A nearly cylindrically symmetric source in the Brans-Dicke gravity as the generator of the rotational curves of the galaxies

Observation shows that the velocities of stars grow by approximately 2 to 3 orders of magnitude when the distances from the centers of the galaxies are in the range of $0.5$ kpc to $82.3$ kpc, before they begin to tend to a constant value. Up to know, the reason for this behavior is still a matter for debate. In this work, we propose a model which adequately describes this unusual behavior using a (nearly) cylindrical symmetrical solution in the framework of a scalar-tensor-like (the Brans-Dicke model) theory of gravity.


I. INTRODUCTION
Brans-Dicke (BD) gravitation [1] describes gravitational phenomena with the aid of a scalar field φ in addiction to the usual tensorial field in General Relativity (GR). The gravitational action of the BD theory, which gives rise to the dynamic equations, is given by (in the Jordan-Fierz physical The dimensionless parameter ω, called the BD coupling parameter, determines the deviation of the results obtained in this theory from those in GR. The lower the value of ω, the more different are the correspondent results. For instance, for the Solar System, where GR does extremely well, |ω| > 40000 [2]. Thus, one could think of discarding a scalar-tensor-like theory. Nevertheless, during the investigation of the cosmological evolution of the Universe with a scalar-tensorial theory it was concluded that GR is an attractor of the scalar-tensor theory, because the scalar field dynamics would have been gradually suppressed [3][4][5][6]. Therefore, a scalar-tensor gravitation theory seems to be most appropriate for describing sources that originated shortly after the Big Bang, when scalar field strength and its variation were still important.
It is now known that only less than 5% of what makes up the Universe is well understood, consisting of baryonic matter and radiation. Dark matter and dark energy, which dominate our Universe, have not yet their origin and evolution satisfactorily explained. Dark matter, supposedly present in the halo region of galaxies, seems to be the mechanism that causes the unusual behavior of the tangential velocity of the stars which is superior to that predicted by interaction with visible matter [7][8][9][10][11]. Moreover, in addition to the velocity larger than expected, as the distance of the stars from the center of the galaxy grows, their velocities tend to a constant value [9]. An alternative explanation to that behavior is to consider a scalar-tensor theory which might solve the rotational curves paradigm [12][13][14]. The results of the abovementioned references do reproduce the large distance behavior of the velocity with the expected order of magnitude. However, they give no information on how the intensity of that velocity evolves.
The stars of the galaxies studied in Ref. [9] are distributed along an approximately flat region.
Therefore, it is possible to suppose that a symmetrically cylindrical spacetime could describe the rotation curves. Symmetrically cylindrical sources are widely studied in several areas. Among these sources, some prominence can be given to the spinning cosmic strings, which would have originated from a spontaneous symmetry breaking, occurring from a phase transition of the primordial Universe . They have an angular velocity about the longitudinal axis of symmetry and, being straight strings, they preserve the Lorentz invariance along its symmetry axis. Many studies in the GR theory indicate that there are closed timelike curves (CTC's) at least in part of the spacetime around these strings [15]. As they would have formed in the primordial Universe, a recent work has demonstrated what occurs when, instead of GR, the description of these defects is analysed in the framework a BD gravity [16]. As in GR, the results indicated that CTC's are also present in the spacetime around these strings. There is a great deal of discussion as to whether CTC's exist or not in the Universe, since their confirmation would violate the causality principle.
It is argued that some unknown form of matter could allow the existence of CTC's [17], which can not be ruled out, since we do not have a full understanding of what most of the Universe is made up. The works [16,18] analyzed the energy quantization of a particle with a non inertial motion around a straight spinning cosmic string and concluded that it would be unstable, because it would have an angular velocity much higher than would be physically acceptable. Thus, if there was the formation of this type of string in the primordial Universe, it should not have survived.
