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Noether and Space-Time Symmetries in Physics

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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: closed (28 February 2023) | Viewed by 21607

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Special Issue Editors

Prof. Dr. Ugur Camci

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Guest Editor

Department of Chemistry and Physics, Roger Williams University, One Old Ferry Road, Bristol, RI 02809, USA
Interests: cosmology; mathematical physics; general relativity; modified gravity; symmetries; collineations
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Bobomurat Ahmedov

E-Mail Website
Guest Editor

Laboratory of Theoretical Astrophysics, Ulugh Beg Astronomical Institute, Tashkent 100052, Uzbekistan
Interests: relativistic astrophysics; general relativity; neutron stars and black holes
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Ashfaque H. Bokhari

E-Mail Website
Guest Editor

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
Interests: general relativity; Lorentzian geometry; symmetry methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Symmetry is the most common and important principle which guides to construct realistic theories in science. The notion of symmetry is fundamental not only in cosmological theories but also in quantum theory, thermodynamics, statistical physics, etc.

Most of the equations for dynamical systems in physics, such as the field equations of any gravity theory, are a system of non-linear ordinary/partial differential equations and are generally difficult to solve. In order to solve these complicated systems of ordinary/partial differential equations, Noether and space–time symmetries are some of the tools which can be used to find their exact solutions. Symmetries of Lagrangians are of great interest on account of Noether’s theorem, which has been widely used in cosmology and gravity theories. Space–time symmetries such as isometries and collineations can reduce the number of unknown functions in space–time metric components. The main aim of this Special Edition is to invite researchers working in theoretical and mathematical physics to submit their work, in which Noether and the space–time symmetry approach are used to find exact solutions for the corresponding equations representing certain dynamical systems.

In this Special Issue, original research articles, reviews, communications and concept papers are welcome. We look forward to receiving your contributions.

Prof. Dr. Ugur Camci
Prof. Dr. Bobomurat Ahmedov
Prof. Dr. Ashfaque H. Bokhari
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

Noether symmetry
Space–time symmetry
Exact solutions
Cosmology
Gravity
Modified gravity
Conserved quantity
Astrophysics solutions

Published Papers (14 papers)

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Editorial

Jump to: Research, Review

3 pages, 160 KiB

Open AccessEditorial

Noether and Space-Time Symmetries in Physics

by Ugur Camci, Bobomurat Ahmedov and Ashfaque H. Bokhari

Symmetry 2023, 15(4), 933; https://doi.org/10.3390/sym15040933 - 18 Apr 2023

Viewed by 1029

Abstract

Symmetry is the most common and important principle of those which guide efforts to construct realistic theories in science [...] Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

Research

Jump to: Editorial, Review

13 pages, 315 KiB

Open AccessArticle

Killing Vector Fields of Static Cylindrically Symmetric Spacetime—A Rif Tree Approach

by Maryam Khalid Albuhayr, Ashfaque H. Bokhari and Tahir Hussain

Symmetry 2023, 15(5), 1111; https://doi.org/10.3390/sym15051111 - 19 May 2023

Viewed by 985

Abstract

In this paper, we use an algebraic approach to classify cylindrically symmetric static spacetimes according to their killing vector fields. This approach is based on using a maple algorithm to re-cast the Killing’s equations into a reduced involutive form and integrating the Killing’s equations subject to the constraints given by the algorithm. It is shown that this approach provides some additional spacetime metrics, which were not provided previously by solving the Killing’s equations using a direct integration technique. To discuss some physical implications of the obtained spacetime metrics, we use them in the Einstein equations and discuss their significance. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

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31 pages, 482 KiB

Open AccessFeature PaperArticle

Monstrous M-Theory

by Alessio Marrani, Mike Rios and David Chester

Symmetry 2023, 15(2), 490; https://doi.org/10.3390/sym15020490 - 13 Feb 2023

Cited by 1 | Viewed by 4574

Abstract

26 + 1

space–time dimensions, we introduce a gravity theory whose massless spectrum can be acted upon by the Monster group when reduced to

25 + 1

dimensions. This theory generalizes M-theory in many respects, and we name it Monstrous M-theory, or [...] Read more.

