Self-similar scalar field collapse

1.1 MB·

by  Narayan Banerjee

Self-similar scalar field collapse
Narayan Banerjee and Soumya Chakrabarti
∗ †
Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata,
Mohanpur Campus, Nadia, West Bengal 741246, India.
(Dated: January 17, 2017)
A spherically symmetric collapsing scalar field model is discussed with a dissipative fluid which
includes a heat flux. This vastly general matter distribution is analysed at the expense of a high
degreeofsymmetryinthespacetime,thatofconformalflatnessandselfsimilarity. Indeedcollapsing
models, terminating into a curvature singularity can be obtained. The formation of black holes or
theoccurrence of nakedsingularities depend on theinitial collapsing profiles.
PACSnumbers: 04.50.Kd;04.70.Bw
7
1
0 I. INTRODUCTION investigation regarding various aspects of a scalar field
2 distribution.

n The problem of gravitational collapse of massive
a stars received a substantial amount of attention for Some studies are however, already there in the liter-
J ature in connection with exact solutions of Einstein’s
many years, ever since the first significant work in

6 this connection by Oppenheimer and Snyder [1]. A equations when a scalar field is coupled to gravity
1 and collapsing modes of such solutions. Christodoulou
spacetimesingularityisexpectedtoformasanend-state
ofacontinualgravitationalcollapse. Thoughthe general established the global existence and uniqueness of the
]
c speculation is that an event horizon would form and solutions of Einstein-scalar field equations [9]. A suffi-
q cient condition for the formation of a trapped surface
shield this singular state from a distant observer, there
- in the evolution of a given initial data set was also
r are examples with quite reasonable physical conditions,
g where no horizon can form and the singularity remains studied by the same author [10]. Goldwirth and Piran
[ showed that a scalar field collapse leads to a singularity
exposed, maybe forever, giving rise to what is known as
which is cut-off from the exterior observer by an event
1 a naked singularity. Therefore it is of great importance
v to investigate what an unhindered gravitational collapse horizon [11]. Scalar field collapse have been analytically
5 can lead to. For a systematic exposition of various as- studied by Goswami and Joshi [12], Giambo [13] quite
3 recently. Various massive scalar field models exist as a
pects of gravitational collapse and its possible outcome,
2 possible driverof the recentacceleratedexpansionofthe
we refer to the discussions by Joshi [2, 3].
4
universe. A few examples of collapse scenario of such
0
type of scalar fields are already in literature [8, 13–18].
. A recent upsurge of interest in scalar field cosmology
1
is driven by the observational evidence of accelerated
0
Numerical investigations have also provided useful
7 expansion of the universe and thus the need for an
insights into black hole formation in a scalar field
1 exotic matter component, called dark energy, which can
collapse. In their extensive works,Choptuik, Brady,and
: provide an effective negative pressure [4–6]. A scalar
v Gundlach [19–21] considered the numerical evolution of
field with a positive potential can indeed serve this
i
X purpose. The distribution of the dark energy vis-a-vis collapsing profiles characterised by a single parameter
(p) and showed that the behaviour of the resulting
r the distribution of the fluid remains unclear except
a families of solutions depends on the value of p; there
perhaps the dogma that it does not cluster at any scale
is a critical evolution with p = p , which signals the
below the Hubble scale. An exhaustive investigation ∗
transition between complete dispersal and black hole
regarding a scalar field collapse might lead to some
formation. For a comprehensive review of the critical
hint towards the possible clustering of the dark energy.
phenomena associated with a scalar field collapse, we
For instance, the spherical collapse model was studied
refer to the work of Gundlach[22].
by Mota and Bruck [7] in the context of dark energy
cosmologies, in which dark energy was modelled as
The aim of the present work is to look at the collapse
a minimally coupled scalar field for different popular
of a massive scalar field along with a fluid distribution
interaction potentials. Moreover, formally a scalar field
which is locally anisotropic and contains a radial heat
works well as a description of many perfectly reasonable
flux. The potential is taken to be a power law function
matter distributions as discussed by Goncalves and
of the scalar field or suitable combinations of power-law
Moss [8]. This servesas anadditionalmotivation for the
terms. Power law potentials are quite relevant and
well-studied in a cosmological context. Many realistic
matter distribution can be modelled with power law
∗Electronicaddress: [email protected] potentials. For instance, a quadratic potential, on the
†Electronicaddress: [email protected] average, mimics a pressureless dust whereas the quartic
2
potential exhibits radiation like behavior [23]. For a ofananharmonicoscillatorequationis reviewedinbrief.
potential with a power less than unity results in an in- Weinvestigatethecollapseofascalarfieldwithapower-
verse power-law self interaction as in the wave equation, law potential in detail in section five. The discussion on
the dV term introduces a term with inverse power of acombinationofaquadraticandanarbitrarypower-law
dφ
φ. Inverse power law potentials were extensively used potential is given in section six. The seventh and the
for the construction of tracker fields[24, 25]. We study final section includes a summary of the results obtained.
