On"Rotating charged AdS solutions in quadratic $f(T)$ gravity": New rotating solutions

We show that there are two or more procedures to generalize the known four-dimensional transformation, aiming to generate cylindrically rotating charged exact solutions, to higher dimensional spacetimes . In the one procedure, presented in Eur. Phys. J. C (2019) \textbf{79}:668, one uses a non-trivial, non-diagonal, Minkowskian metric $\bar{\eta}_{ij}$ to derive complicated rotating solutions. In the other procedure, discussed in this work, one selects a diagonal Minkowskian metric ${\eta}_{ij}$ to derive much simpler and appealing rotating solutions. We also show that if ($g_{\mu\nu},\,\eta_{ij}$) is a rotating solution then ($\bar{g}_{\mu\nu},\,\bar{\eta}_{ij}$) is a rotating solution too with similar geometrical properties, provided $\bar{\eta}_{ij}$ and ${\eta}_{ij}$ are related by a symmetric matrix $R$: $\bar{\eta}_{ij}={\eta}_{ik}R_{kj}$.

We show that there are two or more procedures to generalize the known four-dimensional transformation, aiming to generate cylindrically rotating charged exact solutions, to higher dimensional spacetimes .In the one procedure, presented in Eur.Phys.J. C (2019) 79:668, one uses a non-trivial, non-diagonal, Minkowskian metric ηij to derive complicated rotating solutions.In the other procedure, discussed in this work, one selects a diagonal Minkowskian metric η ij to derive much simpler and appealing rotating solutions.We also show that if (g µν , η ij ) is a rotating solution then ( ḡµν , ηij ) is a rotating solution too with similar geometrical properties, provided ηij and η ij are related by a symmetric matrix R: ηij = η ik R kj .

PRELIMINARIES
In this work we will use the notation of [1] with a slight difference.Instead of taking f (T) = T + αT 2 with α < 0 we will take f (T) = T − αT 2 with α > 0.
Another different choice, which will be made clearer later, is the signature of the N-dimensional Minkowski spacetime: (+, −, −, −, • • • ).Most of the other notations will be almost similar to that of [1].
As a first comment we state that there are some sign mistakes in the definition of K αµν of [1].We use the following definitions 1 : It is obvious from these definitions that the global sign of T would depend on the signature of the metric.For a static metric with signature (+, −, −, −, • • • ) where n is the number of angular coordinates, N is the dimension of spacetime and l is related to the cosmological constant by We obtain 1 S αµν may be given in a more compact form as: Had we reversed the signature of the metric we would obtain the same expression with the two '+' sings changed to '−' sings.A second comment is also in order: The expression of T given in [1] has an extra factor 2 in the term including A ′ .A final comment: The last term in Eq. ( 14) of [1] should have the opposite global sign.Using our metricsignature choice, Eq. ( 14) of [1] takes the form where (ω 1 , ω 2 , • • • , ω n ) are the rotation n parameters, (φ 1 , φ 2 , • • • , φ n ) are the n angular coordinates and Ξ = 1 + Σ n i=1 (ω 2 i /l 2 ).Note that the last term, −(r 2 /l 2 ) ∑ n i<j (ω i dφ j − ω j dφ i ) 2 , vanishes identically if the spacetime has only one angular coordinate.
The field equations of Maxwellf (T) gravity are given in Eq. (3) of [1], which we rewrite here for convenience where e ≡ |g| and T (em is the energy-momentum tensor of the electromagnetic field.Here the ratio (N − 1)(N − 2)/l 2 is proportional to the cosmological constant Λ (3).It is obvious from the shape of Eqs.(6) that we are dealing with a spin-zero (pure tetrad) f (T) gravity.The general field equations including spin connection terms are provided in [2].

