Self-force calculations for Kerr black hole inspirals A Review of Recent Progress

Transcription

1 Self-force calculations for Kerr black hole inspirals A Review of Recent Progress Sam Dolan University of IST, Lisbon, Jan 2012 Sam Dolan (Southampton) Self-force Calculations Lisbon 1 / 98

2 Talk Overview 1 Introduction to Gravitational Self-Force Motivations Key ideas Calculation methods 2 Recent Progress Gauge-invariant comparisons with PN, NR & EOB First self-forced evolutions Resonances in EMRIs 3 The frontier: Kerr spacetime The m-mode regularization method Non-radiative modes & linearly-growing gauge modes Latest results 4 Conclusion Sam Dolan (Southampton) Self-force Calculations Lisbon 2 / 98

3 Motivation I: EMRIs Sam Dolan (Southampton) Self-force Calculations Lisbon 3 / 98

4 Motivation I: EMRIs A typical galaxy contains: 1 A massive central BH (M) 2 A population of compact objects (µ) within cusp (rcusp 1pc). Extreme mass-ratio q = µ/m = Two-body scattering of objects into nearly-parabolic orbits Highly eccentric 1 e , p 8 100M Inspiral and capture if t gw (1 e)t relax Radiation reaction reduces eccentricity, but inspiral orbits are still typically: 1 moderately eccentric even up to plunge 2 non-equatorial (not aligned with BH spin) Sam Dolan (Southampton) Self-force Calculations Lisbon 4 / 98

5 Motivation II: LISA? Sam Dolan (Southampton) Self-force Calculations Lisbon 5 / 98

6 Motivation II: re-scoped LISA! The new [LISA] configuration should detect thousands of galactic binaries, tens of (super)massive black hole mergers out to a redshift of z=10 and tens of extreme mass ratio inspirals out to a redshift of 1.5 during its two year mission. Karsten Danzmann, Aug Sam Dolan (Southampton) Self-force Calculations Lisbon 6 / 98

7 Motivation III: the general 2-body problem in relativity Sam Dolan (Southampton) Self-force Calculations Lisbon 7 / 98

8 Motivation III: the general 2-body problem in relativity Effective One-Body (EOB) formulation of Damour et al. provides a possible analytic fitting framework Sam Dolan (Southampton) Self-force Calculations Lisbon 8 / 98

9 Ideas I: Radiation Reaction in Electromagnetism An accelerated charge emits radiation Loss of energy force acting on charge Interpretation: the accelerated charge interacts with its own field Point charge infinite field... mathematical problems? A regularization method is needed. Sam Dolan (Southampton) Self-force Calculations Lisbon 9 / 98

10 Ideas I: Radiation Reaction in Electromagnetism Dirac split the electromagnetic potential A µ into S and R parts: A µ S = 1 2 A µ R = 1 2 ( A µ ret + ) Aµ adv ( A µ ret ) Aµ adv S for symmetric / singular R for radiative / regular Self-Force from F µ = ν F R µν, where F R µν = µ A R ν ν A R µ Sam Dolan (Southampton) Self-force Calculations Lisbon 10 / 98

11 Ideas II: Self-Force in Curved Spacetime Flat Curved In flat spacetime, Green function has support on light-cone only. In curved spacetime, Green function also has a tail within the light cone. Also, the light cone intersects itself (light ring at r = 3M) Dirac s radiative potential becomes non-causal in curved spacetimes. Sam Dolan (Southampton) Self-force Calculations Lisbon 11 / 98

12 (Intersecting Light Cone) See e.g. V. Perlick s Living Review on lensing. Sam Dolan (Southampton) Self-force Calculations Lisbon 12 / 98

13 Ideas II: SF in Curved Spacetime: Electromagnetism DeWitt & Brehme (1960) derived the EM SF in curved spacetime ( ma µ = f µ 2 ext + e2 (δ ν µ + u µ df ext u ν ) + 1 ) 3m dτ 3 Rν λu λ +2e 2 u ν lim [µ G ν] ( ret λ z(τ), z(τ ) ) u λ dτ ɛ 0 τ ɛ Tail integral over past history of motion is v. difficult to compute! Need practical regularization schemes that avoid tail integral in this form Sam Dolan (Southampton) Self-force Calculations Lisbon 13 / 98

14 Ideas II: SF in Curved Spacetime: Gravitational Charge e mass µ, field A µ metric perturbation h µν Equations for (first-order in µ) Gravitational Self-Force (GSF) Formally derived via Method of Matched Asymptotic Expansions Obtained by Mino, Sasaki & Tanaka, and Quinn & Wald (1997). Known as the MiSaTaQuWa equation MiSaTaQuWa equation still features a tail integral Need regularization schemes for practical calculations. Sam Dolan (Southampton) Self-force Calculations Lisbon 14 / 98

15 Ideas II: Regularization Method Dirac s split into R and S fields was acausal Alternative Detweiler-Whiting split (2003) into S and R fields is causal Correct MiSaTaQuWa self-force is recovered from R part. S part not known exactly (global existence questionable), but it can be computed in vicinity of worldline via series expansions. Sam Dolan (Southampton) Self-force Calculations Lisbon 15 / 98

