Anisotropic star in (2+1)-dimension with linear or nonlinear equation of state

“International Conference on
Recent Advances in
Mathematics(ICRAM-2014)”

“Anisotropic star in (2+1)

dimension with linear or
nonlinear equation of state”
Ayan Banerjee
Jadavpur University

“Anisotropic star in (2+1) dimension with
linear or nonlinear equation of state”
Gravitational analyses in lower dimensions has
become a field of active research interest, ever
since Ba˜nados, Teitelboim and Zanelli (BTZ)
proved the existence of a black hole solution in
(2 + 1) dimensions. The BTZ metric has
inspired many investigators to develop and
analyze circularly symmetric stellar models
which can be matched to the exterior BTZ
metric. We have obtained two new classes of
solutions for a (2 + 1)-dimensional anisotropic
star in anti-de Sitter space-time

which have been obtained by assuming that
the equation of state (EOS) describing the
material composition of the star could either
be linear or non-linear in nature. By
matching the interior solution to the BTZ
exterior metric, we have demonstrated that
the solutions provided here are regular and
well-behaved at the stellar interior.

INTERIOR SPACE-TIME
We write the line element for a static circularly
symmetric star with zero angular momentum
in the form
where
and
are yet to be determined.
The Einstein’s field equations for an anisotropic
fluid in the presence of a negative cosmological
constant (Λ < 0) are then obtained as

(we set G = c = 1)

where,
is the energy density,
the radial pressure and
is the
tangential pressure

is

may be combined to yield

• which is analogous to the generalized TolmanOppenheimer-Volkoff (TOV) equation in
(3 + 1) dimensions. Defining the mass within a
radius r as

using the Einstein’s field Eqs. yields
• where C is integrating constant. We set C = 1
and assume 2 (r) =
so as to ensure
regular behavior of the mass function m(r) at
the centre. The energy density is then
obtained as

The constant A can be determined from the
central density
• To determine the unknown metric potential,
we prescribe an EOS corresponding to the
material composition of the star in the form
• where
and
are two positive arbitrary
constants constraining the EOS.

• The physical radius R of the star can be
obtained by ensuring that
• This can be either linear or non-linear in
nature and accordingly we consider the two
possibilities separately.
Case I: Solution admitting a linear EOS
Let us first assume a linear EOS of the form

For the choice of the linear Eq. and solving the
system analytically, we get

where
is integrating constant. Also the the
measure of anisotropy is given by

]
Note that the measure of anisotropic vanishes
at the centre which is desirable feature of a
realistic star.

Case II: Solution admitting a non-linear EOS
Assuming a non-linear EOS of the form :
where and are constants parameters, we
solve the system analytically and obtain

and the tangential pressure is given by

is an integration constant same as linear case.

The measure of anisotropy in this case turns

• Note that for both cases the anisotropy vanishes at
the centre which is a desirable feature of a realistic
star.

Exterior space-time and boundary conditions
• We assume that the exterior space-time of our
circularly symmetric star is described by the
BTZ metric
• where
is the conserved charge associated
with asymptotic invariance under time
displacements.

At the boundary r = R, continuity of the metric
potentials yield the following junction
conditions:

• Moreover, the radial pressure must vanish at
the boundary, i.e., (r = R) = 0. These three
conditions can be utilized to fix the values of
the constants A, C and R in different case as
give below respectively

Case I : linear EOS
The unknown terms are

Case II : non-linear EOS
We have

IV. PHYSICAL ACCEPTABILITY AND
REGULARITY OF THE MODEL

For a physically acceptable model, we impose
the following restrictions:
• Energy-density and pressure should be
“monotonically” decreasing functions of r.
• Radial sound velocity and transverse sound
velocity should be less than unity i.e.

Here, we are trying to discuss all the restriction
for both cases step by step as given below:
Case I: Linear EOS

• show that, for >0, both energy-density and
radial pressure decrease from their maximum
values at the centre. We are trying to describe
the variations of the energy-density and the two
pressures have been shown in figs.
Case II: Non-linear EOS
For non-linear equation of state:

and

• Though it is not straight forward, we note that
the inequality holds for appropriate choices of
the values of and . show that both
energy-density and radial pressure decrease
from their maximum values at the centre.

Behavior of two pressures:

• It is known that an important “physical
acceptability conditions” for anisotropic
matter are the squares of radial and tangential
sound speeds should be less than the speed of
light. In the following we are trying to find
whether our models follow this important
property.
Case I. Linear EOS
radial pressure is

and the transverse sound speed is

-

case II. Non-linear EOS

shows regular behavior of radial and transverse
sound speeds in the model with non linear EOS.

V. SOME FEATURES OF THE MODEL
• The total gravitational mass M(r = R) in our
model can be obtained by plugging C = 1 and
2 (r) =
in the mass Eq. we get
Which implies the mass-radius relation

for our stellar configuration of radius R = 2:5 K.m,

the compactness M/R is found to be 0:176 for
Λ = -0.005 and the corresponding surface
red-shift is

turns out to be 0.234.

VI. DISCUSSIONS
• In this work, we have generated new analytic
solutions for a circularly symmetric star
which admits a linear or non-linear equation
of state. The matter composition of the star
has been assumed to be anisotropic in nature.
The values of the constants in our solution
have been fixed by matching the interior
solution to the BTZ exterior metric.

The cosmological constant Λ remains a free
parameter in our construction and for some
specific choices of the cosmological constant,
we have shown that our solutions are well
behaved and, therefore, can be utilized to
develop physically acceptable model of a static
circularly symmetric star in Ads space-time.


thank you

Anisotropic star in (2+1)-dimension with linear or nonlinear equation of state

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