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
1
Department of Mathematics, Jadavpur University, Kolkata 700 032, West Bengal, India
2
Indian Institute of Engineering Sceince and Technology Shibpur (Formerly, Bengal Engineering and Science University Shibpur),
Howrah 711 013, West Bengal, India
3
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
© The Author(s) 2014. This article is published with open access at Springerlink.com
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
a
e-mail: farook_rahaman@yahoo.com; rahaman@iucaa.ernet.in
b
e-mail: piyalibhar90@gmail.com
c
e- mail: biswas.ritabrata@gmail.com
d
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].
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