Lower dimensional gravity has the potential of providing non-trivial and
valuable insight into some of the conceptual issues arising in four dimensional
relativistic gravitational analysis. The asymptotically anti-de Sitter (2+1)
dimensional spacetime described by Ba$\tilde{n}$ados, Teitelboim and Zanelli
(BTZ) which admits a black hole solution, has become a source of fascination
for theoretical physicists in recent years. By suitably choosing the form of
the mass function $m(r)$, we derive a new class of solutions for the interior
of an isotropic star corresponding to the exterior (2+1) asymptotically anti-de
Sitter BTZ spacetime. The solution obtained satisfies all the regularity
conditions and its simple analytical form helps us to study the physical
parameters of the configuration in a simple manner.Comment: 5 pages, 5 figures. Accepted for publication in Phys. Lett. B, some
  typos are correcte

Karar, Indrani

Rahaman, Farook

Sharma, Ranjan

English

arXiv

AbstractLower-dimensional gravity has the potential of providing non-trivial and valuable insight into some of the conceptual issues arising in four-dimensional relativistic gravitational analysis. The asymptotically anti-de Sitter (2+1)-dimensional spacetime described by Bañados, Teitelboim and Zanelli (BTZ) which admits a black hole solution, has become a source of fascination for theoretical physicists in recent years. By suitably choosing the form of the mass function m(r), we derive a new class of solutions for the interior of an isotropic star corresponding to the exterior (2+1) asymptotically anti-de Sitter BTZ spacetime. The solution obtained satisfies all the regularity conditions and its simple analytical form helps us to study the physical parameters of the configuration in a simple manner

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A class of interior solutions corresponding to a (2+1)-dimensional asymptotically anti-de Sitter spacetime 

