#### Gravitational perfect fluid collapse in Gauss–Bonnet gravity

Eur. Phys. J. C
Gravitational perfect fluid collapse in Gauss-Bonnet gravity
G. Abbas 0
M. Tahir 0
0 Department of Mathematics, The Islamia University of Bahawalpur , Bahawalpur 63100 , Pakistan
The Einstein Gauss-Bonnet theory of gravity is the low-energy limit of heterotic super-symmetric string theory. This paper deals with gravitational collapse of a perfect fluid in Einstein-Gauss-Bonnet gravity by considering the Lemaitre-Tolman-Bondi metric. For this purpose, the closed form of the exact solution of the equations of motion has been determined by using the conservation of the stress-energy tensor and the condition of marginally bound shells. It has been investigated that the presence of a Gauss-Bonnet coupling term α > 0 and the pressure of the fluid modifies the structure and time formation of singularity. In this analysis a singularity forms earlier than a horizon, so the end state of the collapse is a naked singularity depending on the initial data. But this singularity is weak and timelike, which goes against the investigation of general relativity.
1 Introduction
In recent years great interest has arisen in the study of higher
order curvature theories of gravity [1–4]. Among all these
higher curvature theories of gravity, the most widely studied
theory of gravity is known as Einstein–Gauss–Bonnet (EGB)
gravity. The Lagrangian of EGB gravity is the particular form
of Lagrangian of Lovelock theory [5]. In a 4-dimensional
vacuum theory of gravity, the most general Lovelock theory
is a combination of the zeroth and first Euler density; in other
words, general relativity (GR) with a cosmological constant.
Also, it has been observed that the second Euler density is
known as the Gauss–Bonnet combination, which is a
topological invariant in 4 dimensions and does not contribute to the
dynamical equations of the motion if included in the action
[6]. Further, a simple and natural way to make a GB
combination dynamical in 4-dimensional theory is to couple it to
a dynamical scalar field. The perturbation method has been
used to see the effects of the GB coupling term α > 0 on the
dynamical instability of non-static anisotropic fluid spheres
[7]. The thermal aspects of a gravitating source in 5 D EGB
theory of gravity have been explored in detail by coupling
the heat transport equation with the dynamical equations [8].
Boulware and Deser [4] have formulated the static black hole
exact solutions in greater than or equal to 5-dimensional
theories of gravity with a 4-dimensional GB term modifying the
usual Einstein–Hilbert action [9].
Several LTB-like solutions Einstein field equations have
been explored in higher order theories of gravity, which
include the higher derivative curvature terms in their action.
The higher curvature correction to the action has a great effect
on the topological structure of the singularity appearing
during the gravitational collapse of massive star [10]. In EGB
theory of gravity with α > 0, there exists a massive, naked
and un-central singularity only in 5D, while such a
singularity is disallowed in D ≥ 6 [11]. It has been the subject
of great interest for many theoretical physicists to explore
the slowly rotating BH solutions in Gauss–Bonnet theory of
gravity. However, due to the nonlinearity of the field
equations in GB gravity, it is very hard to obtain the exact
analytic rotating black hole solutions in the framework of this
theory. Therefore by introducing a small angular momentum
as a perturbation into a rotating system, some BH solutions
have been investigated in the past [12, 13]. Also, such slowly
rotating BHs solutions exist in the EGB theory with an AdS
term [14]. Using the linear order of the rotation parameter a,
the slowly rotating charged/uncharged BHs solutions in AdS
third order Lovelock gravity have been investigated in [15],
which have some interesting physical features.
During the most recent decade AdS BHs and especially
their thermodynamics have attracted the attention of many
researchers due to AdS/CFT duality. In the AdS EGB gravity,
the thermodynamical relations such temperature and entropy
of the charged BHs get no corrections from the rotation
parameter a [16]. Zou et al. [17] have explored the
thermodynamics of GB–Born–Infeld BHs in AdS space. Also, they
have calculated the mass, temperature, entropy and heat of
the resulting BHs. In third order Lovelock AdS gravity, the
thermodynamic and conserved quantities of BHs with flat
horizons are independent of the Lovelock coefficients [18].
Such interests in higher order theories of gravity have
motivated us to study the gravitational collapse of a perfect fluid
in EGB with the LTB model.
The stars are composed of some nuclear matter and
gravitational collapse is the phenomenon in which the stars are
continuously gravitating and attracted towards their centers
due to the gravitational interaction of its particles.
