Covariant Evolution of Gravitoelectromagnetism

Round 1

Reviewer 1 Report

In this paper it is reviewed a covariant formalism describing the dynamics of gravito-Mawxell fields, gravitational waves propagation and some limits of Newtonian models without gravitomagnetism.

The review is well organized and a comprehensive discussion of the subject is provided.

I only recommend to update the bibliography including at least some of the following seminal and recent works about gravito-Maxwell fields and equations: 

Phys. Rev. D 15 (1977) 2047
Gen. Rel. Grav. 15 (1983), n. 8, 725
Gen. Rel. Grav. 23 (1991) 1231
Gen. Rel. Grav. 22 (1990), n. 6, 609
Nuovo Cim. B 117 (2002) 743
Eur. J. Phys. 25 (2004) 203
Eur. Phys. J. C 77 (2017), n. 8, 549
J. Phys. A 16 (1983), n. 6, 1331
Symmetry 11 (2019), n. 11, 1341,
Physica C: Superconductivity 339 (2000), n. 2, 120
Phys. Rev. D 43 (1991), n. 2, 457
Phys. Rev. D 39 (1989), n. 10, 2825
Class. Quant. Grav. 37 (2020), n. 21, 217001
Entropy 23 (2021), n. 2, 193
New J. Phys. 23 (2021), n. 5, 053019

Author Response

RESPONSES TO THE REVIEWER'S COMMENTS:

COMMENT: In this paper it is reviewed a covariant formalism describing the dynamics of gravito-Mawxell fields, gravitational waves propagation and some limits of Newtonian models without gravitomagnetism.

The review is well organized and a comprehensive discussion of the subject is provided.

RESPONSE: I appreciate the referee for review of my manuscript, and I am pleased that the referee found “the review is well organized and a comprehensive discussion of the subject is provided.” I also added some of references about observation effects and laboratory tests mentioned by the referee.

COMMENT: I only recommend to update the bibliography including at least some of the following seminal and recent works about gravito-Maxwell fields and equations:

Phys. Rev. D 15 (1977) 2047

Gen. Rel. Grav. 15 (1983), n. 8, 725

Gen. Rel. Grav. 23 (1991) 1231

Gen. Rel. Grav. 22 (1990), n. 6, 609

Nuovo Cim. B 117 (2002) 743

Eur. J. Phys. 25 (2004) 203

Eur. Phys. J. C 77 (2017), n. 8, 549

J. Phys. A 16 (1983), n. 6, 1331

Symmetry 11 (2019), n. 11, 1341,

Physica C: Superconductivity 339 (2000), n. 2, 120

Phys. Rev. D 43 (1991), n. 2, 457

Phys. Rev. D 39 (1989), n. 10, 2825

Class. Quant. Grav. 37 (2020), n. 21, 217001

Entropy 23 (2021), n. 2, 193

New J. Phys. 23 (2021), n. 5, 053019

RESPONSE: Some of these references about observation effects and laboratory tests of gravitomagnetism were cited in a footnote (number 8) in Section 10: “Also see [144–149] for the gravitomagnetic clock effect, and [150–157] for proposed laboratory experiments.”

Reviewer 2 Report

Referee's report on the paper "Covariant evolution of gravitoelectromagnetism"
by A. Danehkar

The author presented a highly technical review of the 3+1 decomposition method
applied to the study of the gravitational field dynamics in terms of electric
and magnetic pieces of the Weyl tensor. The subject is of interest, and the
reference list appears to be fairly complete. However, I cannot recommend this
paper for publication in Universe, at least not in its present form. A careful
revision is needed along the following lines.

1. The 3+1 decomposition formalism overviewed in Sec. 1 is presented in a rather
sloppy way. The velocity field u introduces two types of projectors: on the
"time" direction (along u) and on the "space" (orthogonal to u). In Sec. 1, the
author tacitly speaks only about spatial projectors without however mentioning
this explicitly. For example, in eqs. (4), (8) and (9) the spatial projections
of various objects are given, whereas the author describes these merely as
"projected tensor", "projected vector", "projected 2-tensors", etc. The text
should be rewritten in a more rigorous way. In particular, it should be made
clear that the \bot symbol denotes a spatial projection, not just projection.

2. On page 3, just above eq. (9) we read: "In the covariant formalism, we have
scalars, projected vectors, and PSTF tensors, instead of the spacetime metric."
It is unclear to me why the metric tensor is singled out here? The 3+1 formalism
provides the method of dealing with arbitrary decomposed objects, not only with
the spacetime metric.

3. On page 1, line 19, the author describes the Weyl tensor as a "a traceless
massless part". I wonder, in which sense the Weyl tensor is massless?

