# Quantum singularities in a model of f(R) Gravity

**ABSTRACT** The formation of a naked singularity in a model of f(R) gravity having as

source a linear electromagnetic field is considered in view of quantum

mechanics. Quantum test fields obeying the Klein-Gordon, Dirac and Maxwell

equations are used to probe the classical timelike naked singularity developed

at r=0. We prove that the spatial derivative operator of the fields fails to be

essentially self-adjoint. As a result, the classical timelike naked singularity

remains quantum mechanically singular when it is probed with quantum fields

having different spin structures.

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**ABSTRACT:**We present an exponential $F(R)$ modified gravity model in the Jordan and the Einstein frame. We use a general approach in order to investigate and demonstrate the viability of the model. Apart from the general features that this models has, which actually render it viable at a first step, we address the issues of finite time singularities, Newton's law corrections and the scalaron mass. As we will evince, the model passes these latter two tests successfully and also has no finite time singularities, a feature inherent to other well studied exponential models.General Relativity and Gravitation 04/2013; · 1.90 Impact Factor - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**A definition of quantum singularity for the case of static spacetimes has recently been extended to conformally static spacetimes. Here the theory behind quantum singularities in conformally static spacetimes is reviewed, and then applied to a class of spherically symmetric, conformally static spacetimes, including as special cases those studied by Roberts, by Fonarev, and by Husain, Martinez, and N\'u\~nez. We use solutions of the generally coupled, massless Klein-Gordon equation as test fields. In this way we find the ranges of metric parameters and coupling coefficients for which classical timelike singularities in these spacetimes are healed quantum mechanically.Physical review D: Particles and fields 02/2013; 87(10).

Page 1

arXiv:1205.5125v2 [gr-qc] 25 Jul 2012

Quantum singularities in a model of f(R) gravity

O. Gurtug∗and T. Tahamtan†

Department of Physics, Eastern Mediterranean University,

G. Magusa, north Cyprus, Mersin 10, Turkey.

The formation of a naked singularity in a model of f(R) gravity having as source a linear elec-

tromagnetic field is considered in view of quantum mechanics. Quantum test fields obeying the

Klein−Gordon, Dirac and Maxwell equations are used to probe the classical timelike naked sin-

gularity developed at r = 0. We prove that the spatial derivative operator of the fields fails to

be essentially self-adjoint. As a result, the classical timelike naked singularity remains quantum

mechanically singular when it is probed with quantum fields having different spin structures.

I.INTRODUCTION

In the last decade, there have been extensive studies in Extended Theories of Gravity (ETG) such as the Lovelock

and f(R) gravity theories. The main motivation to study the ETG is to understand the accelerated expansion of

the universe and the issue of dark matter/energy (see [1] and references therein for a general review). One of the

most attractive branches of the ETG is the f(R) gravity theory in which the standard Einstein’s gravity is extended

with an arbitrary function of the Ricci scalar R instead of the linear one [1]. In this model, the Ricci scalar R in

the Einstein−Hilbert action is replaced with f(R) = R + αg(R), where g(R) is an arbitrary function of R so that in

the limit α = 0, one recovers the Einstein limit. Although the majority of researchers prefer to use this ansatz, in

general, finding an exact analytic solution to the field equations is not an easy task. As far as analytic exact solutions

are concerned, static, spherically symmetric models in f(R) gravity have been shown to serve for this purpose [2–6].

In this context of static, spherically symmetric solutions of f(R) gravity, the solutions admitting black holes have

attracted much attention.

In the context of static, spherically symmetric f(R) gravity, it has recently been shown that [7], an exact analytic

solution is also possible if one assumes f(R) to have the form of f(R) = ξ (R + R1) + 2α√R + R0, in which ξ,α,R0

and R1are constants, a priority to secure the Einstein limit by setting the constants R0= R1= α = 0 and ξ = 1.

In this model of f(R) gravity, exact solutions with external electromagnetic sources (both linear and nonlinear) are

found. It was shown that the solution with a linear electromagnetic field does not admit a black hole while the solution

with a nonlinear electromagnetic source admits a black hole solution. The physical properties of the latter solution

are investigated by calculating thermodynamic quantities and it was shown to satisfy the first law of thermodynamics.

