-
GaussBonnet Boson Stars in AdS
Jrgen Riedelin collaboration with
Yves Brihaye(1), Betti Hartmann(2)(3) , and Raluca Suciu(2)
Physics Letters B (2013) (arXiv:1308.3391), arXiv:1310.7223,
andarXiv:1404.1874
Faculty of Physics, University Oldenburg, GermanyModels of
Gravity
IBERICOS 2014Aveiro, Portugal, April 28th, 2014
(1)Physique-Mathmatique, Universit de Mons, 7000 Mons,
Belgium
(2)School of Engineering and Science, Jacobs University Bremen,
Germany
(3)Universidade Federal do Espirito Santo (UFES), Departamento
de Fisica, Vitoria (ES), Brazil
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Agenda
Describe the model to construct non-rotatingGauss-Bonnet boson
stars in AdSDescribe effect of Gauss-Bonnet term to boson
starsolutionsStability analysis of rotating Gauss-Bonnet boson
starsolutions (ignoring the Gauss-Bonnet coupling for
now)Outlook
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Solitons in non-linear field theories
General properties of soliton solutionsLocalized, finite energy,
stable, regular solutions ofnon-linear field equationsCan be viewed
as models of elementary particles
ExamplesTopological solitons: Skyrme model of hadrons in
highenergy physics one of first models and magneticmonopoles,
domain walls etc.Non-topological solitons: Q-balls (named after
Noethercharge Q) (flat space-time) and boson stars(generalisation
in curved space-time)
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Non-topolocial solitons
Properties of non-topological solitonsSolutions possess the same
boundary conditions atinfinity as the physical vacuum
stateDegenerate vacuum states do not necessarily existRequire an
additive conservation law, e.g. gaugeinvariance under an arbitrary
global phasetransformation
S. R. Coleman, Nucl. Phys. B 262 (1985), 263, R. Friedberg, T.
D. Lee and A. Sirlin, Phys. Rev. D 13 (1976) 2739),
D. J. Kaup, Phys. Rev. 172 (1968), 1331, R. Friedberg, T. D. Lee
and Y. Pang, Phys. Rev. D 35 (1987), 3658, P.
Jetzer, Phys. Rept. 220 (1992), 163, F. E. Schunck and E.
Mielke, Class. Quant. Grav. 20 (2003) R31, F. E.
Schunck and E. Mielke, Phys. Lett. A 249 (1998), 389.
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Why study Q-balls and boson stars?
Boson starsSimple toy models for a wide range of objects such
asparticles, compact stars, e.g. neutron stars and evencentres of
galaxiesWe are interested in the effect of Gauss-Bonnet gravityand
will study these objects in the minimal number ofdimensions in
which the term does not become a totalderivative.Toy models for
studying properties of AdS space-timeToy models for AdS/CFT
correspondence. Planar bosonstars in AdS have been interpreted as
Bose-Einsteincondensates of glueballs
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Model for GaussBonnet Boson Stars
Action
S =1
16piG5
d5xg (R 2 + LGB + 16piG5Lmatter)
LGB =(RMNKLRMNKL 4RMNRMN + R2
)(1)
Matter Lagrangian Lmatter = () U()Gauge mediated potential
USUSY(||) = m22susy(
1 exp( ||
2
2susy
))(2)
USUSY(||) = m2||2 m2||4
22susy+m2||664susy
+ O(||8
)(3)
A. Kusenko, Phys. Lett. B 404 (1997), 285; Phys. Lett. B 405
(1997), 108, L. Campanelli and M. Ruggieri,
Phys. Rev. D 77 (2008), 043504
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Model for GaussBonnet Boson Stars
Einstein Equations are derived from the variation of theaction
with respect to the metric fields
GMN + gMN +
2HMN = 8piG5TMN (4)
where HMN is given by
HMN = 2(RMABCRABCN 2RMANBRAB 2RMARAN + RRMN
) 1
2gMN
(R2 4RABRAB + RABCDRABCD
)(5)
Energy-momentum tensor
TMN = gMN[
12gKL (KL + LK) + U()
]+ M
N + NM (6)
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Model continued
The Klein-Gordon equation is given by:( U
||2) = 0 (7)
Lmatter is invariant under the global U(1) transformation ei .
