Quantum probes of timelike naked singularities in $2+1-$ dimensional power - law spacetimes

The formation of naked singularities in $2+1-$ dimensional power - law spacetimes in linear Einstein-Maxwell and Einstein-scalar theories sourced by azimuthally symmetric electric field and a self-interacting real scalar field respectively, are considered in view of quantum mechanics. Quantum test fields obeying the Klein-Gordon and Dirac equations are used to probe the classical timelike naked singularities developed at $r=0$. We show that when the classically singular spacetimes probed with scalar waves, the considered spacetimes remains singular. However, the spinorial wave probe of the singularity in the metric of a self-interacting real scalar field remains quantum regular. The notable outcome in this study is that the quantum regularity/singularity can not be associated with the energy conditions.


I. INTRODUCTION
In recent years, general relativity in 2 + 1− dimensions has been one of the attractive arena for understanding the general aspects of black holes physics. The main motivation of this attraction is the existing tractable mathematical structure when compared with the higher dimensional counterparts. The preliminary works in this field was popularized by the black hole solution of Banados-Teitelboim-Zanelli (BTZ), a spacetime sourced by a negative cosmological constant [1][2][3]. Extension of 2 + 1− dimensional solutions to Einstein-Maxwell (EM) cases followed in [4] and its Massive Gravity version is given in [5]. The static and rotating charged blacks hole in 2 + 1−dimensional Brans-Dicke theory was studied in [6] and rotating black holes with torsion was considered in [7]. The 2 + 1−dimensional charged black hole with nonlinear electrodynamic coupled to gravity has been studied in [8] and with a scalar hair has been given in [9]. The peculiar feature in the aforementioned studies in the EM theory is that the electric field is considered in the radial direction.
In recent decades, the physical properties of the solutions presented in both linear and nonlinear EM theory have been investigated by researchers. The solutions admitting black holes are analyzed in terms of thermodynamical aspects such as temperature and entropy [3,10,11]. Furthermore, the AdS/CFT correspondence which relates thermal properties of black holes in the AdS space to a dual CFT is another important achievement of 2 + 1−dimensional gravity [12].
On the other hand, the solutions admitting naked singularities are not analyzed in details and hence, it requires further care as far as the weak cosmic censorship hypothesis is concerned. Therefore, the resolution of singularities becomes important not in 3 + 1−dimensional gravity, but also in lower or even in higher dimensional gravity as well. Because of the scales where these singularities are forming, their resolutions requires a consistent theory at these small scales. The theory of quantum gravity seems to be the most promising theory, however, it is still "under construction". An alternative method for resolving the singularities is proposed by Horowitz and Marolf (HM) [13] by developing the work of Wald [14]. According to this method, the classical notion of a curvature singularity that is regarded as geodesics incompleteness with respect to point particle probe is replaced by quantum singularity with respect to wave probes.
In 2 + 1−dimensional gravity, the method of HM has been used in the following works to probe the timelike naked curvature singularities: The BTZ black hole is considered in [15]. The EM extension of BTZ black hole both in linear and nonlinear theory and in Einstein-Maxwell -dilaton theory is considered in [16]. The formation of naked singularities for a magnetically charged solution in Einstein-power-Maxwell-theory is considered in [17]. Occurrence of naked singularities in Einstein-nonlinear electrodynamics with circularly symmetric electric field is considered in [18]. In these studies, the timelike naked singularity is probed with waves that differs in spin structure. Namely, the bosonic and the fermionic waves are used that obey the Klein-Gordon and the Dirac equation. The common outcome in these studies is that the naked singularity remains quantum singular when it is probed with bosonic waves. However, probing the singularity with fermionic waves has revealed that only the magnetically charged solution in Einstein-power-Maxwell theory is singular. The other spacetimes considered so far behaves quantum regular against fermionic waves.
The solution in linear EM theory with azimuthally symmetric electric field in 2 + 1−dimension was given in [19]. To our knowledge, the singularity structure of the solution presented in [19] has not been studied so far. Our motivation in this study is to investigate the solutions admitting naked singularity in [19]. In our analysis, the classical naked singularity will be probed with quantum fields obeying the massless Klein-Gordon and Dirac equations.
The electric field component in EM extensions was con-arXiv:1312.0147v2 [physics.gen-ph] 4 Sep 2014 sidered to be radial while the possibility of a circular electric field went unnoticed. Recall from the Maxwell equations, ∇ × E ∼ ∂B ∂t and ∇ × B ∼ ∂E ∂t that the possibility of a constant E (= E 0 = F tθ = constant) and gradient form of B, (i.e. B = ∇b, for b a scalar independent of time) may occur. When confined to 2 + 1−dimensions such E and B satisfy Maxwell's equations trivially with singularity due to a physical source at r = 0. It is this latter case that we wish to point out and investigate in this paper.
The paper is organized as follows. In section II, the solution obtained in [19] is rederived and the structure of the spacetime is briefly introduced. In section III, the definition of quantum singularity is summarized and the timelike naked singularity in the considered spacetime is analyzed with quantum fields obeying the Klein-Gordon and Dirac equations. The paper is concluded with a conclusion in section IV.