The solution having as a source a linear electromagnetic field resulted with a naked curvature singularity at r = 0,

which is a typical central singularity peculiar to spherically symmetric systems. The solution given in [7], is a kind

of extension of a global monopole solution [8] which represents a solution of the Einstein’s equations with spherical

symmetry with matter that extends to infinity. It can also be interpreted as a cloud of cosmic strings with spherical

symmetry [9]. Hence, the spacetime is conical. However, with the inclusion of a linear or nonlinear electromagnetic

field, the spacetime is no more conical in the context of f(R) gravity.

Within the framework of ETG gravity, black hole solutions have been widely studied in the literature (see [1, 10] and

references therein for a complete review). However, the solutions that result with naked singularities have not been

studied in detail. In physics, naked singularities are considered to be a threat to the cosmic censorship hypothesis.

Furthermore, as in classical general relativity, compared to the black hole solutions, naked singularities are not well

understood in the context of f(R) gravity. This still remains a fundamental problem in general relativity as well

as in ETG to be solved. Another important diffıculty in resolving this problem is the scale on which the curvature

singularity occurs. On these small scales, it is believed that the classical methods should be replaced with quantum

techniques in resolving the singularity problems that necessitate the use of quantum gravity. Since the quantum

theory of gravity is still ”under construction”, an alternative method is proposed by Wald [11] which was further

developed by Horowitz and Marolf (HM) [12] in determining the character of classically singular spacetime and to see

if quantum effects have any chance to heal or regularize the dynamics and restore the predictability if the singularity

is probed with quantum particles/fields.

∗Electronic address: ozay.gurtug@emu.edu.tr

†Electronic address: tayabeh.tahamtan@emu.edu.tr

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2

In this paper, we investigate the occurrence of naked singularities in the context of f(R) gravity from the point

of view of quantum mechanics. We believe that this will be the unique example wherein the formation of a clas-

sically naked curvature singularities in f(R) gravity will be probed with quantum fields/particles that obey the

Klein−Gordon, Dirac and Maxwell equations. The criterion proposed by HM will be used in this study to investigate

the occurrence of naked singularities.

This criterion has been used successfully for other spacetimes to check whether the classically singular spacetimes

are quantum mechanically regular or not. As an example; negative mass Schwarzschild spacetime, charged dilatonic

black hole spacetime and fundamental string spacetimes are considered in [12]. An alternative function space, namely

the Sobelov space instead of the Hilbert space, has been introduced in [13], for analyzing the singularities within the

framework of quantum mechanics. Helliwell and Konkowski have studied quasiregular [14], Gal’tsov−Letelier−Tod

spacetime [15], Levi-Civita spacetimes [16, 17], and recently, they have also considered conformally static spacetimes

[18]. Pitelli and Letelier have studied spherical and cylindrical topological defects [19], Banados−Teitelboim−Zanelli

(BTZ) spacetimes [20], the global monopole spacetime [21] and cosmological spacetimes [22]. Quantum singularities

in matter coupled 2 + 1 dimensional black hole spacetimes are considered in [23]. Quantum singularities are also

considered in Lovelock theory [24] and linear dilaton black hole spacetimes [25]. Recently, the occurrence of naked

singularities in a 2+1 dimensional magnetically charged solution in Einstein−Power−Maxwell theory have also been

considered [26].

The main theme 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 solution to be investigated in this paper is a kind of f(R) gravity extension of the analysis presented in [21]

for the global monopole spacetime. The inclusion of the linear Maxwell field within the context of f(R) gravity

affects the topology significantly and removes the conical nature at infinity. Furthermore, the true timelike naked

curvature singularity is created at r = 0 which is peculiar to spherically symmetric systems. We investigate this

singularity within the framework of quantum mechanics by employing three different quantum fields/particles obeying

the Klein−Gordon, Dirac and Maxwell fields with different spin structures.

The paper is organized as follows: In Sec.II, we review the solution found recently in [7], and give the structure of

the spacetime. In Sec.III, first, the definition of quantum singularity for static spacetimes is briefly introduced. Then,

the quantum fields obeying the Klein−Gordon, Dirac and Maxwell equations are used to probe the singularity. The

paper ends with a conclusion in Sec. IV.

