Astrophys Space SciDOI 10.1007/s10509-014-2220-4

O R I G I NA L A RT I C L E

Polytropic thin-shell collapse in non-commutative d-dimensionalReissner–Nordström geometry

M. Sharif · Sehrish Iftikhar

Received: 25 October 2014 / Accepted: 13 November 2014© Springer Science+Business Media Dordrecht 2014

Abstract We study thin-shell collapse in non-commutatived-dimensional Reissner–Nordström geometry with mostgeneral polytropic equation of state. We formulate equa-tion of motion for the shell using Israel junction condi-tions and express it in terms of non-commutative factorthrough smeared Gaussian distribution. It is found that non-commutative collapsing thin-shell in the presence of extradimensions leads to the formation of either a black hole ornaked singularity.

Keywords Gravitational collapse · Non-commutativegeometry · Higher dimensional theory · Junction conditions

1 Introduction

One of the fascinating topics in general relativity (GR) isthe formation as well as nature of singularities from gravi-tationally collapsing matter. A remarkable development thatemerged from such studies is a singularity theorem (Pen-rose 1965) according to which gravitational collapse al-ways leads to the formation of spacetime singularity. It isbelieved that singularities are either hidden within eventhorizon or can be visible to a distant observer. Penrose

M. Sharif (B) · S. IftikharDepartment of Mathematics, University of the Punjab,Quaid-e-Azam Campus, Lahore 54590, Pakistane-mail: msharif.math@pu.edu.pk

S. Iftikhare-mail: sehrish3iftikhar@gmail.com

S. IftikharDepartment of Mathematics, Lahore Collegefor Women University, Lahore 54000, Pakistan

(1969) presented a hypothesis known as cosmic censor-ship hypothesis (CCH) according to which singularities ap-pearing in the process of collapse must be hidden withinevent horizon and hence invisible for the distant observer.This hypothesis is of fundamental importance but remainsunresolved, thus people have presented counter examplesagainst CCH (Eardley and Smarr 1979; Ori and Piran 1990;Szekeres and Iyer 1993; Harada et al. 1998; Ghosh andDadhich 2001; Joshi et al. 2002; Virbhadra and Ellis 2002;Virbhadra and Keeton 2008; Patil and Joshi 2010; Jhinganand Kaushik 2014).

It order to study gravitational collapse, it is necessary todescribe appropriate conditions which enable matching ofinterior and exterior regions. The most commonly methodfor matching the two regions separated by a boundary sur-face is developed by Israel (1966) which has wide applica-tions in GR. Kuchar (1968) studied collapse of the chargedspherical shell and found that charge is not able to preventthe shell from collapse. Chase (1970) was among the pio-neers to study collapse of a charged shell using equation ofstate. Boulware (1973) investigated the formation of nakedsingularity (NS) from charged collapsing shell. Peleg andSteif (1995) explored collapse of dust thin-shell and foundthat the nature of singularity depends upon the cosmologi-cal constant. Pereira and Wang (2000) investigated collapseof a cylindrical thin-shell composed of counter rotating dustparticles and found that angular momentum of dust particlescan halt the collapse. Mann and Oh (2006) considered dif-ferent collapsing shell models (with and without pressure)with generalized Chaplygin gas which leads to either blackhole (BH) or NS.

It is well-known that the concept of smooth spacetimemanifold in classical relativity breaks down at short dis-tances. Non-commutative (NC) geometry gives a significantapproach to investigate short distance spacetime dynamics.

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In this approach, there exists a universal minimal lengthscale (equivalent to Planck length). In GR, the effects ofnon-commutativity can be taken into account by keep-ing the standard form of the Einstein tensor and modify-ing the energy-momentum tensor using smeared objects.Non-commutative BHs require an appropriate frameworkin which the non-commutativity corresponds to GR.