According to the singularity theorem [19], space-time singularities
are generated during the continual gravitational collapse of
massive stars. During the recent decades, it has been an
interesting problem in astronomy and astrophysics to determine
the final fate of gravitational collapse. In this connection,
Oppenheimer and Snyder [20] were the pioneers who found
the black hole as the end state of the dust collapse by using
the static Schwarzschild space-time as an exterior space-time
and a Friedmann like solution as an interior spacetime. Later
on this work was extended with a positive and negative
cosmological constant [21,22]. Several authors [23–33] have
discussed the phenomena of gravitational collapse using the
dissipative and viscous fluid in general relativity. The lack
of analytical consequences has lead to unproven conjectures
namely, the cosmic censorship conjecture (CCC) [34], the
hoop conjecture (HC) [35], and Seifert’s conjecture [36].
Oppenheimer and Snyder [20] considered a homogeneous
spherical star with zero rotation and vanishing internal
pressure under these ideal conditions, the cloud collapses
simultaneously to a space-time singularity, which is enclosed by
an event horizon. Further, it is interesting to study the
gravitational collapse of stars with some realistic matter and
geometry.
Recently, Jhingan and Ghosh [37], have studied the dust
spherical collapse in 5 dimensions with GB term. They found
the exact solutions in closed form with the marginally bound
conditions. In this paper, we extended the work of Jhingan
and Ghosh [37], to the case of a perfect fluid with marginally
bound conditions. The paper has been arranged as follows:
in Sect. 2, we present the exact solution of the field equation.
Section 3 deals with a singularity analysis. In the last section,
we summarize the results of the paper.
2 Exact solution of field equations in Einstein
Gauss–Bonnet gravity
Here, the required 5D gravitational action is
S =
d5x √−g
(
2
)
(
3
)
(
4
)
(
5
)
(
6
)
(
7
)
(
8
)
where R and κ5 ≡ √8π G5, α are the Ricci scalar, the
gravitational constant in 5D, and the Gauss–Bonnet coupling
constant, respectively. In this case the Gauss–Bonnet Lagrangian
is
LGB = R2 − 4Rμν Rμν + Rμνγ λ Rμνγ λ.
The above action appears as the low-energy limit of heterotic
super-string theory [4] and α is the inverse string tension,
which is usually taken as positive finite, so we restrict
ourselves to the case when α ≥ 0. The variation of the action
(
1
) yields the following form of the equations of motion in
Einstein Gauss–Bonnet gravity:
Gμν + α Hμν = Tμν ,
where Gμν = Rμν − 21 gμν R is the Einstein tensor and
Hμν = 2[R Rμν − 2 Rμα Rνα − 2 Rαβ Rμανβ + Rαβγ Rναβγ ]
μ
1
− 2 gμν LGB
is the Lanczos tensor. The stress-energy tensor for a perfect
fluid is
Tμν = ρ(r, t ) + p(r, t ) VμVν + pgμν ,
where Vμ = δμt is the 5D velocity, ρ(r, t ) is the energy density
and p(r, t ) is the isotropic pressure due to the fluid
distribution in the interior region of a star. The LTB metric with
co-moving coordinates in the 5D case [37–40] is
ds2 = −dt 2 + B2dr 2 + C 2d 23,
where B = B(r, t ) and C = C (r, t ), and d 23 = (dθ 2 +
sin2 θ (dφ2 + sin2 φdψ 2)).