The main goal of this paper is to present the full solution for a nearly cylindrically symmetric spacetime in the BD gravity and use it, as an alternative to dark matter, to describe the rotational curves of the galaxies in Ref. [9]. We show the solutions for the equations of motion for the action (1) when Lorentz invariance along the z longitudinal axis of symmetry is slightly violated, given that a straight source may be unstable, as happen with the spinning cosmic strings. The relevant invariance when dealing with the gravitational phenomena is to be taken with respect to general coordinate transformations. Since we are interested in the vacuum solution, less importance is devoted to the source of this spacetime. Nevertheless, one may bear in mind a (non-straight) spinning cosmic string with a very large radius. If one thinks of the source as a Lorentz violating object, then it is important to remember that many works have pointed out that the search for a theory of grand unification leads to a spontaneous breaking of Lorentz symmetry at the time of the primordial Universe [19,20], when we consider that the formation of the source has occurred and, therefore, also the BD gravity was dominant over GR. Moreover, in order not to lose the (nearly) cylindrical symmetry adopted, the ratio between the dt 2 and dz 2 coefficients of the line element is approximately unity.
We rescale the equations of motion obtained from the action in the Jordan-Fierz frame (1) to the Einstein (or conformal) frame in which the scalar and the tensor degrees of freedom do not mix. This makes the interpretation and handling of the solutions easier. It is important to strength that the qualitative results obtained in one frame are equivalent to those in the other. In any case, throughout the paper the interpretation of the results will be performed for the two frames, since both cases may be important in the description of a given gravitational system [21,22]. With the solutions of the dynamic equations at hand, we verified that the spacetime is indeed regular as evidenced by the analysis of the Kretschmann scalars. Depending on the values adopted for the integration constants, CTC's may appear or not in certain regions. The resulting spacetime adequately reproduces the motion of the stars of the galaxies studied in Ref. [9], either in the regions where CTC's exist or where the metric is Euclidean. This is the main result of our paper.
We also provide additional argumentation in favor of using BD gravity to address this problem.
This paper is organized as follows. In Section II, we first present the dynamic equations for the BD action in the Jordan-Fierz and Einstein frames. From the equations in the Einstein frame, the exact solution for a nearly cylindrically symmetric spacetime is obtained. In Section III, the physical properties of the spacetime obtained in the previous section are described. In Section IV, we show that the resulting spacetime adequately describes the rotational curves of Sc type galaxies. In Section V, it is discussed that the BD gravitation is essential for the aforementioned application. In Section VI, we summarize our findings and discuss the relevant conclusions. We leave for the Appendix complete tables encompassing relevant data and calculation results of the presented analysis.

BRANS-DICKE GRAVITY
The equations of motion from the action (1) in the Jordan-Fierz frame arẽ There is a more convenient way to handle the BD equations which is through the Einstein frame.
In this frame,g µν andφ from the Eqs. (2) are rescaling by the new dynamical variables g µν and φ, as in [23]:g where κ 2 = 1 2ω + 3 (5) and G is the effective gravitational Newtonian constant. Thus, in the Einstein frame, Eqs. (2) can be written as For the system we are about to consider, since the qualitative behavior is basically the same in Jordan-Fierz and Einstein frames, we shall present and interpret the results in the manageable Einstein frame and only occasionally make reference to the Jordan-Fierz frame.
The most general cylindrically symmetric line element to be addressed in this work is [15] where the angular coordinate varies, in principle, in the range 0 ≤ ϕ < 2π. By means of the cylindrical symmetry we have α, β and M as functions of the, strictly positive, radial distance r only. In order not to completely give up cylindrical symmetry we adopt β α ≈ 2. It is in this sense that we mean nearly cylindrical symmetry. Notice that an eventual source for this spacetime would not be straight and slightly breaking Lorentz invariance. The field equations were obtained in detail previously [16]. Therefore we will suppress the calculations. They read, for the vacuum case, where The prime ( ) in the above equations refers to derivative with respect to r.