Physics Letters B 704 (2011) 1–4Contents lists available at SciVerse ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
A class of interior solutions corresponding to a (2+ 1)-dimensional asymptotically
anti-de Sitter spacetime
Ranjan Sharma a, Farook Rahaman b,∗, Indrani Karar c
a Department of Physics, P. D. Women’s College, Jalpaiguri 735101, India
b Department of Mathematics, Jadavpur University, Kolkata 700 032, West Bengal, India
c Department of Mathematics, Saroj Mohan Institute of Technology, Guptipara, West Bengal, India
a r t i c l e i n f o a b s t r a c t
Article history:
Received 23 July 2011
Received in revised form 19 August 2011
Accepted 19 August 2011
Available online 31 August 2011
Editor: M. Cveticˇ
Keywords:
Einstein’s ﬁeld equations
Stellar equilibrium
Lower-dimensional gravity has the potential of providing non-trivial and valuable insight into some of the
conceptual issues arising in four-dimensional relativistic gravitational analysis. The asymptotically anti-de
Sitter (2 + 1)-dimensional spacetime described by Bañados, Teitelboim and Zanelli (BTZ) which admits
a black hole solution, has become a source of fascination for theoretical physicists in recent years. By
suitably choosing the form of the mass function m(r), we derive a new class of solutions for the interior
of an isotropic star corresponding to the exterior (2+ 1) asymptotically anti-de Sitter BTZ spacetime. The
solution obtained satisﬁes all the regularity conditions and its simple analytical form helps us to study
the physical parameters of the conﬁguration in a simple manner.
© 2011 Elsevier B.V. All rights reserved.1. Introduction
It is well known that lower-dimensional gravity has the po-
tential of providing non-trivial and valuable insight into some of
the conceptual issues arising in four-dimensional relativistic grav-
itational analyses. For example, the (2 + 1)-dimensional space-
time geometry described by Bañados, Teitelboim and Zanelli [1]
(henceforth BTZ), which is asymptotically anti-de Sitter and admits
a black hole solution, has become a source of fascination for theo-
retical physicists in recent years. The advantage here is that though
the system mimics 4-dimensional analysis, it offers less intricate
set of equations to deal with. It is, therefore, crucial to ﬁnd a phys-
ically reasonable interior solution corresponding to the exterior
BTZ spacetime which may give us deep understanding about the
nature of a gravitationally collapsing body and its consequences.
Mann and Ross [2] analyzed the collapse of a (2 + 1)-dimensional
star ﬁlled with dust (p = 0) corresponding to the exterior BTZ
spacetime and showed under what condition it might collapse to
a black hole. For an incompressible ﬂuid (ρ = constant), the inte-
rior solution obtained by Cruz and Zanelli [3] puts a bound on the
maximum allowed mass for such an object. The study also claims
that the collapsed stage would always be covered under its event
horizon. Cruz et al. [4] presented a new solution corresponding
to an exterior BTZ spacetime with the choice ρ = ρc(1 − r2/R2),
* Corresponding author.
E-mail addresses: rsharma@iucaa.ernet.in (R. Sharma), rahaman@iucaa.ernet.in
(F. Rahaman), indrani.karar08@gmail.com (I. Karar).0370-2693/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.physletb.2011.08.054where, ρc is the central density, ρ is the density which is a func-
tion of the radial parameter r, and R is the boundary of the star.
Another solution has been reported by Paulo M. Sá [5] where
a polytropic equation of state (EOS) of the form p = Kρ1+1/n was
assumed, where n is the polytropic index and K is the polytropic
constant.
In this Letter, we report a new class of solutions correspond-
ing to the BTZ exterior spacetime which has been obtained by
assuming a particular form of the mass function m(r). The solution
obtained has been found to satisfy all the regularity conditions and
its simple analytical form helps us to study the physical parame-
ters of the conﬁguration in a simple manner.
2. Interior spacetime
The metric for the interior of a static spherically symmetric dis-
tribution of matter in (2+ 1) dimensions has the standard form
ds2 = −e2γ (r) dt2 + e2μ(r) dr2 + r2 dθ2. (1)
We assume that the energy–momentum tensor for the matter dis-
tribution at the interior of the star is given by
Tij = (ρ + p)uiu j + pgij, (2)
where, ρ represents the energy density, p is the isotropic pressure,
and ui is the 3-velocity of the ﬂuid. The Einstein’s ﬁeld equations
with a negative cosmological constant (Λ < 0), for the spacetime
given in Eq. (1) together with the energy–momentum tensor given
2 R. Sharma et al. / Physics Letters B 704 (2011) 1–4in Eq. (2), rendering G = c = 1, yield the following independent
equations
2πρ + Λ = μ
′e−2μ
r
, (3)
2π p − Λ = γ
′e−2μ
r
, (4)
2π p − Λ = e−2μ(γ ′2 + γ ′′ − γ ′μ′), (5)
where a ‘′’ denotes differentiation with respect to the radial pa-
rameter r. Combining Eqs. (3)–(5), we have
(ρ + p)γ ′ + p′ = 0, (6)