26 + 1

space–time dimensions, we introduce a gravity theory whose massless spectrum can be acted upon by the Monster group when reduced to

25 + 1

dimensions. This theory generalizes M-theory in many respects, and we name it Monstrous M-theory, or M

^{2}

-theory. Upon Kaluza–Klein reduction to

25 + 1

dimensions, the M

^{2}

-theory spectrum irreducibly splits as 1 ⊕ 196,883, where 1 is identified with the dilaton, and 196,883 is the dimension of the smallest non-trivial representation of the Monster. This provides a field theory explanation of the lowest instance of the Monstrous Moonshine, and it clarifies the definition of the Monster as the automorphism group of the Griess algebra by showing that such an algebra is not merely a sum of unrelated spaces, but descends from massless states for M

^{2}

-theory, which includes Horowitz and Susskind’s bosonic M-theory as a subsector. Further evidence is provided by the decomposition of the coefficients of the partition function of Witten’s extremal Monster SCFT in terms of representations of

S O_{24}

, the massless little group in

25 + 1

; the purely bosonic nature of the involved

S O_{24}

-representations may be traced back to the unique feature of 24 dimensions, which allow for a non-trivial generalization of the triality holding in 8 dimensions. Last but not least, a certain subsector of M

^{2}

-theory, when coupled to a Rarita–Schwinger massless field in

26 + 1

, exhibits the same number of bosonic and fermionic degrees of freedom; we cannot help but conjecture the existence of a would-be

N = 1

supergravity theory in

26 + 1

space–time dimensions. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

17 pages, 312 KiB

Open AccessArticle

Classification of the Lie and Noether Symmetries for the Klein–Gordon Equation in Anisotropic Cosmology

by Andronikos Paliathanasis

Symmetry 2023, 15(2), 306; https://doi.org/10.3390/sym15020306 - 22 Jan 2023

Cited by 3 | Viewed by 977

Abstract

We carried out a detailed group classification of the potential in Klein–Gordon equation in anisotropic Riemannian manifolds. Specifically, we consider the Klein–Gordon equations for the four-dimensional anisotropic and homogeneous spacetimes of Bianchi I, Bianchi III and Bianchi V. We derive all the closed-form expressions for the potential function where the equation admits Lie and Noether symmetries. We apply previous results which connect the Lie symmetries of the differential equation with the collineations of the Riemannian space which defines the Laplace operator, and we solve the classification problem in a systematic way. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

22 pages, 410 KiB

Open AccessFeature PaperArticle

Higher-Order First Integrals of Autonomous Non-Riemannian Dynamical Systems

by Antonios Mitsopoulos, Michael Tsamparlis and Aniekan Magnus Ukpong

Symmetry 2023, 15(1), 222; https://doi.org/10.3390/sym15010222 - 12 Jan 2023

Cited by 2 | Viewed by 1171

Abstract

We consider autonomous holonomic dynamical systems defined by equations of the form

{\ddot{q}}^{a} = - Γ_{b c}^{a} (q) {\dot{q}}^{b} {\dot{q}}^{c}

- Q^{a} (q)

, where [...] Read more.

We consider autonomous holonomic dynamical systems defined by equations of the form

{\ddot{q}}^{a} = - Γ_{b c}^{a} (q) {\dot{q}}^{b} {\dot{q}}^{c}

- Q^{a} (q)

, where

Γ_{b c}^{a} (q)

are the coefficients of a symmetric (possibly non-metrical) connection and

- Q^{a} (q)

are the generalized forces. We prove a theorem which for these systems determines autonomous and time-dependent first integrals (FIs) of any order in a systematic way, using the ’symmetries’ of the geometry defined by the dynamical equations. We demonstrate the application of the theorem to compute linear, quadratic, and cubic FIs of various Riemannian and non-Riemannian dynamical systems. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

11 pages, 310 KiB

Open AccessArticle

Noether Symmetries and Conservation Laws in Static Cylindrically Symmetric Spacetimes via Rif Tree Approach

by Muhammad Farhan, Suhad Subhi Aiadi, Tahir Hussain and Nabil Mlaiki

Symmetry 2023, 15(1), 184; https://doi.org/10.3390/sym15010184 - 08 Jan 2023

Cited by 2 | Viewed by 992

Abstract

A new approach is adopted to completely classify the Lagrangian associated with the static cylindrically symmetric spacetime metric via Noether symmetries. The determining equations representing Noether symmetries are analyzed using a Maple algorithm that imposes different conditions on metric coefficients under which static cylindrically symmetric spacetimes admit Noether symmetries of different dimensions. These conditions are used to solve the determining equations, giving the explicit form of vector fields representing Noether symmetries. The obtained Noether symmetry generators are used in Noether’s theorem to find the expressions for corresponding conservation laws. The singularity of the obtained metrics is discussed by finding their Kretschmann scalar. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