the scalar field evolution equation extensively for some
reasonablechoicesofpotentialssothatwehaveexamples
from both positive and negative self interactions. II. CONFORMALLY FLAT SCALAR FIELD
MODEL WITH PRESSURE ANISOTROPY AND
HEAT FLUX
Analytical studies are limited to very special cases
due to the nonlinearity of the system of equations. We
The space-time metric is chosen to have vanishing
start with a conformally flat spacetime. In a very recent
Weyl tensor implying a conformal flatness. Conformally
work, an exact solution for such a scalar field collapse
flat space-times are well-studied and utilised in the
with a spatial homogeneity was given [26], where it was
shown that a massive scalar field collapse without any context of radiating fluid spheres or shear-free radiating
stars [30–37]. The metric can be written as
apriori choice of an equation of state, can indeed lead
to the formation of a black hole. Under the assumption
that the scalar field evolution equation is integrable,
1
the work extensively utilised the integrability conditions ds2 = dt2 dr2 r2dΩ2 , (1)
for a general anharmonic oscillator, developed by Euler A(r,t)2" − − #
[27] and utilized by Harko, Lobo and Mak[28]. For the
sake of simplicity, both the conformal factor and the where A(r,t) is the conformal factor and governs the
scalar field were assumed to be spatially homogeneous, evolution of the sphere.
making the system effectively a scalar field analogue
of the Oppenheimer-Snyder collapsing model. In the The fluid inside the spherically symmetric body is as-
presentwork,the assumptionofhomogeneityis dropped sumed to be locally anisotropic along with the presence
and also the fluid content has a local anisotropy and of heat flux. Thus the energy-momentumtensor is given
also has a radial heat flux. So the system described in by
the present work is a lot more general than the one in
T =(ρ+p )u u p g +(p p )χ χ +q u +q u ,
[26]. However, the price paid is that the present work αβ t α β− t αβ r− t α β α β β α
(2)
assumes the existence of a homothetic Killing vector
where qα = (0,q,0,0) is the radially directed heat flux
implying a self-similarity in the spacetime.
vector. ρistheenergydensity,p thetangentialpressure,
t
p the radial pressure, u the four-velocity of the fluid
Indeed the assumption of self-similarity imposes r α
and χ is the unit four-vector along the radial direction.
a restriction on the metric tensor, but it is not re- α
The vectors u and χ are normalised as
ally unphysical and there are lots of examples where α α
self-similarity is indeed observed. In non-relativistic uαu =1, χαχ = 1, χαu =0. (3)
α α α
Newtonian fluid dynamics, self-similarity indicates that −
the physical variables are functions of a dimensioness A comoving observer is chosen, so that uα =Aδα and
0
variable x where x and t are space and time variables the normalization equation (3) is satisfied (A = 1 ).
l(t) √g00
and l, a function of t, has the dimension of length. It is to be noted that there is no apriori assumption of
Existence of self-similarity indicates that the spatial an isotropic fluid pressure. The radial and transverse
distribution of physical variables remains similar to pressures are different. Anisotropic fluid pressure is
itself at all time. Such examples can be found in strong quite relevant in the study of compact objects and
explosion and thermal waves. In general relativity considerable attention has been given to this in existing
also, self-similarity, characterized by the existence of a literature. Comprehensive review can be found in the
homethetic Killing vector, finds application. We refer work of Herrera and Santos[38] and Herrera et. al.[39].
to the work of Carr and Coley[29] for a review on the In addition to the obvious relevance of an anisotropic
implications of self-similarity in general relativity. pressure in collapsing bodies or in any compact object,
it has also been shown by Herrera and Leon that on
The paper is organized as follows. In the second sec- allowing a one-parameter group of conformal motions,
tion we write down the relevant equations for a general a smooth matching of interior and exterior geometry is
scalarfieldmodelalongwithafluiddistributioninacon- possible iff there is an pressure anisotropy in the fluid
formallyflatsphericallysymmetricspacetime. Thethird description[40].
sectionintroducestheself-similarityinthe systemwhere
we write the relevant equation in terms of a single vari- Wealsoincludeadissipativeprocess,namelyheatcon-
able. In the fourth section, the theorem of integrability duction in the system. In the evolution of stellar bod-
3
ies, dissipative processes are of utmost importance. Par- III. SELF SIMILARITY AND EXACT
ticularly when a collapsing star becomes too compact, SOLUTION
the size of the constituent particles can no longer be ne-
glectedincomparisonwiththemeanfreepath,anddissi- A self-similar solution is one in which the spacetime
pative processescanindeedplay a vitalrole,in shedding admits a homothetic killing vector ξ, which satisfies the
off energy so as to settle down to a stable system. For equation
more relevant details we refer to the works [39, 41].
Whena scalarfieldφ=φ(r,t) isminimally coupledto L g =ξ +ξ =2g , (12)
ξ ab a;b b;a ab
gravity, the relevant action is given by
where L denotes the Lie derivative. In such a case, one
1
A= √ gd4x[R+ φµφµ V(φ)+Lm], (4) can have repeatative structures at various scales. With
− 2 −
Z a conformal symmetry the angle between two curves
where V(φ) is the potential and L is the Lagrangian remains the same and the distance between two points
m
density for the fluid distribution. are scaled depending on the spacetime dependence of
From this action, the contribution to the energy- the conformal factor (1 in the present case). For a
A
momentum tensor from the scalar field φ can be written self-similar space-time, by a suitable transformation of
as coordinates, all metric coefficients and dependent vari-
ables can be put in the form in which they are functions
1
Tφ =∂ φ∂ φ g gαβ∂ φ∂ φ V(φ) . (5) ofasingleindependentvariable,whichisadimensionless
µν µ ν − µν"2 α β − # combinationof space and time coordinates;for instance,
in a spherically symmetric space this variable is t.
