Flat Connection for Rotating Vacuum Spacetimes in Extended Teleparallel Gravity Theoriesâ€
Round 1
Reviewer 1 Report
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The authors investigate a gravitational theory in a Weitzenb\"ock spacetime, i.e., a teleparallel theory of gravity. In the Abstract they claim that they inquire into the antisymmetric part of the of the field equations for the connection. However, they are first [1] not aware of the existing literature on this question and, secondly [2], they care only about the vacuum, whereas for a well-defined gravitational theory a discussion of the coupling of the gravitational field to the material sources is indispensable; otherwise their theory is not well-defined.
Perhaps it is useful to have a look at the history of the teleparallelism. When Einstein had proposed general relativity (GR) in 1915/16, he subsequently turned his attention, inter alia, to the so-called unified field theories. With these models Einstein and collaborators tried to unify gravity and electromagnetism. No other form of matter was considered. One of the first models Einstein considered was {\it teleparallelism} in 1928.
Later mainly Schr\"odinger joint in into these efforts to formulate a unified field theory. Schr\"odinger mentioned also mesonic fields, which he wanted also to understand, but the characteristic feature of all these models---for a historical account compare Goenner [3], see also Schr\"odinger [4]---was that no external matter was allowed. Everything should follow from the geometry alone.
In the 1950s it became clear that all these attempts failed to describe nature and there were given up (more or less). Ever since theoreticians were very careful to include {\bf matter} as extra field into all attempts to set up a geometrical theory generalizing GR. Accordingly, when M{\o}ller restarted teleparallelism as (dualistic) gravitational theory in the 1950s, he was very careful to include external matter fields. His followers, like Pellegrini and Plebanski behaved the same way. They included Dirac fields into these considerations. In other words, this new teleparallelism as gravitational theory (in contrast to a unified theory) was linked to external matter fields. And this in a very successful way. Later in the gauge approach to gravity, Utiyama, Sciama, and Kibble followed the same rationale. This made their respective approaches to a successful theoretical framework with explanatory power.
The decisive element in those theories with matter is the coupling between geometry and matter as exemplified in GR by Hilbert's definition of the material energy-momentum tensor as variational derivative of the matter Lagrangian with respect to the metric: ${\cal T}_{ij}\sim\delta {\cal L}_{\rm mat}/\delta g^{ij}$. In theories with torsion and nonmetricity, this relation has to be suitable generalized, see Kibble [2] and Blagojevic et al. [1f], for example.
This brings us back to the present paper. When the authors speak about the field equations, they only speak about the {\it vacuum} field equations. Matter is not involved in the authors' ansatz. In other words, how matter is related to the geometry is left open. The authors speak of the antisymmetric piece of the field equations, but they forgot to introduce matter and their scheme is left completely unspecified. A gravitational ansatz without specifying the matter which is supposed to be involved, is at most a mathematical scheme, but not a physical theory.
On page 1 in the Introduction we read: ``..., the choice of a connecion is a mathematical convention and not an independent property of spacetime.'' This statement is incorrect. It could be only made since the authors discuss only the vacuum case. A connection can couple to a material current---and the existence of this current is {\it not} a mathematical convention. For the same reason a ``geometrical trinity of gravity'' is a hoax and the corresponding papers Ref.(12) and Ref.(14) arrive at physically untenable conclusions. Nonmetricity and torsion have very distinct geometrical properties and couple in definite and different ways to matter. There is no equivalence between the different theories for nonmetricity and torsion from the point of view of physics.
The oldest reference in the authors' article is of 1999 (which is a highly problematic and, in my opinion, incorrect paper). Apparently the authors don't read books (with exception of Ref.(48)) and they only cite articles published subsequent the year 1998. On the other hand they have a overloaded and highly redundant list\footnote{See in this context G.Pacchioni, ``The Overproduction of Truth,'' Oxford (2018).} of recent papers. Let me inform the authors that their work is partly based on an article which was published in the prehistoric time of 1915.
On page 2 the authors state: ``How to appropriately interpret this additional freedom and determine teleparallel connection for a solution of interest [and?] are among the central issues of the theory.'' This is correct. However, a discussion without matter (see above) is incomplete.
My suggestion to the authors is to correct the Abstract and the Introduction of their paper according to the remaks I made.
In the Sections 2 and 3 once again the formalism of teleparallel gravity is displayed as far as the vacuum field is concerned. Thus, up to including page 5, nothing really new is presented. Eventually four different and known strategies are discussed of how to solve the field equations. In Sec.4 cylidrically symmetric Weyl coordinates are introduced and the corresponding tetrads are determined. In order to be able to compare this with Kerr type solutions, in Sec.5 Boyer-Lindquist coordinates are introduced and the tetrads determined. No exact solution of teleparallelsm is really displayed. The paper terminates when it just begins to become interesting.
