Minkowskian Approach to the Pseudorange Navigation Equations

Round 1

Reviewer 1 Report

This paper is a theoretical study, which studies how to use the Relativity methods to reformulate Bancroft’s solution. The way the authors used are common in Relativity, such as the contravariant vectors, covariant vectors and etc. However, I think the idea of using Relativity methods to recover Bancroft’s solution is new. The authors used the figures and tables to illustrate their results, which I did not see any problems. Although I am not an expert in navigation systems, the methods the authors used in Relativity are in my research fields. I am pleased to see that this method can be used in real applications in navigations. Therefore, I think this paper is interesting and deserved publication in the journal of Universe. 

Author Response

Thank you for your review and the positive feedback. Please see the cover letter on the changes made to the article as suggested by some of the reviewers and the editor.

Author Response File: Author Response.pdf

Reviewer 2 Report

In the manuscript “Minkowskian approach to the pseudorange navigation equations”  the authors address  the question of relating the Relativistic  Positioning Systems (RPS) associated with the Global Navigation Satellite Systems (GNSS) to Bancroft’s classical solution. Indeed  the authors derive Bancroft’s solution from the RPS, resorting to the relativistic  formalism. They show how one bridges the latter to the former classical solution in an elegant and  revealing way. 

The manuscript is well written and well organised, so that I am of the opinion that it adds a significantly useful contribution to previous works that have envisaged  the issue under investigation in the literature.

I therefore recommend its publication in the journal Universe, and I have no changes to suggest. 

Author Response

Thank you for your review and the positive feedback. Please see the cover letter on the changes made to the article as suggested by some of the reviewers and the editor.

Author Response File: Author Response.pdf

Reviewer 3 Report

The manuscript presents a clear introduction to the concept of the Relativistic Positioning System (RPS) in relation to Global Navigation Satellite Systems (GNSS). It includes a precise description of RPS operation, as well as the null propagation equations, which determine the user's emission coordinates. It also explores the connection of RPS to traditional satellite navigation systems, referring to GNSS throughout the text, and subsequently proposes an approach to unify the concepts of these two methods.

The manuscript is well-structured, covering several aspects of RPS and its relation to existing satellite navigation systems. The references are suitable for the subject discussed.

The conclusion is consistent with the content, highlighting the relation between the solution proposed by Bancroft and the RPS solution, and the validity of the RPS approach to distinct emission configurations.

There is a typo just after Eq. (15); "prodcedure" should read "procedure".

In my opinion, the article is suitable for publication as it is.

There is a typo just after Eq. (15); "prodcedure" should read "procedure".

Author Response

Thank you for your review and the positive feedback. Please see the cover letter on the changes made to the article as suggested by the reviewers and the editor.

Author Response File: Author Response.pdf