Einstein field equations (in the units 8πG = 1) can r
thus be written as
It deserves mention that a homothetic Killing vector
4 1 1
3A˙2 3A′2+2AA′′+ AA′ =ρ+ A2φ˙2 A2φ′2 (HKV) is a special case of a conformal Killing vector
− r 2 − 2 (CKV) η defined by L g = η +η = λg , where
η ab a;b b;a ab
+V(φ), (6) λ is a function. Clearly HKV is obtained from a CKV
when λ = 2. For a brief but useful discussion on CKV,
we refer to the work by Maartens and Maharaj[42].
4 1 1
2A¨A 3A˙2+3A2 AA =p + φ2A2+ A2φ˙2 Writing A(r,t)=rB(z), the scalar field equation (11)
′ ′ r ′
− − r 2 2
is transformed into
V(φ), (7)
−
dV
B 2z
2A¨A−3A˙2+3A′2− 2rAA′−2AA′′ =pt− 21φ′2A2 φ◦◦−φ◦"2 B◦ + 1 z2#+ B2(1dφ z2) =0. (13)
− −
1
+ A2φ˙2 V(φ), (8)
2 −
Here, an overhead denotes a derivative with respect
2A˙ q to z = t. ◦
′ = +φ˙φ. (9) r
′
A −A3
Equation(13)canbeformallywritteninageneralclas-
The wave equation for the scalar field is given by
sical anharmonic oscillator equation with variable coeffi-
dV cients as
✷φ+ =0, (10)
dφ
dV
φ +f (z)φ +f (z)φ+f (z) =0, (14)
which, for the present metric (1), translates into ◦◦ 1 ◦ 2 3 dφ
A˙ φ φA 1 dV
φ¨ φ′′ 2 φ˙ 2 ′ +2 ′ ′ + =0. (11) where fi’s are functions of z only.
− − A − r A A2 dφ
ρ, p , p , q, A and φ are unknowns in this system of
r t
equations containing four equations (6 8). The equa- IV. A NOTE ON THE INTEGRABILITY OF AN
tion(9)followsfromthefieldequations−asaconsequence ANHARMONIC OSCILLATOR EQUATION
oftheBianchiidentitiesandthereforeisnotindependent.
Rather than assuming any specific equation of state to A nonlinear anharmonic oscillator with variable coef-
close the system of equation, we probe the system un- ficients and a power law potential can be written in a
der the assumption that the scalar field equation (11) is general form as
integrable; which facilitates the appearence of an addi-
tional differential condition on the conformal factor, as
discussed in the following sections. φ¨+f (u)φ˙+f (u)φ+f (u)φn =0, (15)
1 2 3
4
where f ’s are functions of u and n , is a constant.
i
∈Q
Here u is the independent variable used only to define 2.0´1010 m = -0.6
the theorem. An overhead dot represents a differenti-
z > 1000.0
ation with respect to u. Using Eulers theorem on the 1.5´1010
itniotneg[r2a7b]ialintydorefctehnetrgeesnuelrtaslgaivnehnabrmyoHnairckoosectillaalt[o2r8]e,qtuhais- BzHL 1.0´1010
equationcanbeintegratedundercertainconditions. The
essence is given in the form of a theorem as [27, 28], 5.0´109
Theorem If and only if n / 3, 1,0,1 , and the
coefficients of Eq.(15) satisfy th∈e{d−iffer−ential c}ondition 0
1000 1200 1400 1600 1800 2000
z
1 1 d2f3 n+4 1 df3 2 0.12
(n+3)f3(u) du2 − (n+3)2 (cid:20)f3(u) du(cid:21) 0.10
m = 3.0
n 1 1 df 2 df
+ − 3 f (u)+ 1 0.08 z > 10.0
(2n(n++3)12)(cid:20)f3(u) du(cid:21) 1 n+3 du BzHL 0.06
+ f2(u)=f (u), (16)
(n+3)2 1 2 0.04
0.02
equation (15) is integrable.
0.00
100.000 100.002 100.004 100.006 100.008 100.010
If we introduce a pair of new variables Φ and U given
z
by
FIG. 1: Evolution of the conformal factor B(z) with respect
Φ(U) = Cφ(u)f3n+13 (u)en+23Ruf1(x)dx, (17) toz for z>10. The potentials are takento beV(φ)= 5φ2/5
2
U(φ,u) = C1−2n uf3n+23 (ξ)e(1n−+n3)Rξf1(x)dxdξ,(18) s(gidrea)p.h on the left side) and V(φ) = φ44 (graph on the right
Z
whereC is aconstant,equation(15)canbe writtenas
d2Φ It is indeed difficult to handle the equation (21)
+Φn(U)=0. (19) analytically, but it can be treated numerically so
dU2
as to examine the nature of B as a function of z.
Our aim is to understand whether the system reaches
V. POWER LAW POTENTIAL : V(φ)= φm+1 anysingularityofzeropropervolumeatanyfinitefuture.
(m+1)
Figure (1) showsthe numericalevolutionof B(z) with
For a simple power law potential, for which dV =φm, respect to z for m = 3 and m = 3. Keeping in mind
dφ −5
one can write equation (13) as the fact that the scale factor is proportional to 1 ; it
B(z)
B◦ 2z φm appears that for V(φ) = 5φ22/5, the system undergoes a
φ◦◦ φ◦ 2 + + =0. (20) uniform gravitational collapse, but the rate of collapse
− " B 1−z2# B2(1−z2) dies down eventually, and a zero proper volume singu-
larityis reachedonly forz , i.e. for finite t=t but
s
Comparing with (15), it is straightforward to identify r 0. However, for V(φ)→=∞φ4, the system shrinks to
→ 4
f1(z)=−"2BB◦+1−2zz2#,f2(z)=0andf3(z)= B2(11−z2). ozefroz raanthderthriaspisdinlyg,uwlahreitrye Bis(zn)ot→n∞ece,sasatrailyfinaitecevnatlruael
singularity.
