Self-Force and the m-mode regularization scheme

1. BriXGrav, Dublin 2010 Self-Force and the m-mode regularization scheme Sam Dolan (with Leor Barack) University of Southampton

2. TALK OVERVIEW • Motivation: Gravitational Waves and EMRIs • Introduction to Self-Force • The m-mode scheme • Implementation: scalar charge, circular orbit, Schwarzschild spacetime • Results + Future Directions

3. 1. MOTIVATION Gravitational Waves and Extreme Mass Ratio Inspirals

4. Evidence for Gravitational Waves:The Hulse-Taylor Binary Pulsar • Doppler shift in pulsar period (59ms) • => Orbital period (7.75hr) • Cumulative shift of 40 sec over 30 yr • Energy loss due to GW emission • => Inspiral Figure from Weisberg & Taylor (2004)

5. Electromagnetic Waves Gravitational Waves • Coherent • Bulk dynamics • Amplitude: h ~ 1 / r • Not scattered or absorbed • λ > diameter • “Hearing” • All-sky, stereo • Incoherent, 1023 emitters • Thermodynamic state • Intensity: I ~ 1 / r2 • Scattered and absorbed • λ << diameter • Imaging, focussing • Narrow field vs

6. Sources of Gravitational Waves • Binary neutron star mergers, ~ 1.5 Ms • Stellar mass black hole mergers, ~ 10 Ms • Supermassive black hole mergers, ~ 107 Ms • Extreme Mass Ratio Inspirals (EMRI), ~ 10 Ms and ~ 106 Ms

7. EMRI dynamics and LISA • LISA will see ~10to 1000s inspirals simultaneously • Scattering of NS or BH into highly eccentric orbits • Eccentric until plunge • ~105 wave cycles over final year • Waveforms => Physics (e.g. “map” of near-horizon geometry, no-hair theorem, parameter estimation) t=-106 yr t=-1yr t=0

8. EMRI Timescales • e.g. M ~ 106 Ms and m ~ 10 Ms • To track orbital phase over Tobs, need high accuracy:

9. log(r/M) Post-Newtonian Theory 4 3 2 Numerical Relativity Perturbation Theory 1 log(M/m) 1 2 3 4 Two-Body Problem in GR: regimes

10. 2. SELF-FORCE Radiation Reaction in Curved Spacetime

11. Radiation Reaction in Classical Electromagnetism • Accelerated charge => radiation • Loss of energy => force acting on charge • Self-Force: charge interacts with own field • Point charge  infinite field … interpretation? • Regularization method needed

12. Dirac’s approach: • “S” is infinite but symmetric on worldline => mass renormalisation • “R” is regular on worldline => self-force

13. EM Self-Force in Flat Spacetime

14. Flat Curved Self-Force in Curved Spacetime: Problem of Regularization In flat spacetime, Green’s function has support on light-cone only In curved spacetime, Green’s function also has a tail within the lightcone Difficulty: Local Radiative potential becomes non-causal in curved space!

15. EM Self-Force in Curved Spacetime Local “instantaneous” terms tail integral over past history of motion DeWitt & Brehme (1960)

16. Self-Force Derivations E.M.: DeWitt & Brehme (1960) Gravitational: Mino, Sasaki & Tanaka (1997) Scalar: Quinn (2000) Example: Matched Asymptotic Expansions Near zone: Far zone: Match in buffer zone M >> r >> m, to obtain equation of motion

17. 3. SF CALCULATIONS: m-mode regularization in 2+1D

18. l = 1 l = 2 l = 3 l = 0 + + + + … + + + + … m = 0 m = 1 m = 2 m = 3 Two Mode-Sum Methods • “l-mode” regularization (spher. sym, e.g. Schw.) Decompose field into spherical harmonics, then regularize mode sum over l • “m-mode” regularization (axisymmetric, e.g. Kerr) Decompose field into exp(imϕ), introduce puncture field, then sum over modes.

19. m-mode decomposition • Kerr perturbation not separable in 1+1D • 2+1D decomposition • Evolve in time domain using finite difference scheme. • Problem: each m-mode diverges logarithmically at particle position • Resolution: analytically expand divergence; introduce a puncture function, leaving a regular residual.

20. Puncture Scheme • Idea: The divergence at the particle has a simple logarithmic form; subtract it out and evolve the regular residual field delta-function source on particle worldline Extended source, without distributional component regular on worldline

21. Puncture Function Construction (Barack, Golbourn & Sago 2007) Puncture field: Choose such that: where Σ: t = t0 r=r0 worldline

22. Puncture Function (II) • Some freedom in choice of • We choose • where • m-mode decomposition: Functions of orbital parameters only Elliptic Integrals

23. Puncture and World-tube • Construct world-tube around particle • Inside the tube, solve for • with extended source • Outside the tube, solve vacuum eqns for • The self-force is found from derivative of residual field at particle position: world-tube t θ world-line r mode sum converges

24. First Implementation: Scalar Field, Circular Orbits, Schwarzschild • 2+1D time evolution on 2+1 grid: u = t + r*, v = t - r*, θ • 2nd order accurate finite difference scheme

25. ImplementationDetails • Causal grid, arbitrary i.c. • Boundary condition at poles • No radial boundary condition • High-res runs on Iridis3 HPC (in Southampton). • Stability constraint => max. angular resolution • Resolution x2 => runtime x8, memory x4 u v

26. 4. FIRST RESULTS

27. 1. Puncture Field Evolution m=0 m=1 m=2 r*/M q (Barack & Golbourn 2007)

28. 2. Extrapolate To Infinite Resolution • Extrapolate from results of simulations at range of resolutions: where h = grid spacing.

29. 3. Sum over m-modes Fit tail: Monopole and dipole are negative

30. Scheme of Work: • Scalar SF, circular orbits, Schw. • Scalar SF, circular orbits, Kerr (equatorial). • Gravitational SF, circ. orbits, Kerr. • Eccentric orbits (elliptic orbits, zoom-whirl) References: m-mode regularization • L Barack & D Golbourn, PRD 76 (2007) 044020 [arXiv:0705.3620]. • L Barack, D Golbourn &N Sago, PRD 76 (2007) 124026 [arXiv:0709.4588]. • S Dolan & L Barack, (2010) in progress.