1 Yasushi Mino ( 蓑 泰 志 ) WUGRAV, Washington University at St. Louis 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 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System

2 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System 1: Introduction We want to calculate the 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 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System 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 10^8 rotations for one-year observation of gravitational waves. Beyond 10^3 rotations, the orbit deviates from a geodesic by the secular effect of radiation reaction.

4 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System 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 …

6 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System 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 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System The metric perturbation diverges around the particle. R : spatial distance between the field point and the particle location The geodesic equation in the perturbed metric diverges at R -> 0. Because of the divergence, the linear perturbation becomes invalid. These problems were solved by the mass-regularization, the matched asymptotic expansion method, and an axiomatic approach.

8 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System Self-Force (MiSaTaQuWa Force) : full Metric perturbation induced by a point particle : R-Part of perturbation, regular : S-Part of perturbation, singular

9 5 years ago …

10 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System 3: Regularization of the fields 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 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System 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 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System A self-force calculation is successful in a Schwarzschild background in a simple manner. A calculation in a Kerr background comes to be possible in principle. The practical application is difficult, especially, in calculating the full metric perturbation.

13 1 years ago …

14 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System 4: Radiation Reaction and “Adiabatic” Evolution 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 Poor-man’s method… really poor?

15 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System Geodesics around a Kerr black hole are characterized by 6 constants; 1.We consider an evolution equation of (E,L,C) based on our understanding of the self-force. 2.By integrating the orbital equation perturbatively, we derive the evolution of the rest of constants. 3.We derive the “adiabatic evolution” of the orbit.

16 We consider a family of geodesics bounded by the BH gravitational potential. r Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System

17 r/  -motions are independent periodic motions Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System integral constants; t-motion and  -motion integral constant; A family of geodesics is characterized by 7 constants.

18 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System Because of the periodicity of the orbit, the self-force can be expanded as  E,  L,  C Linear Perturbation becomes invalid

19 , ,  t,  Linear Perturbation becomes invalid Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System In the short time scale (of the order of the dynamical time scale), the orbit just exchanges the energy with radiation. In the long time scale, the orbital energy radiates away, and the orbital energy tends to lose.

20 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System 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 10^9 (10^6) rotation.

21 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. Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System 2. We prove that it is consistent in any gauge condition.

22 Now …

23 Gravitation: A Decennial Perspective (6/8-6/12/2003, PennState) Gravitational Waves from Extreme Mass-Ratio Binary System 5: Summary and Future We now have finished a theoretical foundation to make gravitational wave templates. We need; 1.to make the data analysis strategy. 2.to make a program to generate a template bank ^8? 10^14 templates?... A semi-analytic method?

24 We still have a long way…