Special Relativity in Terms of Hyperbolic Functions with Coupled Parameters in 3+1 Dimensions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I reviewed the paper "Special relativity in terms of hyperbolic functions with coupled parameters in 3+1 dimensions" submitted by Akintsov and collaborators to Symmetry. After going through the manuscript, I reached the conclusion that the paper is not suitable for publication as it contributes very little to the existing literature, some of it by the authors themselves (eg, Refs. 21-22).

Some additional comments:

1. Spacetime coordinates do not need to be associated with particle trajectories. However, one can clearly use such a trajectory to build coordinates. The most natural coordinates associated with a point particle (not necessarily free) are the Fermi coordinates. I think it is necessary for the authors to compare their coordinates with Fermi coordinates and to provide a geometrical interpretation of them. Furthermore, I was expecting a Lie group analizis at some point where the group SO(1,3) would be coordinatized in a new and useful way. This should also be presented.

2. As the authors note, hyperbolic functions are widely used in special relativity, and this has been developed for many decades. Very few is added to this and what it is done is only relevant for very particular electromagnetic interactions, which can be solved in many coordinate systems. In this respect, it is unclear what is the advantage of the presented coordinates. What is more, Figure 1 is presents a comparison in which the corresponding initial conditions must be specified to gain some knowledge. Moreover, the electromagnetic potential, in 3+1 dimensions, ought to have a z component, and the approximation where the rapidities of the particle and the electromagnetic rapidity is generically not valid and it should be justified.

3. The actual Lagrangian/Hamiltonian are not presented.

A minor comment: even though there are symbols that we all use in specail relativity, in a paper that is rewriting the usual manners of working with a theory, it is necessary that every mathematical symbol appearing should be clearly defined. For example, some symbols in Eq. 4 (P,E) are not defined.

Given that much of what is relevant in this topic has been presented before, and because the new results presented here are have an extremly narrow set of applications, I stand by my reccomendation of rejecting this paper. I wish the best to the authors in their future endevors.

 

Comments on the Quality of English Language

No comments.

Author Response

Reviewer 1' comments:

 

I would like to thank reviewer#1 for her/his work and valuable comments that have improved the quality of my initial manuscript. I perused the referees’ comments and considered them precisely. The answers to all the comments are given below.

 

Comments and Suggestions for Authors: I reviewed the paper "Special relativity in terms of hyperbolic functions with coupled parameters in 3+1 dimensions" submitted by Akintsov and collaborators to Symmetry. After going through the manuscript, I reached the conclusion that the paper is not suitable for publication as it contributes very little to the existing literature, some of it by the authors themselves (eg, Refs. 21-22).

 

Comment 1: Spacetime coordinates do not need to be associated with particle trajectories. However, one can clearly use such a trajectory to build coordinates. The most natural coordinates associated with a point particle (not necessarily free) are the Fermi coordinates. I think it is necessary for the authors to compare their coordinates with Fermi coordinates and to provide a geometrical interpretation of them. Furthermore, I was expecting a Lie group analizis at some point where the group SO(1,3) would be coordinatized in a new and useful way. This should also be presented.

 

Response 1: Indeed, in Refs. [4, 25, 26] Lorentz space-time coordinates with coupled parameters were obtained. The main advantage of this work is that “new” Lorentz space-time coordinates ,  and  were obtained, Eqs. (78), (86) and (89), which have an invariant form ,  and  with respect to the Lorentz transformations, Eqs. (82), (87) and (90).

We think this paper is relevant, since today there is no literature in which proper coordinates were introduced for local coordinates with coupled parameters relative to the Lorentz transformation, and when differentiating them by rapidity , an invariant form was fulfilled. This work also includes many examples that describe the advantages of using “new” and “old” Lorentz coordinates. The following Sections 12-15, pp 17-25 have been added in the supplement.

On your recommendation, Section 13 was presented, which included a comparison of “old” and “new” Lorentz space-time coordinates with Fermi coordinates. Section 14 has also been added to compare the proper Lorentz groups of SO(1,3) with coupled parameters.

 

Comment 2: As the authors note, hyperbolic functions are widely used in special relativity, and this has been developed for many decades. Very few is added to this and what it is done is only relevant for very particular electromagnetic interactions, which can be solved in many coordinate systems. In this respect, it is unclear what is the advantage of the presented coordinates. What is more, Figure 1 is presents a comparison in which the corresponding initial conditions must be specified to gain some knowledge. Moreover, the electromagnetic potential, in 3+1 dimensions, ought to have a z component, and the approximation where the rapidities of the particle and the electromagnetic rapidity is generically not valid and it should be justified.

