motion in such a way that ’quantum’ propagation occurs on appropriate scales. This is done by starting in a discrete space and explicitly taking a continuum limit. The argument is direct and illuminates the special tension between ’rest’ and ’uniform motion’ found in Minkowski space, showing how the formal analytic continuations involved in Minkowski space and quantum propagation arise from the same source.

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 On the Nordtvedt effect in Minkowski spacetime with nonlinear connection

2017 ◽  
Vol 6 (4) ◽  
pp. 130
Author(s):  
Kostadin Trenčevski ◽  
Emilija Celakoska
Keyword(s):  
Minkowski Space ◽  
Minkowski Spacetime ◽  
Gravitational Theory ◽  
Chemical Compositions ◽  
Lunar Laser Ranging ◽  
Nonlinear Connection ◽  
The Earth ◽  
Gravitational Theories ◽  
Lunar Laser ◽  
The Mathematical Model

The Lunar Laser Ranging (LLR) experiment provided precise data which brought the possibility to make more stringent conclusions for the foundations of gravitational theories, i.e. the Equivalence Principles. Beside some effects of non - gravitational origin, the LLR data was fitted with the well-known gravitational effects such as the apsidal and geodetic precessions, the time delay, etc. The Nordtvedt effect in General Relativity (GR) vanishes, while the LLR experiment data of the Earth-Moon distance and the laboratory experiments with experimental bodies made of different chemical compositions measured a variation of distance in millimeters. According to the mathematical model of gravitation in Minkowski space endowed with a nonlinear connection we obtained a result closer to the experimental measurements. More precisely, we obtained a difference of 0.17 mm (or 0.28 mm, depending on the value of the scaling factor) from the LLR measurements of the variation of the Earth-Moon distance, while the corresponding result in GR makes a difference from the LLR measurements of 5.7 mm. The gravitational theory with nonlinear connection in Minkowski space gives the same results for the confirmed GR effects, nevertheless it yields some additional variations of the distance concerning the Nordtvedt effect.

scholarly journals Killing tensors and conformal Killing tensors from conformal Killing vectors

2003 ◽  
Vol 20 (11) ◽  
pp. 1929-1942 ◽  
Author(s):  
Raffaele Rani ◽  
S Brian Edgar ◽  
Alan Barnes
Keyword(s):  
Conformal Killing ◽  
Killing Vectors ◽  
Killing Tensors ◽  
Conformal Killing Vectors
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