The Finsler Spacetime Condition for (α,β)-Metrics and Their Isometries
, Annamária Friedl-Szász 1,†, Elena Popovici-Popescu 1,† and Christian Pfeifer 2
Zhongmin Shen
Round 1
Reviewer 1 Report
In this paper under review, the authors propose an (alpha,beta)-metric which defines Finler spacetime. Definition 1 is meaningful. In particular, I like the following assumption:
At each x ∈ M, the cone T_x of L and the future pointing timelike cone T^a_x of the metric a have a non-empty intersection.
The authors have done some fundamental/original work. I suggest to accept this paper for publication.
Some suggestions:
Various notions of curvatures have been discussed in the following book.
Z. Shen, Differential Geometry of Spray and Finsler spaces, Kluwer Academic Publishers, 2001.
It will be an important problem to construct some non-trivial Ricci-flat Finsler spacetimes with some vanishing non-Riemannian quantities. Some interesting Ricci-flat Finsler spacetimes have been constructed although they are not in an (alpha, beta)-form.
P. Marcal & Z. Shen, Ricci flat Finsler metrics by warped product, Proceedings of the American Mathematical Society, 2023, DOI: 10.1090/proc/16217.
Author Response
- Please see the attachment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The manuscript under review concerns Finsler spacetimes and focus on the so-called (α, β) type metrics. This constitutes an important class of Finsler spacetime metric that includes several examples in literature (these are reviewed and examined in Section 4).
The authors' aim is to give precise conditions ensuring that an (α, β)-metric has fundamental tensor of signature (+---) in the cone bundle of the Finsler timelike directions (Theorem 6). They also study Killing vector fields of these metrics. In particular, they give an interesting characterization about (α, β) - metrics that can admit a Killing vector field that is not Killing for the underlying Lorentzian metric and the one form b (Theorem 11).
In my opinion, the manuscript is interesting; it might become an important reference for other studies concerning Finsler spacetimes. I recommend publication on Universe after some revisions.
Some minor points to check are the following:
- rows 42-43: please notice that Randers metric come into play in the descriptions of light rays in stationary spacetime (for the static ones Riemannian metrics are enough), see E. Caponio, M. A. Javaloyes, A. Masiello,
``On the energy functional on Finsler manifolds and applications to stationary spacetimes'', Mathematische Annalen, 351 (2011), 365--392 and E. Caponio, M.A. Javaloyes, M. Sanchez, ``On the interplay between Lorentzian Causality and Finsler metrics of Randers type'', Revista Matematica Iberoamericana, 27 (2011), 919--952 where a systematic study of Randers metrics in this context was performed for the first time (notice that the first version of the first paper above appeared on the arXiv on 2007, several years before [14];
-rows 44-45: I don't think that in the SME genuine Randers metrics comes into play (please check this point);
- Eq. (2): add space between T_2^0 M and ( x, ẋ );
- Figure 1: maybe picture (a) and (c) are misleading because in (a) T_x and T_x^a are not equal while in (c) T_x is not contained in T_x^a
- row 180: ẋ = <b, b> should be ẋ = \tilde b
- proof of Lemma3, p. 7 : the third point the proof of step 1 seems superfluous after points one and two; please check
- proof of Lemma3, step 3: please give some detail about the fact that the matrix $g_{ij}$ there is degenerate (this point is more important than all the other ones);
- row 227: add "sign" after "constant";
- p. 11 row -4 from bottom: $T_x$ should be $T^a_x$;
- proof of the first implication in Prop. 7: can the authors please explain why the hyperplane B=0 cannot intersect $T_x$ (haven't they showed some paragraphs above that B is only \leq 0 on $T_x$?);
-row 315: B^q should be B^{2q};
- section 5> please notice that if \xi is Killing for L and $a$ then it might be that
- Lemma A12 and Corollary A13: please add to the statements the $Q_{ij}$ is symmmetric;
-row below Eq. (A60): "= const." should be "\neq const.".
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