 

Response 2: As an example of the applicability of this commentary, Section 12 was presented, which provides a comparison of the kinetic energies of a particle in the field of a plane circularly polarized laser pulse using and comparing “new” and “old” Lorentz space-time coordinates. Also in Section 12, the advantage of using Lorentz space-time coordinates is indicated due to the fact that all parameters are coupled and there is no need to use initial conditions and the Cauchy problem.

In Fig. 3 shows a comparison of particle velocities depending on the dimensionless field amplitude a. To analyze the dynamics of a particle, there is no need to specify the initial conditions, since the initial conditions are  and . When these initial conditions are specified, the particle velocity is zero. Therefore, the work examines the dynamics of a particle depending on dynamically changing parameters, one of which is the dimensionless field amplitude a.

In the examples given, we considered the interaction of a charged particle in an electromagnetic field in a vacuum, in particular, with a flat laser pulse. We assumed that a flat pulse in a vacuum, having only a transverse component of the electromagnetic field, is described by circular polarization.

On page 11, lines 321-329, the connection between the rapidity  of a free relativistic particle and the rapidity  of a particle located in an electromagnetic field was substantiated in more detail.

To prevent readers from getting the idea that the Lorentz-invariant coordinates used are applicable only for very particular electromagnetic interactions, as an example, we have added Section 15, which considers a relativistic hydrodynamic with coupled parameters in 1+1 dimensions for the coordinate . In the future, the authors of this work plan to consider a relativistic hydrodynamic system in 3+1 dimensions based on the Lorentz space-time coordinates obtained in this work.

 

Comment 3: The actual Lagrangian/Hamiltonian are not presented.

A minor comment: even though there are symbols that we all use in specail relativity, in a paper that is rewriting the usual manners of working with a theory, it is necessary that every mathematical symbol appearing should be clearly defined. For example, some symbols in Eq. 4 (P,E) are not defined.

Given that much of what is relevant in this topic has been presented before, and because the new results presented here are have an extremly narrow set of applications, I stand by my reccomendation of rejecting this paper. I wish the best to the authors in their future endevors.

Response 3: Initially, the Hamiltonian was defined in the introduction for a free relativistic particle, formula (7) as a dimensionless Hamiltonian. According to the reviewer’s request, explanations were made under Equation (7) in lines 134-136.

From Hamiltonian (7) in Equations (29), the components of the particle momentum along the gyrovectors  and  were determined with respect to the perpendicular  and angular  rapidities. Regarding the projections of particle momentum, Euler-Hamilton equations in 3+1 dimensions were obtained in Equations (38) and (40).

For a more general understanding, a dimensionless Lagrangian, Equation (49), of the classical relativistic particle from [28] was added, only expressed as a function of the rapidity .

From formula (51) the connection between the Hamiltonian and the Lagrangian of a relativistic particle in the field of a plane wave and in the field of a plane laser pulse was shown. Sections 8-10 analyze motion in 3+1 dimensions.

As can be seen from the presented version of the article, the special theory of relativity with coupled parameters has wide application both to the description of the dynamics of a particle in an electromagnetic field, and to the dynamics of a particle in relativistic hydrodynamic.

We would like you to once again draw your attention to our edited article.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper presents a new parameterization of the Lorentz group. It explores the role of rapidity in Special Relativity. It introduces coordinates, which are named invariant coordinates with respect to the Lorentz transformation. In addition to angular rapidity and perpendicular rapidity is obtained, and others.

The presentation of the results is not clear. I think that the paper must be revised significantly to be ready for publication. It is very difficult to understand from the Abstract what is really obtained in the paper. Also reading the Conclusion also does not help to understand this. No references to the formulas in the text is given.

Below are by remarks to the first Sections. However, the corrections of this type should be done also on the other sections:

1. The pares is on Special relativity. Most people in SR do not use "Lobachevsky space". Probably, they mean "Minkowski space". If so, they should say it at the beginning, and identify the signature (there are two possibilities)

2. Why a direction of motion 𝒏 is not a vector? Why gyrovector? I understand that gyrovectors have only meaning for velocity addition, which is not relevant here.

3. Formula (3). The meaning of 𝑓 is not defined. The coordinate 𝜉 is not Lorentz-invariant. It is the coordinate with respect to a null-vector 𝑎=(1,±1). Lorentz transformations transform this vector to another vector which is also null, but different from 𝑎. Therefore, the coordinates are not invariant. What is the meaning of 𝜃 for the time and position coordinates?

4. "physical processes proceed in the same way in all inertial systems" is the Galilean principle of relativity

5. Title of Section 3 "Action of Lorentz-invariant transformation with respect to rapidity θ" is not clear. What is the meaning of Lorentz-invariant transformation? A scalar can be Lorentz-invariant, but not transformation. What do you mean by the action of a transformation?

6. Your Lorentz-invariant space–time coordinates are the known coordinates with respect to null tetrads. Pease check the connection to them.