Theintegrabilitycriteriaasmentionedinthelastsection,
gives a second order non-linear differential condition on
B(z) as These plots, though represent the general evolution of
the spherical body, further analysis like formation of an
2 apparent horizon or the nature of curvature scalar may
B (m+1) B 8(2m+3) B z
◦◦ ◦ ◦
3 not be quite straightforward, without any solution for
B − (m+3) B ! − 6(m+3) B(1−z2) B(z) in a closed form. With this in mind, we now inves-
2(1+m)z2 2(m+3) tigate whether the collapsing system reaches any central
− =0. (21)
− 6(m+3)(1 z2)2 singularity at a future time given by t ts and r 0.
→ →
− Forthatpurposewenowlookforanapproximatebutan-
Here [m= 3, 1,0,1] as mentioned before. alytical expression for B(z) for z >> 1. In this domain
6 − −
5
the last term on the LHS of (21), 2((1m++m3))z22(−12(zm2)+23) is of 0.000092
the order of 1 + 1 and hence can be ig−nored with 0.0000915 m = -0.6
∼ z2 z4
respectto the otherterms. This approximationdoesnot 0.000091 z > 10.0
affect the nature of the evolution as shall be seen in the C1 = 100
subsequentanalysis. Thuswewritetheeffectiveequation 1 BzHL 0.0000905
governing the collapsing fluid for z >>1 as 0.00009
0.0000895
2 0.000089
B (m+1) B 8(2m+3) B z
◦◦ 3 ◦ ◦ =0. 0 20000 40000 60000 80000 100000
B − (m+3) B ! − 6(m+3) B(1 z2) z
−
(22)
0.045
A solution of equation (22) can be written in term of
m = -0.6
Gauss’ Hypergeometric function as
0.040 z > 10.0
B1 α 1 3 1 BzHL C1 = 0.01
− = F ,β; ;z2 εz+ε , (23) 0.035
2 1 0
(1 α) 2 2
" #
−
0.030
whereαandβaredefinedintermsofmasα=3(m+1),
(m+3)
β = 8(2m+3) and ε and ε are constants of integration. 0 20000 40000 60000 80000 100000
6(m+3) 0 z
(23) describes the evolution of the spherical body as a
0.000113
function of the self-similarity variable z and the expo-
nentofthe selfinteraction,m. Onecanexpandequation
0.000112
(23)ina powerseriesinthe limitz andwriteB(z) m = -0.6
explicitlyasafunctionofz. Howeve→r,i∞nordertopresent 1 BzHL 0.000111 z > 10.0
some simple examples we choose three values of m such C1 = -100
that the parameter β has the values 1, 2 and 0 respec- 0.000110
tively; namely, m= 3, m=3 and m= 3. Therefore
−5 −2 0.000109
the potentials are effectively chosen as V(φ) = 5φ2/5, 0 20000 40000 60000 80000 100000
2
V(φ)= φ4 and V(φ)= 2 . We note here that a case z
4 −φ1/2
wherem= 3 is alsotakenupsoasto haveanexample FIG.2: EvolutionofB(z)withrespecttozforV(φ)= 5φ2/5,
−2 2
for an inverse power law potential as well. for different choices of the parameter C .
1
1. Case 1 : m= 3, β =1 and α= 1
−5 2
thosecases,nowweexamineequation(24)nearthe
Puttingthesevaluesin(22),asolutioncanbewrit-
central singularity. It is evident that, the singular-
ten as
ity, characterised by B , appears only when
→ ∞
2 z , i.e, either when ts , or when t ts,
1 → ∞ → ∞ →
B(z)=C2"C1+ 2ln z+(z2−1)12 # , (24) rsy→ste0m. Hinownoevtecr,olfloarpsailnlgneagtatailvl;e vraatluheesr,oiftCs1e,tttlhees
(cid:12) (cid:12)
(cid:12) (cid:12) down asymptotically at a finite volume after a pe-
where C1 and C2 are(cid:12)constants of(cid:12)integration. riod of steady expansion with respect to z. Figure
Therefore the inverse of the conformal factor can (2) shows the evolution of B(z) with respect to z.
be written explicitly as a function of r and t as
2. Case2: m=3, α=β =2.
1 2
1 t t2 2
A(z)=rC C + ln + 1 . (25)
2" 1 2 (cid:12)r r2 − ! (cid:12)# With these valuesofthe parameters,equation(22)
(cid:12) (cid:12) can be solved to write B(z) as
(cid:12) (cid:12)
(cid:12) (cid:12)
The evolution with re(cid:12)spect to z and t(cid:12)herefore for-
C
2
mation of a singularity will depend on the integra- B(z)= . (26)
tionconstantsandofcourse,thechoiceofpotential ln C1(1+z)
(1 z)
−
does play a crucial role here. Without any loss of h i
generality we assume C = 1. It must be men- Here, C can again be chosen to be unity. For
2 2
tioned, that for all C < 0, one has either a nega- the solution to be valid in the region z >> 1,
2
tive volume or no real evolution at all. Excluding C must be a non-zero negative number such that
1
6
ln C1(1+z) is real.
