Linearized stability analysis of gravastars in noncommutative geometry

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

In this work, we find exact gravastar solutions in the context of noncommutative geometry, and explore their physical properties and characteristics. The energy density of these geometries is a smeared and particle-like gravitational source, where the mass is diffused throughout a region of linear dimension ... due to the intrinsic uncertainty encoded in the coordinate commutator. These solutions are then matched to an exterior Schwarzschild spacetime. We further explore the dynamical stability of the transition layer of these gravastars, for the specific case of [beta]=M ^sup 2^/[alpha]<1.9, where M is the black hole mass, to linearized spherically symmetric radial perturbations about static equilibrium solutions. It is found that large stability regions exist and, in particular, located sufficiently close to where the event horizon is expected to form.

Published for SISSA by Springer

Received: April 24, 2013 Revised: October 7, 2013

Accepted: November 16, 2013 Published: December 13, 2013

Linearized stability analysis of gravastars in noncommutative geometry

Francisco S.N. Loboa and Remo Garattinib,c

aCentro de Astronomia e Astrofsica da Universidade de Lisboa,

Campo Grande, Ed. C8, 1749-016 Lisboa, Portugal

bFacolt di Ingegneria, Universit degli Studi di Bergamo, Viale Marconi 5, 24044 Dalmine (Bergamo), Italy

cINFN Sezione di Milano, Via Celoria 16, Milan, Italy

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: In this work, we nd exact gravastar solutions in the context of noncommutative geometry, and explore their physical properties and characteristics. The energy density of these geometries is a smeared and particle-like gravitational source, where the mass is di used throughout a region of linear dimension p due to the intrinsic uncertainty encoded in the coordinate commutator. These solutions are then matched to an exterior Schwarzschild spacetime. We further explore the dynamical stability of the transition layer of these gravastars, for the specic case of = M2/ < 1.9, where M is the black hole mass, to linearized spherically symmetric radial perturbations about static equilibrium solutions. It is found that large stability regions exist and, in particular, located su ciently close to where the event horizon is expected to form.

Keywords: Non-Commutative Geometry, Classical Theories of Gravity, Black Holes

ArXiv ePrint: 1004.2520

c

JHEP12(2013)065

SISSA 2013 doi:http://dx.doi.org/10.1007/JHEP12(2013)065

Web End =10.1007/JHEP12(2013)065

Contents

1 Introduction 1

2 Structure equations of gravastars in noncommutative geometry 32.1 Spacetime metric and eld equations 32.2 Gravitational collapse and gravity prole 5

3 Thin-shell formalism 63.1 Exterior spacetime 73.2 Junction interface 83.3 Extrinsic curvature 83.4 Lanczos equation and surface stresses 93.5 Energy conditions on the junction surface 93.6 Conservation identity 10

4 Linearized stability analysis 114.1 Equation of motion 114.2 Parametrization of the stable equilibrium 124.3 Stability regions 13

5 Summary and conclusion 14

1 Introduction

About a decade ago, an alternative picture for the nal state of gravitational collapse has emerged [14]. The latter, denoted as a gravastar (gravitational vacuum star), consists of an interior compact object matched to an exterior Schwarzschild vacuum spacetime, at or near where the event horizon is expected to form. Therefore, these alternative models do not possess a singularity at the origin and have no event horizon, as its rigid surface is located at a radius slightly greater than the Schwarzschild radius. More specically, the gravastar picture, proposed by Mazur and Mottola [14], has an e ective phase transition at/near where the event horizon is expected to form, and the interior is replaced by a de Sitter condensate. This new emerging picture consisting of a compact object resembling ordinary spacetime, in which the vacuum energy is much larger than the cosmological vacuum energy, is also denoted as a dark energy star [5, 6]. In fact, a wide variety of gravastar models have been considered in the literature [717] and their observational signatures have also been explored [1825]. It was argued that the resulting gravitational condensate star conguration resolve all black hole paradoxes, and provides a testable alternative to black holes as the nal state of complete gravitational collapse [26].