II. REDERIVATION OF THE SOLUTION
We start with the Einstein-Maxwell action given by in which R is the Ricci scalar and The circularly symmetric line element is given by where A(r) and B(r) are unknown functions of r and 0 ≤ θ ≤ 2π. The electric field ansatz is chosen to be normal to radial direction and uniform i.e., in which E 0 =constant [18]. The dual field is found as It is known that the integral of F gives the total charge. Let us note that even in a flat space with A = B = 1 we obtain a logarithmic expression for the charge, i.e. Q (r) ∼ ln r. This electric field is derived from an electric potential one-form given by in which a 0 and b 0 are constants satisfying a 0 + b 0 = 1. The Maxwell's equation is trivially satisfied. Note that the invariant of electromagnetic field is given by Next, the Einstein-Maxwell equations are given by in which Having F known one finds and as the only non-vanishing energy-momentum component.
To proceed further, we must have the exact form of the Einstein tensor components given by and in which a 'prime' means d dr . The field equations then read as follows and The above field equations admit the following solutions for A and B : in which χ > 0 is an integration constant and without loss of generality we set it to χ = 1. Hence the line element becomes This is a black point solution with the horizon at the origin which is the singular point of the spacetime with Kretschmann scalar The solution has a single parameter which is the electric field E 0 . Setting E 0 = 0 makes the solution the (2 + 1) −dimensional flat spacetime. Note also that for the choice E 0 = 1, from (18), we obtain a conformally flat metric with conformal factor r 2 . It is observed that the strength of E 0 serves to increase the degree of divergence of the scalar curvature. Based on our energy momentum tensor components one finds that the energy density and the radial and tangential pressures are given by and Therefore the weak energy conditions (WEC) i.e., i) ρ ≥ 0, ii) ρ + p ≥ 0 and iii) ρ + q ≥ 0 all are satisfied (taking into account that χ is positive). The strong energy conditions are also satisfied i.e., the WECs together with iv) ρ + p + q ≥ 0. Dominant Energy Condition (DEC) i.e. p ef f ≥ 0 and Causality Condition (CC) i.e. 0 ≤ p ef f ≤ 1 are also easily satisfied knowing that p ef f = p+q 2 = 0.