which is the conservation equation in (2+1) dimensions. Since the
system has three independent equations involving four unknown
parameters (ρ , p, γ (r), and μ(r)), we will always have one de-
gree of freedom to solve the set of equations. In some previous
works, this freedom has been utilized to model (i) a dust cloud
(p = 0) [2]; (ii) a ﬂuid of uniform density (ρ = constant) [3]; and
(iii) a polytropic EOS p = Kρ1+1/n [5]. In this Letter, we choose a
physically meaningful form of the mass function m(r), expressible
in terms of the metric function μ(r) in the form
2m(r) = C − e−2μ(r) − Λr2, (7)
which can be obtained by integrating Eq. (3). In Eq. (7), C is an
integration constant and m(r) is the mass within a radial distance r
deﬁned as
m(r) =
r∫
0
2πρr˜ dr˜. (8)
We assume that m(r) has the form
m(r) = 1
2
− Λ
2
r2 − e
−Ar2
2
, (9)
which is regular at the centre, i.e., m(r) = 0 at r = 0, A being a
constant. This implies
2μ(r) = Ar2, (10)
where, without any loss of generality, we have set C = 1. Note that,
for grr > 0, we must have
m(r) <
1
2
− Λr
2
2
. (11)
From Eq. (3), the energy density is then obtained as
ρ = 1
2π
[
Ae−Ar2 − Λ]. (12)
Combining Eqs. (4), (5) and (10), we get,
γ ′2 + γ ′′ − Arγ ′ = γ
′
r
, (13)
which on integration yields,
γ ′ = Are
A
2 r
2
B A + e A2 r2
, (14)
where B is an integration constant. Further integrating Eq. (14) we
get
γ = ln[D(B A + e A2 r2)], (15)Fig. 1. Density variation at the stellar interior.
Fig. 2. Pressure variation at the stellar interior.
where D is another integration constant. The spacetime metric
thus obtained is free from any central singularity problem. The
isotropic pressure is then obtained as
p =
A
2π e
− A2 r2
B A + e A2 r2
+ Λ
2π
. (16)
Before we analyze physical features of the model, let us write
Λ = −K 2 to ensure that the cosmological constant always remains
negative. The central density and pressure are then obtained as
ρc = ρ(r = 0) = 1
2π
(
A + K 2), (17)
p(r = 0) = 1
2π
(
A
BA + 1 − K
2
)
. (18)
Thus, the central density and pressure remain positive if
A
(BA+1) > K
2. For a physically meaningful solution, the radial and
tangential pressures should be decreasing functions of r. In our
model, we have
dρ
dr
= − 1
π
r A2e−Ar2 , (19)
dp
dr
= − 1
2π
A2r
(B A + e A2 r2)
[
e−
A
2 r
2 + 1
(B A + e A2 r2)
]
. (20)
Obviously, at r = 0, dρdr = dpdr = 0, and the density and pressure
decrease radially outward as can be seen in Figs. 1 and 2, respec-
tively.
R. Sharma et al. / Physics Letters B 704 (2011) 1–4 3Fig. 3. Variation of mass function at stellar interior.
The radius of the star R can be obtained by letting p(r = R) = 0
in Eq. (16), which gives
R =
√
2
A
ln
[
1
2
(
−B A +
√
B2A2 + 4A
K 2
)]
. (21)
The total mass conﬁned within the radius R is obtained as
MTotal = 12 +
K 2R2
2
− e
−AR2
2
. (22)
3. Exterior spacetime and matching conditions
We match the interior solution to the exterior BTZ metric [1]
given by
ds2+ = −
(−M0 − Λr2)dt2 + (−M0 − Λr2)−1 dr2 + r2 dθ2, (23)
where, the parameter M0 is the conserved charge associated with
asymptotic invariance under time displacements. Continuity of gtt ,
grr and
∂ gtt
∂r across the circular disc joining the two metrics at
r = R , implies(−M0 − ΛR2)= D2(AB + e A2 R2)2, (24)(−M0 − ΛR2)= e−AR2 , (25)
−2ΛR = 2D2(AB + e A2 R2)e A2 R2 AR. (26)
Solving the above set of equations simultaneously, we get
A = − 1
R2
ln
(
K 2R2 − M0
)
, (27)
D = −K
2R2
ln(K 2R2 − M0) , (28)
B =
[
(K 2R2 − M0) + K 2R2ln(K 2R2−M0)
]
K 2
√
(K 2R2 − M0)
. (29)
We demand that the following energy conditions should be
satisﬁed by the distribution: (i) ρ , p  0, (ii) ρ + p  0 and
(iii) ρ + 2p  0. Employing these energy conditions at the cen-
tre (r = 0), we note that the ﬁrst condition will be satisﬁed if
A
(BA+1) > K
2, the second condition will be satisﬁed if A  0, and
the third condition demands that A + 2A
(BA+1) > K
2.
To understand behavior of the physical parameters in our
model, we assume A = 0.5, B = 0.05, Λ = −0.0625 satisfying
the above conditions. Solving Eqs. (27) and (29), we then getFig. 4. The variation of m(r)r is shown against r.
R = 2.035, M0 = 0.1327. With these values, variations of the phys-
ical parameters like energy density, pressure and mass function
have been shown in Figs. 1, 2 and 3, respectively.
4. Some features
4.1. Maximum mass–radius relation
The upper bound on the mass in our model can be obtained
from the observation that
2m(r) ≡ 1+ K 2r2 − e−Ar2  1+ K 2r2 − e−AR2 , (30)
which implies(
m
r
)
max
≡ M
R
 1+ K
2R2 − e−AR2
2R
. (31)
To see the maximum allowable mass–radius ratio in our model,
we plot m(r)r against r (see Fig. 4) which shows that the ratio
m(r)
r
is an increasing function of the radial parameter. We note that a
constraint on the maximum allowed mass–radius ratio in our case
falls within the limit to the (3 + 1)-dimensional case of isotropic
ﬂuid sphere i.e., (mr )max = 0.278 < 49 .
4.2. Compactness
The compactness of the star is given by
u = m(r)
r
= 1
2r
(
1+ K 2r2 − e−Ar2). (32)
The surface redshift (Zs) corresponding to the above compact-
ness (u) is obtained as
