#### Collapsing spherical star in Scalar-Einstein-Gauss-Bonnet gravity with a quadratic coupling

Eur. Phys. J. C
Collapsing spherical star in Scalar-Einstein-Gauss-Bonnet gravity with a quadratic coupling
Soumya Chakrabarti 0
0 Centre for Theoretical Studies, Indian Institute of Technology Kharagpur , Kharagpur, West Bengal 721 302 , India
We study the evolution of a self interacting scalar field in Einstein-Gauss-Bonnet theory in four dimension where the scalar field couples non minimally with the GaussBonnet term. Considering a polynomial coupling of the scalar field with the Gauss-Bonnet term, a self-interaction potential and an additional perfect fluid distribution alongwith the scalar field, we investigate different possibilities regarding the outcome of the collapsing scalar field. The strength of the coupling and choice of the self-interaction potential serves as the pivotal initial conditions of the models presented. The high degree of non-linearity in the equation system is taken care off by using a method of invertibe point transformation of anharmonic oscillator equation, which has proven itself very useful in recent past while investigating dynamics of minimally coupled scalar fields.
1 Introduction
General theory of relativity is widely regarded to be the best
theory of gravity as over the years it has passed many
observational tests with flying colors. Significant observational
aspects such as the perihelion precission of Mercury’s orbit,
gravitational lensing, redshift in the light spectrum from
extragalactic objects are well documented in the framework
of general relativity. However, general relativity can pose
considerable intrigue in certain aspects such as the possibility
of an extremely strong gravitational field imploding without
limit and ending up in a region where the density of matter
and the strength of the gravitational field can in principle,
become infinite. Such a region is called a spacetime
singularity. Aspects of gravitational collapse and the formation of
a spacetime singularity forms an integral part of gravitational
physics today.
In general, a continual gravitational collapse occurs
whenever a massive astronomical body, upon exhausting it’s
nucleur fuel supply, fails to support itself against the force
of gravity. For a simple enough configuration of collapsing
matter, a horizon generally develops prior to the formation
of the singularity, thereby enveloping the singularity
producing a black hole end-state. The first analytic model of
such an unhindered contraction of an idealized star ending
up in a black hole was given by Oppenheimer and Snyder
[
1
] and independently by Dutt [
2,3
] and these serve as the
paradigm of gravitational collapse today. However, whether
or not every sufficiently massive star undergoing
gravitational collapse ultimately ends up in a black hole, remains
an intriguing question even in the current context. In this
regard, Penrose proposed [4] the cosmic censorship
hypothesis (CCH), which roughly states that gravitational collapse
of physically reasonable matters with generic regular initial
data will always end up in a covered singularity.
The CCH, however, remains one of the most
thoughtprovoking problems in gravitational physics today. There is
no general proof of the CCH as yet which applies under
all conditions. Moreover, there are many counterexamples
in general relativity, where it is shown that the singularities
formed in a collapse of reasonable distribution of matter can
stay exposed, giving rise to the concept of a naked, i.e., an
observable singularity (for a brief summary of such
examples, we refer to [
5
] and references therein). This much is
realized that the nature of the final outcome of a collapse depends
on the initial configuration from which the collapse evolves,
and the allowed dynamical evolutions in the spacetime, as
permitted by the non-linear field equations of gravity. For
recent works and different aspects regarding the dynamics of
a continued gravitational collapse we refer to the summary
by Joshi [
6,7
].
The singularity or a spacetime region with infinitely high
curvature can be realized only during the final stage of
gravitational collapse, where the collapsing body almost reaches a
zero proper volume and known laws of physics are expected
to break down. A possibility of naked singularity implies
that one may have a chance to observe indication of quantum
effects of gravity. Amongst the many quantum theories of
gravity proposed, superstring/M-theory is a promising
candidate, which motivates the presence of a higher-dimensional
spacetime [
8–17
]. During the final stages of the stellar
evolution, where the curvature of the central high-density region
is very high, the effects of extra dimensions can perhaps play
a crucial role. From such a perspective, higher-dimensional
gravitational collapse models are studied in general relativity
(we refer to the works of Banerjee, Debnath and Chakraborty
[18], Patil [
19
], Goswami and Joshi [
20,21
] in this regard).
Given that the entire aspects of superstring/M-theory are
not understood completely so far, taking their effects
perturbatively into classical gravity is one possible approach
to study higher curvature effects. The Gauss-Bonnet (GB)
term, G = R2 − 4Rμν Rμν + Rμναβ Rμναβ in the standard
Einstein-Hilbert Lagrangian is the higher curvature
correction to general relativity which finds it’s motivation from
heterotic superstring theory [
22,23
]. Such a theory is called
the Einstein-Gauss-Bonnet gravity.
The additional elegance of including the GB term is that
this is a Lovelock scalar and if included linearly in the action,
the field equations include no higher than second order partial
derivatives of the metric tensor (unlike f (R) gravity whose
field equations are fourth order in metric components). In
a four dimensional spacetime the Gauss-Bonnet term does
not modify the field equations. However, if the non-minimal
coupling of a scalar field with the GB term is considered, the
dynamical equations are quite different from the standard
field equations and the influence of the GB term in a four
dimensional universe is effective.