(8)
Locally conserved Noether current jM
jM = i2
(M M
); jM;M = 0 (9)
The globally conserved Noether charge Q reads
Q =
d4xgj0 . (10)
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Ansatz non-rotating
Metric Ansatz
ds2 = N(r)A2(r)dt2 + 1N(r)
dr2
+ r2(d2 + sin2 d2 + sin2 sin2 d2
)(11)
whereN(r) = 1 2n(r)
r2(12)
Stationary Ansatz for complex scalar field
(r , t) = f (r)eit (13)
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Ansatz rotating
Metric Ansatz
ds2 = A(r)dt2 + 1N(r)
dr2 + G(r)d2
+ H(r) sin2 (d1 W (r)dt)2+ H(r) cos2 (d2 W (r)dt)2+ (G(r) H(r))
sin2 cos2 (d1 d2)2, (14)
with = [0,pi/2] and 1,2 = [0,2pi].Cohomogeneity-1 Ansatz for
Complex Scalar Field
= (r)eit , (15)
with = (sin ei1 , cos ei2)t (16)
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Boundary Conditions for asymptotic AdS space time
If < 0 the scalar field function falls of with
(r >> 1) =r
, = 2 +
4 + L2eff . (17)
Where Leff is the effective AdS-radius:
L2eff =2
1
1 4L2;L2 =
6
(18)
Chern-Simons limit:
=L2
4(19)
Mass for > 0 we define the gravitational mass at
AdSboundary
MG n(r )/ (20)
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Finding solutions: fixing
f
V (f )
4 2 0 2 40 .
0 3
0 .0 1
0 .0 0
0 .0 1
0 .0 2
f(0)
k=0
k=1
k=2
= 0.80 = 0.85 = 0.87 = 0.90V = 0.0
r
0 5 10 15 200 .
10 .
10 .
20 .
30 .
40 .
5
k= 0= 1= 2 = 0.0
r0 2 4 6 8 10
1 .0
0 .
50 .
00 .
51 .
0
Functionsphi(r) & (0) = 0.2phi(r) & (0) = 4.0phi(r)
& (0) = 9.0b(r) & (0) = 0.2b(r) & (0) = 4.0b(r) &
(0) = 9.0f(r) & (0) = 0.2f(r) & (0) = 4.0f(r) & (0) =
9.0w(r) & (0) = 0.2w(r) & (0) = 4.0w(r) & (0) = 9.0
Figure : Effective potential V (f ) = 2f 2 U(f ).Brihaye,
Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in AdS
-
GaussBonnet Boson Stars with < 0
Q
0.8 0.9 1.0 1.1 1.2 1.3 1.4
11 0
1 00
1 00 0
1 00 0
0
= 0.0= 0.01= 0.02= 0.03= 0.04= 0.05= 0.06= 0.075= 0.1= 0.2= 0.3=
0.5= 1.0= 5.0= 10.0= 12.0= 15.0 (CS limit) 1st excited mode =
1.0
Figure : Charge Q in dependence on the frequency for = 0.01, =
0.02 anddifferent values of . max shift: max = Leff
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
GaussBonnet Boson Stars with < 0 Animation
(Loading Video...)
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
HandBrake 0.9.9 2013051800
GBAdSBosonStar.mp4Media File (video/mp4)
-
Excited GaussBonnet Boson Stars with < 0
Q
0.6 0.8 1.0 1.2 1.4 1.6
11 0
01 0
0 00
k=0 k=1 k=2 k=3 k=4
= 0.0= 0.01= 0.05= 0.1= 0.5= 1.0= 5.0= 10.0= 50.0= 100.0= 150.0
(CS limit) = 1.0
Figure : Charge Q in dependence on the frequency for = 0.01, =
0.02, anddifferent values of . max shift: max = +2kLeff
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Summary Effect of Gauss-Bonnet Correction
Boson starsSmall coupling to GB term, i.e. small , we find
similarspiral like characteristic as for boson stars in
pureEinstein gravity.When the Gauss-Bonnet parameter is large
enoughthe spiral unwinds.When and the coupling to gravity () are of
the samemagnitude, only one branch of solutions survives.The single
branch extends to the small values of thefrequency .
