CosmiccensorshipbeyondGeneralRelativity ... · CosmiccensorshipbeyondGeneralRelativity: Collapsingchargedthinshellsinlowenergyeffectivestringtheory Pedro Aniceto1 and Jorge V. Rocha

Cosmic censorship beyond General Relativity:Collapsing charged thin shells in low energy effective string theory

Pedro Aniceto1 and Jorge V. Rocha (Supervisor)21CENTRA, Departamento de Física, Instituto Superior Técnico, Universidade de Lisboa,

Avenida Rovisco Pais 1, 1049 Lisboa, Portugal2Departament de Física Fonamental, Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona,

Martí i Franquès 1, E-08028 Barcelona, Spain

The thin shell collapse in Einstein-Maxwell-dilaton is studied by resorting to solutions whichrepresent four-dimensional, spherically symmetric black holes (or naked singularities) in low energyeffective string theory, with the aim of testing the weak cosmic censorship conjecture. Throughthe Darmois-Israel formalism, we obtain the junction conditions that describe the matching of twospacetimes in this theory through timelike thin shells. The junction conditions, together with theweak and dominant energy conditions, are used to study the allowed positions of static thin shellswhose matter content is a perfect fluid. Moreover, we study the collapse of thin shells made ofdust, whose interior and exterior spacetimes are both described by static Einstein-Maxwell-dilatonsolutions with equal dilaton charge, which is required for motion to be allowed. For this collapse, itis shown that the weak energy condition imposes that cosmic censorship is always satisfied and theshell either bounces or collapses into a black hole. Finally, we investigate the collapse of thin shellsjoining a time-dependent Einstein-Maxwell-dilaton exterior, that is radiating in the form of a nullfluid, with a static interior. In this case, it is shown that if the energy conditions for the null fluidstress-energy tensor are satisfied, then the thin shell either collapses into a future event horizon,bounces back to infinity or otherwise the weak energy condition of the thin shell is violated beforecollapsing into a naked singularity, once more upholding the cosmic censorship conjecture.

I. INTRODUCTION

Cosmic censorship was conjectured by Penrose [1] al-most half a century ago but it is still a subject of cur-rent debate and is considered one of the most importantopen problems in general relativity (GR). It posits, inits weak version, that any curvature singularities thatmight form during the evolution of generic regular initialdata, with physically reasonable matter, always appearhidden behind event horizons, and thus are inside blackholes. The cosmic censorship conjecture (CCC) is a cor-nerstone of classical general relativity. GR is expected tobreakdown near curvature singularities (where quantumphysics kicks in). In the absence of horizons - which es-sentially act as one way membranes, allowing signals topropagate inside but not outside - enclosing such regions,a distant observer would be able to see the quantum na-ture of the high curvature region. This would configure abreakdown of predictability within GR. Hence, CCC pro-vides a safeguard to GR, guaranteeing its self-consistencyas a classical theory.

Since its genesis, the CCC has been continually tested.Some of such tests consist of very simple models suchas the gravitational collapse of fluids and destruction ofhorizons through the use of test particles. One inter-esting setting to use these tests is the spherical gravi-tational collapse of electrically charged thin shells, thatwhile not as realistic as rotating models which are knownto be non-spherical, it provides relatively simple exactsolutions of the field equations. If the collapse were toproceed until formation of a curvature singularity andthe charge of the shell were sufficiently large (comparedto its mass), the end state would feature a naked sin-

gularity, that is, a singularity that isn’t covered by anevent horizon. This would constitute a violation of cos-mic censorship, but it is well known in the literature (e.g.Refs [2–4]) that such collapses never generate naked sin-gularities when one considers the Einstein-Maxwell the-ory which describes electromagnetism coupled to gravity.In that theory, shells with sufficiently large charges toproduce an overcharged configuration either violate ba-sic (physically motivated) energy conditions or simply donot collapse. Instead, they bounce back at some finite ra-dius. Similar studies for modified theories of gravity aremostly unexplored, although some examples do exist, forexample Refs. [5–12].

In this work we will be interested in testing the pos-sibility of formation of naked singularities in a four-dimensional, low energy string theory for which spheri-cally symmetric black hole solutions are known (e.g. [13–16]). The theory is based on the addition of a scalar field(the dilaton), that appears frequently in string theory,to the metric and Maxwell field. This field may reducethe critical value for the electric charge necessary for theappearance of a naked singularity, due to the disappear-ance of the event horizon of the black hole, when com-pared to the value obtained for Maxwell-Einstein theory.The comparison between these two theories will allowus to understand what are the aspects that low-energystring theory adds to the already known description ofcharged black holes. Moreover, the weak cosmic cen-sorship conjecture and, by implication, the viability ofsuch string-inspired models, will then be investigated inthis Einstein-Maxwell-dilaton theory by studying the so-lutions describing the spherical collapse of charged thinshells. While this is a very simple model, its usefulness

2

lies in the fact that it allows us to obtain exact analyticalsolutions. This provides us the means to infer, withoutthe need of a perturbative approach, what are the dy-namics and end results of this type of collapse.

The rest of the paper is organised as follows. In thenext section we briefly summarize the GMGHS solu-tion [13–15] and its time-dependent counterpart obtainedin Ref. [16]. In section III we apply the Darmois-Israelformalism to the solutions we are interested in studyingand we also obtain new junction conditions that arespecific for the Einstein-Maxwell-dilaton theory. Insection IV we present the allowed regions where staticshells of a perfect fluid satisfy the weak and dominantenergy condition. The end result of the collapse of thinshells made of dust and implications towards the CosmicCensorship Conjecture are studied in section V. Weclose in section VI with a discussion of the results, theirinterpretation and a brief outlook.

