Self-force Problem: Past, Present, Future

1
Index 1 Introduction 2 Self-force in a
Linear Perturbation 3 Regularization of the
fields 4 Radiation Reaction and
Adiabatic Evolution 5 Summary and Future
Yasushi Mino (? ??) CGWA, University of Texas,
Brownsville E-mail mino_at_phys.utb.edu
2
1 Introduction
We want to calculate a gravitational wave form
from an extreme mass-ratio binary system for LISA
project.
The central black hole is considered to be a Kerr
black hole. For its extreme mass-ratio, we
expect that a linear perturbation is an effective
method of investigation.
3

• One can calculate the gravitational wave form by
  a linear perturbation, given an orbital evolution
  of the binary system.
• We need the orbital evolution of 106 rotations
  for one-year observation of gravitational waves.

Beyond 103 (104) rotations, the orbit deviates
from a geodesic by the secular effect of
radiation reaction.
4
We want to know the evolution equation of the
orbit, namely, we want to solve The self-force
problem. We can use the linear perturbation
since the instantaneous deviation from a geodesic
is small.
5
10 years ago
South Padre Island
6
2 Self-force in A Linear Perturbation
We consider a linear perturbation induced by a
point mass.
Use of a point particle may be a good
approximation, but, it causes a difficulty.
7
The metric perturbation diverges around the
particle.
R spatial distance between the field point
and the particle location

• Because of the divergence, the linear
  perturbation becomes invalid.
• The geodesic equation in the perturbed metric
  diverges at R -gt 0.
• Because of the divergence of the perturbed
  metric, one cannot define the motion in a usual
  manner.
• These problems were solved by the
  mass-regularization, the matched asymptotic
  expansion method, and an axiomatic approach.

8
Self-Force (MiSaTaQuWa Force)
full Metric perturbation induced by a
point particle
S-Part of perturbation, singular
R-Part of perturbation, regular
9
5 years ago
South Padre Island
10
3 Regularization of the fields
is difficult to derive directly, but, may be
easier to calculate.
We need a regularization calculation.

• A new type of regularization calculation
• A global technique to derive S-part is not known.
• We cannot use the spatial Fourier transformation.

11
We have developed a formula By the local
coordinate expansion of S-part, we can decompose
the divergent and non-vanishing part of it.
If we have the full metric perturbation as a sum
of harmonics, we can derive the self-force.
12
The practical application for LISA
templates is difficult, especially because, we
have to calculate the self-force at each orbital
points of number of orbits.
13
1 years ago
South Padre Island
14
4 Radiation Reaction and Adiabatic Evolution
Poor-mans method really poor?

• A calculation by the energy balance equation
• We approximate the orbit at a given instant by a
  geodesic of constants (E,L,C).
• Instead of integrating the orbital equation, we
  consider the evolution of these constants.

E
L
15
We consider a family of geodesics bounded by the
BH gravitational potential.
r
16

• r/q-motions are independent periodic motions

integral constants

• t-motion and f-motion

integral constant

• A family of geodesics is characterized by 6
  constants.

17
Geodesics around a Kerr black hole are
characterized by 6 constants

• Procedure
• We consider an evolution equation of (E,L,C)
  based on our understanding of the self-force.
• By integrating the orbital equation
  perturbatively, we derive the evolution of the
  rest of constants.
• We define the adiabatic evolution of the orbit.

18
Because of the periodicity of the orbit, the
self-force can be expanded as
By a symmetry of a Kerr BH,
As a result, we have
19
Periodicity of the self-force
Symmetry of BH
l
l
Future Light cone
lr
r
r
Past Light cone
20
One can formally calculate the orbital evolution.
Dominant terms are derived by calculating
21
Adiabatic Approximation
l
Linear Perturbation becomes invalid
Error is always small.
dE, dL, dC
l
Linear Perturbation becomes invalid
Error grows linearly in time.
dl, dl, dt, df
22
By taking the dominant part, we can define the
adiabatic evolution by the self-force.
These equations give a correct prediction of the
orbit up 106 (108) rotation.
23
1. One can calculate the time-averaged radiation
reaction to E,L,C, by using the radiative Green
function. The calculation method of this Green
function is known.
2. We prove that it is consistent in any gauge
condition.
24
Now
Spring Break, South Padre Island
25
5 Summary and Future
Brownsville is a nice place to visit.
26
5 Summary and Future
We now have finished a theoretical foundation
to make gravitational wave templates for LISA.
We are trying to make an efficient algorithm
for the waveform generation. We find that a
semi-analytic technique is useful especially for
an accurate calculation.
27
Real Cocktail for Cocktail Party Problem
Cocktail a la Barack Cutler 0,05 (non
alcohol)
Cocktail a la Hughes, Glampedakis Kennefick
10000,00
Cocktail a la CGWA 0,10