{'fraction_non_alphanumeric': 0.03828975761529583, 'fraction_numerical': 0.013097806995533697, 'mean_word_length': 4.3014099494546425, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Nonsingular Kerr-Sen Black Hole

We present a novel solution describing four dimensional rotating regular charged black hole in the low energy heterotic string theory. This present solution is characterized by its mass, charged, as well as angular momentum. Some aspect including event horizon, ergosphere, and the angular velocity are discussed. It is shown that later, our expression for event horizon and ergosphere cannot be obtained analytically. On the other hand, to ensure whether our solution describe a nonsingular rotating charged black hole, we use two type of scalar invariant; contracted Ricci tensor and Kretschmann scalar. From our result, the inclusion of charge does not alter the regularity of a black hole. We also proposed a natural prescription to have rotating charged black hole solution with a minimal violation of weak energy conditions.


I. INTRODUCTION
String theory, until now, is one of the best candidate to explain quantum gravity and their related phenomenology in extra dimensions. Fortunately, it is always possible to study the nature of string theory from our dimensions point of view. If we have large mass parameter compared to its Planck mass and curvature below the Planck scale, then static charged black hole in low energy string theory exist [1]. Rotating version in the low energy obtained by Sen [2] in 1992 with an extra term that differ from GHS one. He obtained an exact solution solution of rotating charged black hole in four dimensional spacetime by using generating technique [2,3] and employ Kerr [4] as a vacuum solution. Hence the solution in Einstein frame characterized by its mass, spin, and electric charge. Switched off the charge parameter makes the solution goes back to the well known Kerr, whereas for vanisihing spin combined with some rescale, the solution reduces to the Einstein-Maxwell-dilaton (GHS) black hole.
We can infer that the vacuum solution from string theory is the same as ordinary Einstein gravity. In charged static black hole, coupling between the nontrivial dilaton and Maxwell tensor makes the solution differ from Einstein gravity. The only difference is that the appearance of the surface singularity. For rotating case, although we have the same family on charged black hole, Kerr-Sen and Kerr-Newman solution appeared to be different since it has, again, a nontrivial dilaton. There are several feature that distinct between the two solution. One of the example is that the hidden conformal properties that has been studied in [5] do not have Q-picture hidden conformal symmetry compared to the rotating Einstein-Maxwell [6]. Most of them can be done from the black hole (with ring singularity) under the main assumption from validity of the energy condition. Since collapsing matter forms a spacetime singularity at the origin [7]. Inspired by Bardeen [8], regular (nonsingular) black hole have been considered that has a spherically symmetric, static, asymptotically flat spacetime, and regular at the center. While there has been no comprehensive solid theory on quantum gravity, several toy models like black hole without singularity has been reported. The physical source of a regular black hole with magnetic monopole and nonlinear electrodyanmics has been obtained by Ayon-Beato and Garcia [10][11][12]. These spacetime has an event horizon and no pathological singularity, and the magnetic monopole in reference [12] has been verified with the conserved charge evaluated in the boundary. Of course this present spacetime is not a vacuum solution to GR but they can be introduced as exotic field or modified gravity. They could spared the singularity because they meet the weak energy condition, but not the strong one [14]. From this extension literature, one can find a rotating version corresponds to our previous reference on static spacetime via Newman-janis algorithm [13]. A rotating regular black hole was obtained by Bambi-Modesto [14]. They use two type of static regular black hole; Hayward [15] and Ayon-Beato magnetic monopole [12]. Another rotating regular black hole with N-J algorithm has been reported in [16]. In the presence of cosmological constant, Neves-Sa [17] obtained the solution with synge g-method. From the rotating regular literature (both in vacuum and charged case), they all showed that the rotation makes the weak energy condition to be violated, but the violation can be very small depends on how we treat the parameter. The shared feature also applied to another vacuum nonsingular rotating black hole [18].
