Adiabatic radiation reaction to the orbits in Kerr Spacetime
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Transcript of Adiabatic radiation reaction to the orbits in Kerr Spacetime
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