1

JHEP12(2013)065

In this work, we consider a further extension of the gravastar picture in the context of noncommutative geometry. The dynamical stability of the transition layer of these gravastars to linearized spherically symmetric radial perturbations about static equilibrium solutions is also explored. The analysis of thin shells [2734] and the respective linearized stability analysis of thin shells has been recently extensively considered in the literature, and we refer the reader to refs. [3550] for details. Relative to the context of the stability analysis, the radial stability of the continuous pressure gravastar was studied using the conventional Chandrasekhar method, for radial pulsations and small perturbations around a stable equilibrium [51]. The study of the oscillation spectrum was also studied [52] in the context of dark energy stars, where the frequencies of the fundamental mode and the higher overtones are strongly a ected by the dark energy content. It was also argued that this can be used in the future to detect the presence of dark energy in neutron stars and to constrain the dark-energy models.

In the context of noncommutative geometry, an interesting development of string/M-theory has been the necessity for spacetime quantization, where the spacetime coordinates become noncommuting operators on a D-brane [53, 54]. The noncommutativity of space-time is encoded in the commutator [x, x ] = i , where is an antisymmetric matrix which determines the fundamental discretization of spacetime. It has also been shown that noncommutativity eliminates point-like structures in favor of smeared objects in at space-time [55]. Thus, one may consider the possibility that noncommutativity could cure the divergences that appear in general relativity. The e ect of the smearing is mathematically implemented with a substitution of the Dirac-delta function by a Gaussian distribution of minimal length p . In particular, the energy density of a static and spherically symmetric, smeared and particle-like gravitational source has been considered in the following form [56]

(r) = M(4 )3/2 exp

r2 4

JHEP12(2013)065

, (1.1)

where the mass M is di used throughout a region of linear dimension p due to the intrinsic uncertainty encoded in the coordinate commutator.

The Schwarzschild metric is modied when a non-commutative spacetime is taken into account [56, 57]. Although one may consider the analysis in a general static and spherically symmetric line element in the following form

ds2 = A(r)dt2 + A1(r)dr2 + R2(r)(d 2 + sin2 d2) (1.2)

or in isotropic coordinates where the line element is given by by ds2 = e2'(r)dt2 + e (r)[dr2 + r2(d 2 + sin2 d2], where the '(r) and (r) are nite everywhere, we emphasize that physically correct results do not depend on the coordinate system used. In this context, we use Schwarzschild coordinates throughout this work.

Thus, the solution obtained is described by the following spacetime metric

ds2 = f(r) dt2 +

dr2f(r) + r2 (d 2 + sin2 d2) , (1.3)

2

with f(r) = 1 2m(r)/r, where the mass function is dened as

m(r) = 2M

p

and

dtpt exp(t) , (1.5)

is the lower incomplete gamma function [56]. The classical Schwarzschild mass is recovered in the limit r/p ! 1. It was shown that the coordinate noncommutativity cures the

usual problems encountered in the description of the terminal phase of black hole evaporation. More specically, it was found that the evaporation end-point is a zero temperature extremal black hole and there exist a nite maximum temperature that a black hole can reach before cooling down to absolute zero. The existence of a regular de Sitter at the origins neighborhood was also shown, implying the absence of a curvature singularity at the origin. Recently, further research on noncommutative black holes has been undertaken, with new solutions found providing smeared source terms for charged and higher dimensional cases [5862]. Furthermore, exact solutions of semi-classical wormholes [63, 64] in the context of noncommutative geometry were found [65], and their physical properties and characteristics were analyzed.

Despite the fact that both concepts have their own scale of observability, in particular, non-commutativeness manifests itself only at su ciently high energies and small distances, and the gravastar concept is applicable to larger scales, one may argue that due to gravitational instabilities inhomogeneities may arise. Thus, the gravastar solutions outlined in this paper may possibly originate from density uctuations in the cosmological background, resulting in the nucleation through the respective density perturbations.