II. THE METRIC FOR f(R) GRAVITY COUPLED TO MAXWELL FIELDS AND SPACETIME

STRUCTURE

Recently, an exact analytic solution for f (R) gravity coupled with linear and nonlinear Maxwell field in four

dimensions has been presented in [7]. The corresponding action for f (R) gravity coupled with linear Maxwell field in

four dimensions is given by,

S =

?

d4x√−g

?f (R)

2κ

−

1

4πF

?

, (1)

in which f (R) is a real function of the Ricci scalar R, and F =

two-form is given by

1

4FµνFµνis the Maxwell invariant. The Maxwell

F =Q

r2dt ∧ dr + P sinθdθ ∧ dϕ, (2)

in which Q and P are the electric and magnetic charges, respectively. The static spherically symmetric metric ansatz

is

ds2= −B (r)dt2+

dr2

B (r)+ r2?dθ2+ sin2θdϕ2?, (3)

where B (r) stands for the only metric function to be found. The Maxwell equations (i.e. dF = 0 = d∗F) are satisfied

and the field equations are given by

fRRν

µ+

?

?fR−1

2f

?

δν

µ− ∇ν∇µfR= κTν

µ,(4)

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3

in which

fR =

df (R)

dR

1

√−g∂µ

, (5)

?fR =

?√−g∂µ?fR,

(fR),µ,α− Γm

(6)

∇ν∇µfR = gαν?

µα(fR),m

?

, (7)

while the energy momentum tensor is

4πTν

µ= −Fδν

µ+ FµλFνλ. (8)

Furthermore, the trace of the field equation (4) reads

fRR + (d − 1)?fR−d

2f = κT, (9)

with T = Tµ

µ. The non-zero energy momentum tensor components are

Tν

µ=P2+ Q2

8πr4

diag[−1,−1,1,1], (10)

and with zero trace, we have

f =1

2fRR + 3?fR.(11)

With reference to the paper [7], the form of the function f (R) is assumed to be ,

f (R) = ξ

?

R +1

2R0

?

+ 2α

?

R + R0,(12)

which leads to

R =

α2

η2r2− R0,(13)

where α , R0, and ξ are constants. Consequently, the metric function B(r) is obtained for the free parameters α = η

as,

B (r) =1

2−m

r+q2

r2−Λeff

3

r2,(14)

where m =−ξ

this solution does not admit the Reissner−Nordstr¨ om (RN)−de Sitter (dS) limit. However, in the limit ξ = 1 and

P = Q = 0, the solution reduces to the well known global monopole solution reported in [8], which represents a

spherically symmetric, non-asymptotically flat solution with a matter field that extends to infinity. Furthermore, this

solution can also be considered as a spherically symmetric cloud of cosmic string which gives rise to a deficit angle

[9]. Therefore, the solution given in equation (14) , is a kind of Einstein−Maxwell extension of the global monopole

solution in f (R) gravity. One of the striking effects of the additional fields is the removal of the conical geometry

of the global monopole spacetime. The Kretschmann scalar which indicates the formation of curvature singularity is

given by

3η, Λeff=−R0

4

and q2=Q2+P2

ξ

. As was explained in [7], due to the constraints on the free parameters,

K =1

3

8λ2r8+ 4λr6+ 3r4+ 12mr3+ 12r2?3m2− q2?− 144mq2r + +168q4

It is obvious that r = 0 is a typical central curvature singularity. This is a timelike naked singularity because the

behavior of the new radial coordinate defined by r∗=?

Our aim in the next section is to investigate this classically singular spacetime with regard to the quantum me-

chanical point of view.

r8

.

dr

B(r)is finite when r → 0. Hence, the new solution obtained

in [7] and given in equation (14) is classically a singular spacetime.

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III.QUANTUM SINGULARITIES

One of the important predictions of the Einstein’s theory of general relativity is the formation of spacetime sin-

gularities. In classical general relativity, singularities are defined as the points in which the evolution of timelike or

null geodesics is not defined after a proper time. According to the classification of the classical singularities devised

by Ellis and Schmidt , scalar curvature singularities are the strongest ones in the sense that the spacetime cannot

be extended and all physical quantities, such as the gravitational field, energy density and tidal forces, diverge at

the singular point. In black hole spacetimes, the location of the curvature singularity is at r = 0 and is covered by

horizon(s). As long as the singularities are hidden by horizon(s), they do not constitute a threat to the Penrose cosmic

censorship hypothesis. However, there are some cases that the singularity is not hidden and hence, it is naked. In

the case of naked singularities, further care is required because they violate the cosmic censorship hypothesis. The

resolution of the naked singularities stands as one of the most drastic problems in general relativity to be solved.