While there has been extensive literature on vacuum rotating regular black hole, there are no comprehensive study on regular black hole in the low energy string theory. Hence, in this paper, we examine charged rotating regular black hole in the low energy heterotic string theory. We will not follow the Newman-Janis algorithm and synge g-method in this context. Instead, we use generating technique as the reference [2], and using Ghosh regular black hole as a seed solution [18] since it obeys the vacuum solution in large region. The output of this solution is in string frame and we will use the conformal transformation to obtain the metric in Einstein frame. We utilize the conserved quantity in order to obey the three physical parameters (M, Q, J). This paper is structured as follows. Section II, we review charged rotating black hole in the low energy heterotic string theory both in string and Einstein frame combined with the several aspect such as angular velocity and the Hawking temperature. Section III contains of our novel solution by adding the regular factor in the Kerr metric and using generating technique into regular (nonsingular) Kerr-Sen black hole. The vacuum regular solution are already obtained by [18]. After obtaining the solution in the Einstein frame, we analyze several aspect including horizon, ergosphere, as well as angular velocity. Moreover, to test the existence of regular charged black hole, we use two type of scalar invariant that is contracted Ricci tensor as well as the Kretschmann scalar. Section IV, we investigate the weak energy condition and compare it to the other reference. Section V, we conclude of our statement.

II. KERR-SEN BLACK HOLE
Sen [2] in 1992 obtained the exact solution of rotating charged black hole in low energy heterotic string theory. Since He used Kerr black hole as a seed, the solution later commonly referred to as a Kerr-Sen. The corresponding effective action in string frame for this context whereg is a determinant of metric tensor g µν , R is a Ricci scalar, Maxwell field, Φ as a scalar dilaton, and is a 3-form tensor field which contain the antisymmetric 2-form tensor field and Chern-Simons term. By using the conformal transformation we can get the action in Einstein frame When all the non gravitational field is zero, the action reduces to the well known Einstein-Hilbert action. Let us discuss the metric solution in Einstein frame. In Boyer-Lindquist coordinate, Kerr-Sen solution can be written as The complete solution (5) characterized by mass M , angular momentum J, and charge Q.
Rotational parameter a is defined as a ratio between angular momentum and its mass. Their related non gravitational fields are and Where ρ 2 KS = r 2 + 2br + a 2 cos 2 θ, ∆ KS = r 2 + 2br − 2M r + a 2 , and b = Q 2 /2M . To get the solution, one can obtained Hassan-Sen transformation [3] resulting a string frame metric solution. Using conformal transformation combined with the conserved mass, angular momentum, and charge, we obtained the solution in (5). We will discuss the conserved quantity in the next section. However, the black hole's event horizon is obtained when ∆(r ± ) = 0 satisfied. From this condition, we conclude that the Kerr-Sen black hole have at least two radius of horizon where the r ± is the outer and inner horizon respectively. Moreover, we can obtained the angular velocity of a Kerr-Sen black hole by setting the Killing vector (∂ t + Ω∂ φ ), null at the horizon. Thus, we get The Hawking temperature at the horizon also reads When b parameter is vanish, the Einstein frame metric reduces to Kerr black hole and simultaneously we have vanisihing non gravitational field. Furthermore, vanishing rotational parameter combined with rescaled r → r − 2b, gives us the GMGHS black hole describing charged black hole in string theory [1].

III. A NONSINGULAR KERR-SEN BLACK HOLE
In the previous section, we already discuss Kerr where the accent sign ( ) is stands for the new field. The definition for matrix M and Ω in the (11) is as follows and where the K, g −1 , A, and η is the matrix of K µν , g µν , A µ , and η µν . The dot sign in the matrix Υ indicates the zero matrix whereas I 7x7 is an identity matrix 7x7. Important to note that the matrix K µν is given by whereas the flat metric convention is η µν = diag (1, 1, 1, −1). From the definition, we can conlcude that the matrix M contains all of the set field. The idea of this step is that the action (1) will be reduced to the Einstein-Hilbert action when all fields (Φ, A µ , B µν ) vanish (vacuum case as a seed solution). So by using Hassan-Sen transformation (11), we will obtain a new solution with non vanisihing fields {B µν , A µ , φ }. The metric tensor expression that obey these conditions is specifically written as where ρ 2 = r 2 + a 2 cos 2 θ, ∆ = r 2 + a 2 − 2M re −k/r , and k is a free parameter that measure the deviation from the classical rotating Kerr metric and assume to be positive [18]. When k = 0, the solution should reduces to the Kerr black hole solution. However, after finding the non zero matrix M and M , we obtained the solution in string framẽ The corresponding non gravitational field reads These metric and their related matter fields constitute a complete solution for non singular black hole in string frame. By using the conformal transformation (3), we obtain the metric solution in Einstein frame whereρ 2 = ρ 2 + 2me −k/r r sinh 2 (α/2). Although the metric solution in Einstein frame has been successfully obtained, the black hole must also admit the no-hair theorem. The theorem is stated that the black hole solution will be always externally observable by three classical parameters: mass M , charge Q, and angular momentum J. In fact, obtaining those three parameter is not as simple as its stated. One can verify that the three parameters (M, Q, J) living in the asymptotically flat black hole by evaluating Komar integral or ADM formulation. In this paper, we verify the no hair theorem variable with the ADM one by using the gravitational Hamiltonian [19]. With appropriate choices of lapse and shift function, we obtained the mass and angular momentum where S t two surface at constant t, k = σ AB k AB is the extrinsic curvature of S t embedded in hypersurface Σ t , k 0 is the extrinsic curvature embedded in flat space, K ab is the extrinsic curvature in a boundary. The minus sign in angular momentum was added to recover the right hand rule. However, we can obtained the explicit form for mass and angular momentum by setting the Eintein frame solution (19) in the asymptotic region (r m). Straightforward calculation yields M = m cosh 2 (α/2), and J = am cosh 2 (α/2).