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18 pages, 374 KiB

Open AccessArticle

Noether Symmetries and Conservation Laws in Non-Static Plane Symmetric Spacetime

by Muhammad Farhan, Tahir Hussain, Fatima Azmi and Nabil Mlaiki

Symmetry 2022, 14(10), 2174; https://doi.org/10.3390/sym14102174 - 17 Oct 2022

Cited by 2 | Viewed by 981

Abstract

In this paper, we find all nonstatic plane symmetric spacetime metrics whose corresponding Lagrangians possess Noether symmetries. The set of determining equations is analyzed through a Maple algorithm that restricts the metric coefficients to satisfy certain conditions. These restrictions on metric coefficients, while using them to solve the determining equations, give rise to a number of plane symmetric metrics admitting 4-, 5-, 6-, 7-, 8-, 9-, 11-, and 17-dimensional Noether algebras. The Noether theorem is used to find a conserved quantity corresponding to each Noether symmetry. Some physical implications are discussed by finding bounds for different energy conditions for the obtained metrics. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

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28 pages, 435 KiB

Open AccessArticle

Quadratic First Integrals of Constrained Autonomous Conservative Dynamical Systems with Fixed Energy

by Antonios Mitsopoulos and Michael Tsamparlis

Symmetry 2022, 14(9), 1870; https://doi.org/10.3390/sym14091870 - 07 Sep 2022

Cited by 2 | Viewed by 910

Abstract

We consider autonomous conservative dynamical systems which are constrained with the condition that the total energy of the system has a specified value. We prove a theorem which provides the quadratic first integrals (QFIs), time-dependent and autonomous, of these systems in terms of the symmetries (conformal Killing vectors and conformal Killing tensors) of the kinetic metric. It is proved that there are three types of QFIs and for each type we give explicit formulae for their computation. It is also shown that when the autonomous QFIs are considered, then we recover the known results of previous works. For a zero potential function, we have the case of constrained geodesics and obtain formulae to compute their QFIs. The theorem is applied in two cases. In the first case, we determine potentials which admit the second of the three types of QFIs. We recover a superintegrable potential of the Ermakov type and a new integrable potential whose trajectories for zero energy and zero QFI are circles. In the second case, we integrate the constrained geodesic equations for a family of two-dimensional conformally flat metrics. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

12 pages, 302 KiB

Open AccessArticle

Noether Symmetries of Lemaitre-Tolman-Bondi Metric via Rif Tree Approach

by Muhammad Farhan, Tahir Hussain, Nabil Mlaiki and Aiman Mukheimer

Symmetry 2022, 14(9), 1864; https://doi.org/10.3390/sym14091864 - 07 Sep 2022

Cited by 2 | Viewed by 1021

Abstract

In this paper, we have explored Noether symmetries for the Lagrangian corresponding to the Lemaitre-Tolman-Bondi (LTB) spacetime metric via a Rif tree approach. Instead of the frequently used method of directly integrating the Noether symmetry equations, a MAPLE algorithm is used to convert these equations to the reduced involutive form (Rif). The interesting feature of this algorithm is that it provides all possible metrics admitting different dimensional Noether symmetries. These metrics are given in the form of branches of a tree, known as a Rif tree. These metrics are used to solve the determining equations and the explicit form of symmetry vector fields are found, giving 4, 5, 6, 7, 8, 9, 11, and 17-dimensional Noether algebras. To add some physical implications, Einstein’s field equations are used to find the stress-energy tensor for all the explicitly known metrics, and the parameters appearing in the metrics are used to find bounds for different energy conditions. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

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12 pages, 302 KiB

Open AccessArticle

by Andronikos Paliathanasis

Symmetry 2022, 14(8), 1603; https://doi.org/10.3390/sym14081603 - 04 Aug 2022

Cited by 1 | Viewed by 1116

Abstract

We apply the theory of Lie point symmetries for the study of a family of partial differential equations which are integrable by the hyperbolic reductions method and are reduced to members of the Painlevé transcendents. The main results of this study are that from the application of the similarity transformations provided by the Lie point symmetries, all the members of the family of the partial differential equations are reduced to second-order differential equations, which are maximal symmetric and can be linearized. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

