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arXiv:1004.2572v2 [gr-qc] 11 Aug 2010

Quantum singularities in (2 + 1) dimensional matter coupled black hole spacetimes

O. Unver and O. Gurtug∗

Department of Physics, Eastern Mediterranean University,

G. Magusa, North Cyprus, Mersin 10 - Turkey.

Quantum singularities considered in the 3D BTZ spacetime by Pitelli and Letelier (Phys. Rev.

D77: 124030, 2008) is extended to charged BTZ and 3D Einstein-Maxwell-dilaton gravity space-

times. The occurence of naked singularities in the Einstein-Maxwell extension of the BTZ spacetime

both in linear and non-linear electrodynamics as well as in the Einstein-Maxwell-dilaton gravity

spacetimes are analysed with the quantum test fields obeying the Klein-Gordon and Dirac equa-

tions. We show that with the inclusion of the matter fields; the conical geometry near r = 0 is

removed and restricted classes of solutions are admitted for the Klein-Gordon and Dirac equations.

Hence, the classical central singularity at r = 0 turns out to be quantum mechanically singular

for quantum particles obeying Klein-Gordon equation but nonsingular for fermions obeying Dirac

equation. Explicit calculations reveal that the occurrence of the timelike naked singularities in the

considered spacetimes do not violate the cosmic censorship hypothesis as far as the Dirac fields are

concerned. The role of horizons that clothes the singularity in the black hole cases is replaced by

repulsive potential barrier against the propagation of Dirac fields.

I.INTRODUCTION

In recent years, (2 + 1) dimensional, Banados-Teitelboim-Zanelli (BTZ) [1] black hole has attracted much attention.

One of the basic reasons for this attraction is that the BTZ black hole has a relatively simple tractable mathematical

structure so that it provides a better understanding of investigating the general aspects of black hole physics since

the BTZ black hole carries all the characteristic features such as the event horizon and Hawking radiation, it can be

treated as a real black hole. Another motivation to study BTZ black hole is the AdS/CFT correspondence which

relates thermal properties of black holes in the AdS space to a dual CFT. In view of these points, the unresolved black

hole properties belonging to (3 + 1) or higher dimensional black holes at the quantum level, make the BTZ black hole

an excellent background for exploring the black hole physics.

Another interesting subject is the study of naked singularities that can be considered as a threat to the cosmic

censorship hypothesis. Compared to the black holes, the naked singularities are less understood. Today, there is no

common consensus either on the structure or the existence of the naked singularities.

Recently, Pitelli and Letelier (PL) [2] have analysed the occurrence of naked singularities for the BTZ spacetime

from quantum mechanical point of view. In their analysis, the criteria proposed by Horowitz and Marolf (HM)

[3] is used. The classical naked singularity is studied with the quantum test particles that obey Klein-Gordon and

Dirac equations. They confirmed that the naked singularity is ”healed” when tested by massless scalar particles or

fermions without introducing extra boundary conditions. However, for massive scalar particles additional information

is needed. Despite the recent developments on the concept of quantum singularities [4], our understanding of naked

singularities as far as quantum gravity concerned is still far from being complete.

The purpose of this paper is to analyse the naked singularities within the context of the quantum mechanics that

form in the matter coupled 2 + 1 dimensional black hole spacetimes. Our motivation here is to investigate the

effect of the matter fields on the quantum singularity structure of the BTZ spacetime because the surface at r = 0

for the BTZ black hole is not a curvature singularity, but is a singularity in the causal structure. This situation

changes when a matter field is coupled. This is precisely the case that we shall elaborate on in this article. For this

purpose we consider the charged BTZ spacetime both in linear and non-linear electrodynamics. This is analogous

to a kind of Einstein-Maxwell extension of the work presented in [2]. Furthermore, we extend the analysis to cover

2 + 1 dimensional Einstein-Maxwell-dilaton coupled black hole spacetime. The presence of charge both in linear and

non-linear case and also the dilaton field modify the resulting spacetime geometry significantly. Near the origin, the

spacetime is not conic and true curvature singularity develops at r = 0. Consequently, the spacetime geometry that

we have investigated in this study differs when compared with the case considered in [2].