This paper is outlined in the following manner. In section 2, we present the generic structure equations of gravastars, and specify the mass function in the context of noncom-mutative geometry. In section 3, the linearized stability analysis procedure is outlined, and the stability regions of the transition layer of gravastars are determined. Finally in section 5, we conclude. We adopt the convention G = c = 1 throughout this work.

2 Structure equations of gravastars in noncommutative geometry

2.1 Spacetime metric and eld equations

Consider the interior spacetime, without a loss of generality, given by the following metric, in curvature coordinates

ds2 = e2 (r) dt2 +

dr21 2m(r)/r

32,r2 4

, (1.4)

32,r2 4

=

r2/4

Z

0

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+ r2 d 2, (2.1)

where d 2 = (d 2 + sin2 d2); (r) and m(r) are arbitrary functions of the radial coordinate, r. The function m(r) is the quasi-local mass, and is denoted as the mass function.

3

The Einstein eld equation, G = 8T provides the following relationships

m = 4r2 , (2.2)

= m + 4r3pr r(r 2m)

( + pr)(m + 4r3pr) r(r 2m)

2m(r) r

r 2M

where is dened as = M2/ .

Note that three cases need to be analyzed [56]:

a) if < 1.9, no roots are present;

b) if > 1.9, we have two roots, r and r+, with r+ > r;

c) if = 1.9, we have r+ = r, which may be interpreted as an extreme situation, such as the extreme Reissner-Nordstrm metric.

The function f(r) = (1 2m(r)/r) is depicted in gure 1 for these three cases for the

following values = 1.5, = 1.9 and = 3.5, respectively. Note that all the roots lie within the Schwarzschild radius rb = 2M, where M is the total mass of the system.

4

, (2.3)

pr =

which provides the solution given by

(r) = 12 ln

+ 2

r (pt pr) , (2.4)

where the prime denotes a derivative with respect to the radial coordinate. (r) is the energy density, pr(r) is the radial pressure, and pt(r) is the tangential pressure. Equation (2.4)

corresponds to the anisotropic pressure Tolman-Oppenheimer-Volko (TOV) equation.Using the equation of state, pr = , and taking into account the eld equations (2.2)

and (2.3), we have the following relationship

(r) = m rm r (r 2m)

, (2.5)

1

. (2.6)

One now has at hand three equations, namely, the eld eqs. (2.2)(2.4), with four unknown functions of r, i.e., (r), pr(r), pt(r), and m(r). We shall consider the approach by choosing a specic choice for a physically reasonable mass function m(r), thus closing the system.

Despite the fact that one may consider the general line element given by eq. (1.2), and determine a generalized mass function, we emphasize that we have considered the analysis using curvature coordinates. Thus, in this context, we are interested in the noncommutative geometry inspired mass function given by eq. (1.4), in curvature coordinates. The latter is reorganized into the following form

m(r) = 2M

p

3 2,

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2 !, (2.7)

Figure 1. The function f(r) = (1 2m(r)/r) is depicted for the three cases with the following

values = 1.5, = 1.9 and = 3.5, respectively.

Note that for the specic equation of state that yields the solution (2.6), the stress-energy prole is given by the following relationships

(r) = pr(r) =

1 4

JHEP12(2013)065

m(r)

r2 , (2.8)

pt(r) = 1

8

m(r)

r , (2.9) respectively. Taking into account the mass function given by eq. (2.7), these take the following form

(r) = pr(r) =

M (4 )3/2 exp

r2 4

, (2.10)

pt(r) = M

(4 )3/2

1 r2 4

exp

r2 4

. (2.11)

The stress-energy prole is qualitatively represented in gure 2. The plots are depicted in terms of dimensionless quantities, M2pr(r) and M2pt(r), respectively. Note that the radial pressure is essentially negative, taking extremely low negative values for high values of the parameter and low values of the radial coordinate, and tends asymptotically to zero at spatial innity. Note that as the equation of state pr(r) = (r) is assumed, the

energy density is positive and tends to zero as r ! 1. The tangential pressure, depicted

in the right plot of gure 2, possesses a positive patch, but is also essentially negative throughout the spacetime geometry.