Naked singularities that occur at r = 0 are on the very small scales where classical general relativity is expected to

be replaced by quantum theory of gravity. In this paper, the occurrence of naked singularities in f(R) gravity will

be analyzed through a quantum mechanical point of view. In probing the singularity, quantum test particles/fields

obeying the Klein−Gordon, Dirac and Maxwell equations are used. In other words, the singularity will be probed

with spin 0, spin 1/2 and spin 1 fields. The reason for using three different types of field is to clarify whether or not

the classical singularity is sensitive to the spin of the fields.

Our analysis will be based on the pioneering work of Wald, which was further developed by HM to probe the

classical singularities with quantum test particles obeying the Klein−Gordon equation in static spacetimes having

timelike singularities. According to HM, 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 a self-adjoint extension of the operator to the entire Hilbert space. If the extension is unique, it is

said that the space is quantum mechanically regular. A brief review now 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 in this space is

?∇µ∇µ− M2?ψ = 0. (15)

This equation can be written in the form

∂2ψ

∂t2=

?

fDi??

fDiψ

?

− fM2ψ = −Aψ,(16)

in which f = −ξµξµand Diis the spatial covariant derivative on Σ. The Hilbert space H,

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

the spacetime singularities. An appropriate set is C∞

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

The 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 [27–29]. Accordingly, the Klein−Gordon equation for a free

particle satisfies

?L2(Σ)?is the space of

idψ

dt=

?

AEψ,(17)

with the solution

ψ (t) = exp

?

−it

?

AE

?

ψ (0). (18)

If A is not essentially self-adjoint, the future time evolution of the wave function (18) is ambiguous. Then the HM

criterion defines the spacetime as quantum mechanically singular. However, if there is only a single self-adjoint

extension, the operator A is said to be essentially self-adjoint and the quantum evolution described by Eq.(18) is

uniquely determined by the initial conditions. According to the HM criterion, this spacetime is said to be quantum

mechanically non-singular. 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. [13] 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−

(19)

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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 or not 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 is then reduced to counting the number of solutions of A∗ψ = Zψ ; (for λ = 1),

A∗ψ ± iψ = 0(20)

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

A possesses a unique self-adjoint extension and essentially self-adjoint. Consequently, the way to find a sufficient

condition for the operator A to be essentially self-adjoint is to investigate the solutions satisfying Eq. (20) that do

not belong to the Hilbert space.

A.Klein−Gordon Fields

The Klein−Gordon equation for a scalar particle with mass M is given by

?ψ = g−1/2∂µ

?

g1/2gµν∂ν

?

ψ = M2ψ.(21)

For the metric (3), the Klein−Gordon equation becomes

∂2ψ

∂t2

= −B (r)

+B (r)M2ψ.

?

B (r)∂2ψ

∂r2+1

r2

∂2ψ

∂θ2+

1

r2sin2θ

∂2ψ

∂ϕ2+cotθ

r2

∂ψ

∂θ+

?2B (r)

r

+ B

′(r)

?∂ψ

∂r

?

(22)

In analogy with equation (16), the spatial operator A for the massless case is

A = B (r)

?

B (r)∂2

∂r2+1

r2

∂2

∂θ2+

1

r2sin2θ

∂2

∂ϕ2+cotθ

r2

∂

∂θ+

?2B (r)

r

+ B

′(r)

?

(θ,ϕ), we get the radial

∂

∂r

?

,(23)

and the equation to be solved is (A∗± i)ψ = 0.Using separation of variables, ψ = R(r)Ym

portion of equation (20) as

l

d2R(r)

dr2

+

?r2B (r)?′

r2B (r)

dR(r)

dr

+

?−l(l + 1)

r2B (r)

±

i

B2(r)

?

R(r) = 0.(24)

where a prime denotes the derivative with respect to r.

1.The case of r→ ∞

The case r → ∞ is topologically different compared to the analysis reported in [21]. In the present problem the

geometry is not conical. The approximate metric when r → ∞ is

ds2≃ −(R0r2

12R0r2

)dt2+

?