7. Formula (11). I do not get the term 𝑑𝑠/𝑑𝜃. What is 𝑠 in this formula?

Comments on the Quality of English Language

Extensive editing of English language is required

Author Response

Reviewer 2' comments:

 

I would like to thank reviewer#2 for her/his work and valuable comments that have improved the quality of my initial manuscript. I perused the referees’ comments and considered them precisely. The answers to all the comments are given below.

Comments and Suggestions for Authors: This paper presents a new parameterization of the Lorentz group. It explores the role of rapidity in Special Relativity. It introduces coordinates, which are named invariant coordinates with respect to the Lorentz transformation. In addition to angular rapidity and perpendicular rapidity is obtained, and others.

The presentation of the results is not clear. I think that the paper must be revised significantly to be ready for publication. It is very difficult to understand from the Abstract what is really obtained in the paper. Also reading the Conclusion also does not help to understand this. No references to the formulas in the text is given.

Below are by remarks to the first Sections. However, the corrections of this type should be done also on the other sections:

Comment 1: The pares is on Special relativity. Most people in SR do not use "Lobachevsky space". Probably, they mean "Minkowski space". If so, they should say it at the beginning, and identify the signature (there are two possibilities)

 

Response 1: At the request of the reviewer, the article was completely revised. The places where changes have been made are highlighted in yellow. The abstract of the article has been completely revised. In the conclusion, references to formulas were placed and part of the conclusion was significantly revised.

 

To comment 1 in the article on pp. 2-3 the following lines 80-87 have been added.

 

Comment 2: Why a direction of motion ? is not a vector? Why gyrovector? I understand that gyrovectors have only meaning for velocity addition, which is not relevant here.

 

Response 2: The concept of gyrovector is understood as hyperbolic vectors in Lobachevsky geometry, which are characterized by the selected direction ,  or , and which take into account the negative curvature and radius of curvature of space.

Yes, indeed, the basis gyrovectors ,  or , can be used to add and subtract velocities with respect to the rapidity  along the selected directions  , , and , but we do not consider this case in the article.

 

Comment 3: Formula (3). The meaning of ? is not defined. The coordinate ? is not Lorentz-invariant. It is the coordinate with respect to a null-vector ?=(1,±1). Lorentz transformations transform this vector to another vector which is also null, but different from ?. Therefore, the coordinates are not invariant. What is the meaning of ? for the time and position coordinates?

 

Response 3: 1. The answer to this question under Equation (6) p. 3 the following lines 128-131 have been added.

2. Indeed, the coordinates and themselves are space-time coordinates and are not invariant, however, when they are multiplied in the inertial systems  and , an invariant is formed, formula (12),

               .                                                               (12)

3. The coordinates of time and space are expressed in terms of rapidity . When taking the derivative with respect to  from  and  we have

, ,

where the relations between derivatives have the form

.

This means that the proper coordinates  and  are coupled through rapidity . Similarly, the connection between other coordinates can be expressed as the connection between each other through rapidity .

 

Comment 4:  "physical processes proceed in the same way in all inertial systems" is the Galilean principle of relativity

 

Response 4: For this commentary we have reformulated the first postulate of the special theory of relativity on p. 4, lines 171-173.

 

Comment 5: Title of Section 3 "Action of Lorentz-invariant transformation with respect to rapidity θ" is not clear. What is the meaning of Lorentz-invariant transformation? A scalar can be Lorentz-invariant, but not transformation. What do you mean by the action of a transformation?

 

Response 5: Indeed, the previous title of Section 3 led to some confusion and was replaced, p. 5 lines 177-178.

 

Comment 6: Your Lorentz-invariant space–time coordinates are the known coordinates with respect to null tetrads. Pease check the connection to them.

 

Response 6: The concept of complex null tetrads is used in Riemannian geometry. In our representation, we use Lobachevsky geometry, where the following conditions are imposed on the rapidity ?. The rapidity must be real and positive. Therefore, in our opinion, there is no need to complicate the work by introducing the definition of null tetrads, since the representation of space-time coordinates with respect to the Lorentz transformation has the following form, for example for coordinates  and , where we have the following invariant form

.

In addition. In fact, special Lorentz transformations (Lorentz boost) regarding null tetrads have the form

, .

The presented coordinates  and  in Riemann geometry have a complex argument. In the case of Lobachevsky Geometry, local space-time coordinates  and  depend on the rapidity ? which is a real and positive quantity.

 

Comment 7:  Formula (11). I do not get the term ??/??. What is ? in this formula?

 

Response 7: In Equation (15), ? is the space-time interval. ??/?? – this is the derivative of the space-time invariant ? with respect to rapidity ?. In the article for clarification on p. 5 the following explanations for line 187-193 have been added. Also on p. 6, for comparison  and  depending on the rapidity ?, Fig. 1, which explains that for real and positive ? the values of  and  are also equal to some approximation.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Author Response

Reviewer 3' comments:

 

I am very much thankful to the reviewer#3 for her/his positive, deep and thorough review. The answers to all the comments are given below.