The plan of the paper is as follows. In section II, we first review the definition of quantum singularities for general

static spacetimes. In section III, we consider the charged BTZ black hole in non-linear electrodynamics. Klein-Gordon

∗Electronic address: ozlem.unver@emu.edu.tr; Electronic address: ozay.gurtug@emu.edu.tr

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and Dirac fields are used to test the quantum singularity. We also discuss the Sobolev norm which is used first time in

this context by Ishibashi and Hosoya [11]. In sections IV and V, we consider charged BTZ in linear electrodynamics and

dilaton coupled 3D black hole spacetime in the Einstein-Maxwell and Einstein - Maxwell - dilaton theory, respectively.

Dirac and scalar fields are used to judge the quantum singularity. The paper ends with a conclusion in section VI.

II.A BRIEF REVIEW OF QUANTUM SINGULARITIES

In classical general relativity, the spacetime is said to be singular if the evolution of timelike or null geodesics is

not defined after a proper time. Horowitz and Marolf, based on the pioneering work of Wald [9], have proposed the

criteria to test the classical singularities with quantum test particles that obey the Klein-Gordon equation for static

spacetime having timelike singularities. According to this criteria, the singular character of the spacetime is defined

as the ambiguity in the evolution of the wave functions. That is to say, the singular character is determined in terms

of the ambiguity when attempting to find self-adjoint extension of the operator to the entire space. If the extension

is unique, it is said that the space is quantum mechanically regular. The brief review is as follows:

Consider a static spacetime (M,gµν) with a timelike Killing vector field ξµ. Let t denote the Killing parameter and

Σ denote a static slice.The Klein-Gordon equation on this space is

?∇µ∇µ− M2?ψ = 0.

(1)

This equation can be written in the form of

∂2ψ

∂t2=

?

fDi??

fDiψ

?

− fM2ψ = −Aψ,

(2)

in which f = −ξµξµand Di is the spatial covariant derivative on Σ. The Hilbert space

square integrable functions on Σ. The domain of the operator A, D(A) is taken in such a way that it does not enclose

the spacetime singularities. An appropriate set is C∞

Σ. Operator A is real, positive and symmetric therefore its self-adjoint extensions always exist. If it has a unique

extension AE, then A is called essentially self-adjoint [10]. Accordingly, the Klein-Gordon equation for a free particle

satisfies

?L2(Σ)?

is the space of

0(Σ), the set of smooth functions with compact support on

idψ

dt=

?

AEψ,

(3)

with the solution

ψ (t) = exp

?

−it

?

AE

?

ψ (0).

(4)

If A is not essentially self-adjoint, the future time evolution of the wave function ( Eq. (4)) is ambiguous. Then,

Horowitz and Marolf define the spacetime quantum mechanically singular. However, if there is only one self-adjoint

extension, the operator A is said to be essentially self-adjoint and the quantum evolution described by Eq.(4) is

uniquely determined by the initial conditions. According to the Horowitz and Marolf criterion, this spacetime is said

to be quantum mechanically nonsingular. In order to determine the number of self-adjoint extensions, the concept

of deficiency indices is used. The deficiency subspaces N±are defined by ( see Ref. [11] for a detailed mathematical

background),

N+= {ψ ∈ D(A∗),

N−= {ψ ∈ D(A∗),

A∗ψ = Z+ψ,

A∗ψ = Z−ψ,

ImZ+> 0}

ImZ−> 0}

with dimension n+

with dimension n−

(5)

The dimensions ( n+,n−) are the deficiency indices of the operator A. The indices n+(n−) are completely independent

of the choice of Z+(Z−) depending only on whether Z lies in the upper (lower) half complex plane. Generally one

takes Z+ = iλ and Z− = −iλ , where λ is an arbitrary positive constant necessary for dimensional reasons. The

determination of deficiency indices then reduces to counting the number of solutions of A∗ψ = Zψ ; (for λ = 1),

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A∗ψ ± iψ = 0 (6)

that belong to the Hilbert space H. If there is no square integrable solutions ( i.e. n+= n−= 0), the operator A

possesses a unique self-adjoint extension and it is essentially self-adjoint. Consequently, a sufficient condition for the

operator A to be essentially self-adjoint is to investigate the solutions satisfying Eq. (6) that do not belong to the

Hilbert space.

