Solitons in AdS spacetime Sardor Tojiev

Solitons in AdS spacetime
Sardor Tojiev
Jacobs University Bremen, Germany
RTG Spring Workshop
Bielefeld, Germany
7-9 May 2014
Tojiev (JUB)
AdS solitons
7.05.2014
1 / 18
References
Work done in collaboration with:
Yves Brihaye
Université de Mons, Belgium
Betti Hartmann
Jacobs University Bremen, Germany
Papers
1
Yves Brihaye, Betti Hartmann, Sardor Tojiev
Formation of scalar hair on Gauss-Bonnet solitons and black holes.
Phys. Rev. D87 : 024040, 2013
2
Yves Brihaye, Betti Hartmann, Sardor Tojiev
Stability of charged solitons and formation of boson stars in 5-dimensional
Anti-de Sitter space-time.
Class. Quant. Grav. 30:115009, 2013.
3
Yves Brihaye, Betti Hartmann, Sardor Tojiev
AdS solitons with conformal scalar hair.
Phys. Rev. D88 : 104006, 2013.
Tojiev (JUB)
AdS solitons
7.05.2014
2 / 18
Outline
1
Introduction
The model
The ansatz and equations
Boundary conditions
2
Numerical results
m2 > 0
m2 < 0
m2 = 0
3
Summary
Tojiev (JUB)
AdS solitons
7.05.2014
3 / 18
Introduction
Introduction
Motivation
formation of scalar ”hair” -superconductivity.
2 of the scalar field drops below the Breitenlohner effective mass meff
.
Freedman (BF) bound under certain circumstances and the black holes
or solitons - become unstable to the formation of scalar hair.
2
mBF
≥−
d −1
L2
the stability of classical field theory solutions - e.g. black hole
uniqueness.
Tojiev (JUB)
AdS solitons
7.05.2014
4 / 18
Introduction
Boundary conditions
The model
Electrically charged solitons in 5d AdS spacetime. The action reads
R √
S = γ1 d5 x −g (R − 2Λ + γLmatter )
The matter Lagrangian
Lmatter = − 14 FMN F
where FMN = ∂M AN − ∂N AM ,
and
MN
M
− (DM ψ)∗ D ψ − m 2 ψ∗ ψ
DM ψ = ∂M ψ − ieAM ψ
We choose the following spherically symmetric Ansatz for the metric :
ds2 = −a 2 f dt2 + f1 dr2 + r 2 dΩ23 ,
For the electromagnetic field and the scalar field, we have :
AM dx M = φ(r )dt
Tojiev (JUB)
,
AdS solitons
ψ = ψ(r ) .
7.05.2014
5 / 18
Introduction
Boundary conditions
The ansatz and equations
The coupled gravity and matter field equations are :
f
0
a0
φ 00
ψ 00
r2
r
1−f +2 2 −γ
2e 2 φ2 ψ2 + f (2m 2 a 2 ψ2 + φ 02 ) + 2f 2 a 2 ψ 02
2
L
2fa
r (e 2 φ2 ψ2 + a 2 f 2 ψ 02 )
= γ
af 2
0
e 2 ψ2
3 a
−
φ0 + 2
φ
= −
r
a
f
2 2
e φ
m2
3 f 0 a0
+
+
ψ0 −
−
ψ
= −
r
f
a
f 2a 2
f
=
2
r
The system possesses two scaling symmetries:
r → λr ,
φ → λφ ,
Tojiev (JUB)
t → λt , L → λL , e → e/λ
ψ → λψ , e → e/λ , γ → γ/λ2
AdS solitons
7.05.2014
6 / 18
Introduction
Boundary conditions
The metric functions have the following asymptotic behaviour for
r 1
f (r ) =
r2
f2
+ 1 + 2 + ...,
L2
r
c
,
r4
a(r ) = 1 +
M =−
f2
2
Asymptotically, the matter fields behave as follows
φ(r 1) = µ −
Q
ψ−
ψ+
+ . . . , ψ(r 1) = λ− + λ+
r2
r
r
with
λ− = 2 −
p
4 + m 2 L2 , λ + = 2 +
p
4 + m 2 L2
Boundary conditions
To find soliton solution of the equations of motion
f (0) = 1,
ψ(0) = ψ0 ,
φ0 (0) = 0,
ψ0 (0) = 0,
a(r 1) = 1,
ψ− = 0
The solitons can then be characterized by the values of the matter & metric
functions at the origin φ(0), ψ(0), a(0) which depend on the choice of e 2 , Q
and m 2 .