Eqs. (11) and (10) are equal and Eq. (14) has the following solution for the scalar field where C 1 and C 2 are constants. Adding Eqs. (9) and (10) and using our definition of Ω (15) we Subtracting Eqs. (9) from (12) and substituting the term e 2α (M − M α ) 2 by the right-hand side of Eq. (17), we have If α − β = γ, Eq. (18) can be rewritten as which has the solution where C a and C b are constants. Thus, Eq. (12) can be rewritten as Substituting the term e 2α (M − M α ) 2 from the Eq. (22) by the right-hand side of the Eq. (17), When we use Eqs. (16) and (21) in Eq. (23), the solution for α is where C 3 , C 4 and k 1 are constants, with Now, when we use the solutions (16), (21) and (24) in Eq. (17), the solutions for M are where C 5 is a constant. Summarizing, the exact solutions for the system (9)- (14) read where r > 0, β(r) α(r) = C αβ ≈ 2 (e.g., the nearly cylindrical symmetry condition) and, for the solutions to be real, it is necessary that According with the Eqs. (27) and (28), C 3 can be written as where 0 < a C 3 < 1 and r min = min r, a value for the radial coordinate below which the system cannot be associated to the vacuum.
A final remark on recovering GR out from this scalar-tensorial formalism: As we are working with vacuum dynamical equations we cannot implement the condition ω → ∞ in order to keep track of the GR limit. However, it is possible equivalently to suppress the scalar field dynamics taking C 1 = 0. Henceforward, we shall be making contact with GR by means of this (C 1 = 0) necessary condition.

III. SPACETIME PROPERTIES
From the line element (8), with the solutions (21), (24) and (26), the Kretschmann scalars R 2 , R µν R µν , R αβµν R αβµν and C αβµν C αβµν , where C αβµν stands for the Weyl tensor, in the Einstein frame, are functions of α, β, M and their first and second derivatives with respect to r. In the Jordan-Fierz frame, the line element is rewritten as (3) and, with the same solutions (21), (24) and (26) plus the solution (16), the Kretschmann scalars are functions of α, β, M , φ and their first and second derivatives with respect to r, and of the parameter ω. In the two frames, the results show that the spacetime is well behaved for r > 0.
When the ϕ coefficient of the line element (8) is negative, CTC's appear. For this, it is necessary that the following inequality must hold Then, considering that and with the choice of certain constants for the solutions (24) and (26)  Thus, we can observe in Figure 1 that, for a same a C 3 , the region where there is no CTC (when g ϕϕ > 0) is larger when we are in the context of BD theory. This reasoning does not apply to a C 3 = 0.1 because, in this case, there is CTC in both theories for any value of the coordinate r.
For a C 3 = 0.9, there is no CTC up until to the region between r = 10 6 and r = 10 7 , for the BD gravitation, and between r = 10 1 and r = 10 2 , for the GR gravitation. When a C 3 = 0.5, in BD, that region is between r = 10 2 and r = 10 3 , whereas, in GR, that region is between r = 10 0 and r = 10 1 . The fact that the region where there are CTC's, defined from a certain distance from the z-axis (and not up to a certain distance, as with the straight spinning cosmic strings [15,16]) resembles what is predicted in the Gödel solution, when the CTC's appear when the source distance is from a so-called critical radius [24].
The canonical momenta associated with the Lagrangian (32) are where E and L are conserved quantities [25]. By Eqs. (33) and (34), we havė ϕ = e α (L e α +EM ) The Hamiltonian of this particle is Let us consider the stars as truly test particles moving in a circular motion in the equatorial plane around the center of a galaxy. Thus,ṙ = 0 andż = 0. Let us further consider that the Hamiltonian is normalized to be equal to − 1 2 for simplicity 2 [25]. Hence, the system composed by the equations allows solutions to be obtained for the energy E and the moment L: The solutions (42) and (43) applied in Eqs. (37) and (38) provide the angular velocity With the metric (8) and considering the prescription established by Chandrasekhar dτ 2 = −ds 2 [26], we have Considering u 0 = dt dτ , the Eq. (45) can be rewritten as where Therefore, the ratio between the tangential velocity v ϕ and the velocity of light c can be written (48) Figure 2 shows the ratio which is determined when we compute the limit of the ratio vϕ c for large r. This limit is represented in Figure 2 by a solid line. The behavior of the tangential velocities for the cases where Ω = Ω + and M = M + or Ω = Ω − and M = M − is similar to that observed in relation to the velocities of the stars farthest from the centers of the 21 Sc type galaxies studied in Ref. [9]. In Ref. [9], the velocity growth is noticed as the distances from the stars to the centers of the galaxies grow by approximately 2 to 3 orders of magnitude, from 0.5 kpc to 82.3 kpc, until they begin to tend to a constant value. This is similar to what is observed in Figure 2, when the growth and consequent tendency for a constant velocity occurs in the range of about 10 0 -10 3 m. This similarity of behaviors was the main motivation to determine if there would be a function (48) that would fit what was observed in Ref. [9].