III.

(2 + 1)- DIMENSIONAL BTZ SPACETIME COUPLED WITH NON-LINEAR ELECTRODYNAMICS.

The action describing (2 + 1) - dimensional Einstein theory coupled with non-linear electrodynamics is given by

[12],

S =

?

√g

?

1

16π(R − 2Λ) + L(F)

?

d3x.

(7)

The field equations via variational principle read as,

Gab+ Λgab= 8πTab,

(8)

Tab= gabL(F) − FacF

c

bL,F,

(9)

∇a

?FabL,F

?= 0(10)

in which L,F stands for the derivative of L(F) with respect to F =1

the energy momentum tensor (9) has a vanishing trace. The trace of the tensor gives,

4FabFab. The non-linear field is chosen so that

T = Tabgab= 3L(F) − 4FL,F.

(11)

Hence, to have a vanishing trace, the electromagnetic Lagrangian is obtained as

L = c | F |3/4,

(12)

where c is an integration constant. With reference to the paper [12], the complete solution to the above action is

given by the metric,

ds2= −f(r)dt2+ f(r)−1dr2+ r2dθ2,

(13)

where the metric function f(r) is given by,

f(r) = −m +r2

l2+4q2

3r.

(14)

Here m > 0 is the mass, l2= −Λ−1the case Λ > 0 (Λ < 0), that corresponds with an asymptotically de-Sitter (anti de-

Sitter) spacetime, and q is the electric charge. This metric represents the BTZ spacetime in non-linear electrodynamics.

If Λ = 0, we have an asymptotically flat solution coupled with Coulomb-like field. The Kretschmann scalar which

indicates the occurrence of curvature singularity is given by,

K =12

l4+ 6β2

r6.

(15)

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in which β =

and q, this singularity may be clothed by single or double horizons. (See the reference [12] for details).

However, for specific values of Λ , m and q the central curvature singularity becomes naked and it deserves to be

investigated within the framework of quantum mechanics. To find the condition for naked singularities the metric

function is written in the following form,

4q2

3. It is clear that r = 0 is a typical central curvature singularity. According to the values of Λ , m

f(r) = −m

r

?

r +˜Λr3−4˜ q2

3

?

,

(16)

where˜Λ =

the condition for a naked singularity. In order to find the roots, we set f(r) = 0 which yields r3+r

standard procedure is followed for a solution via a new variable defined by r = z −

to 27˜Λ3z6− 36˜Λ2˜ q2z3− 1 = 0. This equation can be solved easily and the final answer is

Λ

mand ˜ q2=

q2

m. Since the range of coordinate r varies from 0 to infinity, the negative root will indicate

˜Λ−4˜ q2

3˜Λ= 0. The

1

3˜Λzthat transforms the equation

r = u1/3−

1

3˜Λu1/3,

(17)

in which u =

with the following equation

12˜ q2˜Λ±2

?

18˜Λ2

3˜Λ(12˜ q4˜Λ+1)

, with a constraint condition 3˜Λ

?

12˜ q4˜Λ + 1

?

> 0. After some algebra, we end up

r = a1/3

??

1 ±b

a

?1/3

+

?

1 ∓b

a

?1/3?

,

(18)

where a =

(18) is always positive. Hence, the only possibility for a negative root is a < 0. This implies˜Λ < 0. Therefore, the

condition 12˜ q4˜Λ + 1 < 0 is imposed from the constraint condition. As a result, for a naked singularity,˜Λ < −

Λ < −m3

Our aim now is to investigate the quantum singularity structure of the naked singularity that may arise if the

constant coefficients satisfy Λ < −m3

2˜ q2

3˜Λand b =

?

3˜Λ(12˜ q4˜Λ+1)

9˜Λ2

. It can be verified easily that the expression inside the curly bracket in Eq.

1

12˜ q4 or

12q4 should be satisfied.

12q4.

