Inductive Rectilinear Frame Dragging and Local Coupling to the Gravitational Field of the Universe
Round 1
Reviewer 1 Report
The paper shows that motion of a body with respect to the background FLRW space-time gives rise to a drag force similar to the typical gravitomagnetic interaction. The result is obtained using a Galilean-like choice of coordinates for the moving observer. The coordinate transformation choice is indeed legitimate and consistent with general relativity, however when coming to measured quantities clocks are affected by motion just like lengths. In other words, the time read on a moving clock corresponds to a Lorentz transformation, i.e.it is not the same as the cosmic time. The authors claim not to be so and deduce that in principle a sort of "bluing" of the Hubble redshift should exist. This point should be better discussed in order to discriminate between measurable quantities and coordinate effects.
Author Response
Dear reviewer,
We agree clocks are affected by motion just like lengths. We also agree the time on the moving clock corresponds to a Lorentz transformation, not the same as cosmic time. So we welcome the chance to clarify this point in 4 ways.
Our discussion of the 3 coordinate transformations in section 3 was predicated on 3 choices of how to treat the time coordinate. The discussion below (29) we hope makes clear what you said, that the cosmic time coordinate does not represent a moving clock. We hope this paragraph is satisfactory to describe the relation of the isotropic and moving coordinate systems, compared to Lorentz transformations. On line 162, below (29), we say that "the significance of the choice (29) is not as a non-relativistic approximation to (28), but as the cosmic time coordinate in a boosted frame"
Our second clarification on this point is to refer to (62) and (63) in the discussion of detectability. Those equations describe the 3-fold effects of relativistic time-dilation, cosmological expansion, and Doppler effect. These equations describe mathematically the 3 redshift effects, and we hope provides the mathematical clarification for your point, regarding the distinction of a Lorentz-boosted time coordinate from the cosmological time coordinate. This section should also address your concern regarding what is measurable and what is a mere coordinate effect.
Our third clarification on this point is to say that the section 3 with the time coordinate in the boosted frame is not essential to our discussion. We can work entirely in the isotropic frame, which is the natural frame to do physics, and is the frame used in the cosmology studies. Our arguments regarding the boost were only to show that the drag force existed in the rest frame of the moving body.
Our 4th comment on this is to go back to your comment, regarding the difference between the two time coordinates. We investigated this among ourselves and so we are not surprised to see the question. The time dilation is actually there in the rest frame and the Galilean boosted frame. Note that the expression for g_{tt} in (34) takes into account time-dilation. This can also be realized by comparing g_{tt} in (34) to the last parentheses in (18). But again, the boosted-frame calculation is only intended to show the force exists in the rest frame of a moving body. We do our detection calculations in the isotropic frame.
We added in a new paragraph below equation (60) to discuss the implications of the momentum transfer for the Equivalence Principle, that was mentioned in the introduction. We also added a new bullet to the conclusions about the EP. We have also made some changes to the abstract and conclusions to help also to improve our clarity.
These changes helped to improve our paper and we hope you find them satisfactory.
Thank you for your consideration.
Reviewer 2 Report
In this manuscript named “Inductive rectilinear frame dragging and local coupling to the gravitational field of the universe”, the authors focus on the problem of the drag force acting universally on moving bodies in the background cosmological metric. The main idea is that, although this force is extremely small over laboratory timescales, its effects can hypothetically be measured in lab-tests through local measurements on moving bodies.
In my opinion the paper is well-motivated and formally well-developed. I checked some equations and they seem to be correct. The result that the cosmological metric can in principle be measured through laboratory-scale dynamical experiments is quite interesting and suitable to stimulate further inquires.
For the above reasons I can recommend the work for publication in Universe. However, before that, it would be nice that the authors made some comments on the following issues:
- What is the role of the equivalence principle in the present analysis? Is there any way to test it through this study?
- Since the potentially measurable quantity is the frequency shift (67), which depends on the scale factor a(t), would it be possible (at least in principle) to favour/disfavour some of the existing models for a(t) through this analysis?
I leave the discussion of these points up to the authors' criteria.
Author Response
Dear reviewer, Thank you for your review and questions.
Regarding the first, we are grateful for the question. We added some detail regarding the Equivalence Principle. We added in a new paragraph below equation (60) to discuss the implications of the momentum transfer for the Equivalence Principle. We also added a new bullet to the conclusions about the EP.
We have also made some changes to the abstract and conclusions to help also to improve our clarity.
Thank you for your second question/suggestion. We had not thought of it before, but it seems possible in principle, given sufficient sensitivity. Therefore we added this sentence to the end of section 6:
"If the slope of the frequency variation could be distinguished, it may be possible to discriminate among cosmological models with such measurements."
Thank you for your consideration.
