Some Interesting Topics on QNM QNM in time-dependent Black hole backgrounds QNM of Black Strings QNM...
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Transcript of Some Interesting Topics on QNM QNM in time-dependent Black hole backgrounds QNM of Black Strings QNM...
Some Interesting Topics on QNM
QNM in time-dependent Black hole backgrounds
QNM of Black Strings
QNM of colliding Black Holes
The perturbation equations
• The perturbation is described by
Incoming wave
transmitted reflected wave
wave
Tail phenomenon of a time-dependent case
• Hod PRD66,024001(2002)
V(x,t) is a time-dependent effective curvatue potential which
determines the scattering of the wave by background geometry
QNM in time-dependent background
• Vaidya metric
• In this coordinate, the scalar perturbation equation is
Where x=r+2m ln(r/2m-1) […]=ln(r/2m -1)-1/(1-2m/r)
the charged Vaidya solution
)sin(2))()(2
1( 222222
22 ddrcdvdrdv
r
vQ
r
vMds
the Klein-Gordon equation
0)( vgg
ml
lm rYvr,
/),(),(
How to simplify the wave equation ?
ttie )(~ Solution of
)(t Is dependent on initial perturbation?
the wave equation
(1) the genericalized tortoise coordinate transformation
02)1(***** ,1,,2 Vc rvrrr
0limlimlim
0
0
0
0
0
0 )(1
)(2
)(
V
vvvrr
vvvrr
vvvrr
0** ),ln(2
1)ln(
2
1vvvrr
krr
krr
0limlimlim
0
0
0
0
0
0 )(1
)(2
)(
V
vvvrr
vvvrr
vvvrr
rckrr
rrc
kr
rcrMr
rckrr
rrc
kr
rcrMr
21112
21112
2
2
the following variable transformation
the wave equation
*** ),,( vvvruu
2
1 2
*
*
dv
dr
0)2( 1
*,
Vr
uuv
1Q
When Q0
invalidate
0** ),ln(2
1)ln(
2
1vvvrr
krr
krr
the wave equation
(2) the genericalized tortoise coordinate transformation
0** ),ln(2
1vvvrr
krr
02)1(***** ,1,,2 Vc rvrrr
)2/(])(22)2(2[ 222 krrrrMrrMrQk
rkr
kMr
rrk
rrk 221
2
2
421
1)(2
)(2
22
21 )1(
)1)(2(
2)21(2
1)(2
)(2
r
ll
rrrk
krk
rrrk
rrkV
)(])()()(2[
)()()(
02
000
2000
vrvQvMvr
vQvMvrk
0limlimlim
0
0
0
0
0
0 )(1
)(2
)(
V
vvvrr
vvvrr
vvvrr
Limit to RN black hole For the slowest damped QNMs
QMq /
2
3
2
2/1
22
1
22
2/1
22
9
81)
9
811(
2
3)
9
811(
2
3
2
3),(
9
81
2
92
2
9)
9
811(
2
3)
2
1(),(
qqqqqlcM
qqqqlqlcM
II
RR
),2( qcR IM ),2( qc Iq - -
0 0.483 0.481 0.0965 0.0962
0.7 0.532 0.530 0.0985 0.0981
0.999 0.626 0.624 0.0889 0.0886
RM
numerical result
linear model
event horizon
)()(
)1()(
11
100
00
vqMvQ
vvm
vvvvm
vvm
vM
02
0
2
1
2
12
1
,21
)21(1
,21
)21(1
1
vvqMMr
vvr
rqMMr
vvr
rqMMr
vvqMMr
1c
5.00 m 002.0 00 v 1501 v 35.01 m
0.0 0.2 0.4 0.6 0.8 1.0-50
0
50
100
150
200
r+
r+r
+
r-
r-
time
v
horizon r+ and r
-
M=0.5-0.001v q=0 q=0.7 q=0.999
20 40 60 80 100-25
-20
-15
-10
-5
0
ln|
|
v
M=0.5-0.001v M=0.5 M=0.5+0.001v
q=0,l=2, evaluated at r=5, initial perturbation located at r=5 00 v
q=0,l=2, r=5 vvM 001.05.0)( 35.05.0:
1500:
M
v
25 50 75 1000.8
0.9
1.0
1.1
1.2
R
v
M=0.5+0.001v q=0 q=0.7 q=0.999
M , the oscillation period becomes longer
q=0,l=2, r=5 vvM 001.05.0)(
25 50 75 100-0.19
-0.18
-0.17
-0.16
-0.15
I
v
M=0.5+0.001v q=0 q=0.7 q=0.999
M , The decay of the oscillation becomes slower
)(
),()(,
)(
),()(
v
qlcvvM
v
qlcvvM
I
I
R
R
25 50 75 1000.50
0.52
0.54
0.56
0.58
0.60
c R(2
,q)/ R
v
M=0.5+0.001v q=0 q=0.7 q=0.999
are nearly equal for different q
The slope of the curveis equal to the
)(
),()(,
)(
),()(
v
qlcvvM
v
qlcvvM
I
I
R
R
25 50 75 1000.50
0.52
0.54
0.56
0.58
0.60
c I(2,q
)/ I
v
M=0.5+0.001v q=0 q=0.7 q=0.999
0 20 40 60 80 100 120 1400.5
0.6
0.7
0.8
0.9
1.0
q=0.999
v'
c R(2
,q)/ R
o
r c I(2
,q)/ I
v
cR/
R for M=0.5+v/300
cI/
I for M=0.5+v/300
cR/
R for M=0.5exp[ln(2)v/150]
cI/
I for M=0.5exp[ln(2)v/150]
the branes are at y = 0, d.Metric perturbations satisfy
Here m is the effective mass on the visible brane of the Kaluza-Klein (KK) mode of the 5D graviton.
Then the boundary conditions in RS gauge are
For this zero-mode, the metric perturbations reduceto those of a 4D Schwarzschild metric, as expected..
For m not 0, the boundary conditions lead to a discretetower of KK mass eigenvalues,
Radial master equations. We generalize the standard 4D analysis to find radial master equations for a reduced set of variables, for all classes of perturbations.
The total gravity wave signal at the observer (x = x_obs)is a superposition of the waveforms ψ(τ) associatedwith the mass eigenvalues m_n.
WE present signals associated with the four lowest massesfor a marginally stable black string.
Can QNM tell us EOS
• Strange star
• Neutron star
Stars: fluid making up star carry oscillations,Perturbations exist in metric and matter quantities over all space of star