#### Scalar field collapse in Gauss–Bonnet gravity

Eur. Phys. J. C
Scalar field collapse in Gauss-Bonnet gravity
Narayan Banerjee 1
Tanmoy Paul 0
0 Department of Theoretical Physics, Indian Association for the Cultivation of Science , 2A and 2B Raja S.C. Mullick Road, Kolkata 700 032 , India
1 Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata , Mohanpur Campus, Nadia, West Bengal 741 246 , India
We consider a “scalar-Einstein-Gauss-Bonnet” theory in four dimension, where the scalar field couples nonminimally with the Gauss-Bonnet (GB) term. This coupling with the scalar field ensures the non-topological character of the GB term. In this scenario, we examine the possibility for collapsing of the scalar field. Our result reveals that such a collapse is possible in the presence of Gauss-Bonnet gravity for suitable choices of parametric regions. The singularity formed as a result of the collapse is found to be a curvature singularity which is hidden from the exterior by an apparent horizon.
1 Introduction
Over a few decades, relativistic astrophysics have gone
through extensive developments, following the discovery of
high energy phenomena in the universe such as gamma ray
bursts. Compact objects like neutron stars have interesting
physical properties, where the effect of strong gravity fields
and hence general relativity are seen to play a
fundamental role. A similar situation, involving a strong gravitational
field, is that of a massive star undergoing a continual
gravitational collapse at the end of its life cycle. This collapse
phenomenon, dominated by the force of gravity, is
fundamental in black hole physics and has received increasing
attention in the past decades. The first systematic analysis
of gravitational collapse in general relativity was given in
1939 by Oppenheimer and Snyder [1]. In this context, we
also refer to the work by Datt [2]. Later developments in the
study of gravitational collapse have been comprehensively
summarized by Joshi [3, 4].
Scalar fields have been of great interest in theories of
gravity for various reasons. A scalar field with a variety of
potential can fit superbly for cosmological requirements such as
playing the role of the driver of the past or the present
accelerated expansion of the universe. Apart from cosmological
aspect, a suitable scalar potential can often mimic various
matter distribution, including fluids with different equations
of state.
Collapsing models with scalar fields are also quite well
studied in general relativity. The collapse of a zero mass
scalar field was discussed by Christodoulou in [5].
Subsequently, the possibility of the formation of a naked
singularity as an end product of a scalar field collapse has also been
explored in [6]. Some variants of scalar field collapse and its
implications are studied in [7–17] (see also [18–23]).
It is well known that the Einstein–Hilbert action can be
generalized by adding higher order curvature terms which
naturally arise from diffeomorphism property of the action.
Such terms also have their origin in string theory from
quantum corrections. In this context F ( R) [24–28], Gauss–
Bonnet (GB) [24, 25, 29–33] or more generally Lanczos–
Lovelock [34–36] gravity are some of the candidates in
higher curvature gravitational theory. The spacetime
curvature inside a collapsing star gradually increases as the
collapse continues and becomes very large near the final state
of the collapse. Thus, for a collapsing geometry where the
curvature becomes very large near the final state of the
collapse, the higher curvature terms are expected to play a
crucial role. Motivated by this idea, the collapsing scenarios in
the presence of F ( R) gravity have been recently discussed
by Goswami et al. [37] and by Chakrabarti and Banerjee
[38, 39].
In the present context, we take the route of Gauss–Bonnet
gravity to investigate the role of the higher order curvature
in a scalar field collapse in four dimension. The advantage
of Gauss–Bonnet gravity is that the equations of motion do
not contain any higher derivative terms (higher than two) of
the metric, which lead to ghost free solution. The particular
questions that we address in this paper are the following:
1. Is the scalar field collapse possible in the presence of
Gauss–Bonnet gravity?
2. If such a collapsing scenario is found, what is then the
end product of the collapse, a black hole or a naked
singularity?
In order to address the above questions, we consider
a “scalar-Einstein–Gauss–Bonnet” theory [40] where the
scalar field is coupled non-minimally to the GB term. The
equations in a relativistic theory of gravity is already highly
nonlinear and the presence of the Gauss–Bonnet terms makes
the situation even more difficult. In order to make the
equations tractable, we start with a spatially homogeneous and
isotropic model, i.e., essentially a Friedmann model. Except
the scalar field, we do not consider any other matter such as
a fluid, but the collapse is homogeneous and thus somewhat
analogous to the Oppenheimer–Snyder collapse [1].