I find that at least some comparisons with the Kerr type solutions of the Poincar\'e gauge gravity could have been provided, which is known since quite some time, see McCrea et al. [5] and Baekler et al. [6]. The limit of vanishing curvature of that solution [5,6] should correspond to the type of solution investigated by the authors. In this way the somewhat indefinite conclusions of Sec.5 could have been made more transparent.
Summing up: I find that the present paper is rather incomplete (matter is excluded) and inconclusive (no exact and explicit exact solution was found). Accordingly, I am sorry, but I cannot suggest publication of the paper. Only after a thorough revision along the lines discussed above, the paper should be reconsidered for publication.
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[1a] C.Pellegrini and J. Plebanski, Mat.Fys.Skr.Dansk Vid.Selsk. {\bf 2}, No.4 (1963).
[1b] M.Schweizer et al., GRG {\bf 12}, 951 (1980).
[1c] M.Schweizer et al., Phys. Lett. {\bf A 71}, 493 (1980).
[1d] J.Nitsch et al,. Phys; Lett. {\bf B 90}, 98 (1980).
[1e] F.W.Hehl, Erice Meeting 1979, Bergmann and de Sabbata (eds.), Plenum Press (1980), pp.5-61 [see Hehl's homepage].
[1f] M.Blagojevic and Hehl, eds., Gauge Theories of Gravitation, Imperial College Press (2013).
[2] T.W.B.~Kibble, J.Math.Phys. {\bf 2}, 212 (1961).
[3] H.~F.~M.~Goenner, ``On the History of Unified Field Theories. Part II. (ca. 1930 - ca. 1965),’' Living Rev.\ Rel.\ {\bf 17}, 5 (2014).
[4] E.~Schr\"odinger, Space-Time Structure, Cambridge (1952).
[5] J.~D.~McCrea et al., Nuovo Cimento B 99, 171 (1987).
[6] P.~Baekler et al., Physics Letters A 128, 245 (1988).
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Author Response
We thank the referee for writing an extensive erudite review and raising several points, to which we briefly respond.
1) Concerning the existing literature on the subject it is quite clear that it is rather voluminous, making a comprehensive citation of all possible references unfeasible in the present short note. For the early work leading to and on teleparallel gravity we had referred the book by Aldrovandi and Pereira [9], and the reviews by Maluf [10] and Cai et al [11]. In the revised version we have added citations to the review by Hehl et al. [12] and to the book edited by Blagojević and Hehl [13], which should cover most of the relevant older publications.
2) The matter coupling problem is indeed an outstanding issue in teleparallel gravity. Including spinless matter in hydrodynamic limit and the electromagnetic field in the theory is quite straightforward as they naturally only couple to the metric (or the Levi-Civita connection), but the prescription of coupling fermions has seen different approaches and conflicting opinions. Since our aim was to investigate a vacuum field configuration in the exterior of a massive body or a black hole, it did not seem necessary to delve into the topic (except for a remark in a paragraph after Eq. (21)) since our result should match with different possible scenarios. In the revised version of the paper we have amended the title and abstract to emphasize that we study the vacuum case, added a paragraph in the Introduction outlining the issue, and an extra clarification in the paragraph after Eq. (10).
3) The aim of the paper was to clarify and illustrate how the connection is consistently treated in the covariant formulation of teleparallel gravity, for there is still some confusion among the researchers. By focussing upon the antisymmetric field equations we were able to derive a new universal class of solutions that can serve as a stepping stone to tackle the symmetric and scalar field equations and obtain a complete solution. In the symmetric and scalar field equations one needs to specify the defining functions f(T,phi) and Z(phi) of the theory, and the treatment would be much longer (already writing down the equations would take almost a page), going beyond the short nature of this conference paper. In the revised version we added phrases and sentences to the last paragraphs of the Introduction and Discussion sections, further stressing the aims and motivations.
4) The papers of McCrea et al and Baekler et al start with a Kerr-de Sitter metric and by employing a clever method supplement it with a connection that has not just nontrivial curvature but also nontrivial torsion, and satisfies the equations of certain Poincaré gauge theories. It is already clear from the setup that the procedure suggested by the referee (imposing vanishing curvature on the connection presented in these papers) can not yield a solution that corresponds to ours, since our configuration is endowed with a condition that makes the metric different from Kerr.
