Curvature Invariants for Lorentzian Traversable Wormholes
, Abinash Kar 1,2, William Julius 1,2, Matthew Gorban 1,2, Cooper Watson 1,2, MD Ali 1,2, Andrew Baas 1,2, Caleb Elmore 1,2, Bahram Shakerin 1,2, Eric Davis 1,3 and Gerald Cleaver 1,2
Richard Conn Henry
Round 1
Reviewer 1 Report
The authors calculate and plot curvature invariants for three different "wormhole" metrics. I do not believe these have been presented before. However, the manuscript does not seem to me to be ready for publication in a journal like Universe. I have three major concerns, discussed below. If the authors can address these concerns, I would be willing to look at a revised version.
First, the Abstract, Introduction and Conclusion (Sections 1 and 6) contain too much discussion on invariants that is overly vague or casual in nature. There are unnecessary citations to the original literature (e.g., "Christoffel proved that ...") but then an almost complete lack of references to the more recent work that has been done computing, plotting, and discussing applications of curvature invariants in the past 30 or 40 years. A search for "curvature invariants" on google scholar will turn up a number of these papers and references therein, which the authors should integrate into their Introduction while also removing or rewriting unclear or overly informal remarks such as that "Curvature invariants are independent of coordinate basis, so the process is free of coordinate mapping distortions and the same regardless of your chosen coordinates" (Abstract) or "analysis of [Riemann tensor] can be misleading because coordinate mapping distortions may arise as an artifact of the coordinate choice," or "Plotting only curvature invariants ... is the best way to visualize curved spacetime phenomena without distortion" (Introduction). What distortions, exactly, do the authors mean? How and why, exactly, do invariants present a superior approach? If the authors are correct, then why do nearly all textbooks and articles in the field of general relativity concentrate on tensors such as Riemann and Ricci, while saying little to nothing about invariants such as Kretschmann? Sweeping assertions like these must have a clear physical or mathematical basis that is absent here. The Conclusion states that invariants would be useful in practice because "their effect on an object's motion can then be analyzed". What does this mean, and how would it work in practice? The meaning of the statement that "The aforementioned tensors lead to a chosen basis of invariants" is not clear to me. If there are many other invariants available, as noted in the Conclusion, then that would seem to undermine the whole rationale for this article. Why focus on this particular set? Questions like this need to be addressed in a clearer, and probably also much shorter revised edition.
Second, the mathematical level of the presentation (beginning in Section 2) is too elementary. It is not necessary to present separate equations defining quantities such as the Christoffel symbol or Riemann tensor, etc. for the readers of Universe (Eqs. 1,2,…). Similarly, Appendix A (Riemann Curvature Invariants) is unnecessary and should be removed (this material can be found in standard references, which can be cited instead). The main metrics being studied here are fairly simple (Eqs. 7, 13, 15, 16). Would it not be possible simply to list these together and then display the relevant curvature invariants, perhaps in tabular form? This would result in a much more concise paper. On this note, much of the length of the expressions in Appendix B derives from the authors’ practice of explicitly including dependencies for each function (e.g., b(r), b’(r), Phi(r), Phi’(r), etc.). This is completely unnecessary since it is clear from the metrics that all functions depend only on r.
Third, there are far too many almost identical plots in this paper (ten). One or two representative plots, together with some discussion, would surely be sufficient given the relatively straightforward mathematical level of the computations.
Author Response
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Author Response File: 
Author Response.pdf
Reviewer 2 Report
This is a healthy and much-needed exploration of possibilities with respect to black holes, which are ubiquitous in our Galaxy. I look at the history of the slow acceptance of the reality of black holes, not just by my fellow astronomers, but even by physicists; and then I note the solidity of almost everyone's present acceptance of their existence: that history sharply inclines me to support technically sound investigations of further possibilities with respect to black holes, such as this paper.
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
I found the paper to be quite interesting and well-written. A use of curvature invariants in order to study a wormhole geometry is new and potentially interesting. I recommend the paper for publication in the journal "Universe".
Author Response
Please see attachment.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The authors' cuts and revisions have considerably improved this article. The other two reviewers also seem to have no criticisms. Thus, I agree that the article is now suitable for publication in Universe.