Our paper is organized as follows. In Sect. 2, we describe
the model. In Sect. 3, we obtain the exact solution for the
metric. Sects. 4 and 5 address the visibility of the singularity
produced as a result of the collapse and a matching of the
solution with an exterior spacetime, respectively. We end the
paper with some concluding remarks in Sect. 6.
2 The model
To explore the effect of Gauss–Bonnet gravity on scalar field
collapse, we consider a “scalar-Einstein–Gauss–Bonnet”
theory in four dimension where the Gauss–Bonnet (GB) term
is coupled with the scalar field. This coupling with the scalar
field ensures the non-topological character of the GB term.
The action for this model is given by
S =
d4x √−g 2Rκ2 − 21 gμν ∂μ ∂ν
where g is the determinant of the metric, R is the Ricci scalar,
1/(2κ2) = M 2p is the four dimensional squared Planck mass,
G = R2 − 4Rμν Rμν + Rμναβ Rμναβ is the GB term,
denotes the scalar field also endowed with a potential V ( ).
The coupling between scalar field and GB term is symbolized
by ξ( ) in the action. Variation of the action with respect to
metric and scalar field leads to the field equations:
1
κ2 [−Rμν
+ 21 gμν R] + 21 ∂μ ∂ν
− 41 gμν ∂ρ ∂ρ
1 gμν
+ 2
− V ( ) + ξ( )G
−2ξ( )R Rμν − 4ξ( )Rρμ Rνρ − 2ξ( )Rμρσ τ Rρνσ τ
+4ξ( )Rμρνσ Rρσ
+2[∇μ∇ν ξ( )]R − 2gμν [∇2ξ( )]R
and
¨ + 3H ˙ + V ( ) + 24ξ ( )(H 4 + H 2 H˙ ) = 0.
To derive these equations, we consider the scalar field to be
homogeneous in space. It is evident that due to the presence
of GB term, cubic and quartic powers of H (t ) appear in the
above equations.
It is well known that Einstein–Gauss–Bonnet gravity in four
dimensions reduces to standard Einstein gravity; the
additional terms actually cancel each other. In the present case,
the non-minimal coupling with the scalar field assists the
contribution from the GB term survive [40]. It is easy to see,
in all the field equations above, that a constant ξ (essentially
no coupling) would immediately make the GB contribution
trivial.
−4[∇ρ ∇μξ( )]Rνρ − 4[∇ρ ∇ν ξ( )]Rμρ
+4[∇2ξ( )]Rμν + 4gμν [∇ρ ∇σ ξ( )]Rρσ
+4[∇ρ ∇σ ξ( )]Rμρνσ = 0
and
gμν [∇μ∇ν ] − V ( ) − ξ ( )G = 0,
where a prime denotes the derivative with respect to . It
may be noticed that the gravitational equation of motion does
not contain any derivative of the metric components of order
higher than two.
The aim here is to construct a continual collapse model
and for this purpose, we consider the following spherically
symmetric non-static metric ansatz for the interior:
ds2 = −dt 2 + a2(t ) dr 2 + r 2dθ 2 + r 2 sin2 θ dϕ2 ,
(
4
)
where the factor a(t ) solely governs the interior spacetime
characterized by the coordinates t , r , θ and ϕ. For such 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˙ ]
with H = a˙ /a and a dot denotes the derivative with respect
to time (t ). Using the metric presented in Eq. (
4
), the field
equations can be simplified and turn out to be
−(3/κ2)H 2 + (1/2) ˙ 2 + V ( ) + 24H 3ξ˙ = 0,
1
κ2 2H˙ + 3H 2
+ (1/2) ˙ 2 − V ( ) − 8H 2ξ¨
−16H H˙ ξ˙ − 16H 3ξ˙ = 0,
(
2
)
(
3
)
(
5
)
(
6
)
(
7
)
3 Exact solutions: collapsing models
In this section, we present a possible analytic solution of the
field equations (Eqs. (
5
), (
6
) and (
7
)) and in order to do this,
we consider a string inspired model [40] with
V ( ) = V0e−√3κ
and
ξ( ) = ξ0e√3κ ,
where V0 and ξ0 are constants. With these forms of V ( )
and ξ( ), Eqs. (
5
), (
6
) and (
7
) turn out be
−(3/κ2)H 2 + (1/2) ˙ 2 + V0e−√3κ