Recently there is an increasing interest in Gauss-Bonnet
theory with a non-minimally coupled scalar field to suffice
for the possible candidature of the late time acceleration
of the universe [
24–28
]. From such a perspective,
spherically symmetric solutions has been studied by Boulware and
Deser[
29,30
] and Gurses [31]. It has also been discussed
that the effective action including correction terms of higher
order in the curvature can perhaps play a significant role in the
dynamics of early universe by Zwiebach[
32
], Zumino[
33
],
Boulware and Deser[
29,30
]. Questions regarding
gravitational instability, cosmological perturbation were also
considered by Kawai, Sakagami and Soda[
34,35
], Kawai and
Soda[
36
]. Observational restrictions over different
cosmological aspects of the scalar field coupled Einstein Gauss
Bonnet gravity were investigated by Guo and Schwarz[
37
],
Koh et. al.[
38
]. Spherically symmetric collapsing solutions of
this theory have also gained some interest quite recently. For
instance, Maeda presented a model of n ≥ 5 dimensional
spherically symmetric gravitational collapse of a null dust
fluid in Einstein-Gauss-Bonnet gravity [
39
] and illustrated
the possibility of a formation of massive naked singularity
in higher dimensions. He comparitively analyzed the results
with the general relativistic cases as well [
40
], which serves
as a higher order generalization of the Misner-Sharp
formalism of the four-dimensional spherically symmetric spacetime
with a perfect fluid in general relativity. Hamiltonian
Formulation of spherically symmetric scalar field collapse in
GaussBonnet gravity was studied in detail by Taves, Leonard,
Kunstatter and Mann [
41
]. They also proved that such a
formulation can readily be generalized to other matter fields
minimally coupled to gravity. Apart from their role in cosmology,
the role of scalar fields in a gravitational collapse is worthy
of attention. It is indeed important to investigate if the CCH
necessarily holds or violates in the collapse of fundamental
matter fields. Moreover, a scalar field alongwith an
interaction potential is known to mimic different kind of
reasonable distribution of matter as discussed by Goncalves and
Moss [
42
]. Numerical simulations of the spacetime
evolution of a massless scalar field minimally coupled to
gravitational field studied by Choptuik [
43
], Brady [
44
] and
Gundlach [
45,46
] hint interesting possibilities, such as, the critical
behavior observed around the threshold of black hole
formation. There is a self-similar solution that sits at the black
hole threshold, dubbed a critical solution, and also a very
interesting mass scaling law for the formed black hole
endstate. Motivated by this, self similar collapsing scenario of
a massive scalar field was analytically studied by Banerjee
and Chakrabarti [47] very recently. Deppe, Taves, Leonard,
Kunstatter and Mann presented a numerical analysis in
generalized flat slice co-ordinates of self-gravitating massless
scalar field collapse in five and six dimensional Einstein
Gauss Bonnet gravity near the threshold of black hole
formation [
48
]. Effects of higher order curvature corrections to
Einstein’s Gravity on the critical phenomenon near the black
hole threshold, were investigated by Golod and Piran [
49
]. In
general relativity also, possibilities and scope of scalar field
collapse have been analytically studied extensively [
50–53
].
Finally, the fact that the distribution of the dark energy
component or the fluid still remains unknown, naturally inspires
a continuing study of scalar field collapse under increasingly
generalized setup (preferably alongwith a fluid distribution)
towards a better understanding of the possible clustering of
dark energy.
In this work, we study aspects of a scalar field
collapse in a Scalar-Einstein-Gauss-Bonnet gravity, where the
self-interacting scalar field φ is non-minimally coupled to
the GB term. Very recently Banerjee and Paul [
54
]
studied such a scalar field collapse where the coupling term
was proportional to e2φ . In the present case the coupling
term is proportional to φ2, i.e., the coupling is quadratic in
φ. Quite recently, Doneva and Yazadjiev showed that in a
very similar setup of Scalar Einstein Gauss Bonnet theory
(the conditons they imposed on the coupling function f (φ)
are f (φ = 0) = 0 and b2 = f (φ = 0) > 0), there
exists new black hole solutions which are formed by
spontaneous scalarization of the Schwarzaschild black holes in
the extreme curvature regime and below a certain mass, the
Schwarzschild solution becomes unstable and new branch
of solutions with nontrivial scalar field bifurcate from the
Schwarzschild one [
55
]. They also proved the existence of
neutron stars in a class of extended scalar-tensor
GaussBonnet theories for which the neutron star solutions are
formed via spontaneous scalarization of the general
relativistic neutron stars [
56
]. Very recently, the spontaneous
scalarization of black holes and compact stars from such a
Gauss-Bonnet coupling has been investigated and dubbed as
the Quadratic Scalar-Gauss-Bonnet gravity by Silva et. al.
[
57
]. In this context, existence of regular black-hole
solutions with scalar hair in the Einstein-scalar-Gauss-Bonnet
theory was investigated by Antoniou, Bakopoulos and Kanti
[
58,59
], with a general coupling function between the scalar
field and the quadratic Gauss-Bonnet term which
highlighted the limitations of the existing no-hair theorems. In
a recent study Kanti, Gannouji and Dadhich have addressed
the importance of such a coupling from a cosmological
purview and proved by some simple analytical calculation
that a quadratic coupling function, although a special choice,
allows for inflationary, de Sitter-type solutions to emerge
[60].
The inclusion of the Gauss-Bonnet terms make the
dynamical field equations even more non-linear. We study a spatially
homogeneous model where the energy momentum tensor is
contributed by the self-interacting scalar field as well as a
perfect fluid. To track down the system of equatons, we use the
method of invertible point transformations and integrability
of anharmonic oscillator equations; an approach which has
been quite useful recently, in the study of minimally coupled
massive scalar field collapse by Chakrabarti and Banerjee
[
47,61
].
The paper is organized as follows. In Sect. 2, we introduce
the action and basic field equations. Section 3 contains the
method of finding the exact solution and in Sect. 4 we study
the evolution of the scale factor for different initial data. The
time evolution of the scalar field is studied in Sect. 5.