12

?

dr2+ r2?dθ2+ sin2θdϕ2?.(25)

For the above metric, the radial equation (24) becomes,

d2R(r)

dr2

+4

r

dR(r)

dr

= 0,(26)

whose solution is

R(r) = C1+C2

r3,

where C1and C2are arbitrary integration constants. It is clearly observed that the above solution is square integrable

as r → ∞ if and only if C1= 0. Hence, the asymptotic behavior of R(r) is given by R(r) ≃C2

r3.

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6

2.The case of r→ 0

Near the origin there is a true timelike curvature singularity resulting from the existence of charge. Therefore, the

approximate metric near the origin is given by

ds2≃ −(q2

r2)dt2+

?r2

q2

?

dr2+ r2?dθ2+ sin2θdϕ2?. (27)

The radial equation (24) for the above metric reduces to

d2R(r)

dr2

−l(l + 1)

q2

R(r) = 0, (28)

whose solution is

R(r) = C3eαr+ C4e−αr

?l(l + 1)

(29)

α =

q

where C3 and C4 are arbitrary integration constants. The square integrability of the above solution is checked by

calculating the squared norm of the above solution in which the function space on each t = constant hypersurface Σ

is defined as H ={R |? R ?< ∞}. The squared norm for the metric (27) is given by,

? R ?2=

?constant

0

|R(r)|2r4

q2

dr.(30)

Our calculation has revealed that the solution above is always square integrable near r = 0, even if l = 0, which

corresponds to the S-wave solutions.

Consequently, the spatial operator A has deficiency indices n+ = n− = 1, and it is not essentially self-adjoint.

Hence, the classical singularity at r = 0 remains quantum mechanically singular when probed with fields obeying the

Klein−Gordon equation.

B.Maxwell fields

The Newman−Penrose formalism will be used to find the source-free Maxwell fields propagating in the space of

f(R) gravity. Let us note that the signature of the metric (3) is changed to −2 in order to use the source-free Maxwell

equations in Newman−Penrose formalism. Thus, the metric (3) is rewritten as,

ds2= B (r)dt2−

dr2

B (r)− r2?dθ2+ sin2θdϕ2?.(31)

The four coupled source-free Maxwell equations for electromagnetic fields in the Newman−Penrose formalism is given

by

Dφ1−¯δφ0= (π − 2α)φ0+ 2ρφ1− κφ2,

δφ2− ∆φ1= −νφ0+ 2µφ1+ (τ − 2β)φ2,

δφ1− ∆φ0= (µ − 2γ)φ0+ 2τφ1− σφ2,

Dφ2−¯δφ1= −λφ0+ 2πφ1+ (ρ − 2ǫ)φ2,

where B(r) is the metric function given in Eq.(14), φ0, φ1and φ2are the Maxwell spinors, ǫ,ρ,π,α,µ,γ,β and τ are

the spin coefficients to be found and the bar denotes complex conjugation. The null tetrad vectors for the metric (31)

are defined by

(32)

la=

?

?1

1

√2

1

B(r),1,0,0

2,−B(r)

?

?

,(33)

na=

2

,0,0

?

i

,

ma=0,0,1

r,rsinθ

?

.

Page 7

7

The directional derivatives in the Maxwell’s equations are defined by D = la∂a,∆ = na∂aand δ = ma∂a. We define

operators in the following way

D0 = D,

D†

0= −

2

B (r)∆,

√2r δ and L†

(34)

L†

0=

1= L†

0+cotθ

2

,

L0 =

√2r¯δ and L1= L0+cotθ

2

.

The non-zero spin coefficients are

µ = −1

r

B(r)

2

,ρ = −1

r,

γ =1

4B

′(r),β = −α =

1

2√2

cotθ

r

.(35)

The Maxwell spinors are defined by [30]

φ0= F13= Fµνlµmν

1

2(F12+ F43) =1

φ2= F42= Fµνmµnν,

(36)

φ1=

2Fµν(lµnν+ mµmν),

where Fij(i,j = 1,2,3,4) and Fµν(µ,ν = 0,1,2,3) are the components of the Maxwell tensor in the tetrad and tensor

bases, respectively. Substituting Eq.(34) into the Maxwell’s equations together with non-zero spin coefficients, the

Maxwell equations become

?

?

D0+2

r

?

?