Nicolini et al. (2006) studied NC radiating SchwarzschildBH and showed that NC effects can cure singularity prob-lems. Ansoldi et al. (2007) found a solution of the coupledEinstien–Maxwell field equations in NC geometry describ-ing a variety of charged self-gravitating objects. Sadeghi(2008) studied NC spaces in two-dimensional BH and ob-tained a lower bound for NC parameter. Alavi (2009) in-vestigated stability of the NC radiating Reissner Nordstöm(RN) BH and found an upper bound for NC parameter.Bertolami and Zarro (2010) discussed the implication ofnon-commutativity on astrophysical objects (white dwarfsand neutron stars). Oh and Park (2010) investigated col-lapsing shell in NC geometry with polytropic equation ofstate and Chaplygin gas which leads to either BH or NS.Sharif and Abbas, Sharif and Abbas (2012, 2013) exploredthin-shell collapse with polytropic matter in NC backgroundand concluded that non-commutativity leads to the validityof CCH. They also formulated non-compact charged objectmodel in NC field and found that singularities are hidden in-side the event horizon. Nicolini et al. (2013) discussed col-lapse of thick matter layers of a shell in NC geometry andfound the existence of BH for heavy shells. Gorji and Nozari(2014) studied effect of non-commutativity in Synder spaceand resolved the big-bang singularity in their framework.

In recent years, higher dimensions (d > 4) in GR has at-tracted many researchers. Antoniadis (1990) suggested thatgenerally perturbed string theories predict the existence ofextra dimension. Arkani-Hamed et al. (1998) are the firstwho proposed that the universe is a 4-dimensional branewith large extra spatial dimensions which can be the promis-ing alternative to the Planck scenario. Ghosh and Beesham(2001) studied higher dimensional spherically symmetricdust collapse and concluded that higher dimensions favorBH formation. Kanti (2004) found the existence of higherdimensional spacetime during high energy collisions. Gaoand Lemos (2008) explored charged thin-shell in higherdimensions and found that the presence of horizon leadsto oscillatory shell while the shell is in equilibrium inthe absence of horizon. Spallucci et al. (2009) obtained ahigher-dimensional charged BH in NC background. Shi-mano and Miyamoto (2014) investigated the formation ofNS in higher-dimensional dust collapse. Ahmad and Haseeb(2014) studied spherically symmetric collapse in higher-dimensions and found the validity of CCH.

In this paper, we study the effects of NC parameter ond-dimensional RN BH using Israel thin-shell formalism

with polytropic equation of state. We use the mass densityof a static spherically symmetric particle like gravitationalsource in d-dimensional spacetime given by a Gaussian dis-tribution of minimal width

√Θ (Nozaria and Mehdipour

2009)

ρΘ(r) = M

(4πΘ)d−1

2

e− r24Θ . (1)

The paper is organized as follows. In Sect. 2, we present Is-rael thin-shell formalism in higher dimensions and formu-late equation of motion for d-dimensional thin-shell withpolytropic equation of state. Section 3 investigates collapsescenario in NC geometry using modified matter density. Inthe last section, we conclude our results.

2 Thin-shell formalism in d-dimensions

We consider static spherically symmetric d-dimensionalspacetime is divided into two regions V − (interior) and V +(exterior) by a timelike (d − 1)-dimensional boundary sur-face Σ . The metric in both regions is given by Gao andLemos (2008)

ds2 = f±(r)dt2 − f −1± (r)dr2 − r2(dΩ2d−2

), (2)

where dΩ2d−2 = dθ2

1 + sin2 θ1dθ22 + sin2 θ1 sin2 θ2dθ2

3 +· · · + ∏d−3

μ=1 sin2 θμdθ2d−2 is the line element on the (d − 2)-

dimensional unit sphere and d is the dimension of space-time. Let M± and Q± represent mass and charge of the in-terior and exterior of BH. We assume that the interior re-gion contains more mass than the exterior, i.e., gravitationalmasses are unequal M− �= M+ while charge is same in bothregions, i.e., Q = Q− = Q+. The geometry of the shell canbe described by the RN metric with

f (r)± = 1 − 2M±rd−3

(8πGd

(d − 2)Ωd−2

)+ εdQ2±

r2(d−3), (3)

where Gd is the gravitational constant in d-dimensions andεd is a constant proportional to d-dimensional vacuum per-meability. For a given dimension d , units for mass andcharge can be chosen as(