Evolution of the curvature scalars and the strong energy condition
is studied in Sect. 6. The physical nature of the singularity is
addressed in 7. We complete the model by matching the
solution with an exterior Vaidya solution in Sect. 8 and conclude
in Sect. 9.
2 Action and basic equations
The relevant action for a four dimensional action containing
the Einstein-Hilbert part, massive scalar field and the
GaussBonnet term coupled with the scalar field. The corresponding
action is given by
where R is the Ricci scalar, 1/(2κ2) = M 2p is the four
dimensional squared Planck scale and G = R2 − 4Rμν Rμν +
Rμναβ Rμναβ is the GB term. φ and V (φ) denote the scalar
field and the self interaction potential respectively. ξ(φ)
defines the coupling between scalar field and GB term.
Variation of the action with respect to metric and scalar field leads
to the field equations as follows,
1
κ2 [−Rμν + (1/2)gμν R] + (1/2)∂μφ∂ν φ
− (1/4)gμν ∂ρ φ∂ρ φ + (1/2)gμν
− V (φ) + ξ(φ)G
− 2ξ(φ)R Rμν − 4ξ(φ)Rρμ Rνρ − 2ξ(φ)Rμρσ τ Rρνσ τ
+ 4ξ(φ)Rμρνσ Rρσ + 2[∇μ∇ν ξ(φ)]R − 2gμν [∇2ξ(φ)]R
− 4[∇ρ ∇μξ(φ)]Rνρ − 4[∇ρ ∇ν ξ(φ)]Rμρ + 4[∇2ξ(φ)]Rμν
+ 4gμν [∇ρ ∇σ ξ(φ)]Rρσ + 4[∇ρ ∇σ ξ(φ)]Rμρνσ = 0,
(2)
and
gμν [∇μ∇ν φ] − V (φ) − ξ (φ)G = 0,
where a prime denotes the derivative with respect to φ.
The metric for the interior is assumed to be a spatially flat
Friedmann-Robertson-Walker metric and is given by
ds2 = −dt 2 + a2(t ) dr 2 + r 2dθ 2 + r 2 sin2 θ dϕ2 ,
where the scale factor a(t ) governs the time evolution of the
interior spacetime. For such a metric, the expression of Ricci
scalar R and GB term G take the following form,
R = 6[2H 2 + H˙ ],
G = 24H 2[H 2 + H˙ ],
where H = a˙ /a and dot denotes the derivative with respect
to time (t ).
Here, we have assumed the interior of the collapsing star
to be consisting of the scalar field as well as a perfect fluid.
Therefore, using the metric in eqn. (4), the field equations
can be written as,
−(3/κ2)H 2 + (1/2)φ˙ 2 + V (φ) + 24H 3ξ˙ + ρ = 0,
1
κ2 2H˙ + 3H 2 + (1/2)φ˙ 2 − V (φ) − 8H 2ξ¨
−16H H˙ ξ˙ − 16H 3ξ˙ + p = 0,
and
φ¨ + 3H φ˙ + V (φ) + 24ξ (φ)(H 4 + H 2 H˙ ) = 0.
(1)
(3)
(4)
(5)
(6)
(7)
ρ and p signifies the density and pressure of the constituent
fluid inside the collapsing star.
3 Exact solution
It can be easily noted that, due to the inclusion of the
additional fluid component (two additional unknown functions
ρ and p), the system of equations becomes even more
difficult to deal with analytically, even more so without
assuming any particular equation of state. However, we try to do
away with the difficulty by incorporating a strategy, where
the scalar field evolution Eq. (7) is identified as an
anharmonic oscillator equation and integrated straightaway
without any apriori assumptions regarding the equation of state
or the scale factor. In this way, the other field equations can
be used to study the evolution of the fluid/scalar field
distribution in a general manner. However, the only assumption
being implemented here is that the Eq. (7) is integrable. The
criterion for such an integrability can be defined in terms of
an invertible point transformation, first worked out by Duarte
et. al. [
62
], Euler et al. [
63
]. Although the main motivation
over this assumption of integrability is extracting
information from the field equations at any cost without resorting to
any assumption of equation of state, this assumption is by no
means unphysical. In recent past, this approach has proved
to be very useful to produce interesting solutions depicting
dynamics of minimally coupled massive scalar field [
61
],
self-similar solutions [
47
] and also showed promise towards
being a tool for reconstructing modified gravity lagrangians.
Using this approach, Chakrabarti, Said and Farrugia have,
quite recently studied a reconstruction method for
teleparrallel gravity [
64
]. This work therefore, carries a subtle
motivation of testing the scope of this approach in the domain of
Scalar-Einstein-Gauss-Bonnet gravity.
While a simple linear harmonic oscillator has a
straightforward sinusoidal solution, an anharmonic oscillator has more
contributing terms and can represent more physical features
of a dynamical system. This takes the form of a nonlinear
second order differential equation with variable coefficients as
φ¨ + f1(t )φ˙ + f2(t )φ + f3(t )φn = 0,
(8)
where fi are functions of t and n ∈ Q is a constant.
Overhead dot represents differentiation with respect to cosmic
time, t . Using Euler and Duartes [
62,63,65
] work on the
integrability of the general anharmonic oscillator equation
and the more applicable reproduction by Harko et al. [
66
],
this equation can be integrated under certain conditions. The
essence of the integrability criterion is that, an equation of
the form Eq. (8) can be point transformed into an integrable
form iff n ∈/ {−3, −1, 0, 1}, provided the coefficients of Eq.