Zs = (1− 2u)− 12 − 1, (33)
where
Zs =
[
1− 1
r
(
1+ K 2r2 − e−Ar2)]−
1
2
− 1. (34)
Thus, the maximum surface redshift of our (2 + 1)-dimensional
star of radius R = 2.035 turns out to be Zs = 0.51.
4 R. Sharma et al. / Physics Letters B 704 (2011) 1–4Fig. 5. The variation of radial sound speed v2s is shown against r.
4.3. Sound speed
For a physically acceptable model, one expects that the velocity
of sound vs = (dp/dρ) 12 should be within the range 0 v2s < 1. In
our model
v2s =
1
2
eAr
2
(B A + e A2 r2)
[
e−
A
2 r
2 + 1
(B A + e A2 r2)
]
. (35)
We plot the radial sound speed in Fig. 5 and observe that this
parameter satisﬁes the inequalities 0 < v2s < 1.
4.4. Equilibrium conﬁguration
In Section 3, we have matched the interior solution to the ex-
terior BTZ metric [1] across the junction surface S where r = R .
In our case, the junction surface is a one-dimensional ring of mat-
ter. Though the metric coeﬃcients are continuous across the sur-
face, their derivatives may not be continuous at the surface. In
other words, the aﬃne connections may be discontinuous at the
boundary surface. This can be taken care of if we consider the
second fundamental forms of the boundary shell. Let, η denotes
the Riemann normal coordinate at the junction which has posi-
tive signature in the manifold described by exterior BTZ spacetime
and negative signature in the manifold described by the interior
spacetime. Mathematically, we have xμ = (τ ,φ,η) and the normal
vector components ξμ = (0,0,1) with the metric
gηη = 1, gηi = 0 and gij = diag
(−1, r2).
The second fundamental forms associated with the two sides of
the shell [6–12] are then given by
K i±j =
1
2
(
gik
∂ gkj
∂η
)
η=±0
= 1
2
[
∂r
∂η
gik
∂ gkj
∂r
]
r=R
. (36)
The discontinuity in the second fundamental forms is denoted byκi j = K+i j − K−i j . (37)
Now, from Lanczos equation in (2+1)-dimensional spacetime, one
can obtain the surface stress energy tensor Sij = diag(−σ ,−v)
where, σ and v are line energy density and line tension, given
by respectively [12]
σ = − 1
8π
κ
φ
φ , (38)
v = − 1
8π
κττ . (39)
Employing Eq. (36), we get
σ = − 1
8π R
[√
K 2R2 − M0 + e− 12 AR2
]
, (40)
v = − 1
8π
[
K 2R√
K 2R2 − M0
+ AR
B A + e 12 AR2
]
, (41)
where, we have set r = R . We note that the line tension is negative
which implies that there must be a line pressure as opposed to the
line tension. As we match the second fundamental forms, a crucial
question arises in the form of the star’s stability against collapse.
Therefore, we must have a thin ring of matter component with
above stresses so that the outer boundary exerts outward force
to balance the inward pull of BTZ exterior. The energy density
is negative in this junction ring which is similar to the (3 + 1)-
dimensional case [13].
5. Discussion
We have obtained a new class of solutions corresponding to the
BTZ exterior spacetime by assuming a particular form of the mass
function m(r). The solution obtained is regular at the centre and
it satisﬁes all the physical requirements except at the boundary
where we propose a thin ring of matter content with negative en-
ergy density so as to prevent collapsing. The discontinuity of the
aﬃne connections at the boundary surface provide the above mat-
ter conﬁned to the ring. Such a stress–energy tensor is not ruled
out from the consideration Casimir effect for massless ﬁelds. The
solution obtained here has a simple analytical form which can be
used to study a full collapsing model of a (2+ 1)-dimensional star
which is beyond the scope of this analysis. This will be taken up
else where.
References
[1] M. Bañados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849.
[2] R.B. Mann, S.F. Ross, Phys. Rev. D 47 (1993) 3319.
[3] N. Cruz, J. Zanelli, Class. Quant. Grav. 12 (1995) 975.
[4] N. Cruz, M. Olivares, J.R. Villanueva, Gen. Relativ. Grav. 37 (2004) 667.
[5] Paulo M. Sá, Phys. Lett. B 467 (1999) 40.
[6] W. Israel, Nuovo Cim. B 44 (1966) 1;
W. Israel, Nuovo Cim. B 48 (1967) 463 (Erratum).
[7] A.A. Usmani, Z. Hasan, F. Rahaman, Sk.A. Rakib, S. Ray, P.K.F. Kuhﬁttig, Gen.
Relativ. Grav. 42 (2010) 2901.
[8] F. Rahaman, K.A. Rahman, Sk.A. Rakib, P.K.F. Kuhﬁttig, Int. J. Theor. Phys. 49
(2010) 2364.
[9] F. Rahaman, et al., Gen. Relativ. Grav. 38 (2006) 1687.
[10] F. Rahaman, et al., Acta Phys. Polon. B 40 (2009) 1575.
[11] F. Rahaman, P.K.F. Kuhﬁttig, M. Kalam, A.A. Usmani, S. Ray, Class. Quant.
Grav. 28 (2011) 155021.
[12] G.P. Perry, R.B. Mann, Gen. Relativ. Grav. 24 (1992) 305.
[13] M. Visser, Nucl. Phys. B 328 (1989) 203.


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A class of interior solutions corresponding to a $(2+1)$ dimensional asymptotically anti-de Sitter spacetime

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A class of interior solutions corresponding to a (2+1)(2+1)(2+1) dimensional asymptotically anti-de Sitter spacetime

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A class of interior solutions corresponding to a $(2+1)$ dimensional asymptotically anti-de Sitter spacetime