 

Comments and Suggestions for Authors: The manuscript "Special relativity in terms of hyperbolic functions with coupled parameters in 3+1 demensions" investigates a new method to parametrize the Lorentz group based on coupled parameters by considering the role of symmetry in special relativity regarding rapidities, the authors also explore the new method to evaluate some aspects of Physics.

In order to improve the manuscript some regarded comments might be considered:

 

Comment 1: The first letter of each word in the title should be capitalized.

 

Response 1: Thanks for the note - it's been fixed.

 

Comment 2: There is a serious problem with understanding the manuscript, following the steps, and deriving the equations, in this regard the following matters are recommended

1. The English of the manuscripts should be improved more comprehensively, also some grammatical issues should be fixed.
2. Referring to any specific equation and figure in the text should be in the form of “Eq. ()” and “Fig. ()” not only the number of equations in a single paranthesis.

iii. In some parts the derivations of the equations are not clear.

1. It would be much better if the manuscript were arranged independently from Ref. [22], clearly there are relations between them but the manuscript should be understandable individually.

 

Response 2: To answer this question, we added new lines 38-62 to the introduction. In the text, the conclusion of all formulas was double-checked. In the formulas where it seemed to us that it would be difficult for the reader to reproduce the results obtained, Fig. 1 and Fig. 2 were additionally constructed. In those places where, in our opinion, it was difficult for the reader to understand, for example, formulas (41) and (45) in the previous version of this manuscript were combined into one formula (46) of the new version of the article. Comments and explanations were also added to formula (46) lines 288-303. The conclusion contained all the main results with references to formulas and figures. All other comments were taken into account and corrected in the manuscript, highlighted in yellow. The content of article is change 81%. The word count increase is 79%.

 

Comment 3: Please recheck the equations such as Eq. (8) which does not fit with the result in Eq. (9). Seems the Eq. (8) should be fixed as

 

Response 3: Thanks for the note - it's been fixed.

 

Comment 4: It would be useful if the authors discuss the laws of conservation of energy and momentum such what they did in the Ref. [22].

 

Response 4: Sections 12 and 15 have been added to this article as a response to this comment.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The paper deals with a specific representation of the particle dynamics wirthin the special relativity framework, that is, the hyperbolic functions representation. This description is well-known to be very convenient within studies of various papers related to particle dynamics.

In the paper, the authors give an introduction to hyperbolic function framework, apply this approach within the Hamiltonian formalism,  and applly their formalism to dynamics of a relativistic particles in field of circularly polarized electromagnetic wave. It is important to note that this problem is usually considered as a more complicated one in comparision with the movement of particles in homogeneous fields. Further, an experimentally testable application of the study is given, that is, description of motion of particle in a field of plane laser pulse. The oscillations of particles are studied in details.

By my opinion, this paper is very interesting, contributing not only to theoretical sytudies of particles in electromagnetic fields, but also in studies with technical applications, and being very useful from the didatic viewpoint. So, by my opinion, the paper can be published.

Author Response

Reviewer 4' comments:

 

I am very much thankful to the reviewer#4 for her/his positive, deep and thorough review. The answers to all the comments are given below.

 

Comments and Suggestions for Authors: The paper deals with a specific representation of the particle dynamics wirthin the special relativity framework, that is, the hyperbolic functions representation. This description is well-known to be very convenient within studies of various papers related to particle dynamics.

Comment: In the paper, the authors give an introduction to hyperbolic function framework, apply this approach within the Hamiltonian formalism,  and applly their formalism to dynamics of a relativistic particles in field of circularly polarized electromagnetic wave. It is important to note that this problem is usually considered as a more complicated one in comparision with the movement of particles in homogeneous fields. Further, an experimentally testable application of the study is given, that is, description of motion of particle in a field of plane laser pulse. The oscillations of particles are studied in details.

By my opinion, this paper is very interesting, contributing not only to theoretical sytudies of particles in electromagnetic fields, but also in studies with technical applications, and being very useful from the didatic viewpoint. So, by my opinion, the paper can be published.

Response: Thank you for your valuable comment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I read the new version of the paper. It has addressed most of my comments (I am still eager to see a GEOMETRICAL construction of the coordinates!). In general, given that coordinates have no physical meaning, I am hesitant to recommend a paper about coordinates being accepted. However, given that the authors have done a huge effort to clarify the potential role of their coordinates in some physical problems, I would recommend this paper for publication. 

Comments on the Quality of English Language

English is understandable!

Reviewer 2 Report

Comments and Suggestions for Authors

After the revision, the paper is ready for publikation

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript is now appropriate to be accepted by the Journal.