2.2 Gravitational collapse and gravity prole

In this work we analyse the linearized stability analysis around a stable solution, but it is also important to consider the instabilities which arise from the gravitational collapse of

5

JHEP12(2013)065

Figure 2. The stress-energy prole is represented in the plots, in terms of dimensionless quantities, M2pr(r) and M2pt(r), respectively. The radial pressure, depicted in the left plot, is essentially negative, taking extremely low negative values for high values of the parameter and low values of the radial coordinate. The tangential pressure, depicted in the right plot, possesses a positive patch, but is also essentially negative throughout the spacetime geometry. See the text for more details.

the star. Note that, in particular, an expression for m(r) is deduced, and it is not clear whether such a mass distribution can be maintained. It is generally believed that any star which crosses its Schwarzschild radius collapses due to gravitational attraction. Thus, it is important to comment on the mechanism which prevents such a collapse.

To this e ect, consider the locally measured acceleration due to gravity, given by the following relationship: A =

p1 2m(r)/r (r) [17], where the factor (r) may be considered the gravity prole. Now, the convention used is that (r) is positive for an inwardly gravitational attraction, and negative for an outward gravitational repulsion. The gravity prole (r) is plotted in gure 3. Note that for large values of the radial coordinate and large values of the parameter , the gravity prole is positive, implying an attractive nature of the geometry.

However, one encounters a repulsive nature of the spacetime geometry, as < 0,

for the following cases: (i) low values of the radial coordinate, especially in the range of 0.5 < 1.9; (ii) for arbitrary values of r, in the range 0.5. In this context, one

may argue that due to the gravitational collapse of the star, the matter does not cross the Schwarzschild horizon due to the repulsive character of the spacetime, and it is possible that an equilibrium stability region is attained. Thus, it is the repulsive character of the geometry that stops the gravitational collapse from crossing the Schwarzschild horizon and that sustains this gravastar conguration.

3 Thin-shell formalism

The thin-shell is not necessary for all solutions. In fact, the original Maur-Mottola gravastar picture considered a nite thick shell of sti matter, p = , situated near where the event

6

Figure 3. Depicted is the dimensionless quantity for the gravity prole, M (r). The latter

is positive for an inwardly gravitational attraction and negative for an outwardly gravitational repulsion. One veries qualitatively from the plot that > 0 for values of 0.5 < 1.9, rendering

the geometry attractive in this range. The geometry possesses a repulsive character for a wide range of values of the radial coordinate and for 0.5. In this context, we argue that this repulsive

nature of the geometry stops the gravitational collapse from crossing the Schwarzschild horizon. Thus, it is possible that an equilibrium stability region is attained. See the text for details.

horizon is expected to form. However, considering an idealization of a thin shell, one may simplify considerably the dynamic stability analysis of the setup, in particular, consider the linearized stability analysis outlined in the present work. Indeed, the simplied model considered shares the key features of the Mazur-Mottola scenario, and is su ciently simple to be amenable to a full dynamical analysis.

3.1 Exterior spacetime

We shall model specic gravastar geometries by matching an interior gravastar geometry, given by eq. (2.1), where the metric functions are given by eqs. (2.6) and (2.7), with an exterior Schwarzschild solution

ds2 =

1

at a junction interface , situated outside the event horizon, a > rb = 2M. We emphasize that the larger root r+ lies inside the Schwarzschild event horizon. More specically, if one considers the case > 1.9, one would have two roots, i.e., two event horizons lying within the Schwarzschild radius. Thus, in this work we are only interested in the case of < 1.9, which corresponds to the absence of event horizons for the inner solution. This is in order to have a gravastar solution without an event horizon, as its rigid surface is located at a radius slightly greater than the Schwarzschild radius.