4. On page 5, the author calls (28)-(30) "algebraic identities". Why? These are
equations relating geometric and physical variables, NOT identities.

5. In (143)-(145) it is better to give the sound speed and the barotropic index
explicitly. I mean, a PHYSICAL review should be as informative and complete as
possible, it's not a telephone directory.

6. On page 19, two lines below (150) we read D_{(a} E_{ab)} . Is here a summation
over the index "a" assumed, or it is a mistake?

7. Also on page 19, just above eq. (152), I am puzzled by the sentence "The
distortion also supports the spatial Laplacian". What does it mean? One should
avoid using jargonisms in a review intended for a wide audience.

8. Similar comment applies to page 24, line 436, where we read something unclear
"(see [98,99] for non-covariant)".

9. On page 31, eq. (238) displays the same object "e" on left- and right-hand
sides. In which sense this is a "transformation" then?

10. Also on page 31, three lines above (241), the angle brackets < , > are not 
explained. I wonder, why not using the standard geometrical language and write
this as a value of a 1-form on a vector instead, i.e. \omega^a ( e_b ) ?

11. The manuscript contains a lot of errors (spelling and not only). For example
on page 3, below (7) "donated" should be replaced by "denoted"; on page 18, line
278 a comma should be removed after "once"; on page 18, line 291, "conditions
(146) is" should be replaced by "conditions (146) are"; on page 20, line 327
"is sometimes be called" one should remove "be"; on page 22, just above (165),
"interpenetration" (!!) should be replaced by "interpretation"; on page 40,
line 632 "fram-dragging" should be fixed, etc etc. The list can be continued.

12. Moreover, the reference list requires improvements. In items [55] and [56],
the name "Zel'Manov" should be spelled as "Zel'manov", whereas in item [57] one
has to replace "Gemetrie" with "Geometrie" and "tensurs" with "tenseurs". In
addition, in items [60,129,130] the titles are missing, which is apparently
inconsistent with the general bibliographic style of providing the full titles
of the cited papers. 

Author Response

RESPONSES TO THE REVIEWER'S COMMENTS:

COMMENT: The author presented a highly technical review of the 3+1 decomposition method

applied to the study of the gravitational field dynamics in terms of electric

and magnetic pieces of the Weyl tensor. The subject is of interest, and the

reference list appears to be fairly complete. However, I cannot recommend this

paper for publication in Universe, at least not in its present form. A careful

revision is needed along the following lines.

RESPONSE: I appreciate the referee for very careful read of my manuscript, and I am pleased that the referee found the subject is of interest, and the reference list appears to be fairly complete. I revised the manuscript according to the referee's comments, and I hope that the paper is now at a state that warrants publication. My detailed list of changes in response to the referee's comments follows:

1. COMMENT: 1. The 3+1 decomposition formalism overviewed in Sec. 1 is presented in a rather

sloppy way. The velocity field u introduces two types of projectors: on the

"time" direction (along u) and on the "space" (orthogonal to u). In Sec. 1, the

author tacitly speaks only about spatial projectors without however mentioning

this explicitly. For example, in eqs. (4), (8) and (9) the spatial projections

of various objects are given, whereas the author describes these merely as

"projected tensor", "projected vector", "projected 2-tensors", etc. The text

should be rewritten in a more rigorous way. In particular, it should be made

clear that the \bot symbol denotes a spatial projection, not just projection.

1. RESPONSE: Added: “perform a 1 + 3 decomposition of spacetime g ab into the time direction and the 3-dimensional space...Accordingly, the metric g ab is projected parallel and orthogonal to u a as follows:” The “spatially” word is also added before Eqs. (4), (8), and (9).

2. COMMENT: 2. On page 3, just above eq. (9) we read: "In the covariant formalism, we have

scalars, projected vectors, and PSTF tensors, instead of the spacetime metric."

It is unclear to me why the metric tensor is singled out here? The 3+1 formalism

provides the method of dealing with arbitrary decomposed objects, not only with

the spacetime metric.

2. RESPONSE: To avoid confusion, the sentence was removed.

3. COMMENT: 3. On page 1, line 19, the author describes the Weyl tensor as a "a traceless

massless part". I wonder, in which sense the Weyl tensor is massless?

3. RESPONSE: “massless” removed.

4. COMMENT: 4. On page 5, the author calls (28)-(30) "algebraic identities". Why? These are

equations relating geometric and physical variables, NOT identities.

4. RESPONSE: "algebraic identities" changed to “relations”

5. COMMENT: 5. In (143)-(145) it is better to give the sound speed and the barotropic index

explicitly. I mean, a PHYSICAL review should be as informative and complete as

possible, it's not a telephone directory.