8πGd

(d − 2)Ωd−2

)= 1, εd = 1. (4)

By employing the intrinsic coordinates (τ, θ1, θ2, . . . ,

θd−2) on the hyper surface (Σ) at r = r(τ ), the metric (2)becomes

(ds)2±Σ =[f±(r) − f −1± (r)

(dr

dτ

)2(dτ

dt

)2]dt2

− r2(τ )(dΩ2

d−2

). (5)

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We take g00 > 0 such that t (τ ) is a timelike coordinate. Theinduced metric on Σ is given as

(ds)2 = dτ 2 − α2(τ )(dΩ2

d−2

). (6)

From the continuity of spacetimes, it follows that

[f±(r) − f −1± (r)

(dr

dτ

)2(dτ

dt

)2] 12

dt = (dτ)Σ,

r(τ ) = α(τ)Σ . (7)

The outward unit normals η±μ in V± coordinates are calcu-

lated as

η±μ = (−r(τ ), t ,0, . . . ,0

).

Here dot represents differentiation with respect to τ . Thesurface stress energy-momentum tensor is defined as

Sμν = 1

κ

([Kμν] − γμν[K]), (8)

where γμν is the induced metric on Σ , κ is the couplingconstant and

[Kμν] = K+μν − K−

μν, [K] = γ μν[Kμν]. (9)

The extrinsic curvature on the both sides of the hypersurfaceis

K±μν = −η±

σ

[∂2χσ±∂ξμξν

+ Γ σαβ

∂χα±∂ξμ

∂χβ±

∂ξν

]. (10)

The perfect fluid surface stress energy-momentum tensorfollows

Sμν = (ρ + p)uμuν − pγμν. (11)

Using Eqs. (8)–(11), the energy density and isotropic pres-sure become

ρ = − (d − 2)

κr

(√r2 + f±

), (12)

p = 1

κ

(d

dr

(√r2 + f±

) + (d − 3)√

r2 + f±r

). (13)

Inserting Eq. (12) into (13), we obtain

dρ

dr+ (d − 2)

r(ρ + p) = 0. (14)

The equation of state for the polytropic matter is given as

p = kρ1+ 1n , (15)

where k and n are polytropic constant and index, respec-tively. The polytropic constant can take positive values for

the stiff fluid and radiation, negative values for the vacuumfluid and zero for the dust (Mukhopadhyay and Ray 2008).Some specific values of the polytropic index correspond todifferent stellar models, e.g., neutron stars for n ∼ 0.5–1,degenerate star cores of white dwarfs, brown dwarfs, andgigantic planets such as Jupiter (n = 1.5), main sequencestars like Sun (n = 3), Chaplygin gas fluid by choosingn, k < 0 (Oh and Park 2010), n = −0.5 is related to stringtheory (Setare et al. 2013) and it describes a perfect fluid forn → ∞.

The positive energy density for finite and infinite n is ob-tained by solving Eqs. (14) and (15) as

ρ =[(

k + ρ− 1

n

0

)( r

r0

) d−2n − k

]−n

, (16)

ρ = ρ0

(r0

r

)(d−2)(k+1)

, (17)

where r0 is the initial position of the shell at τ = τ0 andρ0 is the matter density of the shell at r = r0. It is men-tioned here that energy density for finite n diverges at r ≡r0(

k

k+ρ−1n

0

)n

d−2 . The equation of motion of the thin-shell is

obtained from Eq. (12) as follows

r2 + Veff(r) = 0, (18)

where

Veff(r) = 1

2(f+ + f−) − (d − 2)2(f+ + f−)2

4(κρr)2

− 1

4(d − 2)2(κρr)2. (19)