φ1−

1

r√2L1φ0= 0,

1

r√2L0φ1= 0,

?

1

r√2L†

(37)

D0+1

r

φ2−

(38)

B (r)

2

?

D†

0+B

′(r)

B (r)+1r

φ0+

1

r√2L†

0φ1= 0,(39)

B (r)

2

?

D†

0+2

r

?

φ1+

1φ2= 0.(40)

The equations above will become more tractable if the variables are changed to

Φ0= φ0eikt,Φ1=

√2rφ1eikt,Φ2= 2r2φ2eikt.

Then we have

?

?

D0+1

r

?

?

′(r)

B (r)+1

Φ1− L1Φ0= 0,(41)

D0−1

r

Φ2− L0Φ1= 0,

?

?

(42)

r2B (r)

?

D†

0+B

r

Φ0+ L†

0Φ1= 0,(43)

r2B (r)

?

D†

0+1

r

Φ1+ L†

1Φ2= 0.(44)

Page 8

8

The commutativity of the operators L and D enables us to eliminate each Φi from above equations, and hence we

have

?

L†

0L1+ r2B (r)

?

D0+B

′(r)

B (r)+3r

??

??

D†

0+B

′(r)

B (r)+1r

??

Φ0(r,θ) = 0, (45)

?

L0L†

1+ r2B (r)

?

D†

0+1

r

D0−1

??

r

??

Φ2(r,θ) = 0,(46)

?

L1L†

0+ r2B (r)

?

D†

0+B

′(r)

B (r)+1r

D0+1

r

??

Φ1(r,θ) = 0.(47)

The variables r and θ can be separated by assuming a separable solution in the form of

Φ0(r,θ) = f0(r)Θ0(θ),Φ1(r,θ) = f1(r)Θ1(θ),Φ2(r,θ) = f2(r)Θ2(θ).

The separation constants for Eq. (45) and Eq. (46) are the same, because Ln= −L†

operator L†

for Eq. (47) we will assume another separation constant. Furthermore, by defining R0(r) =

R2(r) =f2(r)

r

, the radial equations can be written as

n(π − θ), or, in other words, the

0L1acting on Θ0(θ) is the same as the operator L0L†

1acting on Θ2(θ) if we replace θ by π −θ. However,

f0(r)

rB(r), R1(r) =f1(r)

r

and

f

′′

0(r) +2

rf

′

0(r)+(48)

?

−iω

?

2

rB (r)−B

′(r)

B2(r)

?

+

ω2

B2(r)−

ǫ2

r2B (r)

?

f0(r) = 0,

f

′′

2(r) −2

?

rf

ω2

B2(r)−

′

2(r)+ (49)

?

iω

?

2

rB (r)−B

′(r)

B2(r)

+

ǫ2

r2B (r)

?

f2(r) = 0,

f

′′

1(r) +B

′(r)

B (r)f

η2

r2B (r)

′

1(r)+(50)

?

ω2

B2(r)−

?

f1(r) = 0,

where ǫ and η are the separability constants.

1. The case r→ ∞

For the case r → ∞, the corresponding metric is given in Eq.(25). Hence, the radial parts of the Maxwell equations,

(48) , (49) and (50), become

f

′′

j(r) +2

rf

′

j(r) = 0,j = 0,1(51)

f

′′

2(r) −2

rf

′

2(r) = 0(52)

Thus, the solutions in the asymptotic case are

Rj(r) = C1+C2

r,

j = 0,1(53)

R2(r) = C3+C4

r3,(54)

in which Ci are integration constants. The solution above is square integrable if C1 = C3 = 0. Therefore, the

asymptotic form of the solutions behaves as Rj(r) ∼C2

r,j = 0,1 and R2(r) ∼C4

r3.

Page 9

9

2. The case r→ 0

The metric near r → 0 is given in Eq.(27). Hence, the radial parts of the Maxwell equations (48) , (49) and (50)

for this case are given by

R

′′

j(r) −2

rR

′

j(r) −α2

0(r) −η2

q2Rj(r) = 0 , j = 1,2 (55)

R

′′

0(r) +2

rR

′

q2R0(r) = 0(56)

whose solutions are obtained as,

Rj(r) = C3e

α

qr(αr − 1) + C4e−α

sinh

qr

qr(αr + 1),

?η

j = 1,2, (57)

R0(r) =

C5

r

?η

?

