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Eur. Phys. J. C (2014) 74:2845

DOI 10.1140/epjc/s10052-014-2845-z

Regular Article - Theoretical Physics

Exact interior solutions in 2 + 1-dimensional spacetime

Farook Rahaman

1,a

, Piyali Bhar

1,b

, Ritabrata Biswas

2,c

,A.A.Usmani

3,d

Department of Mathematics, Jadavpur University, Kolkata 700 032, West Bengal, India

Indian Institute of Engineering Sceince and Technology Shibpur (Formerly, Bengal Engineering and Science University Shibpur),

Howrah 711 013, West Bengal, India

Department of Physics, Aligarh Muslim University, Aligarh 202 002, Uttar Pradesh, India

Received: 28 February 2014 / Accepted: 1 April 2014 / Published online: 18 April 2014

Abstract We provide a new class of exact solutions for

the interior in 2 + 1-dimensional spacetime. The solutions

obtained for the perfect ﬂuid model both with and without

cosmological constant () are found to be regular and sin-

gularity free. It assumes very simple analytical forms that

help us to study the various physical properties of the con-

ﬁguration. Solutions without  are found to be physically

acceptable.

1 Introduction

The study of exact solutions of Einstein’s ﬁeld equations is

an important part of the theory of general relativity. This

importance is not only concerned with more formal mathe-

matical aspects associated with the theory (e.g. the classiﬁ-

cation of spacetimes) but also with the growing importance

of the application of general relativity to astrophysical phe-

nomena. For example, exact solutions may offer physical

insights that numerical solutions cannot. The present time

trend of analyzing different aspects of black hole (BH) solu-

tions did lead us to direct our interests to cleaner 2 + 1-

dimensional gravity. The discovery of BTZ BH [1] ignited

the light ﬁrst. Through this 2 + 1-dimensional model if we

need to explore the foundations of classical and quantum

gravity we would not ﬁnd any Newtonian limit and no prop-

agating degrees of freedom will arise. In the literature, very

easy-to-ﬁnd works in this aspect comprise the study of quasi-

normal modes of charged dilaton BHs in 2 + 1-dimensional

solutions in low energy string theory with asymptotic anti-

de Sitter spacetimes [2]. Hawking radiation from covariant

anomalies in 2 +1-dimensional BHs [3] is another beautiful

e-mail: farook_rahaman@yahoo.com; rahaman@iucaa.ernet.in

e-mail: piyalibhar90@gmail.com

e- mail: biswas.ritabrata@gmail.com

e-mail: anisul@iucaa.ernet.in

example. Lastly, we must also point out the study of branes

with naked singularities analogous to linear or planar defects

in crystals and showing that zero branes in AdS spacetimes

are ‘negative mass BHs!’ [4]. Taking charged gravastars as

an alternative to charged BHs in (2 +1) AdS spacetimes has

already been investigated [5]. Extensions of BTZ BH solu-

tions with charge are also available in the literature. These

are obtained by employing nonlinear Born–Inﬁeld electrody-

namics to eliminate the inner singularity [6]. The non-static

charged BTZ like BHs in (N +1) dimensions have also been

studied [7] which in its static limit, for N = 2, reduces to

(2 + 1) BTZ BH solutions.

The study of interior solutions in (2 + 1) dimensions [8]

shows that even the noncommutative-geometry-inspired

BTZ BH is not free from any singularity. The study of inte-

rior solutions is rarely found in the literature. For example,

solutions of Wolf [9] and Yazadjiev [10], solutions in the

framework of the Brans–Dicke theory of gravity by Kozyrev

[11] and a new class of solutions corresponding to BTZ

exterior spacetime by Sharma et al. [12], which is regular

at the center and it satisﬁes all the physical requirements

except at the boundary where the authors propose a thin

ring of matter content with negative energy density so as

to prevent collapsing. The discontinuity of the afﬁne con-

nections at the boundary surface provide the above matter

conﬁned to the ring. Such a stress-energy tensor is not ruled

out from the consideration of the Casimir effect for massless

ﬁelds.

The purpose of the present work is to ﬁnd exact interior

solutions for a perfect ﬂuid model both with and without

cosmological constant, . The motivation for doing so is

provided by the fact that the assumption of the equation of

state (EoS) is p = mρ, which seems to be very reasonable for

describing the matter distribution in the study of relativistic

objects like stars [13,14], wormholes [15,16] and gravastars

[5,17].

123

2845 Page 2 of 7 Eur. Phys. J. C (2014) 74:2845

The structure of our work is as follows: In Sect. 2,we

derive the required Einstein equations. Section 3 contains

different interior solutions for various cases of the EoS.

Lastly, in Sect. 4 a brief conclusion is provided.

2 Einstein ﬁeld equations in (2 + 1) dimensions

We take the static metric to describe the interior region of a

2 + 1-dimensional spacetime as follows:

=−e

2ν(r)

+ e

2μ(r)

dθ

, (1)

where ν(r) and μ(r) are the two unknown metric functions.

We take the perfect ﬂuid form of the energy momentum ten-

sor,

= diag(−ρ, p, p), (2)

where ρ is the energy density and p is the pressure. Einstein’s

ﬁeld equations with a cosmological constant, ,forthe

spacetime metric (see (1)) together with the energy momen-

tum tensor given in (2) may be written as

2πρ +  = μ



−2μ(r)

/r, (3)

2π p −  = ν



−2μ(r)

/r, (4)

2π p −  = e

−2μ





+ ν

2

− ν





. (5)

Here the superscript ‘’ denotes the derivative with respect

to r. Assuming G = c = 1, the generalized Tolman–

Oppenheimer–Volkov (TOV) equation may be written as

(

ρ + p

)



+ p



= 0, (6)

which represents conservation equations in (2 + 1) dimen-

sions.

We take the EoS of the f orm

p = mρ, (7)

where m is EoS parameter.

3 Interior solutions

We ﬁrst obtain interior solutions without any cosmological

constant, thereby taking  = 0. Later on, we generalize our

study to non-zero value of . We choose various cases of the

EoS parameter for both choices of .

3.1 No cosmological constant ( = 0)

3.1.1 0 < m < 1

For  = 0, the ﬁeld equations (3)–(6) become

2πρ = μ



−2μ

/r, (8)

2πmρ = ν



−2μ

/r, (9)

2πmρ = e

−2μ

(ν

2

+ ν



− μ



). (10)

The TOV equation (11) takes the form

(ρ + mρ)ν



+ mρ



= 0, (11)

Equation (11) yields

(1+m)ν

= C. (12)

Solving (8) and (9), we get

ν = mμ + A. (13)

Equating (9) with (10), we get

= e



/r. (14)

Now, solving (13) and (14), we obtain

ν =

1 − m

−

1 − m



1 − m



B −



, (15)

μ =

1 − m



A −ln



1 − m



B −



, (16)

ρ =

2πm

−

1−m



1 − m



B −



1+m

1−m

. (17)

Here C, A, and B are integration constants.

For consistency of the solutions, the constants should fol-

low the constraint equation,

A = m ln(2πm) + ln C. (18)

These solutions are regular at the center. The central density

is given by

= 1/(2π m)e

−

1−m

[

B(1 − m)/m

]

1+m

1−m

. (19)

The interior solution is valid up to the radius r <

√

2B.For

a physically meaningful solution the radial and tangential

pressure should be a decreasing function of r. From (17), we

ﬁnd

dρ

< 0, (20)

which gives the density and the pressure as decreasing func-

tions of r. At r = 0, one can get

= 0,

dρ

= 0 and



(1 − m)B



1−m

< 0,

(21)

123

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