Yasushi Mino ( 蓑 泰志) Center for Gravitational Wave Astronomy University of Texas at Brownsville

1. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Index 1: Introduction: LISA project 2: Self-force calculation 3: Radiation Reaction Formula 4: Gauge Problem 5: Adiabatic Expansion 6: Conclusion Yasushi Mino(蓑 泰志) Center for Gravitational Wave Astronomy University of Texas at Brownsville E-mail : mino@phys.utb.edu

2. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem 1: Introduction: LISA Project • LISA is a project for GW detection in space, proposed by ESA and NASA, starting 201X.. • Because of the long baseline and because it is free from the seismic noise, it is sensitive to the low-frequency gravitational waves. See LISA project homepage http://lisa.jpl.nasa.gov/ See also the upcoming file Intersteller by S.Spielberg.

3. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Promising Targets of LISA: Massive objects… supermassive Black Hole, Primordial GWs, QSO, Among them, GWs from the so-called EMRIs are the primary target of LISA.

4. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem LISA primary target: EMRI (= Extreme-Mass Ratio Inspirals) the inspiralling binary of a supermassive BH (10^5-7M) and a compact object (1-10M) • Event rate is expected to be more than one per year. • “Waveforms are calculable.” • Strong evidence for the presence of black holes • Standard candles for Observational Cosmology (Observable area is almost the entire universe) For the GW detection by matched filtering and for the extraction of the astrophysical information, it is necessary to establish a theoretical method to predict waveforms.

5. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Problem: How do we calculate the gravitational waveforms? Extreme mass ratio Black hole perturbation formalism Metric: Einstein Eq. m Perturbed Einstein eq. M

6. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Due to the BH uniqueness theorem, the black hole must be a Kerr black hole. Due to the radiation reaction, the orbital motion is dissipative. • Dynamical time • Radiation reaction time • Dephasing time • LISA observation time To predict the waveform during the LISA observation time, it is necessary to consider the radiation reaction effect to the orbit.

7. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem 2: Self-force calculation By the post-Newtonian calculation, we usually calculate the gravitational energy flux and calculate the orbital evolution by Balance formula We cannot use the same approach for the orbit around the Kerr black hole. OK! Kerr geodesic; NO!

8. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem 2: Self-force calculation By the post-Newtonian calculation, we usually calculate the gravitational energy flux and calculate the orbital evolution by Balance formula We cannot use the same approach for the orbit around the Kerr black hole. OK! Kerr geodesic; NO! We use the MiSaTaQuWa self-force.

9. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem MiSaTaQuWa self-force; A general formalism for the self-force acting on a particle: • The particle moves in a curved vacuum background. • It is only for the leading order. (linear perturbation) • If the particle is small enough, it can be represented by a “point” particle. • The metric perturbation induced by the particle is divergent along the orbit and the regularization to remove the divergence is prescribed within the classical framework. • It is formulated under the harmonic gauge condition. • The singular part is derived only around the orbit. • The tail effect of the wave propagation induces the force.

10. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem MiSaTaQuWa self-force; Barack-Mino-Nakano-Ori-Sasaki method; full metric perturbation …? Schwarzschild BH; convenient in Regge-Wheeler gauge possible in the harmonic gauge, but, need numerics. Kerr BH; “convenient” in Radiation gauge in the harmonic gauge … in progress (2D numerics)

11. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem We have a good progress in the self-force calculation, however, it will need too much computational power: • Even possible, the calculation of the metric perturbation in the harmonic gauge is difficult. A present idea is to implement 2D code. • One has to calculate the self-force at each orbital point. • The convergence is not good. We may need as much as 10^18 templates!

12. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem 3: Radiation Reaction Formalism What is the relation b/w Balance formula & MiSaTaQuWa self-force? Hint by Galt’sov : radiative field time average Geodesics around the Kerr black hole have symmetry. A geodesic is transformed into itself. The self-force changes the sign.

13. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem t t Past Light cone x x Future Light cone

14. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem The radiative self-force is expressed with the two-point average of the retarded self-force. l l r r

15. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Time averaged self-force is derived by the radiative field. Easy to calculate: • regularization without regularization • most efficient convergence • gauge invariant exponential convergence There is a convenient method even for a Kerr black hole.

16. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Is the time averaged self-force sufficient to predict the orbital evolution? The evolution of (E,L,C) by the entire self-force under a reasonable gauge; E,L,C t E,L,C Under the radiation reaction formula; t

17. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem E,L,C E,L,C “Radiation reaction approximation” t t F We fail the prediction!! t

18. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem E,L,C E,L,C Radiation reaction formalism t t F We succeed the prediction!! t Radiation gauge condition; it exists only when

19. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem 4: Gauge Problem We use the “Adiabatic Approximation” for the orbital prediction. t At each instant, the orbit is a almost background geodesic. Using this geodesic, we calculate the self-force at each instant. But, at an instant, the self-force is entirely gauge dependent…. Can this problem be solved by higher order perturbation expansion? x

20. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Fools’ derivation of the self-force; The linear metric perturbation requires the geodesic as the source. By the standard metric perturbation, we expand the metric and the orbit with the geodesic as a background. The expansion is valid only when the orbital deviation is small. t zgeodesic z x

21. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem By the gauge transformation, one can always bring the full orbit into the background geodesic during the entire region where the perturbative expansion is valid. By choosing this gauge condition, the self-force is vanish to the full order of the perturbative expansion. t x Note: Gauge is a freedom to assign the coordinates to a perturbed geometry. It has nothing to do with the causality or hyperbolicity of the Einstein equation.

22. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Fools’ derivation of the self-force casts two questions: Q-1) Should we use some specific gauge condition for the self-force? Q-2) The standard perturbation expansion is valid only in the dephasing time. Can we define a consistent perturbation expansion valid long enough for the waveform prediction?

23. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Q-1) Should we use some specific gauge condition for the self-force? … Basically NO! The waveform is observable and gauge invariant. But, the orbit is not directly observable, therefore, it can be gauge dependent. The modulation of the waveform due to radiation reaction also comes from the non-linear term.

24. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem There is no physical reason to use some specific gauge condition for the self-force. MiSaTaQuWa self-force has no direct physical meaning by itself. However, Practically YES! By using a specific gauge condition, the self-force can have a physically meaningful information, which is the phase modulation. Paper in progress... If the metric perturbation is obtained in the form, the self-force carries the physical information of the phase modulation.

25. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem 5: Adiabatic Expansion A new metric perturbation scheme for the waveform calculation; • It must not expand the orbit from the background geodesic. • It must have the picture of “adiabatic approximation” to the leading order. One can formulate a new metric perturbation scheme by two-scale expansion.

26. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem 1st step … the source term We use the point source. Instead of the orbital coordinates, we consider the evolution of the orbital elements. 2nd step … the foliation We introduce the foliation and use it for the orbital parameter. 3rd step … the metric perturbation The metric perturbation is defined to the function of the orbital elements. ( ; Source term for a geodesic)

27. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Ansatz of the two-scale (derivative) expansion; For example, a partial derivative becomes C is effectively regarded as constant. Higher order term 0th order equation becomes which is the linear perturbation in the picture of the adiabatic approximation

28. t x Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Adiabatic Approximation; • We assign the orbital elements, C(t), at each instant. • On the foliation, f, we use the linear metric perturbation induced by the geodesic of C(f). C6 h=h(C6) C5 h=h(C5) C4 h=h(C4) For the purpose of the self-force, the choice of the foliation function is not relevant. h=h(C3) C3 C2 h=h(C2) h=h(C1) C1

29. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem 1st post-adiabatic equation becomes schematically Effect of the orbital deviation from the geodesic

30. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem 6: Conclusion Self-force calculation Possible, but, the technique is not yet established not an effective calculation Radiation reaction formalism well established, most effective One can use a semi-analytic method, but, not yet applied.

31. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Theoretical question: Is the adiabatic expansion convergent? What is the “right” choice of the foliation? What is the “right” constraint on the gauge? Astrophysical Question: What is the validity in using the adiabatic approximation? … PN estimation says YES for EMRI (circular orbit) Can we have the evidence of the BH uniqueness? How about Intermediate-mass-ratio inspirals? Practical Question: How efficient can we calculate the waveforms? What is the best data analysis strategy?

32. Workshop on EFT Techniques in Grav. Wave Physics, Carnegie Mellon University, July 23-25, 2007 Recent issues of the self-force problem Question to EFT people Adiabatic expansion must be taken into account in NRGR method because it is systematic formulation of the particles & metric. … What foliation do you use?