II. EINSTEIN-MAXWELL-DILATON BLACKHOLE SOLUTIONS

In String Theory the existence of the dilaton, whichcouples to the electromagnetic field, implies that when weconsider String Theory models then Einstein-Maxwell so-lutions are not good approximations for the description ofcharged black holes. A necessity therefore arises to studynew solutions that take into account this scalar field. Onesuch solution, that is of high interest due to representingthe String Theory analog of the Reissner-Nordström solu-tion when the scalar field is considered, was first derivedby Gibbons [14] in 1982 and Gibbons and Maeda (GM)[15] in 1987. Later, Garfinkle, Horowitz and Strominger(GHS) in 1990 [13] independently obtained a new solu-tion that was the same as the one of Gibbons and Maedaonly with a rescaling of the metric by a conformal factor.Overall, these solutions enabled us to understand howstatic, charged black holes behave in four-dimensionallow energy string theory.

In this paper we are interested not only in the GMGHSsolution but also in its time-dependent extension. Thisextension was recently obtained in Ref. [16], where theVaidya [17, 18] and Bonner-Vaidya [19] solutions wereextended to their Einstein-Maxwell-dilaton counterparts.Due to the fact that this time-dependent solution be-comes the GMGHS solution when taking the static limit,we will present them jointly and point out the relevantdistinctions between them where necessary.

We consider the following Lagrangian for Einstein-Maxwell-dilaton theory (henceforth we set G = c = 1),

L =

√−g

16π

[R− 2(∇φ)2 − e−2φF 2 + 16πAµJ

µ]

+ Lm ,

(1)where g is the determinant of the metric gµν , Aµ is aMaxwell field with field strength Fµν = ∂µAν−∂νAµ, andφ is a scalar field (the dilaton), which is coupled to the

field strength. In the action above we have also includedthe minimal coupling of the Maxwell field to a currentJµ and an extra matter Lagrangian which will accountfor the radiation in the time-dependent case fluid. Thefield equations arising from Eq. (1) read

∇µ(e−2φFµν

)= −4πJν , (2a)

∇2φ+1

2e−2φFµνF

µν = 0 , (2b)

Gµν = 8πTµν ≡ 8π(T (dil)µν + T (EM)

µν + T (fluid)µν

),(2c)

where Gµν := Rµν − 12Rgµν is the Einstein tensor.

We split the total stress energy tensor Tµν into threepieces according to their different origins: a contribu-tion from the dilaton, 8πT

(dil)µν := 2∇µφ∇νφ− gµν(∇φ)2,

a contribution from the electric field, 8πT(EM)µν :=

e−2φ(2FµαF

αν − 1

2gµνF2), and the charged fluid energy-

momentum tensor,

T (fluid)µν = Tm

µν + gµνAσJσ − 2A(µJν) , (3)

where Tmµν := − 2√

−g∂Lm

∂gµν . We recover the Einstein-Maxwell theory by consistently setting the dilaton andcoupling constant both to zero, φ = a = 0.

Static solutions of the field equations (2) in the absenceof the source terms Jµ and Tm

µν were found in Refs. [13–15]. The electrically charged solution, which we willhenceforth refer to as the GMGHS solution, reads [13]

ds2 = −(

1− r+

r

)dt2 +

dr2

1− r+r

+ r(r − r−)dΩ2 ,(4)

F = −Qr2dt ∧ dr , (5)

e2φ(r) = e2φ0(1− r−/r) , (6)

where φ0 is the asymptotic value of the dilaton field, thephysical massM , the electric charge Q, and the dilatoniccharge D are related to r± by

M =r+

2, Q2 = e2φ0

r+r−2

, D =r−2. (7)

Note that the surface r = r− is singular, since its areagoes to zero and r+ corresponds to the event horizon.The absence of a naked singularity for this solution im-poses r+ > r−.

While the above solution (4-6) refers to electricallycharged black holes, it is easy to obtain magneticallycharged solutions via electric-magnetic duality [13]. Itturns out that the metric remains unaltered while thedilaton field flips sign.

The generalization to the time-dependent radiating so-lution was found in [16], and reads

ds2 = −(

1− r+

r

)du2 − 2dudr + r (r − r−) dΩ2 , (8)

F = −Q(u)

r2du ∧ dr , (9)

e2φ = e2φ0 (1− r−/r) , (10)

Jν = −e−2φ

4πr2Q′(u)δνr , (11)

3

where the prime denotes a derivative with respect to u(the outgoing Eddington-Finkelstein null coordinate) andnow r± = r±(u) and φ = φ(u, r). From Eq. (11) we con-clude that as long as the electric charge Q is not constantwe have a radial current, which is nevertheless divergencefree, i.e. ∇µJµ = 0. Furthermore, it is also shown thatthe null fluid is pressureless and its stress-energy tensoris given by

T (fluid)µν =

µ`µ`ν + ρ(`µwν + `νwµ)

8π, (12)

µ =

[(r+r−)′ − 2rr′+

]+ 2r2r′′−

2r2[r − r−], (13)

ρ = −r′−

r(r − r−), (14)

where we have defined the future pointing null-vectors`µ = −∂µu = −δuµ and wµ = 1

2guuδuµ − δrµ. The quantity

ρ is a generalization of the energy density of a perfectfluid to the null fluid case and µ is an energy density thatreceives contributions from both the matter Lagrangian,Lm, and from the current terms, A(µJν), in Eq. (3). Forthese solutions, the term AσJ

σ in Eq. (3) vanishes as Aµonly has a uu-component while from Eq. (11), Jµ onlyhas an rr-component. Moreover, we also note that theterm A(µJν) only has a uu-component, which contributesto the energy density µ but not to ρ.

As shown in [16] the weak and dominant energy con-ditions of the fluid stress-energy tensor, for the casewe are studying, reduce to r′−(u) ≤ 0 and 2r2r′′− +[(r+r−)′ − 2rr′+

]≥ 0. Additionally, in that paper it is

also shown that these conditions are sufficient for thetotal stress-energy tensor Tµν to satisfy the energy con-ditions too. Consequently, we will restrict our study tothe cases in which these conditions always hold true.