+24√3κξ0 H 3e√3κ ˙ = 0,
1
κ2 2H˙ + 3H 2
+ (1/2) ˙ 2 − V0e−√3κ
−8√3κξ0 H 2e√3κ
−16√3κξ0 H H˙ e√3κ
¨ + √3κ ˙ 2
˙ − 16√3κξ0 H 3e√3κ
and
¨ + 3H ˙ −
+ 24√3κξ0e√3κ
√3κ V0e−√3κ
H 4 + H 2 H˙ = 0,
respectively. Here we are interested on the collapsing
solutions where the radius of the two sphere r a(t ) decreases
monotonically with time. Keeping this in mind, the above
three equations (Eqs. (
9
), (
10
), (
11
)) are solved for H (t ),
(t ) and the solutions are as follows:
p
H (t ) = − (t0 − t ) ,
and
2
3κ
1
κ
(t ) = √
ln
t0 − t ,
where t0 is a constant of integration and the constant p is
related to V0 and ξ0 through the following two relations:
p2 2
−3 κ2 + 3κ2 + V0κ2 + 48 p3ξ0/κ2 = 0,
2 p 2
− κ2 + 3κ2 + V0κ2 − 24 p3( p − 1)ξ0/κ2 = 0.
From the above relations, it can easily be shown that in the
absence of the coupling parameter ξ0 (i.e., if ξ0 = 0), p takes
the value as p = 23 . Therefore, this particular value of the
power exponent p (= 23 ) is not possible in the present
context where the spacetime geometry evolves in the presence
of “Gauss–Bonnet” term. However, by substituting V0 from
Using the solution of a(t ) (see Eq. (
18
)), the above
expression of K can be simplified to
K = 6 p2 ( p − 1)2 + p2
1
t0 − t
It is clear from Eq. (
20
) that the Kretschmann scalar
diverges at t → t0 and thus the collapsing sphere discussed
here ends up in a curvature singularity.
From Eq. (
18
), we obtain the plot (Fig. 1) between a(t ) and
t for two different values of p.
Figure 1 clearly demonstrates that the spherical body
collapses with time almost uniformly until t approaches a value
close to t0, where it hurries towards a zero proper volume
singularity. This qualitative behavior is almost not affected
Clearly p = 23 when ξ0 becomes zero. Therefore in order
to better understand the deviation of the present model from
the usual Einstein-scalar theory (i.e ξ0 = 0), we construct a
perturbative expansion of p in powers of ξ0 by using Eq. (
16
).
Up to the second order in ξ0, such an expansion of p takes
the following form:
Hence the presence of Gauss–Bonnet coupling takes the
parameter p away from the value 23 . Stronger the Gauss–
Bonnet coupling, more the deviation of p from 2/3.
Equation (
12
) immediately leads to the evolution of scale
factor a(t ):
a(t ) = a0(t0 − t ) p,
where a0 is an integration constant. The expression of a(t )
(see Eq. (
18
)) clearly reveals that r a(t ) decreases
monotonically with time for p > 0. Therefore, the volume of the sphere
of scalar field collapses with time and goes to zero at t → t0,
giving rise to a finite time zero proper volume singularity. It
is interesting to note that without the non-minimal coupling,
the solution reduces to the time-reversed standard spatially
flat Friedmann solution for a dust distribution (pressure = 0),
here the solution is contracting rather than expanding.
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 Kretschmann curvature (K )
scalar at t → t0. For the metric presented in Eq. (
4
), K has
the following expression:
K = 6 a¨ (t )2 a˙ (t )4
a(t )2 + a(t )4 .
˙ = 0,
(
10
)
(
11
)
(
8
)
(
9
)
(
12
)
(
13
)
(
14
)
(
15
)
(
16
)
(
17
)
(
18
)
(
19
)
(
20
)
2.0
1.5
by different choices of p, only the rates of the changes. We
have presented here two specific examples with p = 0.5 and
p = 0.2.
4 Visibility of the singularity
Whether the curvature singularity is visible to an exterior
observer depends on the formation of an apparent horizon.