(1 z) 0.0009
−
h i 0.0008 m = -1.5
z > 10.0
0.0007
C1 = 10
0.08 m = 3 1 BzHL 0.0006 Zs ® ¥
z > 10.0 0.0005
0.06
C1 = -0.9 0.0004
1 BzHL 0.04 Zs = 19.0 0.0003
0 200000 400000 600000 800000 1´106
0.02 z
0.00 9.´10-8
10 12 14 16 18 8.´10-8 m = -1.5
z
z > 10.0
2.3050 7.´10-8
C1 = 0.001
1 BzHL 6.´10-8 Zs ® ¥
2.3045 m = 3 5.´10-8
z > 10.0
4.´10-8
1 BzHL 2.3040 C1 = -10 3.´10-8
Zs ® ¥
2.3035 0 2´1017 4´1017 6´1017 8´1017 1´1018
z
2.3030
FIG.4: EvolutionofB(z)withrespecttozforV(φ)=− 2
0 2000 4000 6000 8000 10000 φ1/2
for different positive values of C . The qualitative behaviour
z 1
remains the same overdifferent choices of theparameter.
FIG. 3: Evolution of B(z) with respect to z for V(φ) = φ4
4
for different negative valuesof theparameter C .
1
A. Expressions for scalar field, physical quantities
and curvature scalar
From(26)onecanfindthatthesystemreacheszero
propervolumesingularityatafinitevalueofzgiven
Pointtransformingthe scalarfieldequationusing (17)
by
and (18), equation (13) can be written in an integrable
1 C1 form as (19). Using the exact form of the coefficients
z = − . (27)
s 1+C1 f1(z)andf3(z),aftersomealgebra,weexpressthescalar
field as a function of z as
However, for a collapsing model, the scale factor
must be a decreasing function throughout. This
means ddzln (1(+1z)zC)1 < 0. On simplification, this φ(z)=φ0B(m6+3)(1−z2)−(m1+3)"C(1−2m)
−
yields 2 h < 0, wihich holds true iff z > 1. So
zs1, d<efiC(n1e−dz<2b)y0,zwsh=ich11+−iCCs,11 tihsegrreefoarteerctohnasnist1e,notnwlyitihf Z B2((mm+−33))(1−z2)−((mm++13))dz−Φ0#. (29)
1
−
thrrequirementofanegativeC . Thediscussionis
1 Here, C is an integration constant coming from the
supported graphically in figure (3).
definition of the point transformation and φ is defined
0
3. Case 3 : m= 3 α= 1 and β =0. in terms of C. Φ0 is a constant coming from integration
−2 − over z. For all choices of m discussed in the previous
6
section, B(m+3) is an increasing function w.r.t z, and
Equation (22) is simplified significantly and solu-
diverges when B(z) . Thus it is noted that the
tion of B(z) may be written as → ∞
scalar field diverges at the singularity.
1
B(z)=[2C1(z+z0)]2. (28) From(6), (7), (8) and (9), the density, radialand tan-
Here, C > 0 and z are constants of integration. gential pressure and the heat flux can be expressed gen-
1 0
ThescalefactorisdefinedasY(r,t)= 1 . erally as
Now, Y →0 only when z →∞, i.e., t→[2C1t(sr,t+rz→0)]120. ρ=3A˙2 3A′2+2A′′A+ 4A′A 1A2φ˙2+ 1A2φ′2
Thus the system reaches a central singularity ar − r − 2 2
t ts, as shown in figure (4) for different choices φm+1
→ . (30)
of initial conditions defined by the choice of C .
1 −(m+1)
7
whereR(r,t)istheproperradiusofthecollapsingsphere,
which is rB(z) in the present work.
4 1 1
p =2A¨A 3A˙2+3A2 AA φ2A2 φ˙2A2 Writing the derivative in terms of z = t, one can ex-
r − ′ − r ′ − 2 ′ − 2 r
press (36) as
φm+1
+ . (31) (1 z2)B 2 =0. (37)
(m+1) ◦
−
Weinvestigatetheformationofanapparenthorizoninall
therelevantcasesdiscussed,forV(φ)= 5φ2/5,V(φ)= φ4
pt =2A¨A−3A˙2+3A′2−2A′′A− 2rA′A+ 21φ′2A2 and V(φ) = −φ12/2, with the assumption2 C2 = 1 an4d
1 φm+1 z >>1.
φ˙2A2+ . (32)
−2 (m+1) 1. Case 1: m= 3
−5
The equation (37) gives the condition
q = 2A˙′A2+φ˙φ′A3. (33) z+(z2−1)1/2 =e−2(C1−δ01/2) =eγ, (38)
−
1/2
where we have defined γ = 2(C δ ) and δ
OnecanusetheapproximatesolutionforB fromequa- − 1 − 0 0
is a constant of integration. The above equation
tions (25), (26) and (28) and use the fact that A(r,t) =
can be simplified to find the time of formation of
rB(z)inordertocheckthenatureofthefluidvariablesat
apparent horizon as
singularity. Itcanbefoundthatwhenthesphereshrinks
to zero volume, all these quantities diverge to infinity, re γ
t = − (e2γ 1)=rsinhγ. (39)
confirming the formation of a singularity. To comment ap 2 −
on the nature of singularity we write the kretschmann Since e γ is always positive, the structure of the
−
curvature scalar from (1) as spacetime depends on (e2γ 1) and therefore on
−
the parameter γ.