(8) satisfy the differential condition
(9)
(10)
(11)
(12)
(13)
(14)
= 0,
(15)
(16)
1 1 d2 f3 n + 4
n + 3 f3(t ) dt 2 − (n + 3)2
n − 1 1 d f3
f1 (t )
+ (n + 3)2 f3(t ) dt
+ n + 3 dt + 2(n(n++3)12) f12 (t ) = f2(t ).
2 d f1
1 d f3 2
f3(t ) dt
Introducing a pair of new variables and T given by
T (φ, t ) = C 1−2n
1 2 t f1(x)dx ,
(T ) = C φ (t ) f3n+3 (t ) e n+3
t 2 1−n ξ f1(x)dx dξ,
f3n+3 (ξ ) e n+3
where C is a constant, Eq.(8) can then be written in an
integrable form as
d2
d T 2 +
n (T ) = 0.
We focus our investigation on a particular case of
polynomial coupling, i.e., where ξ(φ) = ξ0 φ22 . We also take the
selfinteraction potential V (φ) = V0 φ(n(n++11)) . Both positive powers
and inverse powers of φ are very useful in the cosmological
setting, in particular, inverse power law models are extremely
useful as quintessence fields among other interesting
properties. With this assumption, the scalar field evolution equation
becomes
φ¨ + 3H φ˙ + 24ξ0φ (H 4 + H 2 H˙ ) + V0φn = 0.
One can easily identify f1(t ) = 3H , f2(t ) = 24ξ0(H 4 +
H 2 H˙ ) and f3 = V0.
Provided n ∈/ {−3, −1, 0, 1}, the integrability criterion
produces a differential equation for H = aa˙ given by
24ξ0 H 4 − 18
(n + 1) H 2 + H˙ 24ξ0 H 2
(n + 3)2
6
− (n + 3)
which we rewrite in the form
H˙ =
18 ((nn++31))2 − 24ξ0 H 2
6
24ξ0 − (n+3)H2
.
During the final stages of the evolution of the collapsing
scalar field, it is expected that the proper volume is very
small and sharply decreasing in nature. Moreover, the rate of
collapse must also be increasing rapidly. With that in mind
one can say, H = aa˙ >> 0 and is a sharply increasing
function of time. Therefore, H12 can be neglected compared to
other term on the denominator on the RHS. With that
simplification and the fact that a˙ < 0 for a collapsing model,
Fig. 1 Evolution of the scale factor with time for V (φ) = φ44 and
for different positive values of a0. (Colour Code : Blue → a0 = 0.6,
Y ellow → a0 = 0.5, Gr een → a0 = 0.4 and Red → a0 = 0.3).
For all values of the initial parameter a0, the collapse reaches a zero
proper volume, however the rate of collapse depends on the choice of
the parameter
we can write a solution for the scale factor, defining the time
evolution of the collapsing scalar field as
1
a(t ) = 6(1 + n)
Here both t0 and a0 are constants of integration and serves
as the initial condition of the model alongwith the choice of
self-interaction potential (value of n). In order to have a real
time evolution one must enforce the restrictions n > −1 and
ξ0 > 0.
The time of reaching the zero proper volume can be
calculated from Eq. (17) as
ts = t0 −
2(3 + n)√ξ0 ln[2
√3(1 + n)
3a0ξ0(1 + n)(3 + n)].
(18)
4 Evolution of the scale factor for different initial conditions
4.1 Evolution of the scale factor for a0 > 0
We present different possible outcomes of the collapse
graphically and study the evolution varying different initial
conditions.
In Fig. 1, the scale factor is plotted as a function of time
for a fixed value of t0, for different positive values of a0
and n = 3, i.e., V (φ) = φ44 . The evolution shows a rapid
collapsing behavior For all values of a0, and an ultimate zero
proper volume end-state, however, the rapidity of the collapse
1500
1000
and the time of formation of zero proper volume depends on
the choice of the parameter.
An important note to make here is that, for the particular
4
case of n = 3, i.e., V (φ) = φ4 , t0 = 20 and as long as a
positive choice of a0 is made, ξ0 must be taken in between 0 and 1.
For ξ0 < 0, there is no real evolution and for ξ0 > 1, the
evolution becomes negative. Therefore a condition of 0 < ξ0 ≤ 1
must be enforced upon the strength of the coupling. However,
this range can differ for a different set of initial conditions,
i.e., for a different set of n and t0. We have presented a
particular case with the note that for any such model, the strength
of the coupling is very important and therefore the restriction
over the allowed domain of ξ0 must be accounted for.
In Fig. 2, we plot the collapsing behavior for ξ0 = 0.4
and for different choices of n, i.e., for different choice of self
interaction potential defined by n = 3, 1.5, 0.5 and 0.001.
For all the cases, the evolution starts at a finite value of the
scale factor (as shown in the first figure) and a zero proper
volme is reached at a finite future. The time of formation of
singularity changes depending on the choice of n (as shown
by the figure below).
4×106
t() 3×106
a
In Fig. 3, we show the evolution of the scale factor with
time for different choice of the coupling ξ0, fixing other initial
parameters. For different value of ξ0 the collapse ends up in
a zero proper volume afterall, but for different ts , which is
also evident from Eq. (18).
4.2 Evolution of the scale factor for a0 < 0
However, depending on the initial conditions, the collapse
may not always lead to a zero proper volme. As shown in
Fig. 4, for all negative values of a0, the system experiences
a collapse initially, but only until a critical point (a non-zero
minimum radius) after which it can not shrink further and
experiences rapid expansion. The plot here is for V (φ) = φ44
and ξ0 = 0.4. This behavior is valid for all negative values
of a0.
We present the evolution graphically for different choices
of the self-interaction potential (depending on the choice of
n) and for a particular choice of negative a0, and fixed ξ0.