The condition for such a surface is given by
gμν Z,μ Z,ν
rah,tah
= 0
where Z is the proper radius of the two sphere, given by
r a(t ) in the present case, rah and tah being the comoving
radial coordinate and time of formation of the apparent
horizon, respectively. Using the form of gμν presented in Eq. (
4
),
the above expression can be simplified and turns out to be
ra2h a˙ (tah )2 = 1
Due to the solution of a(t ), Eq. (
22
) takes the following
form:
t0 − tah
2 p−2
1
= (ra2h a02 p2)
.
The above expression clearly reveals that tah is less than t0
(i.e. tah < t0). Therefore the apparent horizon forms before
the formation of singularity. Thus, the curvature singularity
discussed here is always covered from an exterior observer
by the formation of an apparent horizon. It may be mentioned
that the singularity is independent of the radial coordinate r
and it is covered by a horizon. This result is consistent with
the result obtained by Joshi et al. [41] that unless one has a
central singularity, it cannot be a naked singularity.
5 Matching of the interior spacetime with an exterior
geometry
To complete the model, the interior spacetime geometry of
the collapsing sphere needs to be matched to an exterior
geometry. For the required matching, the Israel conditions
are used, where the metric coefficients and extrinsic
curvatures (first and second fundamental forms, respectively) are
matched at the boundary of the sphere. Following references
[17,42], we match the interior spacetime with a generalized
Vaidya exterior spacetime at the boundary hypersurface
given by r = r0. The metrics inside and outside of are
given by
ds−2 = −dt 2 + a2(t ) dr 2 + r 2dθ 2 + r 2 sin2 θ dϕ2
and
ds+2 = − 1 −
2M (rv, v)
rv
+rv2 sin2 θ dϕ2,
dv2 + 2dvdrv + rv2dθ 2
respectively, where rv, v, θ and ϕ are the exterior
coordinates and M (rv, v) is known as generalized mass function.
The same hypersurface can alternatively be defined by the
exterior coordinates as rv = R(t ) and v = T (t ). Then the
metrics on from the inside and outside coordinates turn
out to be
ds−2, = −dt 2 + a2(t )r02d 2
and
ds+2, = −
1 −
2M (t )
R(t )
T 2
˙ − 2T˙ R˙ dt 2 + R(t )2d 2
where M (t ) (= M (R(t ), T (t ))) is the generalized mass
function on , d 2 denotes the line element on a unit two
sphere and dot represents ddt . Matching the first fundamental
form on (i.e. ds−2, = ds+2, ) yields the following two
conditions:
dT (t )
dt
and
=
R(t ) = r0a(t )
= r0a0(t0 − t ) p
1
1 − 2MR(t()t) − 2 ddTR((tt))
In order to match the second fundamental form, we calculate
the normal of the hypersurface from inside (n− = nt−, nr−,
nθ , nϕ ) and outside (n+ = nv , nrv , nθ , nϕ ) coordinates as
− − + + + +
follows:
nt− = 0, nr− = a(t ), nθ− = nϕ− = 0,
(
24
)
(
25
)
(
26
)
(
27
)
(
28
)
and
nv+ =
nr+v =
,
The above expressions of n− and n+ lead to the
extrinsic curvature of from interior and exterior coordinates,
respectively, and they are given by
Kt−t = 0,
Kθ−θ = r0a(t ),
from the interior metric, and
Kt+t =
∂ M (t )
∂ R(t )
M (t )
− r0a(t ) − r02a(t )a¨ (t ),
Kϕ−ϕ = r0a(t ) sin2 θ ,
Kθ+θ = R(t )
Kϕ+ϕ = R(t ) sin θ 2
1 − 2MR(t()t) − ddTR((tt))
1 − 2MR(t()t) − 2 ddTR((tt))
,
1 − 2MR(t()t) − ddTR((tt)) ,
1 − 2MR(t()t) − 2 ddTR((tt))
from the exterior metric.
The equality of the extrinsic curvatures of from both
sides is therefore equivalent to the following two conditions:
r0a(t ) = R(t )
and
1 − 2MR(t()t) − ddTR((tt))
1 − 2MR(t()t) − 2 ddTR((tt))
∂ M (t )
∂ R(t )
M (t )
= r0a(t ) + r02a(t )a¨ (t )
By using Eqs. (
26
), (
27
) and (
32
), the mass function on
(i.e. M (t )) can be determined and is given by
1
M (t ) = 2 R(t )R˙ 2(t )
∝ (t0 − t )3 p−2
∝ ρa3(t )
where ρ (= 21 ˙ 2 + V ( )) is energy density of the scalar
field . Moreover, with Eqs. (
26
) and (
34
), one finally ends
with the following expression:
d T (t )
dt
1
= 1 + r0a0 p(t0 − t ) p−1 .