A. Klein-Gordon Fields:

Using seperation of variables, ψ = R(r)einθ, we obtain the radial portion of Eq. (6) as

R′′

n+(fr)′

fr

R′

n−

n2

fr2Rn−M2

f

Rn±

i

f2Rn= 0,

(19)

where a prime denotes derivative with respect to r.

1. The case of r → ∞ :

The Coulomb-like field in metric function (14) becomes negligibly small and hence, the metric takes the form

ds2≃ −

?r2

l2

?

dt2+

?r2

l2

?−1

dr2+ r2dθ2.

(20)

This particular case overlaps with the results already reported in [2]. Hence, no new result arises for this particular

case. This is expected because the effect of source term vanishes for large values of r.

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2. The case of r → 0 :

The case near origin is topologically different compared to the analysis reported in [2]. Here, the spacetime is not

conic. The approximate metric near origin is given by,

ds2≃ −

?β

r

?

dt2+

?β

r

?−1

dr2+ r2dθ2.

(21)

This metric can also be interpreted as the 2 + 1 dimensional topological Schwarzchild-like black hole geometry.

For the solution of the radial equation (19), we assume that a massless case (i.e. M = 0), and ignore the term

±iRn

f2 ( since it is negligible near the origin). Then it takes the form,

R′′

n−n2

βrRn= 0,

(22)

whose solution is

Rn(r) = C1n

√rI1(k) + C2n

√rK1(k).

(23)

where I1(k) and K1(k) are the first and second kind modified Bessel functions and k =

modified Bessel functions for real ν ≥ 0 as r → 0 are given by;

?

4n2r

β. The behaviour of the

Iν(x) ≃

1

Γ(ν + 1)

?−?ln?x

2

?x

?+ 0.5772...?,

2

?ν

,

(24)

Kν(x) ≃

2

ν = 0

Γ(ν)

?2

x

?ν,ν ?= 0

?

,

thus I1(k) ∼

behaviour of the integral for I1(k) ≈

combination is also square integrable. It follows the solution (23) belonging to the Hilbert space H and therefore

the operator A described in Eq.(6) is not essentially self-adjoint. So, the naked singularity at r = 0 is quantum

mechanically singular if it is probed with quantum particles.

Another approach to remove the quantum singularity is to choose the function space to be the Sobolev space

(H1) which is used first time in this context by Ishibashi and Hosoya [11]. Here, the function space is defined by

H = {R |? R ?< ∞}, where the norm defined in 2 + 1 dimensional geometry as,

1

Γ(2)

?k

2

?and K1(k) ∼

Γ(1)

2

?2

k

?. Checking for the square integrability of the solution (23) requires the

?r4dr and K1(k) ≈

?dr which are both convergent as r → 0. Any linear

? R ?2∼

?

rf−1| R |2dr +

?

rf |∂R

∂r|2dr,

(25)

which involves both the wave function and its derivative to be square integrable. The failure in the square integrability

indicates that the operator A is essentially self-adjoint and thus, the spacetime is ”wave regular”. According to this

norm, the first integral is square integrable while the second integral behaves for the functions I1(k) as ≈?

mechanically wave singular. It should be noted that the Sobolev space is not the natural quantum mechanical Hilbert

space.

0dr and

K1(k) integral vanishes. As a result, the wave functions are square integrable and thus the spacetime is quantum

B.Dirac Fields:

We apply the same methodology as in [2] for finding a solution to Dirac equation. Since the fermions have only one

spin polarization in 2 + 1 dimensions [13], Dirac matrices are reduced to Pauli matrices [14] so that,

γ(j)=

?

σ(3),iσ(1),iσ(2)?

,

(26)

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where latin indices represent internal (local) indices. In this way,

?

γ(i),γ(j)?

= 2η(ij)I2×2,

(27)

where η(ij)is the Minkowski metric in 2 + 1 dimensions and I2×2is the identity matrix. The coordinate dependent

metric tensor gµν(x) and matrices σµ(x) are related to the triads e(i)

µ (x) by

gµν(x) = e(i)

σµ(x) = eµ

µ(x)e(j)

ν (x)η(ij),

(28)

(i)γ(i),

where µ and ν are the external (global) indices.