it is shown that the corresponding ADM mass and angular momentum is the same as Kerr-Sen black hole [2]. For total charge Q, one can obtained by using Maxwell equation and employ Gauss theorem to make the total charge evaluated in a boundary. The formalism where dS µν = 1 4 ε µναβ dx α ∧ dx β , which gives the same electric charge to that of Kerr-Sen solution.
After obtaining all of these conserved quantity, it will be more convenient to express m, and α in terms of physical parameters M , J, and Q obtained from equation (22) and (23).
After some algebra, we get where b = Q 2 /2M . The rotational parameter a have the same result with KS black hole; ration between angular momentum and its mass. By inserting relation (24) into equation (19) combined with the other hyperbolic identities, we find a new class of solution describing non singular rotating charged black hole in the low energy heterotic string theory; and the matter fields using (24) where ϕ 2 = r(r + 2be −k/r ) + a 2 cos 2 θ, and Ξ = r(r + 2be −k/r ) + a 2 − 2M re −k/r . The field both metric and matter solution we obtained constitute a complete solution of the non singular Kerr-Sen black hole system in string and Einstein frame. Thus, the ordinary Kerr-Sen black hole can be obtained by switched off the free parameter k, where (k > 0). When we switched off the b parameter, the solution should reduce to vacuum case [18]. It is also shown that from the metric part, the black hole is asymptotically-flat. From the matter part, the field are vanish in large r. In this paper, we only analyze the aspect only in the Einstein frame.
Now, let us calculate the event horizon of the black hole. The radius can be obtained by the condition Ξ(r ± ) = 0. The equation is given by r 2 ± + 2br ± e −k/r ± + a 2 − 2M r ± e −k/r ± = 0.
As we can see that the equation above contains transcendent functions. This expression, unfortunately, cannot solve r analytically with general parameter (b, M, k, a). Thus, numerical methods is the best way we can achieve by plotting the equation (27)  static limit surface which g tt (r sl ) = 0 where the issue is the same as equation (27)  The region between the r sl and the horizon r + is called ergosphere. Penrose [20] stated that enery can be extracted from blackhole's ergoregion. The shape of this region is strongly depends on free parameter, charge, and rotational parameter. The exact horizon and static limit radius are shown in Table. I. The first table is for ergosphere with fix rotational and charge parameter, and the second table represent the ergosphere with rotational and free parameter as a fix value. The shared feature with ref. [18] is that the ergosphere region grows as free parameter k gets larger. By adding some charge into equation (14) (which is our solution in equation (25)), we still have a same properties (see Table. I). To analyze further, consider a nonsingular Kerr-Sen spacetime with constant space coordinate (r, θ, φ) and the ergosphere is located at r < r sl + . It is shown that the metric become positive (ds 2 > 0) when r 2 + 2bre −k/r + a 2 cos θ − 2M re −k/r < 0. Hence, the observer follow an orbit of asymptotic time translation Killing field ξ a = ∂ ∂t a resulting the angular velocity that can be written as where the angular velocity of a black hole is Ω H (r + ) when Ξ(r + ) = 0. As an observer shift further (r → ∞), the angular velocity goes to zero, which means an observer does not feel the frame dragging effect from the vicinity of a black hole. When k vanish, the expression (30) reduces to the well known Kerr-Sen black hole's angular velocity [2]. On the other hand, what we mention earlier is that the space time is regular everywhere when the radius shift into the origin of a black hole in Boyer-Lindquist coordinate. In order to ensure whether the metric (25) has a regularity or not, we approch this problem by looking at the behavior of the scalar invariant; R 2 = R µν R µν as well as the Kretschmann scalar K = R µναβ R µναβ . This present scalar invariant can be obtained with a non vanisihing Ricci and Riemann tensor. Unfortunately, their both expression is quite complicated and hence we would not show it here. Both scalar quantity (R 2 and K) are shown in Fig. 3. We can infer that the addition of charge does not alter the regularity of a black hole. Our scalar invariant plot for vanishing charge is the same as [18].