The set of independent field equations in Einstein–Gauss–
Bonnet gravity for the metric (
6
) and stress-energy tensor (
5
)
are
C B + B2C˙ B˙ − BC
α
12(C 2 − B2(1 + C˙ 2))
C 3 B5
3
− B3C 2
B3(1 + C˙ 2)
C 2
+3 B2C 2 −
4α
B4C 2
+B2CC˙ B˙ + CC B − B(CC + C 2) = −ρ ,
−12α
C13 − BC2C2 3 + CC˙ 23
C
¨
3 1 + C˙ 2 + CC¨
C 2
= p,
(
1
)
− 2B B C + B2 B˙ C˙ − BC
C
¨
+B C 2 − B2(1 + C˙ 2) B¨ + 2 B˙ C − BC˙
1
− B3C 2
+2CC B − 2B CC + C 2
= p,
B3 1 + C˙ 2 + 2CC¨
+ B2C 2C˙ B˙ + C B¨
12α
B5C 3
B˙ C − BC˙
B2 1 + C˙ 2
− C 2
− 3
BC˙ − B˙ C
B3C
= 0,
where a dot = ∂t and a prime = ∂r . After some simplification
Eq. (
10
) yields two families of solutions in the following
form:
B(t, r ) =
B(t, r ) = ±
C
Z
,
2√αC
C 2 + 4α(C˙ 2 + 1) 1/2 ,
where Z = Z (r ) is a function of integration. The solution
for B(r, t ) in Eq. (
10
) is similar to 5D-LTB solution [38–40],
while the solution in Eq. (
11
) is trivial for α → 0, hence we
take the non-trivial form of B(r, t ) given in Eq. (
10
). Now
Eq. (
8
) with Eq. (
10
) gives
C
¨ =
C
C˙ 2 − (Z 2 − 1) + C32 p
4α(Z 2 − 1 − C˙ 2) − C 2
.
Now we have to solve the above equation analytically for
C (r, t ), this requires that we have to integrate the above twice
with respect to t . Since this equation involves the unknown
function p(r, t ), when we try to integrate Eq. (
12
), we cannot
get the exact solution because there is the unknown function
p(r, t ); to get rid of it, we try to make it at least independent
of t . To this end, we apply the conservation of the
energymomentum tensor which makes it independent of r . But our
purpose is to make it independent of t , and for this purpose we
take p(t ) as a polynomial in t , then after some simplification,
we make it constant (as in [41]). This whole procedure is
explained as follows.
The conservation of the energy-momentum tensor gives
∂ p
∂r
= 0,
⇒ p = p(t ).
p(t ) = p0
t
T
−c
,
We consider p as a polynomial in t as given by [41]
(
9
)
(
10
)
(
11
)
(
12
)
(
13
)
(
14
)
(
15
)
(
16
)
(
17
)
(
18
)
(
19
)
(
20
)
(
21
)
where T is the constant time introduced in the problem due
to physical reasons by re-scaling of t ; p0 and c are positive
constants. Further, for the integration of Eq. (
12
), we take
c = 0 in Eq. (
14
), so that
There are many other choices for p(t ), so Eq. (
14
) is not
always a unique choice for p(t). For example it can be treated
with c = 0, but such choices do not provide the results
in closed form which reduce to a 5D dimensional perfect
fluid collapse in the limit α → 0 [33,39,40], and we do not
recover the results of 5D dust collapse in Einstein–Gauss–
Bonnet gravity as p → 0 [37]. That is why we have taken the
pressure as constant which is a better choice in the present
situation. Further, some other choice of p(t ), as defined in
Refs. [41,42], may produce interesting numerical results but
such solutions may be considered explicitly in a future
investigation by taking an anisotropic fluid.
Using the above equation in Eq. (
12
) and intergrading, one
get
C 2 1 − 4α
˙
Z 2 − 1
C 2
= (Z 2 − 1) + Cζ2 − 61 C 2 p0
C 4
−2α C˙ 2 .
where ζ = ζ (r ), is integration function and assumed as a mass
function. Equation (
16
) is the main equation of the system.
By using Eqs. (
10
) and (
16
) into Eq. (
7
), we get
ζ = 2 C 3C (ρ + p0).
3
The integration of the above equation yields
Here, we have used ζ (0) = 0. The validity of the energy
conditions is necessarily a requirement for a physically
reasonable energy-momentum tensor. The energy conditions for
a perfect fluid in this case are the following:
N E C : ρ + p0 ≥ 0,
W E C : ρ ≥ 0, ρ + p0 ≥ 0,
S E C : ρ + p0 ≥ 0, ρ + 3 p0 ≥ 0.
From Eq. (
17
), we see that, for ζ (r ) > 0, C > 0 and
C > 0, one gets ρ + p0 > 0 and the null energy condition
is valid as shown in Fig. 1. Using Eq. (
11
), the marginally
bound condition Z = 1, C˙ < 0, C˙ < 0 along with the above
mentioned conditions in the field equations (
8
) and (
10
), we
get ρ ≥ 0 as shown in Fig. 2. Also, the field equations (
8
),
(
10
) and (
15
) along with the above mentioned conditions
yield ρ + 3 p0 ≥ 0 as shown in Fig. 3. Hence null, weak and
corresponding to ∓ sign in Eq. (
23
), there are two families of
solutions. In the limit α → 0 for the minus solution, we get
a 5D − L T B solution [40]; however, no results are available
in the literature if one applies the limit α → 0 to the plus
branch solution; yet such solutions are very interesting as
these provide the contribution of pure Gauss–Bonnet terms,
here we discuss both these solutions and their singularity
structure explicitly.