As we will see in the next section, we are interestedin the case r−(u) = 2D(u) = const. Recall thatr+(u) = 2M(u) and Q2(u)/M(u) = 2e2φ0D = const.Hence, the previous impositions coming from theenergy conditions further reduce to D = const andM ′ ≤ 0 which implies that the apparent horizon isnon-increasing.

III. JUNCTION CONDITIONS

Having reviewed the solutions that will be consideredwe are now interested in gluing two such spacetimesalong a hypersurface by applying the junction conditionsobtained by Darmois [20] and further developed by Is-rael [21]. Through this procedure we intend to determinethe potential that governs the radial dynamics of a thinshell located at a timelike hypersurface separating thetwo different spacetimes.

We start by considering a four dimensional spacetimewhich is partitioned in two regions V+ and V− by atimelike hypersurface. Each region, is defined by their

coordinate patch xα± and metric g±αβ where the suffix +

corresponds to the region V+ whereas − corresponds toV−. With this setup defined we now wish to determinethe conditions that allow for a smooth junction of bothregions at Σ. We further parametrize the hypersurfacethrough the coordinate system ya1 on both sides of thehypersurface. Moreover we also consider a system of localcoordinates xα on the neighborhood of Σ which evidentlyoverlaps with both xα± on an open region of V±.

The vectors tangent to Σ are defined by eαa = ∂xα

∂ya .Note that these vectors can be used to project quantitiesonto Σ. Additionally, we define the normal to Σ as nαwith n · n = 1 and we use the notation [A] ≡ A(V+)|Σ −A(V−)|Σ to represent the jump of the quantity A acrossΣ. Note that by definition [eαa ] = 0.

Using these definitions, the conditions that theDarmois-Israel formalism provides in the presence of athin shell of matter are

[hab] = 0 , (15)

[Kab] = −8π

(Sab −

S

d− 2hab

), (16)

where hab = gαβeαaeβb is the induced metric on the hyper-

surface, Sab is the stress-energy tensor of the thin shell,S = Sabh

ab is its trace and Kab = ∇βnαeαaeβb is the ex-

trinsic curvature of the shell.Due to the presence of the Maxwell field and the scalar

field, on top of the junction conditions we just presentedwe also need to determine the junction conditions thatare imposed by both these fields so that they are prop-erly defined as distributions on both spacetimes joinedby the thin shell. Following the procedure to obtain theDarmois-Israel junction conditions detailed for examplein [22] we conclude that

[φ] = 0 , (17)[φ,⊥] = −4πρ(dil) (18)

where we used the notation Aαnα ≡ A⊥ and ρ(dil) cor-

responds to a scalar charge density on the shell. Thissource term in fact would correspond to having in Eq. (1)a 16πρ(dil)φδ(l), where l defines the position of the shell,and would consequently modify Eq. (2b) to also haveon its right-hand side the −4πρ(dil)δ(l) term. RecallingEqs. (6) and (10) we reach the conclusion that we canonly have a non-static shell if D has the same value inboth the interior and exterior spacetimes.

The Maxwell field on the other hand imposes the fol-lowing conditions

[Fab] = 0 , (19)e−2φ [Fa⊥] = −4πsa , (20)

1 Henceforth we shall use greek indices when considering the ddimensional spacetime whereas latin indices will refer to the d−1dimensional hypersurface.

4

where we have defined the surface charge current as sα =σeu

α|Σ, σe is the surface density of electric charge on theshell and uα is the 4-velocity of the current. Note alsothat Eq. (20) reduces to the classical result obtained inRef. [23] when φ = 0.

There is one final result related to the junction condi-tions of the shell and which will be used throughout thispaper which relates the non-conservation of the shell’sstress-energy tensor with the energy flow across the shelland reads (Ref. [22])

Sba|b + [Tα⊥eαa ] = Sba|b + [ja] = 0 , (21)

where ja is the stress-energy density current of the shell.To study static shells with an empty interior and

a GMGHS exterior and also to study dynamical thinshells that have GMGHS solutions as their interior andexterior, it is preferable to use a new coordinate sys-tem such that r(r − 2D) = %2. In this new coordi-nate system we define the the shell as the hypersurfaceΣ = xµ : t = T (τ), % = R(τ) which is parametrized bythe coordinates ya = τ, θ, ϕ, with τ corresponding tothe proper time of the shell. In this coordinate systemthe GMGHS solution becomes

ds2 = −Adt2 + (AC)−1d%2 + %2dΩ2 , (22)

where we defined the metric functions

A(%) ≡ 1− 2M

D +√D2 + %2

, C(%) ≡ 1+

(D

%

)2

. (23)

We conclude that the coordinate % is precisely the arealradius. Note that in this coordinate system, the eventhorizon is located at % = 2

√M(M −D).

Denoting the overdot as the derivative with respectto τ we have that the shell’s 3-velocity is u = ∂

∂τ=

T ∂t + R∂%, with u · u = −1. With these definitionswe can determine the %-component of nα and obtainn% = ±

√AC + R2. Additionally, we define the shell

to be made of a perfect fluid, thus Sab = σuaub +p (hab + uaub), where σ is the shell’s suface energy den-sity and p is a transverse pressure which we will considerto be null for dynamical shells.

Equipped with these definitions and Eqs. (15) and (21)we obtain respectively

3ds2 = −dτ2 +R2dΩ2 , (24)

2 (σ + p) R+Rσ = −R [∂RCn%/(8πC)] . (25)

where 3ds2 is the induced metric on the hypersurface.Additionally, Eq. (16) yields

Kθθ = Rn% , Kττ = −√C

Rd

dτ

(Kθθ

R√C

), (26)

[Kθθ] = −4πR2σ , [Kττ ] = −4π (σ + 2p) . (27)

Eqs. (24-27), together with Eqs (17-20) provide us allthat we need to proceed with the study of static anddynamical thin shells.

