This paper constructs charged thin-shell wormholes in (2+1) dimensions by cutting and pasting charged BTZ black holes. The surface stresses are determined using the Darmois-Israel formalism. The stability of the shell is analyzed considering phantom energy and generalized Chaplygin gas equations of state. Linearized stability is also discussed around the static solution. Graphs show regions of stability for different parameters.

1. NATIONAL
CONFERENCE
ON
NON-LINEAR
DYNAMICS,
ANALYSIS AND
OPTAMIZATION
( NDAO-2014 )

2. “Stability of char ged
thin-shell wor mholes
in (2 + 1) dimensions”
Ayan Banerjee
Jadavpur University

3. “Stability of charged thin-shell wormholes in (2 +
1)dimensions”
In this paper we construct charged thin-shell
wormholes in (2+1)-dimensions applying the
cut-and paste technique implemented by Visser,
from a B Z black hole which was discovered by
T
B
a˜nados, T
eitelboim and Zanelli , and the
surface stress are determined using the DarmoisIsrael formalism at the wormhole throat. W
e
analyzed the stability of the shell considering
phantom-energy or generalized Chaplygin gas
equation of state for the exotic matter at the
throat. W also discussed the linearized stability
e
of charged thin-shell wormholes around the static
solution.

4. “Stability of char ged thin-shell wor mholes in (2 + 1)
dimensions”
CONSTRUCTION OF CHARGED THIN-SHELL
WORMHOLE
T charged B Z black hole with a
he
T
negative cosmological constant Λ =
-1/ {^2} is a solution of (2+1)-dimensional
L
gravity. T metric is given by
he

5. “Stability of char ged thin-shell wor mholes in (2 +
1)dimensions”
and Q are mass and electric charge of the
B Z black hole.
T
we take two identical copies from B Z
T
black hole with r : a
and stick them together at the junction
surface
to get a new geodesically complete
manifold . T minimal surface area ,
he
referred as a throat of wormhole where we

6. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
At the junction surface , the stress
energy components are non zero can be
evaluated using the second fundamental
forms and the energy density and
pressure is given by
and

7. “Stability of char ged thin-shell wor mholes in (2
+1)dimension”
F the static configuration of radius a.
or
 T energy condition demands, if σ > 0
he
and
σ + p >0 are satisfied, then the weak
energy condition (W C) holds and by
E
continuity, if
σ + p> 0 is satisfied, then the null
energy condition (NE holds. M
C)
oreover,
the strong energy (SE holds, if σ + p >0
C)
and σ + 2p >0 are satisfied.

8. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
T E GRAVIT IONAL F L
H
AT
IE D
In this section we analyze the attractive
and repulsive nature of the wormhole.
Only non-zero component for the line
element , is given by
A test particle when radically moving
and initially at rest, obeys the equation
of motion

9. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
which gives the geodesic equation if
=
0.
Also, we observe that the wormhole is
attractive when
and repulsive
when
, which is shown in fig.

10. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
T E T AL AM
H OT
OUNT OF E
XOT
IC
M TR
AT E
T construct such a thin-shell wormhole,
o
we need exotic matter. In this section,
we evaluate the total amount of exotic
matter for the shell which can be
quantified by the integral
where g represents the determinant of
the metric tensor. Now, by using the
radial coordinate R = r - a, we have

11. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
F the infinitely thin shell it does not
or
exert any radial pressure i.e.
=0
W the help of graphical
ith
representation , we are trying to
describe the variation of the total
amount of exotic matter on the shell
with respect to the mass and the charge.

12. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
The ToTal amounT of exoTic
maTTer

13. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
Stability
Stability is one of the important issue for
the wormhole. H
ere we analyze the
stability of the shell from various angle.
Our approaches are
 phantom-like E
OS
 generalized chaplygin gas E
OS
 linearized radial perturbation, around
the static solution.

14. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
A. phantom-like equation of
state
H
ere, we are trying to describe the
stability of the shell considering the
equation of state when the surface
energy density and the surface pressure
are taken into account. W set an
e
equation w = p/ i.e. p = w σ known as
σ
P
hantom-like equation of state when w
< 0. T surface pressure and energy
he
density obey the conservation equation

15. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
after differentiating w.r.t τ , one can get
Now, consider the static solution with
radius
a = , we have
the equation of motion of the shell, where
the potential V(a) is defined as

16. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
E
xpanding V (a) around the static
solution i.e. at
, we have
where the primes denote the derivative
with respect to a. T wormhole is stable
he
if and only if
has local minimum
at
and
>0.
Now ,using the conditions
=0,

17. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
is given by
W have verified that the inequality
e
> 0, holds for suitable choice of Q, M and
L when w takes the different values. W
e
are trying to describe the stability of the
configuration with help of graphical
representation.

18. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
Graphs for different values of w:

19. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
B. Generalized Chaplygin gas equation of state
H
ere we are trying to check the stability of
the shell considering Chaplygin gas
equation of state, at the throat, which is a
hypothetical substance satisfying an
equation of state:
where σ is surface energy density and p is
surface pressure with A positive constants
and 0 < α ≤ 1.

20. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
using the same procedure ,we find the potential
V (a),
which takes of the form :
W
here
T above solution gives a stable configuration
he
if the second order derivative of the potential is
positive for the static solution and V ( )
posses a local minimum at .T find the range
o
of
for which V′′ ( ) > 0 , we use graphical
representation due to complexity of the

21. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”

22. V
V
“Stability of char ged thin-shell wor mholes in (2 +
)>
1)dimension”
C. Liberalized stability analysis
Now we are trying to find the stability of
the configuration around the static solution
at a = under radial perturbation.
As we previously discussed about the
stability criteria , we found the expression
for V ′′ ( )
where

23. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
T configuration is stable if and only if V ′
he
′ ( ) > 0. So starting with V′′ ( ) = 0 and
solve for
we get
H
ere we observe that
And
T value of
he
which represent the velocity of
sound for ordinary matter,

24. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
So, it should not exceed the speed of
light and lies in the region of 0< < 1.

25. “Stability of char ged thin-shell wor mholes in (2 +
1)dimension”
Discussion
In this work we construct charged thinshell wormhole from B Z black hole
T
using the cut-paste technique and by
employing different equation of state
we find stable wormhole solution in all
cases.

26. Thank
You