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

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

1. MOTIVATION Gravitational Waves and Extreme Mass Ratio Inspirals

Evidence for Gravitational Waves: The Hulse-Taylor Binary Pulsar Figure from Weisberg & Taylor (2004) 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 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

Coherent Bulk dynamics Amplitude: h ~ 1 / r Not scattered or absorbed λ > diameter “Hearing” All-sky, stereo Electromagnetic WavesGravitational Waves Incoherent, emitters Thermodynamic state Intensity: I ~ 1 / r 2 Scattered and absorbed λ << diameter Imaging, focussing Narrow field vs

Sources of Gravitational Waves Binary neutron star mergers, ~ 1.5 M s Stellar mass black hole mergers, ~ 10 M s Supermassive black hole mergers, ~ 10 7 M s Extreme Mass Ratio Inspirals (EMRI), ~ 10 M s and ~ 10 6 M s

EMRI dynamics and LISA LISA will see ~10 to 1000s inspirals simultaneously Scattering of NS or BH into highly eccentric orbits Eccentric until plunge ~10 5 wave cycles over final year Waveforms => Physics (e.g. “map” of near-horizon geometry, no-hair theorem, parameter estimation) t=-10 6 yr t=-1 yr t=0

EMRI Timescales e.g. M ~ 10 6 M s and m ~ 10 M s To track orbital phase over T obs, need high accuracy:

Two-Body Problem in GR: regimes log(M/m) log(r/M) Post-Newtonian Theory Numerical Relativity Perturbation Theory

2. SELF-FORCE Radiation Reaction in Curved Spacetime

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

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

EM Self-Force in Flat Spacetime

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! FlatCurved

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

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

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

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(im ϕ ), introduce puncture field, then sum over modes. l = 0 l = 1l = 2l = … m = 0m = 1m = 2m = …

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.

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 regular on worldline Extended source, without distributional component Extended source, without distributional component

Puncture Function Construction Σ: t = t 0 r=r 0 worldline (Barack, Golbourn & Sago 2007) Puncture field: Choose such that: where

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

Puncture and World-tube world-line world-tube t r θ 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: mode sum converges

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

Implementation Details 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 uv

4. FIRST RESULTS

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

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

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

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) [arXiv: ]. L Barack, D Golbourn &N Sago, PRD 76 (2007) [arXiv: ]. S Dolan & L Barack, (2010) in progress.