The Dirac equation in 2 + 1 dimensional curved spacetime for a free particle with mass M becomes

iσµ(x)[∂µ− Γµ(x)]Ψ(x) = MΨ(x),

(29)

where Γµ(x) is the spinorial affine connection and is given by

Γµ(x) =1

4gλα

?

e(i)

ν,µ(x)eα

(i)(x) − Γα

νµ(x)

?

sλν(x),

(30)

sλν(x) =1

2

?σλ(x),σν(x)?.

(31)

The causal structure of the spacetime indicates that there are two singular cases to be investigated. The asymptotic

case, r → ∞ has already been analysed by PL. The case of r → 0 is not conical so there is a topological difference in

the spacetime near r = 0 . Hence, the suitable triads for the metric (21) are given by,

e(i)

µ(t,r,θ) = diag

??β

r

?1

2

,

?r

β

?1

2

,r

?

,

(32)

The coordinate dependent gamma matrices and the spinorial affine connection are given by

σµ(x) =

??r

?

β

?1

2

σ(3),i

?β

?β

r

?1

?1

2

σ(1),iσ(2)

r

?

,

(33)

Γµ(x) =

−βσ(2)

4r2

,0,i

2

r

2

σ(3)

?

.

Now, for the spinor

Ψ =

?ψ1

ψ2

?

,

(34)

the Dirac equation can be written as

i

?r

?r

β

?1

?1

2∂ψ1

∂t

−

?β

?β

r

?1

?1

2∂ψ2

∂r

+i

r

∂ψ2

∂θ

−1

4

?β

?β

r3

?1

?1

2

ψ2− Mψ1= 0,

(35)

−i

β

2∂ψ2

∂t

−

r

2∂ψ1

∂r

−i

r

∂ψ1

∂θ

−1

4

r3

2

ψ1− Mψ2= 0.

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The following ansatz will be employed for the positive frequency solutions:

Ψn,E(t,x) =

?

R1n(r)

R2n(r)eiθ

?

einθe−iEt.

(36)

The radial parts of the Dirac equation for investigating the behaviour as r → 0, are

R′′

1n+α1

√rR′

1n+

α2

r3/2R1n= 0,

α4

r3/2R2n= 0.

(37)

R′′

2n+α3

√rR′

2n+

where α1=2M−E

2M√β, α2=−7E+4M(4n+1)

16M√β

, α3=2M+E

2M√βand α4=7E−4M(4n+3)

16M√β

. Then, the solutions are given by;

R1(r) = e−b

R2(r) = e−b′

2ρ{C1√ρWhittakerM(a,1,bρ) + C2√ρWhittakerW(a,1,bρ)},

2ρ{C3√ρWhittakerM(a′,1,b′ρ) + C4√ρWhittakerW(a′,1,b′ρ)}

,a′=9E−8M(1+2n)

4(2M+E)

,b = 2α1,and b′= 2α3.

When we look for the square integrability of the above solutions, we obtained that both functions WhittakerMand

WhittakerW are square integrable near ρ = 0 (or r = 0) for both R1(r) and R2(r). One has,

where ρ =√r, a =−9E+8M(1+2n)

4(2M−E)

?

rf−1| R |2dr ≈

?

ρ6e−bρ[WhittakerM(a,1,bρ)]2dρ < ∞,

(38)

and

≈

?

ρ6e−bρ[WhittakerW(a,1,bρ)]2dρ < ∞.

(39)

We note that these results are verified first by expanding the Whittaker functions in series form up to the order of

O(ρ6) and then by integrating term by term in the limit as r → 0.

The set of solutions for the Dirac equation for the spacetime (21) is given by

Ψn,E(t,x) =

?

e−b

2ρ?C3n√ρWhittakerM(a′,1,b′ρ) + C4n√ρWhittakerW(a′,1,b′ρ)?eiθ

and an arbitrary wave packet can be written as

2ρ?C1n√ρWhittakerM(a,1,bρ) + C2n√ρWhittakerW(a,1,bρ)?

e−b′

?

einθe−iEt,

Ψ(t,x) =

+∞

?

n=−∞

Cn

?

e−b

2ρ√ρ(WhittakerM(a′,1,b′ρ) + WhittakerW(a′,1,b′ρ))eiθ

2ρ√ρ(WhittakerM(a,1,bρ) + WhittakerW(a,1,bρ))

e−b′

?

einθe−iEt

(40)

where Cnis an arbitrary constant. Hence, initial condition Ψ(0,x) is sufficient to determine the future time evolution

of the particle. The spacetime is then quantum regular when tested by fermions.