IV. SELF-GRAVITATING STATIC SHELLS

In this section we analyse the allowed regions for staticthin shells whose matter content is a perfect fluid andwhich have an empty interior and GMGHS exterior. Inthis case, for Eq. (17) to be satisfied,R has to be constantand is given by R = D/ sinh(φ0), where both D and φ0

correspond to the dilaton charge and asymptotic valueof the dilaton field of the GMGHS exterior. ApplyingR = 0 in Eqs. (26) and (27) and solving for σ and pyields

σ =1

4πR

(1−

√(1− 2M

D +√R2 +D2

)(1 +

D2

R2

)), (28)

p = −σ2

+M

8π(D +

√R2 +D2

) 32

√(D − 2M +

√R2 +D2

) . (29)

Interestingly, if we were to consider a shell made of dust,i.e. p = 0, then Eqs. (28) and (29) would impose D = M .This means that a static shell made of dust that joins aMinkowski interior with a GMGHS exterior would nec-essarily have to be extremal. Moreover, it is importantto state that no impositions on R appear. This is physi-cally sensible because for extremal black holes the gravi-tational and scalar attractive forces are exactly cancelledby the electric repulsion, for example enabling the exis-

tence of static configurations of multiple black holes [13].These static solutions are an extension of studies suchas the one present in Ref. [24] applied to the Majumdar-Papatetrou solutions [25, 26] when one adds the dilatonscalar field.

We wish to know what are the physically allowed radiiwhere these shells can be positioned. This may be de-termined by the energy conditions. In our case, we willfocus on verifying in which regions the weak and domi-

5

0 0.5 1 1.5

1

2

3

ϱ

FIG. 1: The values of R for which the DEC is satisfied cor-respond to the blue region and the ones where the WEC issatisfied correspond to the union between the blue and orangeregion. The white region below the curve R = 2

√M(M −D)

corresponds to the interior of the event horizon. The orangeregion above it corresponds to another region, this time out-side the horizon, which is forbidden by the DEC.

nant energy conditions (WEC and DEC respectively) aresatisfied. Recall, that for a perfect fluid the WEC statesσ ≥ 0 and σ + p ≥ 0 while the DEC further imposesσ ≥ p [22, 27]. The conditions that are imposed by theWEC and DEC can be seen in Fig. 1.

There are several points worth of notice in Fig. 1.When D → 0, for the DEC to be fulfilled we have a lowervalue of % that is higher than 2

√M(M −D). While this

may appear strange, it is agreement with previous workson this subject concerning the Schwarzschild case suchas Refs. [28, 29] where they show that the DEC is onlysatisfied if 1−2M/% ≥ 1/5. This last one being the resultwe also obtain for D = 0. For a subextremal shell, wemay never have a timelike static shell inside of the BHhorizon, which is in accordance with the fact that insideof this horizon the coordinates t and % change their char-acter. Moreover, we notice that the value of % = 0 is onlyallowed for an extremal shell as it had been previouslymentioned. Finally, for an overcharged shell, a minimum% in which it can be placed, always exists.

Regrettably, with this analysis we are not able tostudy the stability of these static solutions. Such astudy relies on the dynamic properties of these shells,specifically on the sign of the second derivative of thepotential that governs their motion. However, in the

approach followed, the junction conditions forbid anykind of motion whatsoever. Note that we have restrictedour study to R = 0 from the beginning and so we did notextract the radial potential. Therefore, we do not haveinformation about (the signs of) its derivatives and areunable to retrieve through them a physical descriptionof the stability of these static shells.

V. DYNAMICAL SHELLS AND COSMICCENSORSHIP

In this section we will analyse the behavior of the po-tential for thin shells of dust (p = 0) in two situations.First we will consider thin shells that join two GMGHSsolutions of equal dilaton charge. Afterwards, we willconsider thin shells that have a GMGHS interior with aradiating exterior. In both cases, we shall adopt as no-tation the subscripts i and o to denote the quantities ofthe inner and outer spacetimes respectively.

The analysis of the potential will be used to determinethe behavior of the shells in both cases. It will be centeredon determining which are the regions where the potentialis greater or equal to zero and through it infer what arethe radii at which movement of the shell is allowed. Thepotential that governs the dynamics of a shell is given by

R2 = V (R). (30)

This equation shows that if V (R) were to be smaller thanzero then it’s kinetic energy would be negative. As this isnot physically realistic we conclude that the points wherethe potential vanishes portray boundaries that representturning points of a moving shell on which it will bounceback.

A. Collapse with static spacetimes

Recalling that n% = ±√AC + R2 it is straightforward

to see that Eq. (27), specifically the θθ-component ofthe second junction condition, may be used to obtainthe potential we wish to study. Nonetheless, to fullydescribe the potential, we first need to solve Eq. (25)and determine σ(τ). As in this case we are interested innon-static shells, Eq. (17) imposes Di = Do and, conse-quently, Ci = Co ≡ C. Thus, Eq. (25) reads

dσ

dR+ 2

σ

R=σ

R+ 2

σ

R=

1

8πR∂RC

C

(√AiC + R2 −

√AoC + R2

). (31)

It is possible to convert this equation into an ordinary dif-ferential equation. Using Eq. (27), definingM≡ 4πR2σ

and multiplying both sides by 2M we obtain

d(M2

)dR

=∂RC

CM2 , (32)

6

and a simple solution can be given in closed form, yieldingM =

√Cm, where m is an integration constant. Note

that, when D = 0 we just get M = m. It is worthstressing that m, the value that we are imposing to bethe constant of the general solution of these differentialequations, comes directly from Israel’s study in which heconsidered a Schwarzschild exterior and an empty interior(Ref. [21]). There, it is proven that M ≥ m ≥ 0 for the

shell to collapse from infinity.From Eqs. (26) and (27) we can now obtain the poten-

tial of these shells which reads

V (R) = −AoC +R2

4M2

(AiC −AoC −

M2

R2

)2

, (33)

which, when applied to our solutions yields

V (R) =

(R2 +D2

)( (m2+2(√R2+D2−D)(Mo−Mi))