7

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2M r

dt2 +

1 2M r

1dr2 + r2 (d 2 + sin2 d2) , (3.1)

3.2 Junction interface

Consider the junction surface as a timelike hypersurface dened by the parametric equation of the form f(x(i)) = 0. i = (, , ) are the intrinsic coordinates on , where is the proper time on the hypersurface. The three basis vectors tangent to are given by e(i) = @/@i, with the following components e(i) = @x/@i. The induced metric on the junction surface is then provided by the scalar product gij = e(i) e(j) = g e(i)e (j). Thus, the intrinsic metric to is given by

ds2 = d2 + a2 (d 2 + sin2 d2) . (3.2)

Note that the junction surface, r = a, is situated outside the event horizon, i.e., a > rb, to

avoid a black hole solution, and we are only interested in the case of < 1.9, of eq. (2.7), as emphasized above.

For the specic cases considered in this work, namely, the interior and exterior space-times given by eqs. (2.1) and (3.1), respectively, the four-velocity of the junction surface x(, , ) = (t(), a(), , ) is given by

U =

dt d ,

dad , 0, 0

=

q1 2ma + a2 1 2ma

, (3.3)

where the overdot denotes a derivative with respect to the proper time, . The () su

perscripts correspond to the exterior and interior spacetimes, respectively, so that m are dened as m = m(a) and m+ = M, respectively.

The unit normal 4vector, n, to is dened as

n =

g

@f

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, a, 0, 0

@x

@f @x

1/2 @f

@x , (3.4)

with n n = +1 and ne(i) = 0. The Israel formalism requires that the normals point from the interior spacetime to the exterior spacetime [6669]. Thus, for the interior and exterior spacetimes given by the metrics (2.1) and (3.1), respectively, the normals may be determined from eq. (3.4), or from the contractions Un = 0 and nn = +1, and are provided by

n =

a,

q1 2ma + a2 1 2ma

, (3.5)

respectively, with m dened as m = m(a) and m+ = M, as before.

3.3 Extrinsic curvature

The extrinsic curvature is dened as Kij = n; e(i)e (j). Di erentiating ne(i) = 0 with respect to j, we have n @2x

@i @j = n, @x@i @x @j , so that the extrinsic curvature is nally

, 0, 0

Kij = n

@2x

@i @j +

given by

@x

@i

@x

@j

. (3.6)

8

Note that, in general, Kij is not continuous across , so that for notational convenience, the discontinuity in the extrinsic curvature is dened as ij = K+ij Kij.

Taking into account the interior spacetime metric (2.1) and the Schwarzschild solution (3.1), the non-trivial components of the extrinsic curvature are given by

K + =

Ma2 + a

q1 2Ma + a2, (3.7)

K =

m a2

ma + a

q1 2m(a)a + a2, (3.8)

and

JHEP12(2013)065

K + =

1 a

r1 2Ma + a2 , (3.9)

K =

r1 2m(a)a + a2 , (3.10)

respectively. The prime henceforth shall denote a derivative with respect to a.

3.4 Lanczos equation and surface stresses

The Einstein equations may be written in the following form,

Sij =

1 a

18 ( ij ij kk) , (3.11)

denoted as the Lanczos equations, where Sij is the surface stress-energy tensor on . Considerable simplications occur due to spherical symmetry, namely ij = diag( , , ).