+C6

r

cosh

qr

?

(58)

where Ciare constants. The above solution is checked for square integrability. Calculations have revealed that

? Ri?2=

?constant

0

|Ri(r)|2r4

q2

dr < ∞,

which indicates that the obtained solutions are square integrable. The definition of the quantum singularity for

Maxwell fields will be the same as for the Klein−Gordon fields. Here, since we have three equations governing the

dynamics of the photon waves, the unique self-adjoint extension condition on the spatial part of the Maxwell operator

should be examined for each of the three equations. As a result, the occurrence of the naked singularity in f(R)

gravity is quantum mechanically singular if it is probed with photon waves.

C.Dirac Fields

The Newman−Penrose formalism will also be used here to find the massless Dirac fields (fermions) propagating in

the space of f(R)-gravity. The Chandrasekhar-Dirac (CD) equations in the Newman−Penrose formalism are given

by

(D + ǫ − ρ)F1+?¯δ + π − α?F2 = 0,

(∆ + µ − γ)F2+ (δ + β − τ)F1 = 0,

(D +¯ ǫ − ¯ ρ)G2− (δ + ¯ π − ¯ α)G1 = 0,

(∆ + ¯ µ − ¯ γ)G1−?¯δ +¯β − ¯ τ?G2 = 0,

(59)

where F1,F2,G1and G2are the components of the wave function, ǫ,ρ,π,α,µ,γ,β and τ are the spin coefficients to

be found. The non-zero spin coefficients are given in Eq.(35). The directional derivatives in the CD equations are the

same as in the Maxwell equations. Substituting non-zero spin coefficients and the definitions of the operators given

in Eq.(34) into the CD equations leads to

?

D0+1

r

?

′(r)

F1+

1

r√2L1F2= 0,

?

1

r√2L†

?

−B (r)

2

?

?

D†

0+B

2B (r)+1r

F2+

1

r√2L†

1F1= 0,

D0+1

r

?

G2−

1G1= 0,

B (r)

2

?

D†

0+B

′(r)

2B (r)+1r

G1+

1

r√2L1G2= 0.(60)

Page 10

10

For the solution of the CD equations, we assume a separable solution in the form of

F1 = f1(r)Y1(θ)ei(kt+mϕ),

F2 = f2(r)Y2(θ)ei(kt+mϕ),

G1 = g1(r)Y3(θ)ei(kt+mϕ),

G2 = g2(r)Y4(θ)ei(kt+mϕ),

(61)

where m is the azimuthal quantum number and k is the frequency of the Dirac fields, which is assumed to be positive

and real .Since {f1,f2,g1,g2} and {Y1,Y2,Y3,Y4} are functions of r and θ, respectively, by substituting Eq.(61) into

Eq.(60) and applying the assumptions given by

f1(r) = g2(r)

Y1(θ) = Y3(θ)

and

and

f2(r) = g1(r) ,

Y2(θ) = Y4(θ),

(62)

(63)

the Dirac equations transform into Eq.(64). In order to solve the radial equations , the separation constant λ should

be defined. This is achieved by using the angular equations. In fact, it is already known from the literature that

the separation constant can be expressed in terms of the spin-weighted spheroidal harmonics. The radial parts of the

Dirac equations become

?

D0+1

r

?

f1(r) =

λ

r√2f2(r),(64)

B (r)

2

?

D†

0+B

′(r)

2B (r)+1r

?

f2(r) =

λ

r√2f1(r).

We further assume that

f1(r) =

Ψ1(r)

r

Ψ2(r)

r

,

f2(r) =

,

then Eq.(64) transforms into,

D0Ψ1=

λ

r√2Ψ2,

?

(65)

B (r)

2

?

D†

0+B

′(r)

2B (r)

Ψ2=

λ

r√2Ψ1.

Note that

?

B(r)

2D†

0

?

B(r)

2

= D†

0+

B

2B(r)+1

′(r)

r, and using this together with the new functions

R1(r) = Ψ1(r),

R2(r) =

?

B (r)

2

Ψ2(r),

and defining the tortoise coordinate r∗as

d

dr∗

= Bd

dr,

(66)

Eqs.(65) become

?

?

d

dr∗

+ ik

?

?