19 pages, 325 KiB

Open AccessArticle

Conformal Symmetries of the Energy–Momentum Tensor of Spherically Symmetric Static Spacetimes

by Ugur Camci and Khalid Saifullah

Symmetry 2022, 14(4), 647; https://doi.org/10.3390/sym14040647 - 22 Mar 2022

Cited by 5 | Viewed by 1571

Abstract

Conformal matter collineations of the energy–momentum tensor of a general spherically symmetric static spacetime are studied. The general form of these collineations is found when the energy–momentum tensor is non-degenerate, and the maximum number of independent conformal matter collineations is 15. In the degenerate case of the energy–momentum tensor, it is found that these collineations have infinite degrees of freedom. In some subcases of degenerate energy–momentum, the Ricci tensor is non-degenerate, that is, there exist non-degenerate Ricci inheritance collineations. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

13 pages, 287 KiB

Open AccessArticle

Does the Cosmological Expansion Change Local Dynamics?

by Marcelo Schiffer

Symmetry 2021, 13(8), 1417; https://doi.org/10.3390/sym13081417 - 03 Aug 2021

Cited by 1 | Viewed by 1215

Abstract

It is a well-known fact that the Newtonian description of dynamics within Galaxies for its known matter content is in disagreement with the observations as the acceleration approaches

a_{0} \approx 1.2 \times 10^{- 10}

m/s

^{2}

(slighter larger for clusters). Both [...] Read more.

It is a well-known fact that the Newtonian description of dynamics within Galaxies for its known matter content is in disagreement with the observations as the acceleration approaches

a_{0} \approx 1.2 \times 10^{- 10}

m/s

^{2}

(slighter larger for clusters). Both the Dark Matter scenario and Modified Gravity Theories (MGT) fail to explain the existence of such an acceleration scale. Motivated by the closeness of the acceleration scale and the Hubble constant

c H_{0} \approx 10^{- 9}

h m/s

^{2}

, we are led to analyze whether this coincidence might have a Cosmological origin for scalar-tensor and spinor-tensor theories by performing detailed calculations for perturbations that represent the local matter distribution on the top of the cosmological background. Then, we solve the field equations for these perturbations in a power series in the present value of the Hubble constant. As we shall see, for both theories, the power expansion contains only even powers in the Hubble constant, a fact that renders the cosmological expansion irrelevant for the local dynamics. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

16 pages, 305 KiB

Open AccessArticle

Projective Collineations of Decomposable Spacetimes Generated by the Lie Point Symmetries of Geodesic Equations

by Andronikos Paliathanasis

Symmetry 2021, 13(6), 1018; https://doi.org/10.3390/sym13061018 - 06 Jun 2021

Cited by 3 | Viewed by 1707

Abstract

We investigate the relation of the Lie point symmetries for the geodesic equations with the collineations of decomposable spacetimes. We review previous results in the literature on the Lie point symmetries of the geodesic equations and we follow a previous proposed geometric construction [...] Read more.

(n + 1)

-dimensional decomposable Riemannian space are the Lie point symmetries for geodesic equations of the n-dimensional subspace. We demonstrate the application of our results with the presentation of applications. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

Review

Jump to: Editorial, Research

26 pages, 6147 KiB

Open AccessFeature PaperReview

Observational and Energetic Properties of Astrophysical and Galactic Black Holes

by Bakhtiyor Narzilloev and Bobomurat Ahmedov

Symmetry 2023, 15(2), 293; https://doi.org/10.3390/sym15020293 - 20 Jan 2023

Cited by 7 | Viewed by 1792

Abstract

The work reviews the investigation of electromagnetic, optical, and energetic properties of astrophysical and galactic black holes and surrounding matter. The astrophysical applications of the theoretical models of black hole environment to the description of various observed phenomena, such as cosmic rays of the ultra-high-energy, black hole shadow, gravitational lensing, quasinormal modes, jets showing relativistic effects such as the Doppler beaming, thermal radiation from the accretion discs, quasiperiodic oscillations are discussed. It has been demonstrated that the observational data strongly depends on the structure and evolution of the accretion disk surrounding the central black hole. It has been shown that the simulated images of supermassive black holes obtained are in agreement with the observational images obtained by event horizon telescope collaboration. High energetic activity from supermassive black holes due to the magnetic Penrose process discussed in the work is in agreement with the highly energetic cosmic rays observed. The astronomical observation of black holes provides rich fundamental physics laboratories for experimental tests and verification of various models of black hole accretion and different theories of gravity in the regime of strong gravity. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

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