The surface stress-energy tensor may be written in terms of the surface energy density, , and the surface pressure, P, as Sij = diag(, P, P). Thus, the Lanczos equation,

eq. (3.11), then provide us with the following expressions for the surface stresses

=

1 4a

r

1

2Ma + a2

r1 2ma + a2

, (3.12)

P =

1 8a

1 Ma + a2 + aa

q1 2Ma + a2

1 ma m + a2 + aa

q1 2ma + a2

. (3.13)

3.5 Energy conditions on the junction surface

It is interesting to consider whether the surface stresses, given by eqs. (3.12)(3.13) satisfy the energy conditions on the thin-shell. We shall only consider the weak energy condition (WEC) and the null energy condition (NEC). The WEC implies 0 and +P 0, and

by continuity implies the null energy condition (NEC), + P 0. These will be evaluated

for a static solution at a0.

Taking into account eqs. (3.12)(3.13), then the NEC is given by

+ P =

1 8a0

1 3m0a0 + m(a0)

q1 2m(a0)a0

1 3Ma0

q1 2Ma0

, (3.14)

9

Figure 4. The weak energy condition (WEC) and the null energy condition (NEC) proles are represented in the plots, in terms of dimensionless quantities, i.e., M(a0) and M((a0) + P(a0)).

We verify that both quantiites are positive, consequently satisfying the WEC and NEC. See the text for more details.

where the mass function, evaluated at the junction interface a0, is given by eq. (2.7). In order to verify whether the NEC and the WEC are satised, we plot the latter using the following dimensionless quantities: M(a0) and M[(a0)+P(a0)]. We verify from gure 4,

both quantities are positive throughout the spacetime, thus satisfying the WEC and NEC on the thin-shell.

3.6 Conservation identity

We also use the conservation identity given by Sij|i =

T e(j)n

+, where [X]+ denotes the discontinuity across the surface interface, i.e., [X]+ = X+| X| . The conservation identity is given by

T e() n

+ = 0. We refer the reader to refs. [70, 71] for a more detailed analysis. Thus, the conservation identity reduces to the following conservation law Sij|i = 0.

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+ = [T U n ]+ =

(T tt + T rr)

a

q1 b(a)a + a2

1 b(a)a

+

, (3.15)

where T tt and T rr are the stress-energy tensor components, in the interior and exterior spacetimes. The momentum ux term in the right hand side corresponds to the net discontinuity in the momentum ux F = T U which impinges on the shell. The conservation identity is a statement that all energy and momentum that plunges into the thin shell, gets caught by the latter and converts into conserved energy and momentum of the surface stresses of the junction.

For the present case, note that T tt + T rr = (r) + pr(r) = 0 in the interior spacetime and T tt = T rr = 0 in the exterior vacuum Schwarzschild geometry, so that the momentum ux is zero, i.e.,

T e(j)n

Note that Si|i = [

+2a( +P)/a], so that the conservation identity provides us with

=

2a ( + P) . (3.16)

This relationship will be used in the linearized stability analysis considered below.

Consider that A = 4a2 is the surface area of the thin shell, so that the conservation equation provides the following relationship

d(A)d + P

dAd = 0 . (3.17)

The rst term represents the variation of the internal energy of the shell, the second term is the work done by the shells internal force.

4 Linearized stability analysis

4.1 Equation of motion

Equation (3.12) may be rearranged to provide the thin shells equation of motion given by the following relationship

a2 + V (a) = 0 . (4.1)

The potential is given by

V (a) = F (a)

ms(a) 2a

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2

aG(a) ms(a)

2, (4.2)

where, for notational convenience, the factors F (a) and G(a) are dened as

F (a) = 1

m(a) + Ma , (4.3)

G(a) = M m(a)

a . (4.4)

Linearizing around a stable solution situated at a0, we consider a Taylor expansion of V (a) around a0 to second order, given by

V (a) = V (a0) + V (a0)(a a0) +

1

2V (a0)(a a0)2 + O[(a a0)3] . (4.5)

The rst and second derivatives of V (a) are given by

V (a) = F 2

ms

2a

ms

2a

2

aG ms

aG ms

(4.6)

V (a) = F 2

ms

2a

ms

2a

2

aG ms

aG ms

, (4.7)

2 2

ms 2a

2 2

aG ms

respectively.

