General Relativistic Aberration Equation and Measurable Angle of Light Ray in Kerr–de Sitter Spacetime
Round 1
Reviewer 1 Report
The author considers measurable angle of light ray in the KdS spacetime by using the aberration equation. The results are interesting and could be published, but not in the present form. It is possible after proper reaction of the author to the following comments and some improvements of the English, and giving correct numbers to Eqs on p.7.
- The author omits some relevant papers related to the photon motion in the KdS spacetimes, specially the paper that is explicitly considering the radial motion of the observer EUROPEAN PHYSICAL JOURNAL C Volume: 78 Issue: 3 Article Number: 180 Published: MAR 3 2018, or CLASSICAL AND QUANTUM GRAVITY Volume: 17 Issue: 21 Pages: 4541-4576 Published: NOV 7 2000. These works should be cited.
- There is crucial point that has to be addressed. In fact the authors is consider some kind of vacuola with the KdS interior and the expanding universe with the related cosmological constant. Without rotation this is described by the Einstein-Straus-de Sitter vacuola model of the Universe (see
BULLETIN OF THE ASTRONOMICAL INSTITUTES OF CZECHOSLOVAKIA Volume: 35 Issue: 4 Pages: 205-215 Published: ( 1984. Then it is clear that the velocity applied by the author for the observer velocity corresponds to the expanding boundary of the vacuola. However, the observer can be moving inside the vacuola with different velocity. Of special interest is the so called static radius, where the gravitational attraction of the central object is just balanced by the cosmic repulsion, for its definition and special character see the papers
BULLETIN OF THE ASTRONOMICAL INSTITUTES OF CZECHOSLOVAKIA Volume: 34 Issue: 3 Pages: 129-149 Published: 1983
PHYSICAL REVIEW D Volume: 60 Issue: 4 Article Number: 044006 Published: AUG 15 1999. Its astrophysical relevance is explained in
UNIVERSE Volume: 6 Issue: 2 Article Number: 26 Published: FEB 2020.
The author has to explicitly state the message of his work related to the specific point related to the radially moving observer, and he should mention the role of the static radius where the free observer can be stationary, starting to be accelerated by the cosmic repulsion to the boundary of the vacuola where the velocitycan be related to the Hubble constant.
After required modification the publication of the paper can be reconsidered.
Author Response
Dear referee,
Thank you very much for your careful reading and
useful comments on our paper.
We would like to respond to your comments as follows.
Because the track changes command caused a compilation error
when main text contain $...$, \cite{...} and so on,
we have marked the changes with blue font.
Best regards,
Hideyoshi ARAKIDA
----------------------------------------------------------------------
0. A mistake in the equation numbers on page 7 has been corrected.
Since we were referring to an expression number that did not actually
exist,
we inserted the expression of the initial condition itself in the text
instead of the expression number.
1. According to your suggestion, on page 3, we cited the following two
papers;
40. Stuchlík, Z.; Hledík, S. Equatorial photon motion in the
Kerr–Newman spacetimes with a non-zero cosmological constant, Class.
Quantum Grav., 2000, 17, 4541-4576.
41. Stuchlík, Z.; Charbulák, D.; Schee, J. Light escape cones in local
reference frames of Kerr–de Sitter black hole spacetimes and
related black hole shadow, Eur. Phys. J. C, 2018, 78, id.180.
2. In response to your second suggestion, we have responded as follows;
a) From the end of page 11 to the beginning of page 12, we mentioned
that we were considering gravitational lensing at cosmological scales
in this paper, and then added a footnote on page 12 about static radius.
b) In the footnote on page 12, we cited three papers;
46. Stuchlík, Z.; The Motion of Test Particles in Black-Hole
Backgrounds with Non-Zero Cosmological Constant, Bull. Astron. Inst.
Czechosl., 1983, 34, 129-149.
47. Stuchlík, Z.; Hledík, S. Some properties of the Schwarzschild–de
Sitter and Schwarzschild–anti-de Sitter spacetimes, Phys. Rev. D,
1999, 60, id.044006.
48. Stuchlík, Z.; Kološ, M.; Kovárˇ, J.; Slaný, P.; Tursunov,
A. Influence of Cosmic Repulsion and Magnetic Fields on Accretion
Disks Rotating around Kerr Black Holes, Universe, 2020, 6, id.26.
c) Also in the footnote on page 12, we cited the following paper and
described that KdS spacetime becomes the Einstein-Strauss-de Sitter
vacuola cosmological model when a = 0.
----------------------------------------------------------------------
Reviewer 2 Report
In this work the author extends his previous research results on Kerr spacetime (ref.[37]), by investigating the general relativistic aberration equation and measurable angle of light ray in the case that the Kerr spacetime is endowed with a positive cosmological constant (KdS). For this purpose he applies perturbation theory techniques for the computation of light trajectory in KdS spacetime. The theme of the manuscript is interesting and falls within the scope of the Journal .
However before the paper is ready for publication there are some points that need clarification by the author:
1.) In eqn.(13), it appears that there are some mistakes.
a) the fifth term on the RHS of (13) I believe must be: -2 a^2 u^4
Also the last term in the same equation involving the cosmological constant, has a wrong sign. The author must recheck his calculations for eqn(13).
b) In the spirit of improving the presentation of the perturbative methods used, the author must provide details for the derivation of eqn.(20) from plugging eqn.(19) into eqn.(13).
After these modifications take place the paper may be reconsidered for publication the Universe
Author Response
Dear referee,
Thank you very much for your careful reading and
useful comments on our paper.
We would like to respond to your comments as follows.
Because the track changes command caused a compilation error
when main text contain $...$, \cite{...} and so on,
we have marked the changes with blue font.
Best regards,
Hideyoshi ARAKIDA
----------------------------------------------------------------------
1. a) We have corrected the typos in equation (13) on page 4. As you
pointed out, +2 a^2 u^2 should be -2 a^2 u^4, and the sign before
the last term multiplied by the cosmological constant \Lambda should
be minus.
b) As you pointed out, we described the derivation of equation (20) at
the bottom of page 5, and commented on why the two constants B and b
appear in equation (20) on page 6 under equation (20).
----------------------------------------------------------------------