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
AdS Space-time
(4)
Figure : (a) Penrose diagram of AdS space-time, (b) massive
(solid) and massless(dotted) geodesic.
(4)J. Maldacena, The gauge/gravity duality,
arXiv:1106.6073v1Brihaye, Hartmann, Riedel, and Suciu GaussBonnet
Boson Stars in AdS
-
Stability of AdS
It has been shown that AdS is linearly stable (Ishibashiand Wald
2004).The metric conformally approaches the static
cylindricalboundary, waves bounce off at infinity will return
infinite time.general result on the non-linear stability does not
yetexist for AdS.It is speculated that AdS is nonlinearly unstable
since inAdS one would expect small perturbations to bounce ofthe
boundary, to interact with themselves and lead
toinstabilities.Conjecture: small finite Q-balls of AdS in(3 +
1)-dimensional AdS eventually lead to theformation of black holes
(Bizon and Rostworowski 2011).Related to the fact that energy is
transferred to smallerand smaller scales, e.g. pure gravity in AdS
(Dias et al).
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Erogoregions and Instability
If the boson star mass M is smaller than mBQ onewould expect the
boson star to be classically stable withrespect to decay into Q
individual scalar bosons.Solutions to Einsteins field equations
with sufficientlylarge angular momentum can suffer from
superradiantinstability (Hawking and Reall 2000).Instablity occurs
at ergoregion (Friedman 1978) where therelativistic frame-dragging
is so strong that stationaryorbits are no longer possible.At
ergoregion infalling bosonic waves are amplifiedwhen reflected.The
boundary of the ergoregion is defined as where thecovariant
tt-component becomes zero, i.e. gtt = 0Analysis for boson stars in
(3 + 1) dimensions hasbeen done (Cardoso et al 2008, Kleihaus et al
2008)
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Stability of Boson Stars with AdS Radius very large
Q
0.2 0.4 0.6 0.8 1.0 1.21e +
0 21 e
+ 04
1 e+ 0
61 e
+ 08
Y= 0.00005Y= 0.00001
& Y & k0.1 & 0.00005 & 00.01 & 0.00005 &
00.001 & 0.00005 & 00.01 & 0.00001 & 20.001 &
0.000007 & 00.001 & 0.000007 & 4
Figure : Charge Q in dependence on the frequency for different
values of andY = 6.
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Stability for Boson Stars in AdS
Q
0.5 1.0 1.5 2.01e +
0 21 e
+ 04
1 e+ 0
61 e
+ 08
Y= 0.1Y= 0.01Y= 0.001
0.1,0.01,0.001,0.00010.1,0.01,0.001,0.00010.1,0.05,0.01,0.005,0.001
Figure : Charge Q in dependence on the frequency for different
values of andY = 6.
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Stability Analysis for Radial Excited Boson Stars inAdS
Q
0.4 0.6 0.8 1.0 1.2 1.41e +
0 21 e
+ 03
1 e+ 0
41 e
+ 05
k=0 k=1 k=2 k=3 k=5
& Y0.01 & 0.001
Figure : Charge Q in dependence on the frequency for different
excited modes k .Brihaye, Hartmann, Riedel, and Suciu GaussBonnet
Boson Stars in AdS
-
Summary Stability Analysis
Strong coupling to gravity: self-interacting rotatingboson stars
are destabilized.Sufficiently small AdS radius: self-interacting
rotatingboson stars are destabilized.Sufficiently strong rotation
stabilizes self-interactingrotating boson stars.Onset of
ergoregions can occur on the main branch ofboson star solutions,
supposed to be classically stable.Radially excited self-interacting
rotating boson starscan be classically stable in aAdS for
sufficiently largeAdS radius and sufficiently small
backreaction.
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Outlook
Analysis the effect of the Gauss-Bonnet term onstability of
boson stars.To do a stabiliy analysis similar to the one done
fornon-rotating minimal boson stars by Bucher et al
2013([arXiv:1304.4166 [gr-qc])See whether our arguments related to
the classicalstability of our solutions agrees with a full
perturbationanalysisGauss-Bonnet Boson Stars and
AdS/CFTcorrespondanceBoson Stars in general Lovelock theory
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS
-
Thank You
Thank You!
Brihaye, Hartmann, Riedel, and Suciu GaussBonnet Boson Stars in
AdS