2

m2 − 4(R2 − 2

(√R2 +D2 −D

)Mi

))4R4

. (34)

Our interest relies on the collapse of shells initially formedat infinity, which is equivalent to imposing V (R →∞) ≥0. Expanding V (R) at infinity yields

V (R)R→+∞−−−−−→ −1 +

(Mo −Mi)2

m2+O

(1

R

). (35)

We conclude that the condition for these thin shells to beformed at infinity isMo−Mi ≥ m which is what we wouldexpect as it states that the energy of the shell correspondsto the difference between the energy of both spacetimes.Consequently, the case Mo − Mi = m corresponds tohaving the shell initially at rest. Note that if Mi−Mo ≥m this would imply, from Eq. (27) that m ≤ 0 whichwould obviously violate the weak energy condition.

To understand the end result of the gravitational col-lapse of thin shells described by the potential in Eq. (33)we need to understand at which points it becomes neg-ative. Normalizing all the variables in terms of Mo weend up with three independent parameters, m, Mi andD. From Eq. (33) it is clear that the relevant term thatdescribes the behavior of these shells is the second termof the numerator. Depending on its sign and value wemay have shells bouncing (if it becomes null) or col-lapsing either into a black hole or a naked singularityin case it remains positive throughout the entirity of thecollapse. To study this problematic term we will onceagain change the radial coordinate to the one used in theGMGHS solution by using the following transformationR2 = R(R−2D). In these new coordinates the only termthat might be problematic when it is negative, which wename V = V (R), reads

V =

(m+ 2(R− 2D)

Mo −Mi

m

)2

−4(R−2D) (R− 2Mi) .

(36)We will start by focusing on the overcharged exteriorcase, i.e. D ≥ Mo. If the shell is at rest at infin-ity, i.e. Mo − Mi = m, we obtain V ≤ 0 for R ≥2D+m2/(8D− 8Mi − 4m), which would actually implythat the shell cannot be formed at infinity and bounces

at a maximum radius. If on the other hand we have anextremal exterior but have no further impositions on mwe obtain V ≤ 0 for 2D +m2/(2D − 2Mi + 2m) < R <2D+m2/(2D−2Mi−2m). The general overcharged casemay be deduced from the expansion of Eq. (36) in powersof R − 2D. Doing that, it is straightforward that, whenboth spacetimes are overcharged, V always has two sep-arate roots, one of which is always higher than R = 2D.Hence, for the overcharged case we will also always findregions of the potential where V is negative. Therefore,when considering an overcharged thin shell of dust col-lapsing from infinity, this shell will always bounce andconsequently will never collapse into a naked singularity.

Finally, for the subextremal case, i.e. Mo > Mi ≥ D,the possibly negative terms of V are (R − 2D)(Mo −Mi)

2/m2 − (R−Mo −Mi). As in this case we have thatMo + Mi ≥ 2D it is straightforward to see that V ispositive for all values of R. We are led to the conclusionthat when both the interior and exterior are subextremal,these thin shells always collapse into a black hole.

We end this section by presenting a few of the dif-ferent types of collapse we just studied in Fig. 2. Notethat while the purple curve seems to hint that in thatsituation the shell may collapse into a naked singular-ity, it would violate energy conditions since it assumesMi > Mo which from Eq. (27) would mean that the sur-face energy density σ is negative.

One of the main issues that was borne out of this studyis that it is not possible to tackle the interesting case of asubextremal solution that is being overcharged simply bymatching two GMGHS spacetimes. This would imply inthis context Do > Mo ≥Mi > Di. However, this last ex-pression is not possible in a dynamical framework due tothe impositions brought by the junction conditions of thescalar field. Therefore, we are here restricted to study-ing collapses from shells that have at most an extremalinterior if we are to obey the energy conditions. Nonethe-less, one can test CC in this framework. Since Do = Di

one can get an overextremal exterior (Mo < Do) from anunderextremal interior only if Mo < Mi. However, this

7

∞ℛ

∞(ℛ )

m/Mo = 0.4, Mi/Mo = 0.6, D/Mo = 0.5

m/Mo = 0.45, Mi/Mo = 0.35, D/Mo = 0.2

m/Mo = 0.2, Mi/Mo = 0.4, D/Mo = 1.3

m/Mo = 0.6, Mi/Mo = 0.4, D/Mo = 0

m/Mo = 0.2, Mi/Mo = 1.4, D/Mo = 1.1

m/Mo = 0.4, Mi/Mo = 0.7, D/Mo = 1.0

FIG. 2: In blue we have an example of the potential when Mo > D > Mi and Mo −Mi = m, in this case the shell is formed atinfinity and collapses into a black hole. The orange curve represents a situation in which collapse occurs from infinity, althoughthe shell initially is not at rest and both spacetimes are subextremal. The potential represented by the green curve correspondsto a collapse with an overcharged interior and exterior, note that the shell bounces before it reaches the singularity. The redcurve represents collapse with an interior and exterior Schwarzschild solution and with the shell initially at rest at infinity. Thepurple curve represents complete collapse into the singularity when Mi > D > Mo, i.e. the interior spacetime is subextremaland the exterior is overcharged. Finally, the brown curve represents the collapse of a shell with an overcharged interior and anextremal exterior.

would imply that the shell had negative energy and con-sequently violate the energy conditions. Therefore, CCis satisfied in this context.

A possible alternative to violating CC comes about byusing the radiating solutions obtained in Ref. [16] andwhich were described in section II. In this manner wemay end up having the energy of the spacetimes Mo orMi varying with time. By using these time-dependentradiating solutions we may start with both an interiorand exterior that are subextremal and that, as time ad-vances, become overcharged due to having the energiesthat describe the spacetimes decreasing. This study ofcollapsing thin shells in this time dependent context willbe tackled in the next subsection.