IV. WEAK ENERGY CONDITION
In this subsection, we study weak energy condition for nonsingular Kerr-Sen spacetime.
In order to investigate the matter associated with (25), we employ an orthonormal tetrads in which the energy momentum tensor is diagonal [14,[16][17][18]21] where this present tetrads formalism corresponds to the locally non-rotating frame According to the matrix, we must assures that the nonzero components of our tetrads formalism has a real root. Notice that g xx > 0 and also ensures a regular and positive metric. However, from g tt − Ωg tφ = − (r 2 + 2bre −k/r + a 2 x 2 )(r 2 + a 2 + 2bre −k/r − 2M re −k/r ) (r 2 + a 2 + 2bre −k/r )(r 2 + 2bre −k/r + a 2 x 2 ) + 2M it is shown that the equation above is negative. Thus, the sign from matrix (31) must be selected depending on the considered region.
2k r a 2 x 2 + r 2 a 2 kx 2 + r 2 (k − 2r) −2b 2 r 2 (b − M ) −k 2 + 2kr + r 2 r 3 e k/r a 2 x 2 + r 2 + 2br 3 , and ρ + P 2 = 1 Where we switched off the charge parameter b, we have the same expression as vaccum nonsingular rotating black hole obtained by Ghosh [18]. However, it is not quite simple to investigate whether our solution satisfied the energy condition or not by only considering three equation above. So we have plotted weak energy condition in Fig. 4 with respect to radius and several parameter (k and x = cos θ). What we can conclude of our result and figure is that the violation of weak energy condition cannot be prevented if we have nonzero rotational and free parameter, since with b = 0, the ρ + P 2 appeared to be violated as in ref. [18] (also see [17], for the solution in ref. [14] with a presence of cosmological constant).
The inclusion of charge in the low energy string theory does not makes the energy condition satisfied, but the violation can be very small depending on how we treat the parameter [14,[16][17][18].

V. CONCLUSION
In this paper, we examine a rotating charged regular black hole in the low energy heterotic string theory in four dimensional spacetime. We using generating technique inspired by [2] and [3] in vacuum nonsingular rotating black hole [18], where k in the literature referred as free parameter that determine the deviation from Kerr  Moreover, our novel solution reduces to Minkowksi spacetime in large r and seems regular at the origin in Boyer-Lindquist coordinate. To assure this behavior, we employ two type of scalar invariant; contracted Ricci tensor (R 2 ) and Kretschmann scalar (K) that build from contracted Riemann tensor. Since the expression for both scalar invariant are complicated, we instead using numerical plot to study the behavior of invariant quantity. We have plot it with two different parameter (k and b). It has been reported that the addition of charge does not alter the regularity of a black hole. Thus we still have a charged regular black hole in the low energy string theory. The event horizon of this object can be obtained by solving Ξ(r + ) = 0. Since we have a transcendental function, the radius cannot solved analytically and unfortunately we have the same story for static limit radius. So we shift our way by obtaining the radius from plot. As we can see the horizon shrinks faster than the static limit radius as free and charge parameter gets larger. At the same time, a black hole has larger area for energy extraction in the ergopshere as k and b gets larger. The observer that stand still in the ergosphere becomes spacelike and hence should follow an orbit of asymptotic time translation Killing vector resulting the angular velocity as a function of radius and polar angle. On the other hand, to analyze the matter content of our solution (25), we investigate the weak energy condition. We introduce the usual orthonormal tetrads where corresponds to the locally non-rotating frame inspired by [14,17,18,21]. We already shown that the rotation lead to violation of WEC, as in ref. [14,17,18] . Thus we can infer that by adding some charge in Ghosh solution, does not make the WEC satisfied, but the violation can be very small depending on how we set the charge or free parameter.
With our solution, it is tempting to investigate some phenomenological aspect like geodesics, deflection, and thermodynamics quantity. These several points are now under