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?

For the charged Vaidya black hole, horizons r± can be inferred from the null hypersurface condition

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

40 60 80 100 1201.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

R

v

r=10 r=20 r=30

q=0,l=2 vvM 001.05.0)(

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]

QNM in Black Strings

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.

Colliding Black Holes

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

Thanks!!