Adiabatic radiation reactionto the orbits in Kerr Spacetime

Norichika Sago (Osaka)

qr-qc/0506092 (accepted for publication in PTP)NS, Tanaka, Hikida, Nakano

andpreparing paper

NS, Tanaka, Hikida, Nakano and Ganz

8th Capra meeting, 11-14 July 2005at the Coesner’s House, Abingdon, UK

Talk Plan

1. Introduction

2. Strategy

3. Background

4. Calculation of dE/dt, dL/dt and dQ/dt

5. Easy case

6. Summary and Future works

adiabatic approximation, radiative field

geometry, constants of motion, geodesics

evaluation of dE/dt, dL/dt, dQ/dt, consistency check

slightly eccentric, small inclination case

1. Introduction

We consider the motion of a particle orbiting the Kerr geometry.

))/(( 0MO • test particle case [ ]

A particle moves along a geodesic

characterized by E, L, Q.

In the next order, the particle no longer movesalong the geodesic because of the self-force!

GW

SMBHsolar mass CO M

Evolution of E,L,Q Orbital evolution

2. Strategy

• Adiabatic approximation

T : orbital period

: timescale of the orbital evolutionRR

RRT

At the lowest order, we assume that a particle movesalong a geodesic.

• Evolution of constants of motion

Use the radiative field instead of the retarded field.Take infinite time average.

T

T

i

T

iDD hF

u

IdI ][lim

1 rad

2 },,,{ advretrad

hhhQLEI i

• For E and L

The results are consistent with the balance argument.

Gal’tsov, J.Phys. A 15, 3737 (1982)

• For Q Mino, PRD 67,084027 (2003)

This estimation gives the correct long time average.

Key point

Radiative field is a homogeneous solution.

It has no divergence at the location of the particle.

We do not need the regularization!!

Justification

3. Background

• Kerr geometry (Boyer-Lindquest coordinate)

2222

222

22

22

dsinsin2

sin421

rMaar

drdtdMarMr

d

dtds

22222 2 , cos aMrrar

• Killing vector :

)1,0,0,0( , )0,0,0,1( )((t) μμ

0)(

• Killing tensor : 0)( K

grnlK 2

)( 2

sin 1, 0, ,sin

2 ,0 ,

2 ,

2 ,0 ,1 ,

cos2

1 , ,

2222 iia

aaraar

iarmnl

where

Constants of motion for geodesics

zz

zz MarMr

tuE

z

2(t) sin22

1

zz

zz aarMar

tuL

z

2z222222

)( sinsin)(

sin2 z

22222

22

cossin

)sin(zz

z

za

aELuuKQ

• Constants of motion

• Particle’s orbit and 4-velocity

d

dzu

rtz zzzz

)()()()()( , , , : proper time

d

d

There are three constants of motion for Kerr geometry.

In addition, we define another notation for the Carter constant:2)( LaEQC

Geodesic equations

• New parametrization

)(/ zrdd

• Equations of motion

)(coscos

),(

)(sin

)()sin(

22

2

222

zz

zz

zz

z

zz

zz

d

drR

d

dr

rPaL

aEd

d

rPar

LaEad

dt

r- and - componentare decoupled.

4222222

2222

cos)1(cos])1([)(cos

)()()( ],)([)(

EaLEaCC

QrrPrRaLarErP

(Mino ’03)

r - and - components of EOM

In this case, we can expand in the Fourier series.

n

nz

n

rrnz inzinrrr

r ]exp[~)(cos ,]exp[~)(

where

1

2

1

cos

0 4222222

2222

)1(])1([4

)(])([2

zEazLEaCC

dz

QraLarE

drr

rr

1121 , zz rrr

Assume that the radial and motion are bounded:

cosz

t - and - components of EOM

)sin()( 222

LaEarPar

d

dtzz

zz

• t - component of EOM

)sin()sin(

)()(

22)(

2222)(

LaEaLaEadt

rPar

rPar

dt

zz

zz

zzr

divide into oscillation term and constant term

d

dtttt

zrz )()()(

linear termperiodic functions

12 r

12

with period,

with period,

We can obtain - motion in a similar way.

d

d zrz )()()(

4. Evaluation of dE/dt, dL/dt and dQ/dt

T

Tzxt

Td

Td

dE)()(

2

1lim

We can replace –i after mode decomposition.