2.1 Minus branch solution
Maeda [43] found the LTB models near the region r ∼ 0, in
the framework of EGB without finding an explicit solution.
Also, Jhingan and Ghosh [37] explored 5D − L T B − E G B
gravitational collapse with dust matter as the source of
gravity. In this work, we determine the exact solution of the
5D − L T B model in EGB gravity with a perfect fluid in
closed form to see the effect of pressure on the final fate of
gravitational collapse. Hence, we consider the minus branch
solution with the marginally bound case, Z = 1. Integrating
Eq. (
23
), we have
√α
tς (r ) − t = √ tan−1
2 2
3C 2(1− 43 αp0)−
C 4(1− 43 αp0)+8αζ
2√2C (1 − 43 αp0)[ C 4(1 − 43 αp0)+8αζ − C 2]1/2 ⎦
× ⎣
⎡
+
αC 2(1 − 43 αp0)2
C 4(1 − 43 αp0) + 8αζ − C 2
(
22
)
(
23
)
⎤
(
24
)
where tς (r ) is a function of integration, which is related to
the time of formation of the singularity. Without loss of
generality, we consider t = 0, r coincides with the area radius,
C (0, r ) = r.
The above two equations lead to
√α
tς = √ tan−1
2 2
× ⎣
⎡
+
3(1 − 43 αp0) −
1 − 43 αp0 + 8αζ˜
⎤
,
where ζ˜ = ζ /r 4. It is the time of formation of a singularity
which is affected by pressure and α.
The Kretschmann scalar K = Rμνγ λ Rμνγ λ, given (
6
) with
Eq. (
10
), takes the following form:
K = 12 CC¨ 22 + 12 CC˙44 + 4 CC¨ 2
2
+ 12
C˙2C˙ 2
C 2C 2
.
It is finite on the initial data surface. From the field equations
the energy density is
ρ(t, r ) =
3ζ
2C 3C
− p0.
Hence, it is clear that if ζ is finite and strictly positive in
the entire domain, then ρ → ∞ when C = 0 and C = 0.
The shell crossing and shell focusing singularities occur for
C = 0 and C = 0, respectively [37]. Also, the Kretschmann
scalar diverges at t = tc(r ), this implies the existence of a
curvature singularity [19]. Using Eq. (
24
), the shell focusing
singularity curve is
π √α
tc(r ) = tς (r ) + √ . (
29
)
4 2
The trapped surfaces are surfaces whose outward normals
are null,
C 2
gμν C,μC,ν = −C˙2 + B2 = 0.
Now Eqs. (
16
) and (
30
) yield the horizon radius
1
C (tAH (r ), r ) = √ p0 6 p0(ζ − 2α) + 9 − 3.
In the above equation, the location of apparent horizons is
affected by α and p0. Simplifying Eqs. (
24
) and (
29
), we get
the horizon curve
tc(r) − t =
π√α
4√2 +
αC2(1 − 43 αp0)2
C4(1 − 43 αp0) + 8αζ − C2
√α ⎡
+ √ tan−1
2 2
3C2(1− 43 αp0)− C4(1− 43 αp0)+8αζ
⎣ 2√2C(1− 43 αp0)[ C4(1 − 43 αp0)+8αζ − C2]1/2 ⎦ .
⎤
(
26
)
(
27
)
(
28
)
(
30
)
(
31
)
(
25
)
This implies that horizons form after the formation of
a singularity, hence the singularity is uncovered due to the
where Q = √6 p0ζ − 12αp0 + 9.
It is to be noted that, for α > 0, the time difference between
the formation of central singularity and apparent horizon is
affected by the pressure. We would like to mention that all
the results reduced to the dust case [37], when p0 = 0.
2.2 Plus branch solution
The plus branch solution of Eq. (
23
) when subjected to the
marginally bound condition is given by
1 √α 4
tc(r ) − t (r ) = log 8αζ − √ log C 2(1 − 3 αp0)
2 2 2
(
33
)
− C 4(1 − 43 αp0) + 8αζ + √
4
2C (1 − 3 αp0)
× ⎝
⎛
+
4
C 4(1 − 3 αp0) + 8αζ + C 2⎠
αC 2(1 − 43 αp)2
C 4(1 − 43 αp) + 8αζ − C 2
.
