A formulation for the long-term evolution of EMRIs with linear order conservative self-force

1. A formulation for the long-term evolution of EMRIs with linear order conservative self-force Gravitational waves Takahiro Tanaka 　　(Kyoto university) in collaboration with R. Fujita, S. Isoyama, A. Le Tiec, H. Nakano, N. Sago

2. Order of m in wave form Energy balance argument is sufficient. Wave form for quasi-circular orbits, for example. linear perturbation leading order (O(m -1) phase) next leading order (O(1) phase) only up to here

3. Gauge invariance Perturbation is everywhere small outside the world tube Particle’s trajectory “tube radius” >> m Unavoidable ambiguity in the perturbed trajectory of O(m) h~m/r “Self-force is gauge dependent” has unnecessary information. Source trajectory While, “long term orbital evolution is gauge invariant” There must be a concise description keeping only the gauge invariant information

4. Use of canonical transformation Interaction Hamiltonian It is natural to change the variables to the constants of motion in the background and their conjugates Xa. Generating fn: … H(0)=u2/2, and hence X 0is t and Xi( i=1,2,3 ) are all constant for background geodesics.

5. Radiation reaction to the constants of motion “retarded” = “radiative” + ”symmetric” no need for regularization Geodesic preserving transformation (Mino transformation): t → -t, f→ -f,t→ -t G(ret)→ G(adv) Xm→ -Xm, Only radiative part contributes to the change of “constants of motion” except for resonance orbits. (last Capra) which means “Orbits with different values of X are basically equivalent.” For resonance orbits, X3 =Dl has physical meaning. r l(Mino’s time) q Dl Dl

6. Second canonical transformation {X} is not a good set of variables to see the orbital phase evolution. Small change of P, with fixed X, at a late time large variation of x Further canonical transformation: (X,P) → (q,J) action-angle variables

7. Physical meaning of the angle variables After nrand nq cycles, qr= 2p nr, qq= 2p nq, irrespective of . Small change in J (or in P) with fixed Xi small variation of x t - qt is a periodic function w.r.t. qr,qq. qm is gauge invariant in the context of long term evolution. which allows an O(m ) error at each time, but the error should not accumulate.

8. Phase velocity “averaged change rate of qm ” = “phase velocity” almost directly related to the phase of observed GWs ? differentiation of a fn of Jm averaged after force calculation Mino transformation Jm→Jm Xi( i=1,2,3 ) are all constant for the background geodesic. Here, not but . Can we insist ? This term should be removed

9. Perturbation of generating function Yes! “Jm=constant” can be realized by an appropriate gauge transformation, xm→ xm – x m , without secular growth of x m. The same transformation can be achieved by ∵ ~ Additional dWmakes Jm constant. ~ L.h.s. is gauge inv. (dq caused by dW is purely oscillatory) while r.h.s. may look a gauge-independent fn. of J. J was originally gauge dependent but must be gauge invariant.

10. Why does fix the gauge completely? Let’s consider the above gauge transformation at the reflection point of Mino transformation, where . From the symmetry, at the reflection point. Normalization condition By considering two approximate reflection points corresponding to rmin and rmax, we can conclude that constant shift of Ja is not allowed.

11. Conclusion We discussed the effect of long-term evolution due to first order self-force. Radiative part requires no regularization. The contribution of symmetric part is concisely encoded in gauge-invariant interaction Hamiltonian: Evolution of Q in the resonance case can be also described by a similar quantity: Instead of the direct computation of the self-force, alternative simple regularization based on Hint might be handy. { Scalar quantity. Lower order differentiation. Time integral can be performed first.