B. Collapse with an exterior radiating solutionwith time independent scalar charge

We will now focus on thin shells whose interior met-ric, ds2

i is given by Eq. (4) and the exterior metric byEq. (8). It will prove useful to henceforth, start usingagain the coordinate r = D +

√%2 +D2 to describe the

radial coordinate instead of % which was the one that weused until now.

The shell corresponds to the hypersurface that is de-fined through the usage of its interior metric by Σi =xµ : t = T (τ), r = R(τ) which is parametrized by thecoordinates ya = τ, θi, ϕi, where τ corresponds tothe proper time of the shell. For the exterior space-time, we define the shell as the hypersurface Σo =xµ : u = U(τ), r = R(τ) which is parametrized by thecoordinates ya = τ, θ0, ϕo. In this situation, Eq. (15)tells us that θi = θo, ϕi = ϕo and Ri = Ro (weare assuming that the scalar charge D is the same onboth sides of the shell) and the induced metric becomes

3ds2 = −dτ2 +R(R− 2D)dΩ2.Recalling that the extrinsic curvature is a 3-tensor, i.e.

it is invariant under a xα → xα′transformation and it

behaves like a tensor for ya → ya′, we conclude that it is

invariant under the radial coordinate change. Thus, forthe interior spacetime the extrinsic curvature reads

Kiθθ = Ki

ϕϕ/ sin2 θ = (R−D)

√Ai + R2 , (37)

Kiττ = −

Mi

R2 + R√Ai + R2

= − 1

R

d

dτ

(Kθθ

R−D

), (38)

where we have used A ≡ 1 − 2M/(D +

√R2 +D2

)=

1 − 2M/R and R2 = R(R − 2D)R2/(R − D)2. Thislast equation shows us that there are no qualitative dif-ferences between V (R) = R2 and V (R) as R ≥ 2D. Assuch, we will henceforth study the potential V (R) insteadof V (R).

For the spacetime exterior to the shell it is not sostraightforward to obtain its extrinsic curvature. To doso, we start by looking at the first junction conditionwhich in this case imposes (1− 2Mo/R) U2 + 2UR = 1where Mo = Mo(u), and it is being evaluated at u =

U(τ). Solving this for U , for a shell with decreasing R,i.e. R ≤ 0, yields

U =1

R+Bo=

Bo − R1− 2Mo

R

, (39)

where we have defined B =√A+ R2. With this equa-

tion we are able to determine the extrinsic curvature.The θθ-component of the extrinsic curvature yields thesame as Eq. (37) and the ττ -component reads

Koττ = − Mo

R2Bo+MoURBo

− R

Bo. (40)

8

Let us defineM = 4πσR(R−2D), where σ is once againthe surface energy density of the thin shells of dust weare studying. Using this definition, Eqs. (16) and (21)and the fact that Sab = σuaub we obtain

[Kττ ] = −4πσ , [Kθθ] = −M , (41)

dMdτ

=D2R(Bo −Bi) + (R− 2D)(R−D)UMo

R(R− 2D). (42)

The second junction condition allows us to express Boand Bi without depending on R, specifically Eq. (41)implies

Bo −Bi = − MR−D

,Bo +Bi

2=

m

M(R−D) , (43)

where we have defined m = Mo −Mi.We are interested in following the procedure presented

in for example Refs. [30–32]. These studies were done

with a Schwarzschild interior and a Vaidya exterior sowe expect our equations to reduce to the ones obtainedtherein when we take D → 0. To do so, we define thequantity α = m

M . Inserting the quantites α and m intoEq. (42) yields

U =RM

˙m(R−D)+

D2mR

α(R−D)2(R− 2D) ˙m. (44)

By plugging this result in Eq. (39) and substituting Bowith the value obtained by solving (43) we can obtain adifferential equation for ˙m without any dependence on U .Additionally, we can also determine the potential V (R)directly from Eq. (43). Both of these results can be seenbelow

[αR(R−D)(R− 2D)−D2αR

] [2αR(R−D)R+ 2α2(R−D)2 − mR

]R2(R−D)(R− 2D)

[2α(R−D)R− m

] =˙mα

m. (45)

R2 − Mo +Mi

R− m2

4α2(R−D)2= α2

(R−DR

)2

− 1 . (46)

When we take D to be zero in Eqs. (45) and (46) weobtain the same as what was obtained in Refs. [30–32].

The problem we intend to study depends (assumingwe are normalizing all quantities in regards to D) on thethree unkowns Mo(τ), M(τ) and R(τ). To completelydetermine it, we need three different equations, howeverwe currently have only obtained two of them (Eqs. (45)and 46), which means we need to impose a third one.

We solve this by pursuing a simple assumption which issetting M = 0, i.e. imposingM to be constant. Lookingat Eq. (42) we reach the conclusion that this assump-tion corresponds to considering that all the energy thatis being radiated by the exterior spacetime (leaves theshell) is counterbalanced by the energy received due tothe scalar field radiation originated by the dilaton field2.Consequently, we obtain a simple expression for U whichcan be inserted into Eq. (39), yielding

Mo

R=dMo

dR=

D2M(R−D)2(R− 2D)

(Bo + R) , (47)

where both R and Mo should be negative (assuming that

2 This assumption only holds for D 6= 0, otherwise the counterbal-ancing term vanishes.

U ≥ 0) for collapsing shells from infinity. Note thatEq. (45) reduces to Eq. (47) whenM is taken to be con-stant. As such, they represent the same restriction to thesystem we wish to solve. Thus, we arrive at a system oftwo equations (Eqs. (46) and (47)) with two unknowns(Mo and R).