III. SINGULARITY ANALYSIS
It has been known that the spacetime singularities inevitably arise in the Einstein's theory of relativity. It describes the "end point" or incomplete geodesics for timelike or null trajectories followed by classical particles. Among the others, naked singularities which is visible from outside needs further care as far as the weak cosmic censorship hypothesis is concerned. It is believed that, naked singularity forms a threat to this hypothesis. As a result of this, understanding and the resolution of naked singularities seems to be extremely important for the deterministic nature of general relativity. The general belief in the resolution of the singularities is to employ the methods imposed by the quantum theory of gravity. However, the lack of the consistent quantum theory leads the researchers to alternative theories in this regard. String theory [20,21]and loop quantum gravity [22] constitute two major study fields in resolving singularities. Another alternative method; following the work of Wald [14], was proposed by Horowitz and Marolf (HM) [13], which incorporates the "self-adjointness" of the spatial part of the wave operator. Hence, the classical notion of geodesics incompleteness with respect to point-particle probe will be replaced by the notion of quantum singularity with respect to wave probes.
In this paper, the method proposed by HM will be used in analyzing the naked singularity. This method in fact has been used successfully in (3 + 1) and higher dimensional spacetimes. The complete list in these spacetimes given in [23][24][25][26][27][28][29][30][31][32][33][34][35][36]. The main purpose in these studies is to understand whether these classically singular spacetimes turn out to be quantum mechanically regular if they are probed with quantum fields rather than classical particles. The idea is in analogy with the fate of a classical atom in which the electron should plunge into the nucleus but rescued with quantum mechanics. The main concepts of this method which can be applied only to static spacetimes having timelike singularities is summarized as follows.
Let us consider, the Klein-Gordon equation for a free particle that satisfies i dψ dt = √ A E ψ, whose solution is ψ (t) = exp −it √ A E ψ (0) in which A E denotes the extension of the spatial part of the wave operator. If the wave operator A is not essentially self-adjoint, in other words if A has an extension, the future time evolution of the wave function ψ (t) is ambiguous. Then, the HM method defines the spacetime as quantum mechanically singular. However, if there is only a single self-adjoint extension, the wave operator A is said to be essentially selfadjoint and the quantum evolution described by ψ (t) is uniquely determined by the initial conditions. According to the HM method, this spacetime is said to be quantum mechanically nonsingular. The essential self-adjointness of the operator A, can be verified by using the deficiency indices and the Von Neumann's theorem that considers the solutions of the equation and showing that the solutions of (23), do not belong to Hilbert space H. (We refer references; [23,[37][38][39] for detailed mathematical analysis.)

A. Klein-Gordon Fields
The massless Klein-Gordon equation for a scalar wave can be written as The Klein-Gordon equation can be written for the metric (18), by splitting the temporal and spatial part as This equation can also be written as where A is the spatial operator given by and according to the HM method, it is subjected to be investigated whether its self-adjoint extensions exists or not. This is achieved by assuming a separable solution to equation (23) in the form of ψ(r, θ) = R(r)Y (θ), which yields the radial equation as with c the separation constant. The essential selfadjointness of the spatial operator A requires that neither of the two solutions of the above equation is square integrable over all space L 2 (0, ∞).The square integrability of the solution of the above equation for each sign ± is checked by calculating the squared norm of the obtained solution in which the function space on each t = constant hypersurface Σ is defined as H ={R| R < ∞}. The squared norm for the metric (2) is given by, The squared norm is investigated for three different cases of the value of electric field E 0 .
1. Case 1: For this particular case, equation (28) transforms to whose solution is given by such that,ĉ = c ± i.
2. Case 2: 0 < E 2 0 < 1 In this case, equation (28) becomes, and the solution is given by, 3. Case 3: For this choice of E 0 ,equation (28) remains the same and the solution is given by, Note that J 0 (x) and N 0 (x) in equations (31), (33) and (34) are the first kind Bessel and Neumann functions respectively, with integration constants a i in which i = 1...6. Our calculations have revealed that, in general, in each case for appropriate a i , the squared norm R 2 < ∞, which is always square integrable. Hence, the spatial part of the operator is not essentially self-adjoint. Therefore, the classical singularity at r = 0 remains quantum singular as well when probed with massless scalar, bosonic waves.