In principle we certainly agree that it would be very interesting to compare the full solution (once the symmetric field equations are also solved) to Kerr in the teleparallel setting as well as to the other axially symmetric solutions in Poincare gauge theories and beyond. This task will be left for a future stage of the work, as we mention in the end the revised Discussion section along with the citations of McCrea et al. and Baekler et al.
Reviewer 2 Report
The article is well written and deals with a subject interesting to the community. It studies rotation solutions of f(T,phi) theories, where T is a scalar constructed out of torsion tensor of Weitzenböck geometry. Below I present some point which should be cleared by the authors:
1) Equation (8) implies that T is not invariant under local Lorentz transformations. It should be noted however that T and the total divergence are invariant under global Lorentz transformations and only the sum of both parts is invariant under the local symmetry. How is this compatible with the formalism of spin connection used in the manuscript?
2) The authors’s statement about considering all Lorentz frames on equal footing is problematic for the definition of the energy-momentum vector, Pa. Such a quantity is defined under integration, that means the result is independent of coordinates. Hence only global Lorentz vector is allowed on this approach. It seems that local Lorentz covariance is allowed by the field equations as the authors explore in the manuscript. Thus the authors should make some comments on the reasons they decide to explore this feature at the cost of the gravitational energy definition.
3) The authors should note that f(T) theories are not at the same level of f(R) theories exactly because of the role of the reference frame which is established by the tetrad field. There are good and bad tetrads in the f(T) approach, thus not every solution has physical existence. The authors should make some comments about their rotation solution. For instance if there is any physical constraint that excludes a given solution.
4) The kinematical state of the observer is not only established by the spin connection. In fact there is another approach based on the acceleration tensor which was firstly introduced by Bahram Mashhoon. It can be used even with the vanishing of the spin connection. The choice of the reference frame is compatible with global Lorentz symmetry. In my opinion the authors should make some comments elucidating the relation between their approach and the acceleration tensor, if possible.
Author Response
We thank the referee for raising points that call for clarification.
1) The covariant formalism guarantees that T is invariant under local Lorentz transformations. The definitions (12), (13), and (15) ensure that the torsion tensor with Lorentz indexes transforms covariantly under local Lorentz transformations. Therefore the torsion tensor when written in terms of only spacetime components remains invariant under local Lorentz transformations and so do the objects constructed of it, including the ones in Eq. (8). In the covariant approach T is truly a scalar under local Lorentz transformations as well as under general spacetime coordinate transformations, and introducing a function f(T) in the action makes no problems.
In contrast, in the noncovariant formalism when one sets the spin connection to be identically zero always, i.e. dropping the omega terms from (12) and (13), then the torsion tensor will fail to transform covariantly under local Lorentz transformations. Consequently, in the noncovariant formulation T is invariant only under global Lorentz transformations. We have added a further clarification about this to the last paragraph of Sec. 2.
2) The definition of gravitational energy-momentum is an intricate issue with a number of alternative proposals. It is obviously related to the effects of inertia observers in different frames can experience. In this short conference note we aimed at presenting a new universal solution to the antisymmetric field equations and preferred not to elaborate on the topic of gravitational energy momentum. The latter is certainly important but involves further interpretative questions hard to address without extending the paper at least twice.
3) In the covariant formulation we pursue in this paper, a solution consists of pairs of tetrads and spin connections. There is a freedom to make local Lorentz transformations to go into different frames whereby the tetrad and spin connection transform as (15), but the equations are still satisfied. All such pairs are physically acceptable and we explicitly give several such pairs in Sec. 4 and 5 of the manuscript.
The notions of "good" and "bad" tetrads is an old terminology which pertains to the noncovariant formulation of the theory. In this formulation indeed, if the field equations are solved by one tetrad, they would not be automatically solved by another tetrad which is obtained from the first by a local Lorentz transformation. It happens because the noncovariant formulation misses out the spin connection. To emphasize this we added an extra paragraph between Eqs. (28) and (29).
4) We are aware of the acceleration tensor and see its implementation as one of the prospective steps for further study. We have mentioned this in the last paragraph of the discussion section, discussing the research outlook, and cited the paper 0704.0986 which nicely applies it in the case of TEGR. The acceleration tensor would be an interesting and informative object when we have a full solution at our disposal (i.e. a solution of both the antisymmetric and symmetric equations) with possible ergoregions, horizons, etc.
Round 2
Reviewer 1 Report
The authors improved their manuscript appreciably. I still regret that they couldn't present results re the coupling to matter. Otherwise the paper is OK. Thus, I can now recommend publication.