GENERATING CYLINDRICALLY ROTATING CHARGED EXACT SOLUTIONS
Consider the following substitution where a denotes a rotation parameter There is no claim whatsoever in Refs.[4,5] that the substitution ( 8) is a shortcut or a trick for generating rotating solutions from static ones, however, some authors have applied the substitution (8) as a procedure to generate their supposed-to-be rotating solutions.In this work we present a general comment on the transformation (8) and its generalization to higher dimensions.
Our starting point is the expression of the tetrad e i µ in terms of the static metric (A(r), B(r)), the n rotation parameters denoted by (ω 1 , ).The tetrad expression e i µ is given in Eq. ( 12) of [1].However, in order to evaluate e i µ from e i µ , using the expression e i µ = η ij g µν e j ν , we need an expression for the Minkowskian metric η ij .The authors of Ref. [1] did not provide any expression for η ij they used in their work.An anonymous referee claimed that it is the non-diagonal form of η ij , as given in Eq. (44) of Ref. [8] and Eq.(41) of Ref. [9], that has been used in [1] and that it is the only valid form of η ij to be used.In this work we will use two different expressions for η ij and we shall show that the statement of the referee does not hold true by constructing a new cylindrically rotating charged solution using a diagonal expression for η ij .
From now on we restrict ourselves to N = 5 and consider the cases 1) n = 1 and 2) n = 2.
The tetrad expression (12) of [1] reduces to This is not a proper tetrad as the associated spin connection does not vanish [2,6].To evaluate the associated spin connection we refer to [2,6].Using the terminology of these references, the reference tetrad e i (r)µ is, in this case, given by ( 9) upon setting m = q = 0 (absence of gravity and matter) and N = 5.We find that the nonvanishing components of the spin connection ω a bµ are [the Latin indexes (a, b) in ω a bµ run from 1→5]: ).This fact results in violation of local Lorentz invariance.
Taking a diagonal Minkowskian metric which is the same as the metric suggest in Eq. ( 14) of [1]; in this case (N = 5, n = 1) the last term in Eq. ( 14) of [1] vanishes identically.Now, we evaluate T upon substituting ( 9) and ( 10) into (1) and the resulting expression is identical to (4) taking N = 5.
On substituting ( 9), ( 10) and ( 4) into the field equations (6) and using the static solution (7) we noticed that all the field equations are satisfied.
The tetrad expression (12) of [1] reduces to In order to proof that Eq. ( 14) of Ref. [1], which is Eq. ( 5) of this work (including the global sign correction we made), is a rotating solution one needs an expression for the Minkowskian matrix η ij by which one can evaluate e i µ from e i µ (11), then evaluate all the tensors needed in the field equations (6).We divide this case into two sub-cases a) η ij diagonal and b) η ij non-diagonal.
Knowing the metric we evaluate e i µ by e i µ = η ij g µν e j ν .Next, we evaluate T upon substituting (11) and ( 12) into (1) and the resulting expression is identical to (4) taking N = 5.Now, on substituting (11), ( 12) and (4) into the field equations ( 6) and using the static solution (7) we noticed that all the field equations are satisfied.
We have thus obtained a new rotating solution given by (12), which we rewrite for convenience This is a solution to the field equations ( 6) with e i µ given by (11), i=1 ω i dφ i ), and the r-functions (A, B, Φ) are given in (7).

Case b) η ij non-diagonal
The authors of Ref. [1] did not provide an expression for the Minkowskian metric η ij they used in their work.In our first version of this work we assumed η ij = diag(1, −1, −1, −1, −1) and we reached the conclusion that the metric ( 5) is not a solution to the field equations (6).However, an anonymous referee claimed that a correct expression for η ij would be the matrix (44) of Ref. [8], which is also the matrix (41) of Ref. [9].The rightmost column and the bottom line of that matrix have a common element, which is −1, and the rest of the elements of the rightmost column and the bottom line are 0. In the case of five-dimensional spacetime with 2 angular coordinates (N = 5, n = 2), matrix (44) of Ref. [8], or matrix (41) of Ref. [9], takes the following form using the notation and signature of this work [10] With this η ij matrix and the expression of e i µ given in (11), the formula g µν = η ij e i µ e j ν yields the metric (5).
It is straightforward to show that the metric ( 5), which we rewrite for convenience is a solution to the field equations ( 6) with e i µ given by (11), η ij given by (14 i=1 ω i dφ i ), and the r-functions (A, B, Φ) are given in (7).
It is also straightforward to show that T, upon substituting (11) and ( 15) into (1), has the same expression as in (4) taking N = 5.
In concluding, there are two cylindrically rotating solutions to the field equations (6).The first solution, derived in this work (13), is much simpler and is used with a diagonal Minkowskian metric η ij = diag(1, −1, −1, −1, −1).The second solution (15), derived in Ref. [1] (with the global sign correction of its last term made in this work), includes extra terms, −(r 2 /l 2 ) ∑ n i<j (ω i dφ j − ω j dφ i ) 2 , the number of which depends on the number n of angular coordinates and is used with a non-diagonal Minkowskian metric η ij (14).
It is not clear why the authors of Refs.[1,8,9] used a non-trivial, non-diagonal, Minkowskian metric (14) that they claim to be the 'Minkowskian metric in cylindrical coordinates'.This has nothing to do with cylindrical coordinates!(see [10] for details).Moreover, such a non-diagonal Minkowskian metric has led to a more complicated rotating solution (15).As a consequence, the rotating solutions derived in [8,9] have the same complicated structure as the one derived in [1] and they can be simplified on removing the extra terms ∓(r 2 /l 2 ) ∑ n i<j (ω i dφ j − ω j dφ i ) 2 provided they are used with a diagonal Minkowskian metric A point to emphasize is that when evaluating the metric from the formula g µν = η ij e i µ e j ν one has to use η ij = ±diag(1, −1, −1, −1, • • • , −1) and not a non-diagonal expression.The tetrad defined in (11) forms a trivial pseudo-Cartesian system with metric η ij = diag(1, −1, −1, −1, • • • , −1).Another anonymous referee has supported our claim.