5. RESPONSE: It now gives: “c^2_s = ṗ/ ρ̇ and w = p/ρ in FLRW spacetimes [37]”

6. COMMENT: 6. On page 19, two lines below (150) we read D_{(a} E_{ab)} . Is here a summation

over the index "a" assumed, or it is a mistake?

6. RESPONSE: Corrected.

7. COMMENT: 7. Also on page 19, just above eq. (152), I am puzzled by the sentence "The

distortion also supports the spatial Laplacian". What does it mean? One should

avoid using jargonisms in a review intended for a wide audience.

7. RESPONSE: Revised: “The Laplacian equation, which plays a key role in the wave solution, also emerges from the distortion.”

8. COMMENT: 8. Similar comment applies to page 24, line 436, where we read something unclear

"(see [98,99] for non-covariant)".

8. RESPONSE: Corrected: “(see the review by [98])”

9. COMMENT: 9. On page 31, eq. (238) displays the same object "e" on left- and right-hand

sides. In which sense this is a "transformation" then?

9. RESPONSE: Revised: “The local tetrad transformation” It is described as a transformation in the literature (see 2.10 in Ellis, J. Math. Phys. 1967, 8, 1171).

10. COMMENT: 10. Also on page 31, three lines above (241), the angle brackets < , > are not

explained. I wonder, why not using the standard geometrical language and write

this as a value of a 1-form on a vector instead, i.e. \omega^a ( e_b ) ?

10. RESPONSE: Revised: “Then, the vector fields {e_a} are dual to the 1-form fields {ω^a} such that < …” This notation has widely been used in the literature to describe {e_a} being dual to 1-form {ω^a} fields.

11. COMMENT: 11. The manuscript contains a lot of errors (spelling and not only). For example

on page 3, below (7) "donated" should be replaced by "denoted"; on page 18, line

278 a comma should be removed after "once"; on page 18, line 291, "conditions

(146) is" should be replaced by "conditions (146) are"; on page 20, line 327

"is sometimes be called" one should remove "be"; on page 22, just above (165),

"interpenetration" (!!) should be replaced by "interpretation"; on page 40,

line 632 "fram-dragging" should be fixed, etc etc. The list can be continued.

11. RESPONSE: Fixed.

12. COMMENT: 12. Moreover, the reference list requires improvements. In items [55] and [56],

the name "Zel'Manov" should be spelled as "Zel'manov", whereas in item [57] one

has to replace "Gemetrie" with "Geometrie" and "tensurs" with "tenseurs". In

addition, in items [60,129,130] the titles are missing, which is apparently

inconsistent with the general bibliographic style of providing the full titles

of the cited papers.

12. RESPONSE: [55]-[57] typos corrected, and the title added to [60]. As [129,130] were seminar presentations, they are now mentioned in a footnote (number 7).

Round 2

Reviewer 2 Report

Referee's report on the revised paper "Covariant evolution of gravitoelectromagnetism"
by A. Danehkar

The author amended his manuscript in a satisfactory way, so that the paper
is in a better shape now and can be recommended to publication.

However, one point still needs attention. Namely, as in the original version,
on page 32, eq. (238) displays the same object "e" on left- and right-hand
sides. In his reply, the author referes to eq. (2.10) in J. Math. Phys. 1967,
8, 1171. But please notice that Ellis uses a clever notation by marking with a
prime the indices of a transformed object e_a -> e_{a'} . In contrast, on both
sides of eq. (238) we see the same root symbol with the same type of index which
means that this is the same untransformed object. To make eq. (238) a real
transformation, one should either put a prime on the root symbol e -> e' , or
(as Ellis did) put a prime on the index  e_a -> e_{a'} .

Author Response

COMMENT: The author amended his manuscript in a satisfactory way, so that the paper

is in a better shape now and can be recommended to publication.

RESPONSE: I am thankful to the referee for careful review. I made that minor correction as requested by the referee.

COMMENT: However, one point still needs attention. Namely, as in the original version,

on page 32, eq. (238) displays the same object "e" on left- and right-hand

sides. In his reply, the author referes to eq. (2.10) in J. Math. Phys. 1967,

8, 1171. But please notice that Ellis uses a clever notation by marking with a

prime the indices of a transformed object e_a -> e_{a'} . In contrast, on both

sides of eq. (238) we see the same root symbol with the same type of index which

means that this is the same untransformed object. To make eq. (238) a real

transformation, one should either put a prime on the root symbol e -> e' , or

(as Ellis did) put a prime on the index e_a -> e_{a'} .

RESPONSE: Following the referee’s recombination, the notation was corrected by using a prime in the indices of a transformed object e_a -> e_{a'}.