Define z = ( rr0

)d−3 and t = τ

rd−30

, the above equation is sim-

plified to

z + Veff(z) = 0. (20)

The effective potential for finite n is obtained by insertingEq. (16) in (19)

Veff(z) = 1 − ζ+z

+ Q

z2− ζ 2−ω2

(d − 2)2z4

(z

d−2(d−3)n − e

)2n

− z2

(zd−2

(d−3)n )ω2, (21)

where ζ± = M+±M−rd−30

, Q = Q

rd−30

, e = k

(k+ρ−1n

0 )

and ω =(d−2)2(k+ρ

−1n )n

κrd−30

. For perfect fluid (infinite n), it follows that

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Fig. 1 The effective potential versus z for n = 0.5 (left), n = −0.5 (right), and perfect fluid (lower graph)

Veff(z) = 1 − ζ+z

+ Q

z2− ζ 2−z2[ (d−2)(k+1)

d−3 −2]

(d − 2)2ω2

− ω2

z2[ (d−2)(k+1)d−3 −1]

, (22)

where ω2 = κ2ρ20 r

2(d−3)0

(d−2)4 .We investigate the shell’s motion using Eqs. (20)–(22)

with the initial data, M+ = r0 = Q = ρ0 = 1, M− = 0,k = 2, n = 0.5 and n = −0.5 as shown in Fig. 1. The left plot(upper) indicates oscillatory motion of the shell for d = 4 asVeff coincides with z-axis at two points. The turning point(Veff = 0) yields unstable equilibrium position and after thatthe shell either collapses or expands infinitely depending onthe choice of parameters. For d = 5,6,7,8,10, Veff divergesnegatively starting from the positive region which meansthat the shell starts from a finite positive point, smoothlymoves towards negative region but does not cross z-axis.In this case, the polytropic shell either expands infinitely orcollapses to zero forming BH or NS while for d = 9,11, theshell starts with static configuration showing the same be-havior. The right graph (upper) is plotted for n = −0.5 andthe lower graph (perfect fluid) indicates that Veff → −∞ asz → 0 which means that the shell expands indefinitely orcontracts. The shell is in static position when Veff = 0 for

large values of z and then collapses or expands from staticposition as Veff → −∞ for some values of z.

The numerical results of Eq. (20) for finite and infinite n

describe the shell’s radius shown in Fig. 2, where the upperand lower curves indicate increase and decrease in the shell’sradius. The increasing behavior of the shell’s radius favorsexpansion while the decreasing radius indicates shell’s col-lapse.

3 Non-commutative inspired gravitational collapse

Here, we study the effects of NC parameters on thin-shellcollapse in the presence of extra dimensions and explorethe corresponding equation of motion. For this purpose,we require new matter different from the polytropic matterwhose energy density is in terms of NC parameter. The NCspacetime coordinates are deformation of the usual space-time coordinates with an arbitrary dimension d . Thus thenon-vanishing NC relation is given as follows (Oh and Park2010)[xμ, xν

] = iΘμν,

where xμ is the Hermitian operator and Θμν is the deforma-tion parameter, antisymmetric under μ ↔ ν, has dimensions

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Fig. 2 Behavior of the shell’s radius for n = 0.5 (left), n = −0.5 (right) and perfect fluid (lower graph)

of (length)2 and ordinary spacetime is recovered in the limitΘμν → 0. Physically, Θμν is the smallest patch of the ob-servable area in μν-plane as � (Planck’s constant) illustratesthe smallest fundamental cell of the observable phase spacein quantum mechanics. There are two standard techniques tohandle non-commutativity. One of them is Moyal ∗-product(Chaichian et al. 2001) which commutes coordinates us-ing complex number. This method is valid mathematicallybut physically worthless because it fails to resolve someproblems like loss of unitarity and ultraviolet-divergences inquantum field theory. The other approach is the coordinatecoherent states (Smailagic and Spallucci 2003) in which theeffect of non-commutativity is treated using perturbation ofΘ such as Θμν = Θ diag(ε1, ε2, . . . , ε d