)(),(~),(~)()(

))(()(2

)( (in)

,

(in)

xrururx

zdxd

ix

mm

mm

m

• Formula of dE/dt (Gal’tsov ’82)

where

)(xm : mode function of metric perturbation

Here we define the above vector field as )()(~lim uxuzx

+ (up-field term)

nn

nninnm

ddmddtizm

r

rrrzz eZez ddt

,

)(,)//()( )(

Fourier transformation

After mode decomposition and -integration, we can obtain:

)( ,

,,,

2,

nnm

nnm

nnm

r

r

rZddt

dE

In a similar manner, we can obtain dL/dt:

nnm

nnm

nnm

r

rrZm

ddt

dL

,,,

,2, )(

nnddmddt rrzz

nnm

r 1,

)( ),(

)( ),(

zz

zzt

r: periodic function

: periodic function + linear term

Simplified dQ/dt formula

uuhhuugffuK

d

dQ)2)(( ,2 ;;2

1

)(

;; )~~(~~~2

zx

uKuKuuhuKd

dQ

This term vanishes in this case.

0 ,~~);(;; Kuu

T

Tzx

rz

tzz

Tx

d

draar

rPd

d

dQT

)(

22 )()()(

lim 21

2

~~)(

huux

• Formula of dQ/dt

Using the vector field, , we rewrite this formula.)(~ xu

reduce it by integrating by parts

Further reductionIn general for a double-periodic function,

Total derivative with respect to r ),;(),;()()(

r

r

z

r

z

r

z

rr

rzz d

dtd

dtrd

drdd rr

Using this relations, we can obtain:

nnm

nnm

nnm

rr

T

T zx

rrT

z

r

rrZn

d

dtdLaP

dtdEPar

xindTdt

dQ aPt

Parddt

,,,

,2,

22

)(

221

)(2

2)(

2

)(2

1lim )(

This expression is as easy to evaluate as dE/dt and dL/dt.

Consistency check

circular orbit case: )0(when 0, rnn

m nZ r

dtdLaP

dtdEPar

dt

dQ

2

)(2

22

This agrees with the circular orbit condition.(Kennefick and Ori, PRD 53, 4319 (1996))

equatorial orbit case:

Rewriting dQ/dt formula in terms of C=Q(aEL)2 :

nnm

nnm

nnmzz

r

rrZn

ddtdLL

dtdEEa

dt

dC

,,,

,2,222 )(2cot2cos2

2when 0

dt

dC

)0(when 0 and 2 , nZ nn

mr

An equatorial orbit does not leave the equatorial plane.

5. Easy caseWe consider a slightly eccentric orbit with small inclination.

0 ,0))1((0

0 rrdr

dRerR

Eccentricity e Inclination y2

2)( , LaEQC

L

Cy

e << 1, y << 1

In this case, we obtain the geodesic motion:

n

nz

n

rrnz inzinrrr

r ]exp[~)(cos ,]exp[~)(

)(~ ),(~ 2/

nn

nn yOzeOr r

r

If we truncate at the finite order of e and y,it is sufficient to calculate up to the finite Fourier modes.

The calculation have not been done yet but is coming soon !!

6. Summary and Future works

We considered a scheme to compute the evolution of constants of motion by using the adiabatic approximation method and the radiativefield.

We found that the scheme for the Carter constant can be dramaticallysimplified.

We checked the consistency of our formula for circular orbits andequatorial orbits.

As a demonstration, we are calculating dQ/dt for slightly eccentric orbitwith small inclination, up to O(e2), O(y2), 1PN order from leading term.

nnm

nnm

nnm

rr

r

rrZn

ddtdLaP

dtdEPar

dt

dQ

,,,

,2,22

)(22)(

2

• (more) Accurate estimate of dE/dt, dL/dt and dQ/dt

Analytic evaluation for general orbits at higher PN order(Ganz, Tanaka, Hikida, Nakano and NS)

Numerical evaluationHughes, Drasco, Flanagan and Franklin (2005)Fujita, Tagoshi, Nakano and NS: based on Mano’s method

• Evaluation of secular evolution of orbit

Solve EOM for given constants of motion

},,{ QLE })(cos,,{ 0 ier : orbital parameter

• Calculation of waveformUsing Fujita-Tagoshi’s code

Parameter estimation accuracy with LISA(Barack & Cutler (2004), Hikida et al. under consideration)

Future works