The Orbital and Epicyclic Frequencies in Axially Symmetric and Stationary Spacetime

Round 1

Reviewer 1 Report

This paper presents analytical results for the orbits of a test particle in Kerr and extended Kerr spacetimes. I have only noticed some minor grammatical errors, and the following:

-The references are missing on line 47. (after equation 10)

After correcting these minor issues I recommend publication of the manuscript.

Author Response

We would like to thank the Referee for useful comments on the grammatical errors and missed the reference in a page 3. Taking into account the Referee's suggestions we have fixed issues pointed out by him/her in the revised version of the manuscript.

Reviewer 2 Report

This paper derives the expressions for some orbital and epicyclic frequencies orbiting around Kerr-Newman-NUT black hole. Overall speaking, the objectives of the paper are quite clear. I have some suggestions listed as follows:

1. It is not clear which part is contributed by the authors and which part is already done by others. Some of the references have missed (e.g. line 47, before Eq. (24)).

2. It seems that the derived expressions are useful. However, it has no discussion on how these expressions could be compared with the data. For instance, the expressions contain the spin parameter a. It is degenerate with the mass function. Also, how could the expressions be used to test general relativity? 

Author Response

We also would like to thank the Referee, who was the second reviewer for the manuscript, for valuable comments and suggestions.

 Reply 1: We have separated the  known results in the introduction that were already obtained by others. It is well known that derivation of the Keplerian frequency is trivial and can be obtained from the geodesic equation which we included in the main part of the paper. However, derivation of the epicyclic frequencies are main results which also depend on Keplerian frequency. In the revised version of the manuscript, we have included the missed reference in a page 3 which is pointed out by the Referee.

Reply2: Indeed, it is very important question how to apply the expressions for the fundamental frequencies in an arbitrary spacetime. The main target of the paper was deriving the practical formulae for the frequencies and then apply them for phenomenological calculations. So far there are several models of the QPO and most of them are already applied in the Kerr spacetime, in particular, to constrain  the mass and spin of the black hole. Recently, we are working the phenomenological calculations on the QPO to test the spacetime of the multi parametric rotating black hole including statistical analyses.