2), where Θ is the

real valued constant.The d-dimensional RN metric in NC form is given by

(Nozaria and Mehdipour 2010)

f±(r) = 1 − 2M±rd−3

1

Γ(d−1)

2

γ

(d − 1

2,

r2

4Θ

)

+ (d − 3)2(d − 2)

2πd−3

Q2

r2(d−3)g(r),

g(r) = γ 2(

d − 3

2,

r2

4Θ

)

− 211−3d

2

(d − 3)Θd−3

2

γ

(d − 3

2,

r2

4Θ

)rd−3,

and incomplete gamma function is defined as

γ

(a

b, z

)=

∫ z

0

dτ

ττ

ab e−τ dτ. (23)

The above metric behaves like usual RN metric by apply-ing the commutative limit r√

Θ→ ∞ (Θ → 0). Moreover,

we modify surface stress energy-momentum tensor (11) bymodifying energy density and pressure (Oh and Park 2010)as ρmod = ρ + ρΘ and pmod = p + p⊥. Here ρ and p arethe energy density and pressure of polytropic matter as inthe previous section while ρΘ and p⊥ are the energy den-sity and pressure of the smeared gravitating source in NCbackground. In the context of non-commutativity, Eq. (14)can be written as

dρΘ

dr+ (d − 2)

r(ρΘ + p⊥) = 0, (24)

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Fig. 3 Plots of the effective potential versus z corresponding to n = 0.5 (left) and n = −0.5 (right) with Θ = 3

where

p⊥ = −(

1 − r2

4Θ

)ρΘ, (25)

and the NC energy density is

ρΘ = ρe− r2−r20

4Θ , (26)

ρ is the value of ρΘ at r = r0. This matter behaves like aconstant density matter as r → ∞ or Θ → 0. Thus for fi-nite n, the modified energy density is

ρmod =[(

k + ρ−1n

0

)( r

r0

) d−2n − k

]−n

+ ρe−(r2−r2

04Θ

), (27)

and for perfect fluid (n → ∞), we have

ρmod = ρ0

(r0

r

)(d−2)(k+1)

+ ρe−(r2−r2

04Θ

). (28)

The corresponding equation of motion is given byEq. (20) with

Veff(r) = 1

2(f+ + f−) − (d − 2)2(f+ + f−)2

4(κρmodr)2

− 1

4(d − 2)2(κρmodr)

2. (29)

Using the same values of z and t as defined in the previoussection, ρmod and Veff(z) for finite n take the following form

ρmod = [(k + ρ

−1n

0

)z

(d−2)(d−3)n − k

]−n + ρe−r20 ( z

2d−3 −1

4Θ), (30)

Veff(z) = 1 − ζ+z

+ Q2

z2− (d − 2)2ζ 2−

κr2(d−3)0 z4

× [{(k + ρ

−1n

0

)z

(d−2)(d−3)n − k

}−n

+ ρe−r20 ( z

2d−3 −1

4Θ)]−2 − κ2z2r

2(d−3)0

(d − 2)2

× [{(k + ρ

−1n

0

)z

(d−2)(d−3)n − k

}−n

+ ρe−r20 ( z

2d−3 −1

4Θ)]2

. (31)

The effective potential and the modified energy density forperfect fluid is

ρmod = ρ0

(1

z

) (d−2)(k+1)d−3 + ρe−r2

0 ( z2

d−3 −14Θ

), (32)

Veff(z) = 1 − ζ+z

+ Q2

z2− (d − 2)2ζ 2−

κr2(d−3)0 z4

×[ρ0

(1

z

) (d−2)(k+1)d−3 + ρe−r2

0 ( z2

d−3 −14Θ

)

]−2

− κ2z2r2(d−3)0

(d − 2)2

×[ρ0

(1

z

) (d−2)(k+1)d−3 + ρe−r2

0 ( z2

d−3 −14Θ

)

]2

. (33)