NON-DIAGONAL SOLUTIONS VERSUS DIAGONAL SOLUTIONS
From now on, a non-diagonal Minkowskian metric will be denoted by ηij .Let ηij and η ij be a nondiagonal and a diagonal Minkowskian metrics of dimension N, respectively.These two metrics may be related by a symmetric matrix R (R ij = R ji ) such that ηij = η ik R kj .For instance, ηij given by ( 14) and Let ḡµν and g µν be the corresponding spacetime metrics, respectively.The purpose of this section is to show that if (g µν , η ij ) is a rotating solution then ( ḡµν , ηij ) is a rotating solution too with similar geometrical properties.Using ḡµν = ηij e i µ e j ν and the fact that ḡµσ ḡσν = δ ν µ we obtain where η ik and R kj are the inverse matrices of η ik and R kj , respectively.Next, we evaluate ēi µ = ηij ḡµν e j ν .Using the expression (17) of ḡµν and the fact that R ij is symmetric, we obtain which along with the relation ēi µ = e i µ (true by definition since we are using the same tetrad but different Minkoskian metrics) imply that all the barred relevant entities entering the field equations ( 6) are equal to the non-barred entities.Hence, if the field equations are satisfied for the non-barred entities, they are automatically satisfied for the barred entities.
Our solution (13) includes four terms and the solution derived in Ref. [1], Eq. ( 15), includes the same four terms plus the extra term − r 2 l 2 ∑ n i<j (ω i dφ j − ω j dφ i ) 2 , which in the case N = 5, n = 2 takes the form − r 2 l 2 (ω 1 dφ 2 − ω 2 dφ 1 ) 2 .It is clear that these two solutions are manifestly different.Even if they share some similar geometrical and physical properties they are certainly different solutions because they cannot be related by a global coordinate transformation.

CONCLUDING REMARKS
We have thus shown that a trivial generalization of the transformation (8) to higher dimensional spacetimes is possible.By virtue of such a generalization we derived a simple cylindrically rotating solution of the form (5) with the last term −(r 2 /l 2 ) ∑ n i<j (ω i dφ j − ω j dφ i ) 2 removed.This newly derived metric along with A µ dx µ = Φ(r)(Ξdt − Σ n i=1 ω i dφ i ) is a solution to the field equations ( 6) provided the Minkowskian metric is diagonal η ij = diag(1, −1, −1, −1, • • • , −1) with the tetrad given by the expression (12) of [1].The rfunctions (A, B, Φ) are given in (7).
Another, non-trivial, generalization of ( 8) is also possible yielding a complicated cylindrically rotating solution of the form (5).This metric along with A µ dx µ = Φ(r)(Ξdt − Σ n i=1 ω i dφ i ) is a solution to the field equations (6) provided the Minkowskian metric is nondiagonal of the general form given in Eq. (44) of Ref. [8] and Eq.(41) of Ref. [9] with the tetrad given by the expression (12) of [1].The r-functions (A, B, Φ) are given in (7).
We have also shown that if (g µν , η ij ) is a rotating solution with η ij being diagonal, then ( ḡµν , ηij ) is another rotating solution with ηij = η ik R kj being non-diagonal and R ij is a symmetric matrix.These two rotating solutions have the same geometrical properties.