AA
K = 7A˙ 2A2+4(A˙2 A2+ ′ +A A)2+
− ′ − ′ r ′′ 2. Case 2: m=3
2(A˙2 A′2+A′′A A¨A)2+4(A˙2 A′2+ The equation (37) is simplified to yield the condi-
− − − tion
AA AA
r ′ −A¨A)2+(A˙2−A′2+2 r ′)2. (34) C 1+z =e1/Ψ0. (40)
1
1 z
Forany giventime-evolution,the kretschmannscalardi- (cid:16) − (cid:17)
Here Ψ is a constant of integration over z. One
0
verges anyway when r 0, which indicates that the
→ can further simplify to write the time of formation
central shell focussing ends up in a curvature singular-
of apparent horizon as
ity. We note that a singularitymayalso forminrelevant
1
cases where r = 0. To investigate such a singularity, we eΨ0 C1
6 t =r − =rΓ . (41)
write 34 as a function of z given by ap 1 0
eΨ0 +C1!
K(z)= 7B◦2z2B2+4[B◦2 (B B◦z)2+ We note herethatconsideringanapparenthorizon
− − −
B(B B z) B Bz2]2+2[B 2 (B B z)2+ is not necessary for those cases where singularity
◦ ◦◦ ◦ ◦
− − − − forms only when t , i.e. when there is no real
B B B Bz2]2+4[B 2 (B B z)2 →∞
◦◦ ◦◦ ◦ ◦ singularity at all.
− − −
+B(B B z) B B]2+[B 2 (B B z)2
− ◦ − ◦◦ ◦ − − ◦ 3. Case 3: m= 3
+2B(B B z)]2. (35) −2
◦
−
In a similar manner, for V(φ) = 2 , the condi-
Fromthisaboveexpressionwenotethatthefirsttermon −φ1/2
the RHS of Kretschmann scalar B 2z2B2 diverges any- tion for an apparent horizon may be written as
◦
waywhenB(z) . Thus,thesingularityalwaysturns χ 2
out to be a curv→atu∞re singularity. tap =r 0 z0 , (42)
2τ −
(cid:16) (cid:17)
whereχ andτ areconstantsofintegrationandare
0
dependent on suitable choice of initial conditions.
B. Apparent Horizon
In all the cases we find that an apparent horizon is
Visibilityofthe centralsingularitydependsonthefor- formed at a finite time. As there is no explicit expres-
mationofanapparenthorizon,thesurfaceonwhichout- sion for ts, (the time of formation of the singularity) the
goinglightraysarejusttrapped,andcannotescapeout- visibility ofthe singularitycannotbe ascertainedclearly.
ward. The condition of such a surface is given by Butitisquiteclearthatthesingularitycannotbevisible
indefinitely. Ift >t ,itcanbevisibleforafinitetime
app s
gµνR, R, =0, (36) only.
µ ν
8
C. Matching with an exterior Vaidya Solution their feedback in the interior solutions could not be dis-
cussed. Actually the condition of integrability of the
Generally,in collapsing models, a spherically symmet- scalar field equation has been utilized to find a solution
ric interior is matched with a suitable exterior solution; for the metric but the evolution equation for the scalar
Vaidya metric or a Schwarzschild metric depending on field defined by (29) could not be explicitly integrated.
the prevailing conditions [43]. This requires the conti- Therefore the matching condition could not be solved
nuity of both the metric and the extrinsic curvature on without the expression of scalar field in a closed form.
the boundary hypersurface. As the interior has a heat However,foranyinteriorsolution,thereisarequirement
transportdefined,theradiatingVaidyasolutionischosen of the matching, so we just discuss the method and ob-
as a relevant exterior to be matched with the collapsing tain the relevant equations to be solved.
sphere. The interior metric is defined as
1 D. Nature of the singularity
ds2 = dt2 dr2 r2dΩ2 , (43)
A(r,t)2" − − #
In a spherically symmetric gravitational collapse
and the Vaidya metric is given by leadingtoacentralsingularity,ifthecentregetstrapped
prior to the formation of singularity, then it is covered
2m(v) and a black hole results. Otherwise, it could be naked,
ds2 = 1 dv2+2dvdR R2dΩ2. (44)
" − R # − when non-space like future directed trajectories escape
from it. Therefore the important point is to determine
Thequantitym(v)representstheNewtonianmassofthe whether there are any future directed non-spacelike
gravitating body as measured by an observer at infin- geodesics emerging from the singularity.
ity. The metric (44) is the unique spherically symmetric From this point of view, general relativistic solutions
solution of the Einstein field equations for radiation in of self-similar collapse of an adiabatic perfect fluid was
the form of a null fluid. The necessary conditions for discussed by Ori and Piran [47] where it was argued
the smoothmatching ofthe interiorspacetime to the ex- that a shell-focussing naked singularity may appear
terior spacetime was presented by Santos [44] and also if the equation of state is soft enough. Marginally
discussed in detail by Chan [45], Maharajand Govender bound self-similar collapsing Tolman spacetimes were
[46] in context of a radiating gravitational collapse. Fol- examinedandthe necessaryconditionsforthe formation
lowing their work,The relevant equations matching (43) of a naked shell focussing singularity were discussed by
with (44) can be written as Waugh and Lake [48]. For a comprehensive description
of the mathematical formulations on occurence of naked
r =ΣR, (45) singularity in a spherically symmetric gravitational
A(r,t) collapse, we refer to the works of Joshi and Dwivedi
[50, 51], Dwivedi and Dixit [52]. Structure and visibility
of central singularity with an arbitrary number of
r3 AA dimensions and with a general type I matter field was
m(v)=Σ A˙2 A2+ ′ , (46)
2A3 − ′ r ! discussed by Goswami and Joshi[53]. They showedthat
the space-time evolution goes to a final state which is
and either a black hole or a naked singularity, depending on
the nature of initial data, and is also subject to validity
q
p =Σ , (47) of the weak energy condition. Following the work of
r
A(r,t)
Joshi and Dwivedi [50, 51], a similar discussion was
given on the occurrence of naked singularities in the
where Σ is the boundary of the collapsing fluid.