We use numerical method to solve Eq. (20) correspond-ing to NC parameters Θ = 3,6 and 12 keeping the remain-ing parameters same. The graphical behavior of Veff in NCbackground is shown in Figs. 3–6. Figure 3 (left) shows thatmotion of the shell is almost the same from d = 4 to 11, i.e.,NC shell starts from maximum value and then comes to rest.After the rest position, it diverges negatively and attains itsminimum value, so the shell expands or collapses depend-ing on the initial values of parameters. The right graph ofFig. 3 shows that when d = 4, Veff → −∞ for some initialvalues of z but as z increases Veff = 0. This type of motionindicates that the shell collapses to zero forming a singular-ity (BH or NS). For d = 5,6, Veff starts from its maximum

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Fig. 4 The effective potential versus z corresponding to n = 0.5 (left) and n = −0.5 (right) with Θ = 6

Fig. 5 Plots of Veff versus z corresponding to n = 0.5 (left) and n = −0.5 (right) with Θ = 12

value, reaches to static state and then decreases negativelyto a minimum point which implies the expanding or collaps-ing nature of the NC shell. The shell is in rest position ford = 7–9 while for d = 10,11, the shell either collapses orexpands after the rest state.

Figure 4 (left graph) yields three cases for all dimen-sions, i.e., Veff → −∞ as z → 0, Veff = 0 for increasingz and Veff → −∞ as z > 0. In the right graph, Veff increasesfrom −∞ to 0 and then decreases negatively from 0, leadingthe shell to collapse or expand infinitely. For d = 5–11, theshell diverges negatively after crossing the horizontal axisand then comes to rest. The left graph of Fig. 5 shows thatVeff increases from −∞ to a maximum point and then showsthe same behavior as the left graph (Fig. 3) in all dimensions.The right graph for d = 4 and 11 describes complete bounceof the collapsing shell as Veff cuts the horizontal axis twice,i.e., it has two positive roots. For d = 5–10, Veff increasesfrom −∞ to a maximum value but as z increases, the shellbounces off. Figure 6 describes the shell’s motion for per-fect fluid. We see that for all the selected NC parameters(Θ = 3,6,12), Veff shows the same behavior for d = 5–11,it is always negative for all values of z which indicates sin-

gularity formation either as BH or NS. On the other hand,the condition is different for d = 4. The left graph shows thatVeff tends to zero for large z and then diverges negatively. Inthis case, the shell either expands or collapses while the rightgraph suggests that the shell collapses to zero. The left graphdescribes that the shell bounces back for some initial valuesof z and then collapses to zero for z > 0.

The numerical results of the shell’s radius are shownin Figs. 7, 8 and 9 for d = 4–11. We analyze that non-commutativity cannot change the collapsing behavior of theshell. Although NC effect does not eliminate singularityfrom the effective potential, yet it may be helpful to studythe nature of singularity. This can be investigated by calcu-lating BH horizon zh and the singular point such that

1 − 2M±zhr

d−30

1

Γ(d−1)

2

γ

(d − 1

2,(z

1d−3h r0)

2

4Θ

)

+ (d − 3)2(d − 2)

2πd−3

Q2

z2hr

2(d−3)0

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Fig. 6 Plots of the effective potential for perfect fluid. The graphs are plotted for Θ = 3 (left), Θ = 6 (right) and Θ = 12 (lower panel)

Table 1 Horizon radius zh

d Θ zh Θ zh Θ zh

4 3 1.3174 6 1.2031 12 0.3237

5 3 0.1681 6 0.0343 12 0.0091

6 3 0.0663 6 0.0138 12 0.0288

7 3 1.2868 6 0.0569 12 0.0391

8 3 0.4677 6 0.0876 12 0.1956

9 3 0.1634 6 0.0197 12 0.1827

10 3 0.7410 6 0.1216 12 0.0108

11 3 0.5129 6 0.0594 12 1.1408

× γ 2(

d − 3

2,(z

1d−3h r0)

2

4Θ

)

− 211−3d

2

(d − 3)Θd−3

2

γ

(d − 3

2,(z

1d−3h r0)

2

4Θ

)zhr

d−30 = 0.