11

Evaluated at the static solution, at a = a0, so that a = a = 0, we verify that V (a0) = 0 and V (a0) = 0. From the condition V (a0) = 0, one extracts the following useful equilibrium relationship

ms 2a0

, (4.8)

which will be used in determining the master equation, responsible for dictating the stable equilibrium congurations.

4.2 Parametrization of the stable equilibrium

Using the surface mass of the thin shell ms = 4a2, the conservation law of the surface stresses, i.e., eq. (3.16), can be rearranged to provide the following relationship

ms 2a

= 4 , (4.9)

with the parameter dened as = P/, and for notational simplicity, the function is

given by

=

a0 ms

F 2

a0G

ms

a0G

ms

JHEP12(2013)065

4a ( + P) . (4.10)

Equation (4.9) will play a fundamental role in determining the stability regions of the respective solutions. Note that is used as a parametrization of the stable equilibrium, so that there is no need to specify a surface equation of state. The parameter p is normally interpreted as the speed of sound, so that one would expect that 0 < 1, based on the

requirement that the speed of sound should not exceed the speed of light. We refer the reader to refs. [49, 50] for further discussions on the respective physical interpretation of lying outside the range 0 < 1.

The solution is stable if and only if V (a) has a local minimum at a0 and V (a0) > 0 is veried. Thus, from the latter stability condition, after a rather lengthy but straightforward calculation, one deduces the master equation for the stability regions, given by

0 d2

da

a

0

> , (4.11)

where 0 = (a0). Note that to deduce this inequality, we have used eq. (4.9), and for notational simplicity, we have dened the function by

1

2

+ 12a0 ( 2 ) , (4.12)

with given by eq. (4.8), and the function dened as

= F

2

aG ms

2

aG ms

aG ms

. (4.13)

12

Now, from the master inequality (4.11), we nd that the stable equilibrium regions are dictated by the following inequalities

0 > , if d2 da

a

0 < , if d2 da

d2 da

4.3 Stability regions

We now determine the stability regions dictated by the inequalities (4.14)(4.15). In the specic cases that follow, the explicit form of is extremely messy, so that we nd it more instructive to show the stability regions graphically.

For the case of interest under consideration, namely, < 1.9, the specic expression for d2/da|a0 is given by

d2 da

q1 2Ma q1 2m(a)a

q1 2Ma q1 2m(a)a

[a 3m(a) + am(a)]r1 2Ma (a 3M)

> 0 , (4.14)

< 0 , (4.15)

0

a

1. (4.16)

0

with the denition

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a

0

= 1

8a4

a0

. (4.17)

It is useful to consider the dimensionless quantity M3d2/da|a0, which is depicted in g

ure 5. From the latter, it is transparent that d2/da|a0 < 0, so that the stability regions

are dictated by inequality (4.15).

The respective stability regions are given by the plot depicted below the surface in gure 6. It is interesting to note that the stability regions are su ciently close to where the event horizon is expected to form, which is extremely promising. Indeed, large stability stability regions exist in the neighbourhood of where the event horizon is expected to form, for arbitrary values of the parameter . For low values of the parameter, the stability regions decrease for increasing values of a. For large values of the stability regions decrease for increasing values a, and then increase again as a increases. Thus large stability regions exist also for large values of , for regions su ciently far from the where the event horizon is expected to form.

The message that one may extract, is that the above analysis shows that stable congurations of the surface layer, located su ciently near to where the event horizon is expected to form, do indeed exist. Therefore, considering these models, one may conclude that the exterior geometry of a noncommutative geometry inspired gravastar would be practically indistinguishable from a black hole.

13

r1 2m(a) a

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Figure 5. The sign of the dimensionless quantity M3d2/da|a0 is depicted in the plot, where it is

transparent that d2/da|a0 < 0. Thus, the stability regions are dictated by inequality (4.15). See

the text for more details.