IV.2+1 DIMENSIONAL BTZ SPACETIME WITH LINEAR ELECTRODYNAMICS

A.Klein-Gordon Fields:

The metric for the charged BTZ spacetime in linear electrodynamics is given by [15],

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ds2= −f(r)dt2+ f(r)−1dr2+ r2dθ2,

(41)

with the metric function

f(r) = −m +r2

l2− 2q2ln(r

l),

(42)

where q is the electric charge and m > 0 is the mass and l2= Λ−1. The Kretschmann scalar is given by,

K =12

l4−8q2

r2l2+4q4

r4,

(43)

which displays a power-law central curvature singularity at r = 0. According to the values of m, l and q, this central

singularity is clothed by horizons or it remains naked. Our interest here is to investigate the quantum mechanical

behaviour of the naked singularity. In order to find the condition for naked singularity, we set f(rh) = 0 and the

solution for l = 1 is

rh= exp

?

−m

2q2−1

2LambertW

?

−1

q2e−m/q2??

,

in which LambertW(.) represents the Lambert function [20]. Fig.1 displays (unmarked region) the possible values of

m and q that result in naked singularity.

The causal structure is similar to the case considered in the previous section. There are two singular cases to be

investigated. The case for r → ∞ is approximately the same case considered in [2]. Therefore, the results reported

by PL are valid for this case as well. For small r values, the approximate metric can be written in the following form

ds2≈ −?2q2| ln(˜ r) |?dt2+?2q2| ln(˜ r) |?−1dr2+ r2dθ2,

l<< 1. The radial equation becomes

(44)

in which ˜ r =r

R′′

n+1

˜ r

?

1 +

1

ln ˜ r

?

R′

n+

n2

2q2r2ln ˜ rRn= 0,

(45)

sincer

l<< 1, the solution can be written in terms of zeroth order first and second kind modified Bessel functions,

Rn(x) = C1nI0(

√2n

q

x) + C2nK0(

√2n

q

x),

(46)

where −x2= ln ˜ r. As ˜ r → 0, x → ∞. The behaviour of the modified Bessel functions for x >> 1 are I0(x) ≃

and K0(x) ≃?π

?

These results indicate that charged BTZ black hole in linear electrodynamics is quantum mechanically singular

when probed with quantum test particles that obey Klein-Gordon equation.

If we use the Sobolev norm (25), the second integral which involves the derivative of the wave function I0(x) ≃

becomes ≈?x−2e2x(2x − 1)2dx. Numerical integration has revealed that as x → ∞, ∼?x−2e2x(2x − 1)2dx → ∞.

numerically as x → ∞, ∼

BTZ black hole in linear electrodynamics is quantum mechanically wave regular if and only if the arbitrary constant

parameter is C2n= 0 in Eq.(46).

ex

√2πx

2xe−x. These functions are always square integrable for x → ∞, that is

rf−1| R |2dr ≈

?

xe−2x2f−1| R |2dx < ∞.

ex

√2πx

On the other hand for the wave function K0(x) ≃

?π

2xe−x, the second integral in the Sobolev norm is solved

?x−2e−2x(2x + 1)2dx < ∞ which is square integrable. As a result, charged coupled

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Consequently, if the naked singularity both in linear and non-linear electrodynamics is probed with quantum test

particles, the following results are obtained:

1) In classical point of view, the Kretschmann scalar in non-linear case diverges faster than in the linear case.

2) In quantum mechanical point of view, if the chosen function space is Sobolev space, spacetime remains singular

for non-linear case, but the spacetime can be made wave regular for linear case.

From these results we may conclude that the structure of the naked singularity in the non-linear electrodynamics

is deeper rooted than the singularity in the linear case.