Keeping all the parameters same, the position of the horizonis calculated by iterative technique given in Table 1, whilethe singular points zs are given in Table 2. These tables canbe discussed as follows:

Table 2 Singular points zs = kn

d−2

(k+ρ−1n

0 )n

d−2

d zs for n = 0.5 zs for n = −0.5

4 0.903602004 1.106681920

5 0.934655265 1.069913194

6 0.950579825 1.051989506

7 0.960264501 1.041379744

8 0.966775704 1.034366085

9 0.971453580 1.029385264

10 0.974976833 1.025665396

11 0.977725969 1.022781466

• For Θ = 3, singular points are covered by horizon radiiwhich indicate the formation of RN BH from the col-lapsing shell for d = 4 and d = 7. On the other hand,singularity lies outside the horizon radius for d = 5,

6,8,9,10,11, leading to NS.• For Θ = 6, singularity is hidden inside the horizon for

d = 4, while the singularity is visible for d = 5 to 11. Inthe first case, BH will form and the second case impliesNS.

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Fig. 7 Plots of the shell’s radius for n = 0.5 where green, red, blue curves correspond to Θ = 3,6,12, respectively

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Fig. 8 Plots of the shell’s radius for, n = −0.5 where green, red, blue curves correspond to Θ = 3,6,12, respectively

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Fig. 9 Plots of the shell’s radius for perfect fluid, where, green, red, blue curves correspond to Θ = 3,6,12, respectively

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• For Θ = 12, singular point lies inside the horizon radiusfor d = 11 which leads to the formation of BH while ford = 4 to 10, singular point is visible as it is not includedwithin zh, thus the shell will encounter NS in this case.

4 Concluding remarks

Spacetime non-commutativity and higher dimensions arewell-motivated concepts within the context of GR as well asstring theory. Black holes with large extra dimensions havebeen the subject of intensive research with the developmentof higher-dimensional string theories and brane-world. Non-commutativity is an inherent property of the spacetime man-ifold which can remove divergences arising in GR. Thus it isinteresting to study higher-dimensional BHs in the contextof NC geometry.

In this paper, we have explored the effects of non-commutativity on collapsing polytropic matter thin-shell inhigher-dimensional RN geometry, dimensions fromd = 4 to 11. The equation of motion for the thin-shell ind-dimensions is formulated whose solution helps to studythe contraction and expansion of the shell. We have consid-ered the most general polytropic equation of state as col-lapsing matter and studied motion of the shell for finite aswell as infinite polytropic index n. For positive n, the shellshows oscillatory behavior with d = 4 while for d > 4, theshell either expands, contracts to some finite point or comesto unstable equilibrium condition. The shell shows the samebehavior for finite negative value of n as well as for perfectfluid. In both cases, Veff represents three different types ofmotion (rest position, expansion and collapse) for d = 4–11.

Finally, we have introduced smeared gravitational sourceto examine the effects of NC parameter Θ on the shell’scollapse. In this approach, the equation of motion is ob-tained by using modified matter density. For d = 4–11,NC shell bounces off, collapses to zero forming BH or NSand expands for the initial data. The nature of singularityis checked through horizon radius. It is found that BH isformed for the dimensions d = 4,7,11 corresponding tosome choice of NC parameters which confirms the valid-ity of CCH. The shell forms NS for d = 5,6,8,9,10 de-pending on the initial conditions for all choices of NC pa-rameters while for some values of Θ , NS is also observed ind = 4,7,11 which leads to the violation of CCH. The chargeparameter Q also effects the shell’s motion but cannot pre-vent the shell from collapse implying that the effect of Θ is

stronger than Q. We conclude that non-commutativity doesnot remove singularity due to the polytropic matter but itgives a reasonable approach to examine the validity of CCH.

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