R1 =

√Bλ

r

√Bλ

r

R2,(67)

d

dr∗

− ikR2 =R1,

Page 11

11

In order to write Eq.(67) in a more compact form, we combine the solutions in the following way:

Z+ = R1+ R2,

Z− = R2− R1.

After doing some calculations we end up with a pair of one-dimensional Schr¨ odinger-like wave equations with effective

potentials,

?d2

dr2

?

∗

+ k2

?

± λd

Z±= V±Z±,(68)

V±=

Bλ2

r2

dr∗

?√B

r

??

.(69)

In analogy with equation (16), the radial operator A for the Dirac equations can be written as,

A = −d2

dr2

∗

+ V±,

If we write the above operator in terms of the usual coordinates r by using Eq.(66), we have

A = −d2

dr2−B

′

B

d

dr+

1

B2

?

Bλ2

r2

± λBd

dr

?√B

r

??

, (70)

Our aim now is to show whether this radial part of the Dirac operator is essentially self-adjoint or not. This will be

achieved by considering Eq.(20) and counting the number of solutions that do not belong to Hilbert space. Hence,

Eq.(20) becomes

?

d2

dr2+B

′

B

d

dr−

1

B2

?

Bλ2

r2

± λBd

dr

?√B

r

??

∓ i

?

ψ(r) = 0. (71)

For the asymptotic case, r → ∞ , the above equation transforms to

d2ψ

dr2+2

r

dψ

dr= 0,(72)

whose solution is

ψ (r) = C1+C2

r.

(73)

Clearly the solution is square integrable if C1= 0. Hence, the solution is asmptotically well behaved. Near r → 0 ,

Eq.(71) becomes

d2ψ

dr2−2

r

dψ

dr+σ

σ = ∓2λq,

r3ψ = 0,(74)

whose solution is given by

ψ (r) =

?4σ

x2

?3

2

{C3J3(x) + C4N3(x)},(75)

where J3(x) and N3(x) are Bessel functions of the first and second kind, and x = 2?σ

r. As r → 0, we have x → ∞.

The behavior of the Bessel functions for real ν ≥ 0 as x → ∞ is given by

Jν(x) ≃

?

?

2

πxcos

2

πxsin

?

?

x −νπ

x −νπ

2

−π

−π

4

?

?

,(76)

Nν(x) ≃

24

;

Page 12

12

thus the Bessel functions asymptotically behave as J3(x) ∼

Checking for the square integrability has revealed that both solutions are square integrable. Hence, the radial operator

of the Dirac field fails to satisfy a unique self-adjoint extension condition. As a result, the occurrence of the timelike

naked singularity in the context of f(R) gravity remains singular from the quantum mechanical point of view if it is

probed with fermions.

?

2

πxcos?x −7π

4

?

and N3(x) ∼

?

2

πxsin?x −7π

4

?.

IV.CONCLUSION

In this paper, the formation of the naked singularity in the context of a model of f(R) gravity is investigated within

the framework of quantum mechanics, by probing the singularity with the quantum fields obeying the Klein−Gordon,

Maxwell and Dirac equations. We have investigated the essential self-adjointness of the spatial part of the wave

operator A in the natural Hilbert space of quantum mechanics which is a linear function space with square integrability.

Our analysis has shown that the timelike naked curvature singularity remains quantum mechanically singular against

the propagation of the aforementioned quantum fields. Another notable outcome of our analysis is that the spin of

the fields is not effective in healing of the naked singularity for the considered model of the f(R) gravity spacetime.

Another alternative function space for analyzing the singularity in this context is to use the Sobelov space instead

of the natural Hilbert space [13]. The Analysis in Sobelov space entails square integrability both of the wave function

and its derivative. Although the details are not given in this study, the analysis using the Sobelov space has revealed

that irrespective of the spin structure of the fields used to probe the singularity, the model considered of f(R) gravity

spacetime remains quantum mechanically singular.

Hence, the generic conclusion that has emerged from our analysis is that in the model considered of f(R) gravity,

the formation of a timelike naked singularity is quantum mechanically singular.

It will be interesting for future research to extend the quantum singularity analysis in other ETG models. Further-

more, it will be a great achievement if the criterion proposed by HM is extended to stationary metrics. Although the

preliminary work in this direction is considered in [31], the formulation has not been fully completed.

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