Our interest lies in the attempt of overcharging an ex-tremal or near extremal black hole. For motion to beallowed it is necessary for R2 to be positive so we willstart by determining the asymptotic behavior of R2. Todo so, we will change our notation and will henceforth de-fineM = M(R(τ)). As we will be interested in describingthe motion of a shell collapsing from infinity we will con-sider R(τi)→∞, where τi corresponds to the instant ofthe proper time of the shell at which it starts its collapse.Moreover, assuming the shell reaches the singularity, wehave R(τf ) = 2D, where in this case τf is the instant ofthe proper time of the shell at which the shell reachesthe endpoint of its collapse. Expanding Eq. (46) forlarge values of R reads R2 ≈

(−1 + α2

)+O

(1R

), where

m = Mo(∞)−Mi. Meanwhile, for small radii (near thesingularity R = 2D), now considering m = Mo(2D)−Mi,R2 reads

R2 ≈ 1

4

(M2

D2+

2 (Mo +Mi)

D+m2

M2

)−1+O(R−2D) .

(48)

9

From the expansion at infinity we concludethat for the shell’s collapse to be allowed then(Mo(∞)−Mi)

2 ≥ M2. Moreover, in the extremalcase, the first term of Eq. (48) can be simplified to(D (Mo(2D)−D) +M2

)2/(DM)2 which will also

always be greater or equal to zero.As the shell is collapsing from infinity we know that,

at least initially3, R corresponds to the negative root ofR2 in Eq. (46). Therefore, the sign of Eq. (47) dependsonly on the sign of Bo+ R. If we are to consider that theexterior spacetime is always radiating, i.e., dMo/dR ≥ 0

this imposes Bo ≥ −R ⇔ 1 − 2Mo/R ≥ 0 ⇔ R ≥ 2Mo.Recall that in this coordinate system 2Mo represents theapparent horizon of the exterior spacetime. The impo-sition R ≥ 2Mo leads to the conclusion that for CC tobe tested it is necessary for Mo(2D) ≤ D, otherwise, bycontinuity, there exists an RH > 2D (the apparent hori-zon of the exterior spacetime) which the shell must cross.Note that for this to happen it is not needed for R to bezero, i.e., the shell does not need to bounce at (or before)RH . Moreover, recall that the apparent horizon corre-sponds to a non-timelike hypersurface, that is covered,for a shell collapsing from infinity, by a future null-likeevent horizon. This future event horizon would then needto be crossed for the shell to reach the apparent horizon.

The issue of reaching a future event horizon is prob-lematic since, this metric, like the Vaidya metric, is notgeodesically complete and is not defined beyond this fu-ture event horizon. Several studies on the Vaidya’s metricmaximal extension exist, e.g. Refs. [33, 34], where exten-sions into the black hole region are obtained. Nonethe-less, as this extension is outside of the scope of this paper,we will not tackle the CCC for cases in which the shellcrosses the future event horizon of the exterior spacetime.

Taking this information into account, in the followingsituations which will be analysed in regards of CC viola-tion, we will always consider that the shell never crossesthe future event horizon i.e. 2Mo(R) ≤ R.

An analysis of the collapse with a non overcharged in-terior follows. In this case we have Mi ≥ D and fromEq. (43), assuming that the shell is near the singularity,we have√

1− Mo

D+ R2 −

√1− Mi

D+ R2 = − M

R−D. (49)

For the extremal case, the right-hand side becomes√1−Mo/D + R2 −

√R2, which, unlessM is zero, the

collapse always violates the energy conditions of the shell.Otherwise, if R2 is positive, then the right-hand side ofthe equation is also positive, which would meanM wouldneed to be negative. The only other possibility wouldthen be for Mo = D in which case M could be zero,however this in turn would mean, from Eq. (43) that

3 If there is a bounce then R will change its sign.

Mo = Mi. Consequently, the exterior spacetime wouldnot be radiating and the interior and exterior would beexactly the same static spacetime, impossibilitating thestudy of collapse.

In the subextremal case, both Mi and Mo need to beinitially larger than D. Additionally, for CC to be vi-olated it is necessary for Mo(2D) ≤ D. On account of1−Mo/D ≥ 0 and 1−Mi/D < 0 we conclude that onceagain we violate the energy conditions in this frameworkasM would have to be lower than zero.

Thus, the CCC is also upheld in collapses with ex-tremal and subextremal interiors in this framework. Notethat our argument is quite general as it does not rely onthe fact thatM is constant.

We have shown in this chapter that even when consid-ering shells with exterior radiating solutions with timeindependent scalar charge and a GMGHS interior we can-not violate the CCC if we are to impose the proper massof the shell M to be constant. Moreover, we note thatthe treatment that was used to conclude this fact canalso be generalized for shells with variable proper massas Eq. (49) still imposes the CCC to be upheld for theenergy conditions related toM to be satisfied.

VI. CONCLUSIONS

In this paper, we have shown that when applyingthe thin shell formalism in Einstein-Maxwell-dilaton, thepresence of the dilaton field requires both the exteriorand interior spacetime, that are joined by the thin shell,to have the same dilaton charge for motion to be allowed.Supported by these results we determined static shell so-lutions, with a Minkowski interior. We showed that staticshells with a GMGHS exterior arise as a natural exten-sion of the known results when considering an exteriorSchwarzschild solution. We also concluded that the sta-bility of these solutions cannot be studied due to thedifferent scalar charge between the two spacetimes beingjoined.