The coefficient of determination, R 2 , is defined by the Eq. (53): where v ϕ is the calculated velocity by the Eq. (48), when Ω = Ω + and M = M + . Although the coefficient of determination R 2 is not the most suitable for testing a non linear regression model, it is adopted here as a standard definition for the values of the constants a C 3 , r min and v ϕ cst . The values chosen are those that result in the highest coefficient of determination with the uncorrected total sum of squares.
As for the constant C 5 , it is related to a C 3 , r min and v ϕ cst , and has been defined in such a way that CTC does not occur in the range of 0.5-82.3 kpc. For this to happen, the function M = M + , Eq. (26), was rewritten as where Thus, with a C 5 = 10 6 . When the best fit of the function (48) is sought for the 21 galaxies of Ref. [9], if the constant C 5 is calculated according to the Eq. (57), the first term of the Eq. (31) is always larger than the second term, and, therefore, there will be no CTC, because g ϕϕ > 0. However, the C 5 definition leads to v ϕ ≥ v ϕ cst in this case, which runs counter to the observations, since we expect v ϕ ≤ v ϕ cst . For this problem to be solved, simply multiply C 5 by −1, that is, use the same constant C 5 determined before, but with a positive sign. The tangential velocity will have adequate values, but this time CTC's cannot be precluded, since that g ϕϕ < 0. This happens because the values of C 5 for v ϕ ≤ v ϕ cst are out of the range of C 5 where CTC does not appear. With this fitting CTC occurs, because in order to have g ϕϕ > 0 it is necessary that − 4.3267 · 10 −17 < C 5g ϕϕ>0 < −4.3265 · 10 −17 , but this value leads to v ϕ ≥ v ϕ cst .
If we obey Eq. (59), when Figure 3 shows the behavior of the fitted function (48) for the NGC 701, NGC 1087, NGC 3672 and NGC 801 galaxies, when C 5 > 0, that is, when v ϕ takes suitable values with the observations, but g ϕϕ < 0.
The confidence intervals (CI's) of the velocities were calculated according to Eq. (60): where t is the 1 − 1−CI 2 quantile of the Student t-distribution with n − p degrees of freedom, being n the number of measurements taken for the galaxy and p = 1, since all constants of the function (48) have been previously defined. Besides, the standard error s e is with the estimated error variance s 2 equal to CI's equal to 68%, 95% and 99% for the model applied in the four galaxies are represented in the  Table I we see the values of the constants with which the highest coefficient of determination R 2 was obtained, providing the best fit of the function (48) for each one of the four galaxies at hand. Among the 21 studied galaxies, NGC 701 and NGC 3672 are the ones with the lowest coefficients of determination, and NGC 1087 and NGC 801 are those with the highest coefficients. These two last galaxies have smaller standard errors than the two first, ensuring a better accuracy of the calculated velocities.
There is an alternative to the function (48) in order to suitably fit velocities with the observations and also not allow the existence of CTC's. But this implies the need of both e −C 4 and C 5 to Table I: Values of the components g tt and g ϕϕ , the constants and the determination coefficients for the fits of the function (48) made for NGC 701, NGC 1087, NGC 3672 and NGC 801 galaxies, shown in Figure 3.
