1 Yasushi Mino Theoretical AstroPhysics Including Relativity (TAPIR), CalTech Index 1: Introduction: LISA project 2: MiSaTaQuWa Self-force? 3: Adiabatic Metric Perturbation 4: Radiation Reaction Formula 5: Gauge and Validity 6: Conclusion Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula

2 Everyday we experience gravity. Why should we discuss the gravitational physics? Newtonian Gravity : Gravity as a potential No dynamical Freedom Einstein Gravity : Gravity as a geometry Dynamical Freedom in Gravity We want to know the dynamics of the gravitational physics! Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula 1: Introduction: LISA Project

3 The dynamical degree of freedom in Einstein’s Gravity propagates the space-time as gravity waves. When linearized, we have a wave equation of the metric perturbation. Detection of gravitational waves is a strong evidence of the dynamical nature of the gravitational theory. Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula

4 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula Einstein Gravity predicts a very strongly self-gravitating object, called “Black hole”. Newtonian Gravity : It makes a singularity if gravitationally collapsed. Einstein Gravity : The singularity is hidden by the horizon. r  We want to know the nature of the strong gravity! r  rHrH

5 Projects of Gravity Wave detection : Experimental test of the relativistic gravitational theory New observational window to distant astrophysical objects Observation of highly relativistic gravitational phenomena Promising Target: NS/BH Binary system, SuperNova, Primordial GWs, Pulser, GammaRayBurst, ….. Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula

6 Space Project: LISA (joint project by NASA & ESA) See LISA project homepage Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula

7 Space Project: LISA (joint project by NASA & ESA) See LISA project homepage Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula

8 Space Project: LISA (joint project by NASA & ESA) See LISA project homepage Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula

9 LISA primary target A compact object (~10M) is inspiralling into a super- massive blackhole (~10^6M). *extreme mass ratio *eccentric orbit *relativistic motion We need to know what gravitational waves are expected to be detected. Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula

10 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula Theory is challenged by Experiment. Unlike other theoretical physics, we do not (did not?) have a theory to predict the observation until recently!

11 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula 2: MiSaTaQuWa Self-force? It is a good approximation to consider it as a two-body problem in GR. The central black hole is considered to be a Kerr black hole. For its extreme mass-ratio, a linear perturbation might be a good approximation. M m

12 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula Suppose we could use a linear perturbation: We approximate the supermassive black hole by a Kerr black hole, and consider the linear metric perturbation induced by an inspiralling compact object. One can calculate the gravitational waveform by a linear perturbation, being given a orbit. We need to solve the orbital equation.

13 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula A binary system is known to emit gravitational waves, and the gravitational waves carry away the binding energy of the system. As a result, the orbit deviates from the background geodesic. MiSaTaQuWa self-force was derived by a linear perturbation. It is considered to include the radiation reaction effect to the orbit. (MiSaTaQuWa=Mino,Sasaki,Tanaka;Quinn,Wald)

14 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula We come to have a regularization method to evaluate MiSaTaQuWa self-force. … Radiation Reaction Formula (Mino) Mode-decomposition method Barack,Ori; Mino,Nakano,Sasaki; Detweiler,Messaritaki,Whiting,Kim Power-expansion method (Mino,Nakano) but … something weird …

15 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula Do we have to consider the self-force in a certain gauge? Is the orbital evolution gauge-dependent?

16 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula One may choose a gauge condition such that the self-force vanishes, and it is consistent with the linear perturbation! 1) The linear perturbation is derived by solving the linearized Einstein equation. 2) The linearized Einstein equation requires the conservation law in the background, i.e. a geodesic.

17 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula 3) The linear approximation is valid only when the orbital deviation from a geodesic is small. 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. 4) Because the orbital deviation from a geodesic is small, one can bring the orbit’s coordinates back to those of the geodesic.

18 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula Conclusion: There is no self-force. Problem: We have to extend “the linear perturbation formalism” so that it can describe the metric perturbation induced by a non-geodesic orbit. Conclusion: MiSaTaQuWa self-force makes no physically meaningful prediction of the orbital evolution by itself.