The evolution shown in Fig. 5 suggests that the scalar field
experiences a collapse initially, only until a critical point after
which it experiences a bounce. The overall qualitative
behavior remains the same for different n, however, there is an
indication that, depending on n, the bouncing behaviour may be
scaled.
15
t
15
t
5
10
20
25
30
Fig. 4 Evolution of the scale factor with time for V (φ) = φ44 , ξ0 = 0.4
and for different negative values of a0. (ColourCode : Blue → a0 =
−10, Y ellow → a0 = −1, Gr een → a0 = −0.1 and Red → a0 =
−0.00001). The evolution suggests that the scalar field experiences a
collapse initially, only until a critical point after which the collapse
changes into an expanding phase
In Fig. 6, we plot the scale factor as a function of time
4
for V (φ) = φ4 , a0 = −1 and for a different values of the
coupling parameter ξ0. The evolution suggests that the overall
qualitative behavior may change depending on the value of
ξ0, in the sense that, the phase of an initial contraction of the
scale factor depends on the choice of ξ0. For instance, for ξ0 =
0.4, (shown by the blue curve) there is an initial collapsing
phase before reaching an eventual non zero minimum cutoff,
and the expanding phase begins thereafter. However, if one
increases the value of ξ0 gradually, it can be seen (Yellow for
ξ = 1, Green for ξ = 2 and Red for ξ = 3) that the initial
collapsing phase seems to become negligible and the entire
solution turns into an expanding solution.
5 Evolution of the scalar field
Using the defined point transformations (11) and (12), a
general solution of the scalar field can be constructed from the
transformed integrable form of the scalar field evolution Eq.
(13). The solution is given in the form
0 0
2000
1500
.
Fig. 5 Evolution of the scale factor with time for different n, i.e.,
for different choices of V (φ) = φ(n+1)
ular negative value of a0 = −1. (Cn+o1lo,urξ0Co=de0.:4 Balnude fo→r a
particn = 3,
Y ellow → n = 1.5, Gr een → n = 0.5 and Red → n = 0.001)
2φ
φ˙ + (n + 3)
2
=
Here, g1(t ) and g2(t ) are functions of time which we have
written in this form for the sake of brevity.
ALthough it would have been better if a neat and closed
for of the evolution could be written, however, the numerical
integration of the Eq. (21) produces a very large expression
for the scalar field (about 145 terms). We present in brief the
functional form so as to give an idea regarding how the scalar
field can, in principle, evolve.
1
φ (t ) = ψ (t ) 3 ,
g(t )
ψ (t ) = s0e−t + j (t ) ,
j (t ) = s1 1728be √3 m − e d√√3m1
√ m1 t
3
2 F1 1, 2 +
3ξ0; 3 +
1−t0
3ξ0; 1728ξ0a0e √3ξ0
+ci ec j +ckt 2 F1 1, 2 +
3ξ0; 3 +
t−t0
3ξ0; 1728ξ0a0e √3ξ0 .
All the parameteres a1, a2, ai −s, a j −s etc, are infact defined
in terms of the parameters of the theory, i.e., ξ0, n and a0. The
function g(t ) consists of a total of (80+33+31) terms where
there are 80 terms of the form of ai + a j eak +al t , 33 terms
of the form of bi eb j +bk t and 31 terms of the form of ci ec j +ck t .
It is quite obvious that a numerical examination is absolutely
necessary over the time evolution of the scalar field. We note
that, the parameters ξ0 (i.e., the strength of the coupling of
scalar field with the Gauss-Bonnet term) and the constant
of integration C1 play an important part in determining the
behavior of the scalar field afterall. In the next subsection we
present the numericla results of the evolution of the scalar
field as a function of time for different parameters.
Fig. 6 Evolution of the scale factor with time for V (φ) = φ44 , a0 = −1
and for a different values of ξ0. (ColourCode : Blue → ξ = 0.4,
Y ellow → ξ = 1, Gr een → ξ = 2 and Red → ξ = 3)
Calculating the coefficients f1(t ) = 3H , f2(t ) =
24ξ0(H 4 + H 2 H˙ ) and f3 = V0 using the solution for scale
factor (17), one appears at a differential equation governing
the behaviour of the scalar field itself given by
20
25
30
(19)
0
2
4
6
8
10
0
5
10
20
25
30
15
t
Fig. 7 Scalar field as a function of time for V (φ) = φ44 , ξ0 = 0.4
and for different values of C1 (Color Code : Blue ⇒ C1 =
0.001, Y ellow ⇒ C1 = 0.01, Gr een ⇒ C1 = 0.1 and Red ⇒ 1.0)
Fig. 9 Evolution of the scalar field as a function of time for V (φ) = φ44 ,
C1 = 10, ξ0 = 0.4 and a0 = −1
Fig. 8 Evolution of the scalar field as a function of time for V (φ) =
φ4 , C1 = 0.01 and for different choices of coupling parameter ξ0 =
4
0.4(Blue) and ξ0 = 0.5(Y ellow)
5.1 Evolution of the scalar field for a0 > 0, i.e., for
collapsing scale factor
In Fig. 7, the scalar field is plotted as a function of t for
V (φ) = φ44 , ξ0 = 0.4 and for different values of C1
(C1 = 0.001, C1 = 0.01, C1 = 0.1and1.0). The scalar field
diverges around the time of formation of singularity.
However, depending on the value of C1, the nature of the
evolution before reaching singularity may be a little different
but the qualitative behavior is this; the time evolution of the
scalar field starts at a finite value and then decreases
moreor-less steadily, before reaching a minimum critical value.
Thereafter, the scalar field increases with time rapidly and
diverges.