assume pure imaginary values, which will make the solution (26) be pure imaginary number and the solutions (21) and (24) become complex, but the component g ϕϕ (31) and the tangential velocity (48) remain real, as well as the metric (8). So, if we now consider Ω = Ω − instead of Ω = Ω + , and, for example, C 4 = π 2 i and C 5 = i, g ϕϕ will be positive and the tangential velocity will assume values of the expected order of magnitude. The results for the best fits of the NGC 701, NGC 1087, NGC 3672 and NGC 801 galaxies for this last alternative can also be seen in Figure 3 and in the Table I. As for the metric (8), when C 4 = π 2 i, the component g tt reads For the fit of the function (48), the conditions (27) and (28) were maintained for the metric to be real. Thus, C 3 + r 2k 1 > 0, which leads to g tt > 0. Being still g ϕϕ > 0 and the other spatial components g rr and g zz also positive, the metric (8) becomes an Euclidean metric.  As we make use of a nearly cylindrical symmetry, it is necessary that the ratio between the functions (21) and (24) gives β α = C αβ ≈ 2. Since we are interested in the behavior of v ϕ for the r = 0, 5-82.3 kpc range, it is important to calculate the limit for large r of the β α ratio. This limit is constant and can be written as leading to According to Eq. (65), when C αβ cst = 2.002 (that is, 1% bigger than 2) and, for example, v ϕ cst = 200 km/s (this value has the suitable order of magnitude, because, for all 21 studied galaxies, 96 km/s ≤ v ϕ cst ≤ 266 km/s), with c = 3 · 10 5 km/s and v ϕ cst the constant C a equals −1.002. Thus, considering we have C 1 = ±1.001 = 0, that is, the scalar field φ is not constant (see the Eq. (16)), which indicates that the behavior of tangential velocities of the stars seen so far can not be described by GR, and the use of BD gravitation is indeed required.
If the modeling option is in the framework of the GR gravity, i.e., with C 1 = 0, by the Eqs. (25) and (65), with C αβ cst = 2.002, we have By the Eq. (25), k 1 ≥ 0. So, the only valid solution for the system with the Eqs. (66) and (67) is 004. This last result leads to v ϕ cst = 0.707c i, which is not possible, because v ϕ cst must be real.
With the line element in the Jordan-Fierz frame (3), it is also possible to determine the angular velocity 3 : where When we insert the velocity (68) in the Eq. (48), we have the ratio vϕ c in the Jordan-Fierz frame, where the parameter ω now appears explicitly. In this case, the limit of this ratio for large r tends to the single value When ω → ∞, the limit of this ratio is This last result is identical to that one found for the ratio vϕ cst c in the Einstein frame (see Eq. (49)). This happens because when ω → ∞,g µν = g µν . Thus, for a large ω, the behaviors of the velocity v ϕ are the same in the Einstein and Jordan-Fierz frames. For example, the Figure 4 shows As ω grows, the behavior of the velocity v ϕ is close to that one found in the Einstein frame (also shown in Figure 3). For instance, when ω = 10 10 , the constants a C 3 , r min and v ϕ cst coincide in both frames (see the Tables I and II). It is also interesting to note that, for ω = 10 2 or ω = 10 4 , the coefficient of determination in the Jordan-Fierz is somewhat higher than that found when ω = 10 10 or in the Einstein frame.
As usual, when using the Jordan-Fierz frame along with a large ω, the BD gravitation must provide (almost) the same results of GR. Nevertheless, the obtained good results with a small ω ∼ 10 2 attest that such gravitation is the correct option, reinforcing what was already argued in the first part of this Section. In addition, because of the difference between the fitted curves in the different frames, accurate measurements of the stars velocities may indicate which of the two frames would be most appropriate for the quantitative description of the rotational curves of the galaxies.  Table II.
From ω = 10 2 , the fit of the function (48) results in almost identical curves.  The solution presented in this paper is shown to be suitable to generate the rotational curves of the galaxies, because it reproduces the velocity growth behavior in the order of magnitude compatible with the experimental data [9]. For this, the solution indicates that it is necessary to admit the existence of CTC's in a certain range. We notice, however, that this type of possibility is far from consensual. On the one hand the chronological protection conjecture says that the laws of physics prohibit the emergence of CTC's [27]. On the other hand, interesting works point out that these same laws allow the CTC's and they can appear naturally [28][29][30][31].
Returning to the main goal of our paper, we present a model in which CTC's are ruled out, but the metric must be Euclidean. Theories of quantum gravity, which seek to describe phenomena of the time of grand unification theory (GUT), have indeed made use of such a stratagem [32].
Finally, we would like to stress that the BD gravity adequately describes the observational data as an effective theory of gravity coming from an already unknown completely theory of a quantum gravity. r (kpc) [9] v ϕ msr (km/s) [9] v ϕ (km/s) s e (km/s) Confidence interval 68% 95% 99%