gravitational collapse of an adiabatic perfect fluid in
Therelationbetweenradialpressureandtheheatfluxas
self-similarhigherdimensionalspacetimes,byGhoshand
in equation (47) yields a nonlinear differential condition
Deshkar [54]. It was shown that strong curvature naked
between the conformal factor and the scalar field to be
singularities could occur if the weak energy condition
satisfied on the boundary hypersurface Σ. In view of
holds.
equations (31) and (33) the condition can be written as
A¨ A˙2 A2 A A˙ 1 One can write a general spherically symmetric metric
2 2 +3 ′ 4 ′ +2 ′ (φ˙ +φ′)2 as
A − A2 A2 − Ar A!− 2
1 φ(m+1) ds2 =eϑdt2 eχdr2 r2S2dΩ2, (49)
+ =Σ0. (48) − −
A2(m+1)
where ϑ, χ and S are functions of z = t. This space-
r
It deserves mention that we cannot obtain any ana- time admits a homothetic killing vector ξa = r ∂ +t∂ .
∂r ∂t
lyticalexpressionfromthe matchingconditionsandthus Fornullgeodesicsone canwrite KaK =0,where Ka =
a
9
dxa are tangent vectors to null geodesics. Since ξa is a 1. Case 1: m= 3
dk −5
homothetic killing vector, one can write
Using the time evolution (25) and the condition
g =ξ +ξ =2g (50)
ξ ab a;b b;a ab
L that the central shell focusing is at-least locally
where denotes the Lie derivative. Then it is straight- naked for V(z )=0 one finds
0
forwardLto prove that d (ξaK ) = (ξaK ) Kb = 0 (for
mathematicaldetailswdekrefertao[52]). Thaer;aeforeonecan z0+(z02−1)12 =e2C1, (56)
write
for the conformally flat metric, from which it is
ξaK =C, (51)
a straight-forward to write z0 = 12 e2C1 + e−2C1 .
for null geodesics where C is a constant. From this alge-
The visibility of the singularity (cid:16)depends on th(cid:17)e
braic equation and the null condition, one gets the fol-
existence of positive roots to Eq. (56), therefore
lowing expressions for Kt and Kr as [54]
on the positivity of e2C1 + e−2C1 . Since z0 =
Kt = rC[e[χz±eeχϑzΠ2]], (52) leimmet→rgtisnrg→fr0oddmrt,tthheesien(cid:16)qguualtairointyfomratyhb(cid:17)eenwulrlitgteenodaessic
−
1
C 1 zeϑΠ t ts(0)= e2C1 +e−2C1 r. (57)
Kr = ± , (53) − 2
r[eχ eϑz2] (cid:16) (cid:17)
(cid:2) − (cid:3)
where Π = √e χ ϑ > 0. Radial null geodesics, by For all values of C1, e2C1 + e−2C1 is always
− − greaterthanzero. There(cid:16)foreitisalway(cid:17)spossibleto
virtue of Eqs. (52) and (53), satisfy
findaradiallyoutwardnullgeodesicemergingfrom
dt z eχΠ the singularity, indicating a naked singularity.
= ± . (54)
dr 1 zeϑΠ
± 2. Case 2: m=3
The singularity that might have been there is at least
locallynakedifthereexistradialnullgeodesicsemerging
Using (26) in V(z )=0, one obtains
0
fromthe singularity,andif no suchgeodesicsexistit is a
blackhole. Ifthesingularityisnaked,thenthereexistsa
realandpositivevalueofz0 asasolutiontothealgebraic C1(1+z0)
equation =1. (58)
(1 z )
0
−
t dt
z = lim z = lim = lim . (55) For this collapsing model, since C must be a neg-
0 1
tr→→t0s tr→→t0s r tr→→t0s dr ative number, and z0 can be written as z0 = 11−+CC11.
Thus one can indeed have a naked singularity for
Waugh and Lake [49] discussed shell-focussing naked
1 < C < 0. The singularity in this case is re-
singularities in self-similar spacetimes, considering a −alised at1a finite value of z = t, not at r 0 and
general radial homothetic killing trajectory. The la- r →
therefore is not a central singularity. This kind of
grangian can be written in terms of V(z)=(eχ z2eϑ),
− singularities are generally expected to be covered
whose value determines the nature of the trajectory,
by a horizon [55].
for instance, if V(z) = 0, the trajectory is null, and for
V(z)>0thetrajectoryisspace-like. Inthenullcasethe 3. Case 3: m= 3
trajectory can be shown to be geodesic [49]. If V(z)=0 −2
has no roots given by z = z then the singularity is not
0
Equations (28) and the condition V(z ) = 0 lead
naked. On the other hand the central shell focusing is 0
to
at-least locally naked if V(z ) = 0 admits one or more
0
positive roots. The values of the roots give the tangents 1
=0, (59)
oftheescapinggeodesicsnearthesingularity. [50,51,54]. 4C z
1 0
Foraconformallyflatspace-time,themetricisdefined whichcanneverhaveanyfinitepositivesolutionfor
asds2 = 1 dt2 dr2 r2dΩ2 ,sowehaveeχ =eϑ = z0. Therefore the central singularity in this partic-
A(r,t)2 − − ular case, defined at 1 0 at r 0, is always
1 , V(z)= 1 (h1 z2) and Π=Ai2. So the equation for A(z) → →
A2 A2 − hidden to an exterior observer.