⎞
(
34
)
After applying the same procedure as in the previous case,
we get
1
tc(r ) − tAH = 2
√α
log 8αζ + √ log
2 2
1
p0
(Q − 3)
4 4
× 1 − 3 αp0 + (Q − 3)2 1 − 3 αp0 + 8αζ
4
+ 2(Q − 3) 1 − 3 αp0
×
+
4
(Q − 3)4 1 − 3 αp0 + 8αζ + (Q − 3)
α(Q − 3) 1 − 43 αp0 2
(Q − 3)2 1 − 43 αp0 + 8αζ + (Q − 3)
.
(
35
)
absence of event horizons, and the end state of gravitational
collapse is a naked singularity.
3 Summary and conclusion
The Einstein–Gauss–Bonnet theory of gravity is the
lowenergy limit of supersymmetric string theory [44]. Here, we
have investigated the exact solution of the field equations
in the frame work of EGB theory with LTB model which
enclosed the inhomogeneous perfect fluid in 5 D. In order
to do so, the marginally bound condition has been imposed
on the dynamical equation. The conservation of the
energymomentum tensor implies that ∂ p = 0, which produces the
∂r
result p = p(t ). Further, it has been taken as a constant, p0, by
using some physical assumptions as given by Eq. (
14
). The
procedure along with marginally bound condition enables
us to integrate the differential equation Eq. (
8
) analytically
in closed form. It has been found that the resulting solution
implies the spherical inhomogeneous prefect fluid
gravitational collapse. The coupling constant α has direct effect on
the resulting singularity and the end state of gravitational
collapse is reversed. For α > 0, there exists a naked
singularity. The time formation of singularity and horizons is
deeply affected by the pressure term. The presence of
pressure also reduces the total matter density of the gravitating
system.
The position of the apparent horizon in the space-time is
affected by the factor 2α and the pressure p0. Due to the
presence of second order curvature corrections in EGB
gravity the out product collapse is a massive naked singularity,
which is not admissible in 5D − L T B. The prediction as
regards the regular initial data has been made that it results
in the formation of a massive naked singularity, and that is in
contradiction to CCC. The singularity in this case is weaker
as compared to the corresponding 5 D − L T B; therefore the
existence of a naked singularity in the present case is not a
serious contradiction of CCC. According to the Seifert
conjecture [9] when a strictly positive finite amount of matter
undergoes gravitational contraction, then one point is always
hidden, which is a counterexample to the Seifert conjecture.
It has been investigated that singularities are formed rather
than horizons in the marginally bound case (Z (r ) = 1), hence
no BH is formed, and hence they must violate HC. In other
words the present analysis is a counterexample to all three
conjectures. But this analysis does not provide serious threats
to CCC.
Acknowledgements We appreciate the financial support from HEC,
Islamabad, Pakistan under NRPU project with Grant numbers
204059/NRPU/R and D/HEC/14/1217. Also, we appreciate the
constructive comments and suggestions of an anonymous referee.
