1. Adiabatic radiation reaction to the orbits in Kerr Spacetime
Norichika Sago (Osaka)
qr-qc/ (accepted for publication in PTP)
NS, Tanaka, Hikida, Nakano
and
preparing paper
NS, Tanaka, Hikida, Nakano and Ganz
8th Capra meeting, July 2005
at the Coesner’s House, Abingdon, UK

2. Talk Plan Introduction Strategy Background
Calculation of dE/dt, dL/dt and dQ/dt
Easy case
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

3. 1. Introduction [ ] Evolution of E,L,Q Orbital evolution
We consider the motion of a particle orbiting the Kerr geometry. [ ]
test particle case
A particle moves along a geodesic
characterized by E, L, Q. GW
SMBH
solar mass CO
In the next order, the particle no longer moves
along the geodesic because of the self-force!
Evolution of E,L,Q
Orbital evolution

4. 2. Strategy Adiabatic approximation
T : orbital period
: timescale of the orbital evolution
At the lowest order, we assume that a particle moves
along a geodesic.
Evolution of constants of motion
Use the radiative field instead of the retarded field.
Take infinite time average.

5. Radiative field is a homogeneous solution.
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
For E and L
Gal’tsov, J.Phys. A 15, 3737 (1982)
The results are consistent with the balance argument.
For Q
Mino, PRD 67, (2003)
This estimation gives the correct long time average.

6. 3. Background Kerr geometry (Boyer-Lindquest coordinate)
Killing vector :
Killing tensor :
where

7. Constants of motion for geodesics
There are three constants of motion for Kerr geometry.
Particle’s orbit and 4-velocity
: proper time
Constants of motion
In addition, we define another notation for the Carter constant:

8. Geodesic equations New parametrization Equations of motion
(Mino ’03)
Equations of motion
r- and q - component
are decoupled.

9. r - and q - components of EOM
Assume that the radial and q motion are bounded:
In this case, we can expand in the Fourier series.
where

10. t - and f - components of EOM
t - component of EOM
divide into oscillation term and constant term
periodic functions
linear term
with period,
with period,
We can obtain f - motion in a similar way.

11. 4. Evaluation of dE/dt, dL/dt and dQ/dt
Formula of dE/dt
(Gal’tsov ’82)
We can replace –iw after mode decomposition.
where
+ (up-field term)
: mode function of metric perturbation
Here we define the above vector field as

12. Fourier transformation
: periodic function
: periodic function + linear term
Fourier transformation
After mode decomposition and l-integration, we can obtain:
In a similar manner, we can obtain dL/dt:

13. Simplified dQ/dt formula
Formula of dQ/dt
Using the vector field, , we rewrite this formula.
This term vanishes in this case.
reduce it by integrating by parts

14. Further reduction
In general for a double-periodic function,
Total derivative with respect to lr
Using this relations, we can obtain:
This expression is as easy to evaluate as dE/dt and dL/dt.

15. Consistency check
circular orbit case:
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-(aE-L)2 :
An equatorial orbit does not leave the equatorial plane.

16. 5. Easy case e << 1, y << 1
We consider a slightly eccentric orbit with small inclination.
e << 1, y << 1
Eccentricity e
Inclination y
In this case, we obtain the geodesic motion:
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 !!

17. 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 radiative
field.
We found that the scheme for the Carter constant can be dramatically
simplified.
We checked the consistency of our formula for circular orbits and
equatorial orbits.
As a demonstration, we are calculating dQ/dt for slightly eccentric orbit
with small inclination, up to O(e2), O(y2), 1PN order from leading term.

18. Future works (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 evaluation
Hughes, 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
: orbital parameter
Calculation of waveform
Using Fujita-Tagoshi’s code
Parameter estimation accuracy with LISA
(Barack & Cutler (2004), Hikida et al. under consideration)