Tojiev (JUB)
AdS solitons
7.05.2014
7 / 18
Numerical results
m2 = 0
Numerical results
Numerical results
m >0
m <0
m =0
Tojiev (JUB)
AdS solitons
7.05.2014
8 / 18
Numerical results
m2 = 0
m2 = 1
m 2 = 1. The solid lines represent the fundamental solutions, while the
dashed lines correspond to the first excited solutions (here given for
e 2 = 1)
Mass M as function of ψ(0)
Tojiev (JUB)
Mass M as function of Q
AdS solitons
7.05.2014
9 / 18
Numerical results
m2 = 0
m2 = 1
m2 = 1
The profiles of the metric functions: e 2 = 2 and for
ψ(0) = 1.5(black) and ψ(0) = 0.9(red)
Tojiev (JUB)
AdS solitons
7.05.2014
10 / 18
Numerical results
m2 = 0
m2 = 1
The mass M is shown as functions of ψ(0) for m 2 = 1 and several
values of e 2 close to the critical value ecr
Tojiev (JUB)
AdS solitons
7.05.2014
11 / 18
Numerical results
m2 = 0
m 2 = −3
m 2 = −3. The labels A, B and C represent the branches of
fundamental solutions, while A0 corresponds to a branch of first excited
2 ≈ 1.3575
solutions(here given for e 2 = 2): ecr
Mass M as function of ψ(0)
Tojiev (JUB)
Mass M as function of Q
AdS solitons
7.05.2014
12 / 18
Numerical results
m2 = 0
m 2 = −3
The mass M is shown as functions of ψ(0) for m 2 = −3 and several
values of e 2 close to the critical value ecr
Tojiev (JUB)
AdS solitons
7.05.2014
13 / 18
Numerical results
m2 = 0
Dependence on m 2
Here ψ(0) = 5.0, e 2 = 2 and m 2 = −3(dashed lines) and m 2 = 0(solid
lines).
f (r )(black) and a(r )(red) as
function of r
Tojiev (JUB)
ψ(r )(red) and φ(r )(black) as
function of r
AdS solitons
7.05.2014
14 / 18
Numerical results
m2 = 0
m2 = 0
The mass M is shown as functions of ψ(0) for m 2 = 0 and several
values of e 2 close to the critical value ecr
Tojiev (JUB)
AdS solitons
7.05.2014
15 / 18
Numerical results
m2 = 0
All phenomena are summarized
The mass M(black solid), the value φ(0)(black dotted) and the charge
Q(red solid) of the hairy soliton in dependence on m 2 for e 2 = 2,
ψ(0) = 3.5
Gentle (4d),et al. ’12,
Dias (5d),et al. ’12
Tojiev (JUB)
AdS solitons
7.05.2014
16 / 18
Summary
Summary
1
We have studied the formation of scalar hair on charged soliton in
global AdS and the dependence of the solutions on the choice of
the charge e 2 and the mass m 2 of the scalar field.
2
We find that the pattern of solutions depends crucially on the
2 ≈ 2.4 + m 2 /3
choice of e 2 and m 2 with a critical value ecr
dividing this pattern into distinct types.
3
Interestingly, we observe that boson stars in AdS can have arbitrary large M and Q; however, also that a ’forbidden band’ of the
M and Q at intermidate values of the mass and charge exists.
It would be interesting:
4
the effects of Gauss-Bonnet term.
holographic interpretation.
. . .
Tojiev (JUB)
AdS solitons
7.05.2014
17 / 18
Summary
T hank You
for your attention !
Tojiev (JUB)
AdS solitons
7.05.2014
18 / 18

Download Report