radial null geodesic simplifies considerably into dt = 1
dr
(where we have considered only the positive solution as In a recent work Hamid, Goswami and Maharaj [56]
we are looking for a radial outgoing ray). We consider showedthat asphericallysymmetricmatter cloudevolv-
the three cases for which we have found exact collapsing ing from a regular initial epoch, obeying physically rea-
solutions in section (5). sonable energy conditions is found to be free of shell
10
crossingsingularities. Acorollaryoftheirworkisthatfor VI. COMBINATION OF POWER-LAW
a continued gravitational collapse of a spherically sym- POTENTIALS : V(φ)= φ2 + φ(m+1)
2 (m+1)
metric perfect fluid obeying the strong energy condition
ρ 0 and (ρ +3p) 0, the end state of the collapse
In section (5) we studied the scalar field equations
≥ ≥
is necessarily a black hole for a conformally flat space- for a choice of potential for which dV = φm exploit-
time. This idea was also confirmed very recently in a dφ
ing the integrability of the scalar field evolution equa-
massivescalarfieldanalogueofthecollapsingmodelina
tion. This approach, though works only for those cases
conformally flat spacetime [26]. The present work, how-
where the scalar field equation is integrable, is useful to
ever, is quite different, as it deals with matter where the
make some general comments regarding the collapsing
strongenergyconditionis notguaranteed. This canper-
situation. However, the domain of the coordinate trans-
haps be related to the contribution of the scalar field to
formation restricted us not to choose a few values of m
the energy momentum tensor and the dissipation part
(m = 3, 1,0,1). This excluded any chance of study-
of the stress-energy tensor in the form of a heat flux. 6 − −
ing cases with a quadratic potential. In this section we
The choices of potential and the initial conditions in the assume a form for the potential such that dV =φ+φm,
form of the constants of integration can indeed conspire dφ
i.e. V(φ) is a combination of two power-law terms, one
amongst themselves so that the energy conditions are
of them being quadratic in φ. With this choice, the in-
violated which leads to the formation of a naked singu-
tegrability criterion yields a non-linear second order dif-
larity. A violation of energy condition by massive scalar
ferential equation for the conformal factor B(z) as
fields is quite usual and in fact forms the basis of its use
as a dark energy. In the present case, as discussed, we
find various possibilities, and a black hole is not at all 2
B (m+1) B 8(2m+3) B z
the sole possibility, i.e. the central singularity may not ◦◦ 3 ◦ ◦
always be covered, at least for some specific choices of B − (m+3) B ! − 6(m+3) B(1 z2)
−
the self-interacting potential, for instance V(φ) = 5φ2/5 2(1+m)z2 2(m+3) (m+3)
as found out in the present work. 2 − + =0. (62)
− 6(m+3)(1 z2)2 6B2(1 z2)
− −
It is very difficult to find an exact analytical form of
B(z) from (62) and thus any further analytical inves-
E. Non-existence of shear
tigations regarding the collapsing geometry is quite re-
stricted. However, we make use of a numerical method
For a general spherically symmetric metric to analysethe evolutionwhichofcoursedepends heavily
on choice of initial values and ranges of z, B(z) and dB.
dz
Firsttheequation(62)iswrittenintermsofD(z)= 1
B(z)
ds2 =S2dt2 B2dr2 R2dΩ2, (60) as
− −
where S,B,R are functions of r,t, and a comoving 2
D 3(m+1) D 8(2m+3) D z
observer is defined so that the velocity vector is uα = ◦◦ = 2 ◦ + ◦
S−1δ0α, the sheartensor components canbe easily calcu- D " − m+3 # D ! 6(m+3) D(1−z2)
lated. The net shear scalar σ can be found out as 2(1+m)z2 2(m+3) (m+3)D2
− + =0. (63)
− 6(m+3)(1 z2)2 6(1 z2)
− −
1 B˙ R˙ Now equation (63) is solved numerically and studied
σ = . (61) graphically as D(z) vs z in the limit z >> 1. Since
S B − R!
we mean to study a collapsing model, D (z) is always
◦
chosen as negative. The evolution of the collapsing
For the conformally flat metric chosen in the present sphere is sensitive to the factor D(z) , at least
work, S = B = Rr = A(1r,t). It is straightforward to see for some choices of potential as w|illDb◦e(z)sho|wn in the
that σ = 0 in the present case. In the present case, the subsequent analysis. We present the results obtained in
existenceofanisotropicpressureanddissipativeprocesses three different categories.
mightsuggesttheexistenceofshearinthespacetime,but
the situation is actually shearfree. A shearfree motion is
quite commonin the discussionofgravitationalcollapse, 1. Forallm>0,thesphericalbodycollapsestoazero
and it is not unjustified either. But it should be noted proper volumesingularityata finite future defined
that a shearfree condition, particularly in the presence byz =z asshowninfigure(5). Foralargepositive
s
of anisotropy of the pressure and dissipation, leads to value of m, e.g m 3, the singularity is reached
≥
instability. This result has been discussed in detail by more-or-less steadily. However, for choices of m
Herrera, Prisco and Ospino[57]. smaller in magnitude (for example, m = 1.5,m =