B. Dirac Fields
In (2 + 1) −dimensional curved spacetimes, the formalism leading to a solution of the Dirac equation was given in [40]. This formalism has been used in [15] and in our earlier studies [16][17][18]. The Dirac equation in (2 + 1)dimensional curved background for a free particle with mass m is given by where Γ µ (x) is the spinorial affine connection given by Since the fermions have only one spin polarization in (2 + 1) −dimension, the Dirac matrices γ (j) can be given in terms of Pauli spin matrices σ (i) so that where the Latin indices represent internal (local) frame. In this way, where η (ij) is the Minkowski metric in (2 + 1) −dimension and I 2×2 is the identity matrix. The coordinate dependent metric tensor g µν (x) and matrices σ µ (x) are related to the triads e (i) µ (x) by where µ and ν stand for the external (global) indices. The suitable triads for the metric (18) are given by, With reference to our earlier studies in [16][17][18], the following ansatz is used for the positive frequency solutions: The radial part of the Dirac equations near r → 0, is considered as in the case of Klein-Gordon equation for three different values of the electric field intensity E 0 . The behavior of the radial part of the Dirac equation both for 0 < E 2 0 < 1 and E 2 0 > 1 is given by whose solution is For E 2 0 = 1 case the Dirac equation simplifies to where a prime denotes the derivative with respect to r, and the solution is given by where ξ = E 2 − n(n + 1) and b i (i = 1...4), in equations (44) and (46) are the integration constants.
The solutions to equations (43) and (45) are investigated near r = 0. These solutions represents fermionic waves propagating in the considered spacetime where the electric field is constant and azimuthally symmetric. Square integrability of the obtained solutions for (43) and (45) are investigated by calculating the squared norm defined in (29). Our analysis have shown that, irrespective of the value of E 2 0 , the obtained solutions are square integrable. As a consequence, the arbitrary wave packet can be written as and the initial condition Ψ(0, x) is enough to predict the evolution of the fermionic wave. Thus, the considered spacetime behaves nonsingular when probed with fermionic waves.

IV. CONCLUSION
In this paper, the formation of timelike naked singularity in 2 + 1−dimensional linear Einstein-Maxwell theory sourced by azimuthally symmetric electric field is investigated from quantum mechanical point of view. We showed in the considered spacetime that, the spin of the particles/waves is effective in the resolution of the singularity. When probing the singularity with bosonic (spin 0) waves, accumulation of particles occur such that the classically singular spacetime becomes quantum singular. This result seems to be generic for 2 + 1−dimensional spacetimes (similar results are also obtained in [15][16][17][18]). On the other hand, the spacetime is nonsingular when fermionic (spin 1 2 ) fields are used to probe the singularity. That is, the fermionic , i.e. repulsive property of the particles protects the spacetime against singularity formation. This result matches with the earlier studies on BTZ [15], matter coupled BTZ [16] and 2 + 1−dimensional nonlinear Einstein-Maxwell theory sourced by azimuthally symmetric electric field [18]. However, 2 + 1−dimensional magnetically charged solutions in Einstein-power-Maxwell theory, in contrast to the electric case, was shown to be the only quantum singular spacetime against fermionic probe [17]. This result manifest at the same time the distinction between electric and magnetic type Maxwell solutions when we probe the spacetime with fermions.
The result obtained in this paper, and also in earlier studies along this direction has indicated that, the quantum regularity of the classically singular spacetime crucially depends on the wave that we probe the singularity. In order to understand the generic behavior of the 2+1−dimensional spacetimes, more spacetimes should be investigated. So far, vacuum Einstein, Einstein-Maxwell (both linear and nonlinear) and Einstein-Maxwell-dilaton solutions are investigated. As a future research, timelike naked singularities in 2 + 1−dimensional Einstein-scalar and Einstein-Maxwell-scalar solutions should also be investigated from quantum mechanical point of view.