Evolution of the scalar field as a function of time is plotted
φ4 , C1 = 0.01 and for different choices
in Fig. 8 for V (φ) = 4
of ξ0 = 0.4 and 0.5. For different values of the coupling
parameter ξ0, the scalar field diverges at different time. This
φt()0.6
1.2
As discussed by Figs. 3, 4 and 5, the evolution is not always
collapsing forever, depending on the choice of the parameter
a0. For a0 < 0, the scale factor experiences a transition from
a state of contraction into a rapid expansion. Here we show
the behavior of the scalar field with time for such cases, i.e.,
for a0 < 0.
In Fig. 9, we have plotted φ (t ) vs t for V (φ) = φ44 ,
C1 = 10, ξ0 = 0.4 and a0 = −1. The scalar field starts at
some finite value and gradually decreases, exhibiting some
periodic behavior with time. Eventually it decays into a
negligibly small positive value as shown in the figure.
However, the periodic time evolution of the scalar field
is sensitive over the choice of ξ0 as shown in figure 10. The
periodic nature seems to be absent as one gradually increases
the value of ξ0, (here, plotted for ξ0 = 0.4, 1 and 2).
Fig. 11 Evolution of (ρ + 3 p) with time for V (φ) = φ44 , t0 = 20,
a0 = 10, ξ0 = 0.4 and for different choices of C1 (C1 = 0.1(Blue),
C1 = 10(Y ellow), C1 = 100(Gr een) and C1 = 10000(Red))
φ4 , t0 = 20,
Fig. 12 Evolution of (ρ + 3 p) with time for V (φ) = 4
a0 = 10, ξ0 = 0.4 and for very small values of C1 (C1 = 0.001(Blue)
and C1 = 0.0001(Y ellow)
6 Evolution of the strong energy condition
A collapsing perfect fluid is physically reasonable if it obeys
the strong energy condition which is satisfied if for any
timelike unit vector wα and the following inequality holds
(22)
2Tαβ wαwβ + T ≥ 0,
where T is the trace of the energy momentum tensor. The
energy conditions were investigated in details for imperfect
fluids by Kolassis, Santos and Tsoubelis [
67
]. Following their
work, we investigate the validity of the strong energy
condition ((ρ + 3 p) > 0) for our model. The strong energy
condition can be violated only if the total energy density is
negative or if there exists a large negative principal pressure
of T αβ .
Since we have studied the solution for the scale factor
from the scalar field evolution (7) equation straightaway, the
field Eqs. (5) and (6) can be used to study the evolution of
the constituent fluid density and pressure. We numerically
study the strong energy condition and present it’s nature by
the following plots.
In Fig. 11, we plot the evolution of (ρ + 3 p) as a
function of time for a particular choice of potential V (φ) = φ44 ,
positive a0 (ensuring the collapsing nature of the solution),
ξ0 = 0.4 and for different choices of the initial
conditon C1 (C1 = 0.1(Blue), 10(Y ellow), 100(Gr een) and
10000(Red)). The evolution suggests that that (ρ + 3 p) ≥ 0
is satisfied throughout the evolution of the collapsing body,
before reaching the curvature singularity where (ρ + 3 p)
increases sharply, as both pressure and density diverges
eventually.
However, depending on the value of C1, we also find some
cases where the strong energy condition may be violated
during the collapsing evolution, before eventually diverging
at the singular epoch. In Fig. 12, we plot the evolution of
(ρ + 3 p) as a function time for V (φ) = φ44 , positive a0,
ξ0 = 0.4 and for very small positive values of C1 (C1 =
0.001(Blue) and C1 = 0.0001(Y ellow).
However, the evolution of the strong-energy condition in
time leading to the conclusion that this is often violated, is
not really an unexpected outcome in theories of gravity that
contain strong-curvature terms. In the present case one can
argue that the Gauss-Bonnett term in priciple can create an
effective energy-momentum tensor whose contribution to the
total energy-momentum tensor can lead to the violation of
the strong-energy condition. Therefore, it may not be the
nature of the perfect fluid afterall that violates the
strongenergy condition. We also note here that the energy
conditions of general relativity are a mathematical way of making
the notion of locally positive energy density by stating that
various linear combinations of the components of the energy
momentum tensor must stay non-negative and it is
sometimes argued that subtle quantum effects can violate all of
the energy conditions. Moreover, there are examples of
classical systems that violate all the energy conditions as well
(For instance, Lorentzian-signature traversable wormholes
[
70
]). The simplest possible source of classical energy
condition violations is from the contribution of scalar fields, in
particular, non-minimally coupled scalar field contributions,
as worked out in details by Visser and Barcelo [
68,70
],
Flanagan and Wald [69].
In Fig. 13, the evolution of (ρ + 3 p) with time is plotted
4
φ , positive a0, C1 = 10000 and for different
for V (φ) = 4
value of ξ0 = 0.04(Blue) and ξ0 = 0.4(Y ellow). As can be
seen from the graph, (ρ + 3 p) maintains a positive
signature throughout the collapse, however, for different value of
ξ0 the energy condition goes to positive infinity at different
time. This is quite consistent, since we have already discussed
that the strength of coupling with the Gauss-Bonnet term ξ0
plays a crucial role in determining the time of formation of
singularity.
7 Nature and visibility of the singularity
In order to investigate whether the singularity is a curvature
singularity or just an artifact of coordinate choice, one must
look into the behavior of the Kretschmann curvature (K )
scalar at t → ts = t0− 2√(33+(n1+)√nξ)0 ln[2√3a0ξ0(1 + n)(3+n)],
i.e., when the scale factor a(t ) goes to zero. For the metric
presented in Eq. (4), K has the expression,
K = 6 a¨ (t )2 a˙ (t )4
a(t )2 + a(t )4
Using the solution of a(t ) Eq. (17), it is straightforward
to deduce that the Kretschmann scalar diverges at the zero
proper volume and thus the collapsing body discussed here
ends up in a curvature singularity.