Equations (
27
), (
33
), (
34
) and (
35
) completely specify the
matching at the boundary of the collapsing scalar field with
an exterior generalized Vaidya geometry. However, all the
matching conditions are not independent, but Eq. (
33
) can be
derived from the other three conditions.
(
29
)
(
30
)
(
31
)
(
32
)
(
33
)
(
34
)
(
35
)
One important point, to be noted from Eq. (
23
), is that the
horizon radius rah is independent of r0, which is the radial
coordinate defining the separation between the interior and
the exterior of the collapsing object. It also deserves mention
that a horizon, if it forms, is outside the private content of the
collapsing star, and the actual boundary of the matter content
of the star is completely obscure for an observer outside the
horizon.
At this stage, it deserves mention that the interior
spacetime is matched with exterior “Schwarzschild geometry” for
the only possible value of the parameter ’ p’ i.e. p = 23 .
But as discussed earlier, for this particular value of p the
Gauss–Bonnet coupling parameter (ξ0) goes to zero, which
in turn annihilates the GB term from the action. Thus in
the present context, the presence of Gauss–Bonnet gravity
spoils the matching of interior geometry of the collapsing
cloud with an exterior Schwarzschild geometry. This is quite
consistent because the presence of Gauss–Bonnet term
generates an effective energy momentum tensor which cannot
be zero (since it arises effectively from spacetime
curvature) at the outside of and hence no exterior vacuum like
Schwarzschild solution is matched with the interior
spacetime metric in the presence of the Gauss–Bonnet term.
6 Conclusion
We consider a “scalar-Einstein–Gauss–Bonnet” theory in
four dimensions where the scalar field couples non-minimally
with the Gauss–Bonnet term. This coupling with the scalar
field ensures the non topological character of the GB term.
In this scenario, we examine the possibility of collapse of the
scalar field by considering a spherically symmetric spacetime
metric.
With the aforementioned metric, an exact solution is
obtained for the spacetime geometry, which clearly reveals
that the radius of a two sphere decreases monotonically with
time if the parameter p is taken to be greater than zero. This
parameter p is actually determined by the strength of the
coupling of the scalar field to the GB term and a parameter
(V0) by the relations (
14
) and (
15
).
From the behaviour of Kretschmann scalar, it is found that
the singularity formed as a result of the collapse is a finite
time curvature singularity. Moreover, the scalar field energy
density also seems to be divergent at the singularity. The
formation of apparent horizon is investigated and it turns out
that the apparent horizon forms before the formation of
singularity. Therefore the curvature singularity discussed here
is hidden from exterior by an apparent horizon.
Finally, we match the interior spacetime geometry of the
collapsing sphere with generalized Vaidya exterior
geometry at the boundary of the cloud ( ). For this matching, the
Israel junction conditions are used where the metric
coefficients and extrinsic curvatures are matched on . We
determine the matching conditions given in Eqs. (
27
), (
33
), (
34
)
and (
35
). We also investigate whether the interior geometry
can be matched with a Schwarzschild exterior geometry or
not. It is found that the interior spacetime is matched with
the exterior Schwarzschild for the only possible value of ‘ p’
as p = 23 . But for this particular value of p (= 23 ), the
Gauss–Bonnet coupling parameter (ξ0) goes to zero which
in turn vanishes the GB term from the action. Thus in the
present context, the presence of Gauss–Bonnet gravity spoils
the matching of interior geometry of the collapsing cloud
with an exterior Schwarzschild geometry. This result is in
fact quite consistent because the presence of Gauss–Bonnet
term generates an effective energy momentum tensor which
cannot be zero (since it arises effectively from spacetime
curvature) at the outside of and hence no exterior vacuum like
Schwarzschild solution is matched with the interior
spacetime metric in the presence of Gauss–Bonnet term.
Another important point to be mentioned that the
singularity formed is not a central singularity, it is formed at any value
of r within the distribution. Such a singularity in general
relativity is always covered by a horizon [41]. It is interesting to
note that the result obtained in the present work in the
presence of Gauss–Bonnet term is completely consistent with the
corresponding GR result.
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