B. Dirac Fields:

The effect of the charge when r → ∞ does not contribute as much as the term that contains the cosmological

constant. Therefore, we ignore the mass and the charged terms in the metric function (42). This particular case has

already been analysed in [2]. The contribution of the charge is dominant when r → 0. The Dirac equation for the

metric (44) is solved by using the same method demonstrated in the previous section. We obtain the radial equation

in the limit r → 0 as

R′′

j+1

rR′

j−Rj

4r2= 0,j = 1,2 (47)

whose solution is given by

Rj(r) = C1j

√r +C2j

√r.

(48)

where C1j and C2j are arbitrary constants. The solution given in Eq.(48) is square integrable. The arbitrary wave

packet can be written as,

Ψ(t,x) =

+∞

?

n=−∞

??

R1(r)

R2(r)eiθ

??

einθe−iEt

(49)

Thus, the spacetime is quantum mechanically regular when probed with fermions.

V. 2+1 DIMENSIONAL EINSTEIN - MAXWELL - DILATON GRAVITY

In this section, we consider 3D black holes described by the Einstein-Maxwell-dilaton action,

S =

?

d3x√−g

?

R −B

2(▽φ)2− e−4aφFµνFµν+ 2ebφΛ

?

,

(50)

where R is the Ricci scalar, φ is the dilaton field, Fµνis the Maxwell field and Λ, a, b, and B are arbitrary couplings.

The general solution to this action is given by [19],

ds2= −f(r)dt2+4r

4

N−2dr2

N2γ

4

Nf(r)

+ r2dθ2,

(51)

where

f(r) = Ar

2

N−1+

8Λr2

(3N − 2)N+

8Q2

(2 − N)N.

(52)

Here, A is a constant of integration which is proportional to the quasilocal mass (A =

constant and Q is the charge. The dilaton field is given by

−2m

N), γ is an integration

Page 10

10

φ =2k

Nln

?

r

β (γ)

?

(53)

in which β (γ) is a γ related constant parameter. Note that, the above solution for N = 2 contains both the vacuum

BTZ metric if one takes Q = A = 0 and the BTZ black hole if A < 0,Q = 0. However, if the constant parameters are

chosen appropriately, the resulting metric represents black hole solutions with prescribed properties. For example,

when N =6

3, the metric function given in equation (52) becomes

5, A = −5m

f(r) = −5m

3r2/3+25Λ

6

r2+25Q2

3

,

(54)

and therefore the corresponding metric is

ds2= −f(r)dt2+αr4/3dr2

f(r)

+ r2dθ2,

(55)

where α =

25

9γ

10

3

is a constant parameter.

The Kretschmann scalar for this solution is given by

K =

25

?

12m2r

5

3 + 5Λr3?

55Λr

4

3 − 4m

?

+ 40r

1

3Q2?

2

?

5Q2− mr

2

3

?

− 5Λr2??

81α2r7

,

(56)

which indicates a central curvature singularity at r = 0 that is clothed by the event horizon. To find the location of

horizons, gttis set to zero and we have

r2−2m

5Λr

2

3+2Q2

Λ

= 0.

(57)

There are three possible cases to be considered.

Case 1: If

15Λ

hole.

Case 2: If

15Λ

there is only one horizon.

Case 3: If

Λ>

?2m

illustrates the timelike character of the singularity at r = 0. Our aim in this section is to investigate the behaviour of

this naked singularity when probed with Klein-Gordon and Dirac fields in the framework of quantum mechanics.

Q2

Λ<?2m

Q2

Λ=?2m

Q2

?3

?3

2, the equation admits two positive roots indicating inner and outer horizons of the black

2, this is an extreme case and the equation (57) has one real positive root. This means that

15Λ

?3

2, there is no real positive root and the solution does not admit black hole so that the

singularity at r = 0 is naked. With reference to the detailed analysis given in [19], the Penrose diagram of the solution

A.Klein-Gordon Fields:

The radial equation for the metric (55) is obtained for the massles case(M = 0) as,

R′′

n+

?

fr

1

3

?′

fr

1

3

R′

n−αn2

fr

2

3Rn±iαr

4

3

f2Rn= 0.