7.1 Formation of apparent horizon
Whether the curvature singularity is visible to an exterior
observer or not, depends on the formation of an apparent
horizon. The condition for such a surface is given by
gμν R,μ R,ν = 0,
where R is the proper radius of the two-sphere, given by r a(t )
in the present case.
Using the exact solution for collapse given by Eqs. (17)
and (24), we deduce the condition for formation of apparent
horizon as
√3x
12(3 + n)√(1 + n)ξ0 +
√3ξ0(1 + n)a0(3 + n)
x
− λ = 0,
(25)
(23)
(24)
√3(1+nξ)0 (t0−t).
where λ is a constant of separation and x = e 2(3+n)√
One can solve the above equation to exactly deduce the time
of formtion of an apparent horizon as
tap = t0
2(3 + n)√ξ0 ln 2(3+n) 3(1 + n)ξ0(λ ±
− √3(1 + n)
λ2 − a0) .
(26)
(27)
Comparing Eq. (26) with the time of formation of the
singularity (18) we arrive at the very important expression
(ts − tap) =
2(3 + n)√ξ0 ln (λ ±
√3(1 + n)
λ2 − a0)
√a0
.
To comment on the ultimate visibility of the collapse
outcome, it is necessary to see if nonspacelike trajectories
emanate from the singularity and reach a faraway observer.
The singularity is at least locally naked if there are future
directed nonspacelike curves that reach faraway observers.
This is possible if the formation of apparent horizon is
delayed or if there is no formation of horizon at all. In the
present case the time of formation of singularity is
independent of r [given by Eq. (18)] and therefore it is only
natural that the entire collapsing system (scalar field and perfect
fluid) would reach the singularity simulteneously at t = ts .
This kind of singularity is always expected to be covered by
the formation of an apparent horizon as discussed by Joshi,
Goswami and Dadhich [
71
]. Here, from Eq. (27), it can be
said that the formation of an ultimate covered singularity
is dependent over initial conditions such as ξ0, λ and a0 so
that (ts − tap) > 0. However, in principle, one can also ask
about the possible end state if the initial conditions conspire
to make the situation to end up otherwise, i.e., (ts − tap) < 0.
In such a case, there is no formation of apparent horizon at
all, since all the collapsing shells labelled by different values
of r , shrinks to zero proper volume and the physical
quantities diverge at t = ts , which is reached before tap. Therefore
the singularity remains naked and the condition for such an
end-state can be written from Eq. (27) as
8 Matching with an exterior Vaidya Spacetime
For the sake of completeness, proper junction conditions are
to be examined carefully which allow a smooth matching
of an exterior geometry with the collapsing interior. First of
all it was extensively shown by Goncalves and Moss [
42
]
that any sufficiently massive collapsing scalar field can be
formally treated as collapsing inhomogeneous dust in
general relativity. Moreover, astrophysical objects undergoing a
gravitational collapse can be expected to be in an almost
vacuum spacetime, and therefore the exterior spacetime around
a spherically symmetric dying star is well described by the
Schwarzschild geometry. From the continuity of the first
and second differential forms, the matching of the sphere
to a Schwarzschild spacetime on the boundary surface, , is
extensively worked out in literature [
72–75
].
However, conceptually this leads to an inconsistency,
since Schwarzschild has zero scalar field. Therefore, such a
matching would lead to a discontinuity in the scalar field, and
a delta function in the gradient of the scalar field. As a
consequence, there will appear square of a delta function in the
stress-energy, which is definitely an inconsistency. Since we
have a scalar field distribution inside the collapsing sphere,
it is more physical to match the interior with a Vaidya
exterior solution (for a detailed analysis of vaidya matching in a
scalar field collapse we refer to [
76,77
]) across a boundary
hypersurface defined by . The metric just inside is,
ds−2 = dt 2 − a(t )2dr 2 − r 2a(t )2d 2,
and the metric in the exterior of
is given by
ds+2 =
1 −
2M (rv, v)
rv
dv2 + 2dvdrv − rv2d 2. (30)
Matching the first fundamental form on the hypersurface
we get
dv
dt
and
r a(t )
dv
dt
=
1
1 − 2M(rrvv,v)
2drv
+ dv
,
(rv)
= r a(t )
1
= r 6(1 + n)
,
1 √3(1+√nξ)0 (t0−t)
6(1+n) e 2(3+n)
− 2a0(3 + n)2
where a(t )
Equations (32), (33), (34) and (35) completely specify the
matching conditions at the boundary of the collapsing scalar
field.
The present work has been entirely dedicated towards a
deeper understanding of a self-interacting scalar field
collapse in Scalar-Einstein-Gauss-Bonnet gravity. The scalar
field couples non-minimally with the Gauss-Bonnet term by
2
a term quadratic in the scalar field (ξ0 φ2 ). The possibility of
the collapse reaching a zero proper volume singularity is seen
to be dependent on the coupling parameter ξ0, choice of
selfinteraction potential and most importantly the initial
condition a0 which can be related to the initial radius/volume of
the collapsing star. We also comment on the allowed domain
of the choice of the coupling parameter ξ0 to have a real
evolution of the collapse. It is observed that the collapse ends
up in a curvature singularity, where the Kretschmann
curvature scalar blows up, alongwith density and pressure of the
constituent fluid and the scalar field. The strong energy
condition is showed to be valid throughout the collapse. However,
depending on certain initial conditions defining the initial
distribution of the scalar field, the energy conditions may be
violated, which can be attributed to the introduction of a
nonminimal coupling of a scalar field in the lagrangian, which
is known to be a possible classical system that can violate
almost all of the energy conditions [
68,70
].