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1. D.J. Gross , J. Harvey , E. Martinec, R. Rohm , Phys. Rev. Lett. 6 , 502 ( 1985 )
2. P. Candelas , G.T. Horowitz , A. Strominger , E. Witten, Santa Barbara preprint NSF-ITP-84-170
3. Bruno Zumino , Phys. Rev. Lett . 137 , 109 ( 1986 )
4. D.G. Boulware , S. Deser , Phy. Rev. Lett . 55 , 2656 ( 1985 )
5. D. Lovelock , J. Math. Phys. 12 , 498 ( 1971 )
6. V.P.C. Pedro et al., Phys. Rev. Lett. B768 , 373 ( 2017 )
7. G. Abbas , S. Sawar , Astro. Phys. Space Sci . 357 , 23 ( 2015 )
8. G. Abbas , M. Zubair , Mod. Phys. Lett. A 30 , 1550038 ( 2015 )
9. N. Dadhich , A. Molina , A. Khugaev , Phys. Rev. D 81 , 104026 ( 2010 )
10. K. Zhou , Z.-Y. Yang , D.-C. Zou, R.-H. Yue , Int. J. Mod. Phys. D 22 , 2317 ( 2011 )
11. K. Zhou , Z.-Y. Yang , D.-C. Zou, R.-H. Yue , Mod. Phys. Lett. A 26 , 2135 ( 2011 )
12. R.-H. Yue , D.-C. Zou, T.- Y. Yu , Z.-Y. Yang , Chin. Phys. B 20 , 050401 ( 2011 )
13. K. Zhou , Z.-Y. Yang , D.-C. Zou, R.-H. Yue , Chin. Phys. B 21 , 020401 ( 2012 )
14. D.-C. Zou , Z.-Y. Yang , R.-H. Yue , T.-Y. Yu, Chin. Phy. B 20 , 100403 ( 2011 )
15. R.-H. Yue , D.-C. Zou, T.-Y. Yu, P. Li , Z.-Y. Yang , Gen. Relativ. Gravit. 43 , 2103 ( 2011 )
16. Zou De-Cheng, Z.-Y. Yang , R.-H. Yue , Chin. Phys. Lett . 28 , 020402 ( 2011 )
17. De-Cheng Zou , Zhan-Ying Yang , Rui-Hong Yue , P. Li , Mod. Phys. Lett. A 26 , 515 ( 2011 )
18. D.-C. Zou , R.-H. Yue , Z.-Y. Yang , Commun. Theor. Phys . 55 , 499 ( 2011 )
19. S.W. Hawking , G.F.R. Ellis , The Large Scale Structure of Space time (Cambridge University Press, Cambridge, 1979 )
20. J.R. Oppenheimer , H. Snyder, Phys. Rev . 56 , 455 ( 1939 )
21. D. Markovic , S.L. Shapiro , Phy. Rev. D 61 , 084029 ( 2000 )
22. K. Lake, Phys. Rev. D 62 , 027301 ( 2000 )
23. L. Herrera , N.O. Santos , Phys. Rep . 286 , 53 ( 1997 )
24. L. Herrera , A. Di Prisco , J.R. Hernandez , N.O. Santos , Phys. Lett. A 237 , 113 ( 1998 )
25. G. Abbas, Sci. China. Phys. Mech. Astro . 57 , 604 ( 2014 )
26. S.M. Shah , G. Abbas, Eur. Phys. J. C 77 , 251 ( 2017 )
27. G. Abbas, M. Ramzan , Chin. Phys. Lett . 30 , 100403 ( 2013 )
28. G. Abbas, Astrophys. Space Sci . 350 , 307 ( 2014 )
29. G. Abbas, Adv. High Energy Phys . 2014 , 306256 ( 2014 )
30. G. Abbas, Astrophys. Space Sci . 352 , 955 ( 2014 )
31. G. Abbas, U. Sabiullah, Astrophys. Space Sci . 352 , 769 ( 2014 )
32. M. Sharif , Zahid Ahamd, Mod. Phys. Lett. A 22 , 2947 ( 2007 )
33. M. Sharif , J. Zahid Ahamd , Korean Phys. Soc . 52 , 980 ( 2008 )
34. R. Penrose , Riv. Nuovo Cimento 1 , 252 ( 1969 )
35. K.S. Thorne, in Magic Without Magic, ed. by J.R. Klander (Freedman, San Francisco, 1972 )
36. H.J. Seifert , Gen. Relativ. Gravit. 10 , 1065 ( 1979 )
37. S. Jhingan , S.G. Ghosh , Phys. Rev. D 81 , 024010 ( 2010 )
38. A. Banerjee , A. Sil , S. Chatterjee , Gen. Relativ. Gravit. 26 , 999 ( 1994 )
39. S.G. Ghosh , A. Banerjee , Int. J. Mod. Phys. D 12 , 639 ( 2003 )
40. S.G. Ghosh , D.W. Deshkar , N.N. Saste , Int. J. Mod. Phys. D 16 , 53 ( 2007 )
41. S. Chakraborty , S. Chakraborty , D. Debnath , Int. J. Mod. Phys. D 14 , 1707 ( 2005 )
42. S. Nath , U. Debnath , S. Chakraborty , Astrophys. Space Sci . 313 , 431 ( 2008 )
43. H. Maeda , M. Nozawa , Phys. Rev. D 77 , 064031 ( 2008 )
44. J.J. Schwarz , Nucl. Phys. B 226 , 269 ( 1983 )