Additionally, we studied the dynamics of thin shells,when we demand all of the junction conditions to be sat-isfied by assuming the geometries inside and outside ofthe shell to be static. In this context we determined thatthey always collapse into black holes for the subextremalinterior and exterior cases and that collapse into a nakedsingularity is not possible as the shell either bounces be-forehand or violates the weak and dominant energy con-ditions (which are the same on account of only havingstudied shells of dust). Due to the restriction imposedby the junction conditions related to the dilaton field onthe possible methods of overcharging these solutions wefound the need to explore a new class of time-dependentsolutions that are detailed in Ref. [16]. For them, wehave shown that, when considering a non-radiating thinshell which never crosses the future horizon of the ex-terior spacetime, we are not able to violate the CCC.Although we did not explore the radiating thin shell case

10

in this context, the equations we resorted to for our proofappear to hold and suggest that, as long as the energyconditions are satisfied, cosmic censorship is upheld. Wenote that, to have a complete overview of the CCC inthis setting, we would need to extend the spacetime be-yond the future horizon. This would allow us to studycollapse in the cases where the shell crosses the futureevent horizon of the exterior spacetime. Other possiblefuther work on these results that constitutes an interest-ing problem is related to considering thin shells that alsohave pressure, i.e. their matter is a perfect fluid insteadof simply dust.

The results we have obtained throughout this paperhint that string theory models that give rise to the stud-ied solutions seem to be consistent with the CCC beingvalid, and therefore, may be viable descriptions of stringtheory in this context. We note however that further

exploration on this topic is necessary as our focus wasonly on spherically symmetric spacetimes and these rep-resent idealized models, as we expect black holes in ouruniverse to not be spherically symmetric due to havingrotation. That said, a further analysis of GMGHS forrotating solutions is also required to solidify the conclu-sions that we have obtained in our work. Moreover, wenote that while this work only tackled classical aspects,one would expect, when studying gravitational collapse,that for small radii and high curvature regions of the col-lapse, stringy and quantum corrections should be takeninto account that here were disregarded. The conclusionis, even if in our tests the CCC is shown to be valid,these corrections might end up influencing the outcomeof gravitational collapse and the validity of the CCC instring theory.

[1] R. Penrose, Riv. Nuovo Cim. 1, 252 (1969), [Gen. Rel.Grav.34,1141(2002)].

[2] D. G. Boulware, Phys. Rev. D8, 2363 (1973).[3] V. E. Hubeny, Phys. Rev. D59, 064013 (1999), gr-

qc/9808043.[4] S. Gao and J. P. S. Lemos, Int. J. Mod. Phys. A23, 2943

(2008), 0804.0295.[5] H. Maeda, Phys. Rev. D73, 104004 (2006), gr-

qc/0602109.[6] R. Goswami and P. S. Joshi, Phys. Rev. D69, 104002

(2004), gr-qc/0405049.[7] R. Goswami, A. M. Nzioki, S. D. Maharaj, and S. G.

Ghosh, Phys. Rev. D90, 084011 (2014), 1409.2371.[8] S. G. Ghosh and N. Dadhich, Phys. Rev. D82, 044038

(2010), 1009.0982.[9] S. Jhingan and S. G. Ghosh, Phys. Rev. D81, 024010

(2010), 1002.3245.[10] X. O. Camanho and J. D. Edelstein, JHEP 11, 151

(2013), 1308.0304.[11] J. H. Horne and G. T. Horowitz, Phys. Rev. D48, 5457

(1993), hep-th/9307177.[12] T. Jacobson and T. P. Sotiriou, Phys. Rev.

Lett. 103, 141101 (2009), [Erratum: Phys. Rev.Lett.103,209903(2009)], 0907.4146.

[13] D. Garfinkle, G. T. Horowitz, and A. Strominger,Phys. Rev. D43, 3140 (1991), [Erratum: Phys.Rev.D45,3888(1992)].

[14] G. W. Gibbons, Nucl. Phys. B207, 337 (1982).[15] G. W. Gibbons and K.-i. Maeda, Nucl. Phys. B298, 741

(1988).[16] P. Aniceto, P. Pani, and J. V. Rocha, JHEP 05, 115

(2016), 1512.08550.[17] P. Vaidya, Proc. Indian Acad. Sci. A33, 264 (1951).[18] P. C. Vaidya, Phys. Rev. 83, 10 (1951).[19] W. B. Bonnor and P. C. Vaidya, Gen. Rel. Grav. 1, 127

(1970).[20] G. Darmois, Chapitre V, Mémorial de Sciences Mathé-

matiques fascicule XXV (1927).[21] W. Israel, Nuovo Cimento B Serie 44, 1 (1966),

[Erratum-ibid. B 48, 463 (1967)].[22] E. Poisson, A Relativist’s Toolkit: The Mathematics

of Black-Hole Mechanics (Cambridge University Press,2007).

[23] K. Kuchař, Czech. J. Phys. B 18, 435 (1968), ISSN 1572-9486.

[24] A. Das, Proceedings of the Royal Society of London A:Mathematical, Physical and Engineering Sciences 267, 1(1962), ISSN 0080-4630.

[25] S. D. Majumdar, Phys. Rev. 72, 390 (1947).[26] A. Papaetrou, Proc. Roy. Irish Acad.(Sect. A) A51, 191

(1947).[27] S. W. Hawking and G. F. R. Ellis, The Large Scale Struc-

ture of Space-Time, Cambridge Monographs on Math-ematical Physics (Cambridge University Press, 2011),ISBN 9780521200165, 9780521099066, 9780511826306,9780521099066.

[28] J. Frauendiener, C. Hoenselaersm, and W. Konrad,Class. Quant. Grav. 7, 585 (1990).

[29] P. R. Brady, J. Louko, and E. Poisson, Phys. Rev. D44,1891 (1991).

[30] V. H. Hamity and R. J. Gleiser, Astrophys. Space Sci.58, 353 (1978).

[31] M. Castagnino and N. Umerez, Gen. Rel. Grav. 15, 625(1983), ISSN 1572-9532.

[32] V. H. Hamity and R. H. Spinosa, Gen. Rel. Grav. 16, 9(1984), ISSN 1572-9532.

[33] W. Israel, Phys. Lett. pp. 184–186 (1967).[34] F. Fayos, M. M. Martín-Prats, and J. M. M. Senovilla,

Class. Quant. Grav. 12, 2565 (1995).