Figure 6. The respective stability regions are given by the plot depicted below the surface. Large stability stability regions exist in the neighbourhood of where the event horizon is expected to form, for arbitrary values of the parameter . For low values of the parameter, the stability regions decrease for increasing values of a. For large values of the stability regions decrease for increasing values a, and then increase again as a increases. See the text for more details.

5 Summary and conclusion

The gravastar model was proposed as an alternative picture for the nal state of gravitational collapse. It remains an open problem if alternatives to standard black holes as the nal state of gravitational collapse do really exist and only a full quantum theory of gravity

14

could answer this question adequately. This justies the research on black hole mimickers that could explain observational data without the paradoxical problems regarding black holes.

In this work, we have found exact gravastar solutions in the context of noncommutative geometry, and briey explored their physical properties and characteristics. More specically, the energy density of these geometries is a smeared and particle-like gravitational source, where the mass is di used throughout a region of linear dimension p due to the intrinsic uncertainty encoded in the coordinate commutator. Furthermore, the mass function was deduced from the Einstein eld equations, and the equation of state pr(r) = (r) was imposed.

A fundamental aim in this work was to analyse the linearized stability analysis of the transition layer of these noncommutative geometry inspired gravastars around static equilibrium solutions. However, it is also important to consider the instabilities which arise from the gravitational collapse of the star. It is generally believed that any star crosses its Schwarzschild radius collapses due to gravitational attraction. Therefore, we have also commented on a mechanism which prevents such a collapse. We have shown that the spacetime has a repulsive character for a wide range of the parameters of the model. Therefore, we have argued that it is the repulsive nature of the geometry that stops the gravitational collapse of the star, and it is possible that an equilibrium stability region is attained, thus sustaining this gravastar conguration.

We further explored the dynamical stability of the transition layer of these noncommutative geometry inspired gravastars to linearized spherically symmetric radial perturbations about static equilibrium solutions. More specically, we considered the speed of sound parameter dened as = P/, where and P, are the surface energy density and surface

pressure, respectively. The parameter was used as a parametrization of the stable equilibrium, so that there was no need to specify a surface equation of state. Consequently, a master equation was deduced that dictated the stability regions. It was found that large stability regions do exist, which are located su ciently close to where the event horizon is expected to form, for a wide range of the parameters of the model. Thus, it would be di cult to distinguish the exterior geometry of the gravastars, analyzed in this work, from a black hole.

Indeed, in this work, we were only interested in the linearized spherically symmetric radial perturbations of the transition layer around static equilibrium solutions. However, we note that a thorough stability analysis of the whole system is in order to verify the stability of the entire gravastar conguration. To this e ect, one may consider the non-spherically symmetric stability analysis explored in ref. [72], where general axial perturbations and monopole-type polar perturbations were considered in the linear approximation, for two classes of solutions, namely, wormholes with at asymptotic behavior at one end and AdS on the other (M-AdS wormholes) and regular black holes with asymptotically de Sitter expansion far beyond the horizon (the so-called black universes). As a result of the analysis, it was shown that all congurations under study are unstable under spherically symmetric perturbations, except for a special class of black universes where the event horizon coincides with the minimum of the area function. It will be interesting to apply the stability analysis

15

JHEP12(2013)065

outlined in ref. [72] to the entire gravastar conguration, which we leave for a future

publication.

In conclusion, it would be interesting to apply to the present case, the alternative formalism developed in two companion papers [70, 71], where an extremely general and robust framework leading to the linearized stability analysis of dynamical spherically symmetric thin-shell gravastars was developed. In the latter, the logic ow was reversed, where the surface mass as a function of the potential was considered, so that specifying the latter informs on how much surface mass one needs to put on the transition layer. Work along these lines is currently underway.

Acknowledgments

FSNL acknowledges partial nancial support of the Fundao para a Cincia e Tecnologia through the grants CERN/FP/123615/2011 and CERN/FP/123618/2011.

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