(58)

The behaviour of the radial equation as r → 0 is

R′′

n+1

3rR′

n−k2

r

2

3Rn= 0,

(59)

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11

where k =3αn2

25Q2. The solution is given by

Rn(r) = C1ncosh

?3k

2r2/3

?

+ iC2nsinh

?3k

2r2/3

?

.

(60)

Both solutions are square integrable in Hilbert space, that is,

quantum mechanically singular when probed with quantum particles obeying Klein-Gordon equation.

If we use the Sobolev norm,

?rgrr | R |2dr < ∞. Therefore, the spacetime is

? R ?2∼

?

rgrr| R |2dr +

?

rg−1

rr|∂R

∂r|2dr,

although the first integral of the solution is square integrable, the second integral for C1n = 0 fails to be square

integrable and the spacetime is quantum mechanically wave regular.

B. Dirac Fields:

The Dirac equation can be written as,

i

√fψ1,t−

√f

√αr

√f

√αr

2

3ψ2,r+i

rψ2,θ−

?

?

5(−2m + 15Λr

36√αr√f

4

3)

+

√f

2√αr

√f

2√αr

5

3

?

?

ψ2− Mψ1= 0,

(61)

−i

√fψ2,t−

2

3ψ1,r−i

rψ1,θ−

5(−2m + 15Λr

36√αr√f

4

3)

+

5

3

ψ1− Mψ2= 0

where f is given in (54). By using the same anzats as in (36), the radial part of the Dirac equation becomes,

R′′

n+

a1

r1/3R′

n+

a2

r2/3Rn= 0,n = 1,2 (62)

in which a1=3Q√3α−m

15Q2

, a2=

−108Q2αn(1+n)+m(m−6Q√3α)

900Q4

. The solution becomes,

Rn(r) = r1/6e−3a1

4r2/3

C1nWhittakerM(

a1

a2

4√

4√

1−4a2,3

a1

a2

4,3

2r2/3?a2

1− 4a2)+

1− 4a2)

C2nWhittakerW(

1−4a2,3

4,3

2r2/3?a2

,

(63)

which is square integrable. This is verified first by expanding the Whittaker functions in series and then by integrating

term by term in the limit as r → 0. Consequently, the spacetime is quantum mechanically regular when probed with

Dirac fields.

VI.CONCLUSION

Matter coupled 2+1 dimensional black hole spacetimes are shown to share similar quantum mechanical singularity

structure as in the case of pure BTZ black hole. The inclusion of matter fields changes the topology and creates true

curvature singularity at r = 0. The effect of the matter fields allows only specific frequency modes in the solution of

Klein-Gordon and Dirac fields. If the quantum singularity analysis is based on the natural Hilbert space of quantum

mechanics which is the linear function space with square integrability L2, the singularity at r = 0 turns out to be

quantum mechanically singular for particles obeying the Klein- Gordon equation and regular for fermions obeying the

Dirac equation. We have proved that the quantum singularity structure of 2 + 1 dimensional black hole spacetimes

are generic for Dirac particles and the character of the singularity in quantum mechanical point of view is irrespective

whether the matter field is coupled or not. This result suggests that the Dirac fields preserve the cosmic censorhip

hypothesis in the considered spacetimes that exhibit timelike naked singularities. Instead of horizons (that clothes the

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12

singularity in the black hole cases) the repulsive barrier is replaced against the propagation of Dirac fields. However,

for particles obeying Klein - Gordon fields, the singularity becomes worse when a matter field is coupled.

However, we have also shown that in the charged BTZ ( in linear electrodynamics ) and dilaton coupled black hole

spacetimes specific choice of waves exhibit quantum mechanical wave regularity when probed with waves obeying

Klein-Gordon equation, if the function space is Sobolev with the norm defined in (25). The singularity at r = 0

is stronger in the non-linear electrodynamic case. It should be reminded that, one may not feel comfortable to use

Sobolev norm in place of natural linear function space of quantum mechanics.

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Figure Captions

Figure1: Plot of rhfor different values of m and q. Marked region displays the formation of black hole, unmarked

region shows the formation of naked singularity.

Page 14

This figure "ozay.jpg" is available in "jpg"? format from:

http://arxiv.org/ps/1004.2572v2