For the sake of completeness we match the interior
collapsing solution with the an exterior Vaidya geometry on a
boundary hypersurface, since the presence of a Gauss-Bonnet
term non-minimally coupled to a scalar field generates a
nonzero effective energy momentum tensor arising from
spacetime curvature. Therefore matching with an exterior vacuum
solution in the presence of Gauss-Bonnet term may lead to
9 Conclusion
(31)
(32)
(33)
inconsistency. Very recently this has been investigated by
Banerjee and Paul [
54
].
The time of formation of the singularity is independent
of r , which suggests that all the collapsing shells labelled
by different values of r collapses simulteneously when the
zero proper volume is reached. Such a singularity is always
expected to be covered by a horizon as far as similar studies
under the domain of GR are concerned. We note here that
in the present case of Scalar-Einstein-Gauss-Bonnet
gravity with a polynomial coupling, the ultimate end-state of a
covered singularity is conditionally consistent with the
corresponding results in GR. For certain initial conditions [defined
by Eqs. (27) and (28)] there is a probability of an end state
where there is no possibility of formation of horizon at all.
We also observe some interesting results, for instance,
depending on the signature of the aforementioned parameter
a0, the evolution of scale factor suggests that all of the
possible collapsing scenario may not lead to a curvature
singularity. Rather, for a0 < 0, the fluid undergoes contraction only
unto a minimum non-zero cut-off radius, after which it goes
into a rapidly expanding phase. It is also seen that the scalar
field itself in such cases, decreases monotonically with time,
exhibits certain periodic behavior, before becoming
negligibly small. This can be somewhat compared with the
phenomena of collapse and dispersal for a scalar field in general
relativity which have drawn considerable interest in recent
years, mainly from a numerical perspective, subtly pointing
towards the existence of a critical phenomena is the whole
collapsing picture (For details, we refer to the monograph
by Gundlach [
45,46
]). Recently Bhattacharya, Goswami and
Joshi worked out a sufficient condition for the dispersal to
take place for a collapsing scalar field in GR that initially
begins with a contraction and showed that the transition of the
collapsing body into expanding nature is crucially connected
wiith the change of the gradient of the scalar field [78]. In the
present case we can note that the signature of the parameter
a0 > 0 or a0 < 0 is the defining factor of the transition;
i.e., whether or not the collapse will lead into a zero proper
volume or a dispersal of scalar field shall take place after
the scale factor reaches a minimum cut-off volume. Since a0
comes from the expression defining the scale factor (a(t ) =
6(11+n) e 2(√3+3(n1)+√nξ)0 (t0−t) −2a0(3+n)2e− 2(√3+3(n1)+√nξ)0 (t0−t)), the
initial volume of the collapsing system can be predicted as a
defining factor connected to the value of a0.
We conclude with the note that this is indeed a simple case,
in the sense that we have considered spatial homogeneity in
the metric components as well as the scalar field. However,
the solution found is simple enough to encourage further
allied investigations in this direction, such as, the possibility
of a collapse even when the energy conditions are violated
may subtly direct one’s attention towards a possible
clustering of dark energy distribution. Apart from that, this work
also helps further expansion of the usefulness and scope of
the particular method of integrability of anharmonic
oscillator used here. The theorem which inspires this method is
self sufficient as was discussed by Euler [
65
]. The same has
been proved in the context of a massive scalar field
minimally coupled to gravity by Chakrabari and Banerjee [
61
],
where the solutions found by virtue of this theorem indeed
solve the Klein Gordon type evolution equation once they
are put back in the equation. In the current context, the
solutions are far more complicated to say the least, as is
evident from the expression of the scalar field, as worked out
in details in section V . Since the criterion for integrability
was investigated under the expectation that the proper
volume would be very small and sharply decreasing in nature
so that one can neglect the H12 term, to justify such an
arguement one can put the solutions (17) and (21) together into
the equation (14) and study for different initial conditions
defining the scalar field. It can be confirmed that for
relevant cases, the solutions put into the Klein-Gordon type Eq.
(14) yields a number very close to zero (∼ 10−8 or lower).
This approach of point transforming the klein-gordon type
equation for the scalar field, to extract the solution out of a
seemingly impossible non-linear system has worked really
well in the past while investigating different setups of massive
scalar field collapse and also inspired a handy reconstruction
technique which helps one to assess the particular form of the
lagrangian of a modified theory of gravity [
64
]. We note here
that, although the assumption of integrability of the scalar
field evolution equation is inspired only from a
mathematical perspective, solutions found by means of this
assumption are by no means unphysical. Used properly, this method
can potentially be useful for allied investigations as well, for
instance, a detailed definition of the possible bounds over the
choice of coupling function ξ (somewhat similar to a possible
Higgs-Kreschmann invariant coupling in white dwarfs and
neutron stars, as shown by Wegner and Onofrio [
79,80
]).
Further investigation under the setup of a
Scalar-EinsteinGauss-Bonnet gravity in a more generalized scenario
(inclusion of factors spatial inhomogeneity, pressure anisotropy,
heat flux etc) can be done using this method properly and
more rigorously and will be reported elsewhere in future.
Acknowledgements The author would like to thank Professor Sayan
Kar and Professor Narayan Banerjee for useful comments and
suggestions. The author was supported by the National Post-Doctoral
Fellowship (file number: PDF/2017/000750) from the Science and Engineering
Research Board (SERB), Government of India.
Open Access This article is distributed under the terms of the Creative
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