19 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula 3: Adiabatic Metric Perturbation We consider a quasi non-linear extension of the linear metric perturbation so that one can describe the metric perturbation induced by a non-geodesic orbit. For this, we use 1) a physically reasonable class of gauge conditions, 2) the picture of adiabatic approximation. The adiabatic approximation is well known in classical mechanics, but, the application to a classical gauge field is not well known.

20 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula Kerr black hole is a stationary solution of the Einstein equation, thus, the linearized Einstein equation of a Kerr background is time-independent. We call this a physically reasonable class of gauge conditions. One can easily extract the physical information of gravitational waves. Technically feasible metric perturbation formalisms belong to this class.

21 Step 2: We approximate the orbit by a geodesic on each foliation hypersurface. x t Step 1: We consider the spacelike foliation. Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula 66 44 33 22 11 55 Step 3: We patch the linear metric perturbations of geodesics on each foliation surface. h6h6 h5h5 h4h4 h3h3 h2h2 h1h1 Adiabatic extension of the linear perturbation:

22 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula Linear perturbation:  : a geodesic Adiabatic metric perturbation:  (t) : time-evolving geodesic Adiabatic metric perturbation is valid as long as the extra term  is sufficiently small. (around a year) Linear perturbation is valid as long as the orbit does not deviate from a geodesic. (around a week)

23 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula 4: Radiation Reaction Formula Radiation Reaction Formula was originally formulated by a linear perturbation in the physically reasonable gauge. (Strictly speaking, all the formula so far is based on the linear perturbation, and we cannot make any physical interpretation.) For a meaningful discussion of the orbital evolution, we have to consider the adiabatic extension of Radiation reaction formula in a manner consistent with the adiabatic metric perturbation.

24 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula 4-a: “Original” Radiation Reaction Formula We consider a geodesic. Geodesics are characterized by 6 parameters; Primary constants ; Secondary constants ;

25 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula r/  -motiont/  -motion Asymptotic gravitational field in the physically reasonable class of gauge conditions; : three principal frequencies, functions of (E,L,C)

26 We consider the self-force acting on (E,L,C). In the physically reasonable class of gauge conditions, the linear metric perturbation gives the force as Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula It is proven that the zero-mode is gauge-invariant and can be obtained by the radiative part of the field.

27 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula Perturbative evolution of the orbital constants becomes; One can see that the perturbative evolution of the orbit and the linear metric perturbation becomes in valid at the dephasing time scale; (around a week)

28 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula 4-b: “Adiabatic” Radiation Reaction Formula Now the “constants” could evolve non-perturbatively; In the physically reasonable class of gauge conditions, the adiabatic metric perturbation gives the force as (We ignore O(  ^2) terms here.)

29 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula We have the adiabatic evolution equation of the orbit. (See the upcoming paper in detail.) The behavior of these evolution equations; Einstein Equation for the adiabatic metric perturbation;

30 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula Validity of the adiabatic metric perturbation; Validity of the adiabatic orbital evolution; Beyond this time scale, we do not solve the Einstein equation to the accuracy O(  ). Beyond this time scale, the second order effect will change the orbital evolution. This time scale is around several months for LISA.

31 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula 5: Gauge and Validity is proven to be gauge invariant in the physically reasonable class. What about, non-zero components? We found that the non-zero modes are totally gauge dependent. By a special choice of gauge, one can eliminate the non-zero mode.

32 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula We call this gauge condition the radiation reaction gauge. In this gauge, the self-force has only the dissipative term. This DOES NOT mean that the self-force does not have conservative terms. The conservative effect of the self- force is renormalized into the initial conditions.

33 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula General gauge in the physically reasonable class; Radiation reaction gauge; (v) is a typical velocity of the system, and is 0.3 at most.

34 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula Validity of the adiabatic metric perturbation; Validity of the adiabatic orbital evolution; Beyond this time scale, we do not solve the Einstein equation to the accuracy O(  ). Beyond this time scale, the second order effect will change the orbital evolution. This time scale is around several years for LISA.

35 Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004 Self-Force in Radiation Reaction Formula 6: Conclusion A method for an orbital prediction using a linear metric perturbation is established for the first time. The method can predict the orbital evolution for long enough for LISA project. The calculation technique of the method proposed here is already established and the required computational power is minimum. Coding to calculate the waveform is in progress.