16 Ideas III: Dissipative/Conservative Parts of Self-Force Scalar field example: Retarded and advanced fields Φ ret and Φ adv Ret. and adv. R fields, Φ R ret = Φ ret Φ S, Φ R adv = Φ adv Φ S Define conservative and dissipative parts of field Φ cons = 1 ( Φ R 2 ret + Φ R 1 adv) = 2 (Φ ret + Φ adv 2Φ S ) Φ diss = 1 ( Φ R 2 ret Φ R 1 adv) = 2 (Φ ret Φ adv ) Dissipative part does not need regularization! Conservative part needs knowledge of S field. Dissipative part secular loss of energy and angular momentum. Conservative part shift in orbital parameters, periodic. Sam Dolan (Southampton) Self-force Calculations Lisbon 16 / 98

17 Ideas IV: Interpretation of Gravitational Self-Force accelerated motion on a background spacetime µ a g = F self = F diss + F cons geodesic motion in a perturbed spacetime µ a g+h = 0 Sam Dolan (Southampton) Self-force Calculations Lisbon 17 / 98

18 Methods I: l-mode regularization Define F α ret/s µkαµνβ β hret/s µν (as fields), then write F self = (F ret F S ) p ( = Fret l FS) l p (l-mode contributions are finite) = = l=0 l=0 l=0 [ ] Fret(p) l AL B C/L l=0 [ ] FS(p) l AL B C/L [ ] Fret(p) l AL B C/L D (where L = l + 1/2) Regularization Parameters A, B, C, D calculated analytically for generic orbits in Kerr in Lorenz gauge h ;ν µν = 0. Works well for spherically-symmetric spacetimes (e.g. Schw.) which allow decomposition in tensor spherical harmonics Sam Dolan (Southampton) Self-force Calculations Lisbon 18 / 98

19 Methods II: Puncture Schemes Metric perturbation g Schw/Kerr µν + h µν Trace-reversed MP: h µν = h µν 1 2 g µνh Work in Lorenz gauge h ;ν µν = 0. Four gauge constraints. 10 wave equations: (neglecting (µ/m) 2 and higher) Delta-function source, T µν (x α ) = µ h µν + 2R α µ β ν h αβ = 16πT µν ( g) 1/2 δ 4 [x α x α p (τ)]u µ u ν dτ. In 1+1D, MP is C 0 on the worldline. In 2+1D, MP diverges logarithmically. In 3+1D, diverges as 1/distance. Idea: evolve a residual field h res = h ret h punc, where h punc is some local approximation to h S. Sam Dolan (Southampton) Self-force Calculations Lisbon 19 / 98

20 A (selective) review of progress since First comparison of gauge-invariant results with Post-Newtonian theory (PN) and Numerical Relativity (NR): ISCO shift due to conservative part of GSF Perihelion advance of eccentric orbits Benefits of using symmetric mass-ratio 2 Calibration of Effective One-Body (EOB) theory with GSF 3 First self-forced evolutions: 1 via method of osculating geodesics 2 with time domain code (scalar-field) 4 Resonances in EMRIs on Kerr spacetime Sam Dolan (Southampton) Self-force Calculations Lisbon 20 / 98

21 1. Comparisons: (I) The redshift invariant Circular geodesic motion on Schwarzschild at radius r > 3M, E = r 2M r(r 3M) µ, de dt = F t/u t 0 The dissipative components, F t and F r, corresponding to energy and angular momentum loss, are gauge-invariant(*). The conservative component F r is gauge-dependent. Detweiler identified two quantities which are gauge invariant under transforms that respect the helical symmetry of the circular orbit. 1 Orbital frequency Ω radius R (M/Ω 2 ) 1/3 2 Redshift z = 1/u t Both defined w.r.t Schw. t coordinate of background spacetime. z(r) is a gauge-invariant relation. Independent results of Regge-Wheeler and Lorenz gauge calculations compared by Detweiler, and Sago & Barack (2008). Sam Dolan (Southampton) Self-force Calculations Lisbon 21 / 98

22 1. Comparisons: (II) The ISCO shift Innermost stable circular orbit (ISCO) where de/dr = 0. For geodesic motion, r isco = 6M, Ω isco = ( 6 3/2 M) 1. The conservative part of GSF shifts the ISCO by O(µ). Ω isco is invariant under gauge transformations that respect the helical symmetry of the circular orbit. GSF prediction: Ω isco Ω isco = µ/M Barack & Sago, PRL 102, (2009), arxiv: Sam Dolan (Southampton) Self-force Calculations Lisbon 22 / 98

23 1. Comparisons: (II) The ISCO shift GSF prediction must be modified for comparison with PN, because Lorenz gauge is not asymptotically-flat (h tt O(r 0 )). Apply simple monopolar gauge transformation to get: Ω isco Ω isco = µ/m A challenge: can a resummed Post-Newtonian expansion match this strong-field result? Challenge taken up in M. Favata, PRD 83, (2011), arxiv: Sam Dolan (Southampton) Self-force Calculations Lisbon 23 / 98

24 1. Comparisons: (II) The ISCO shift Table 1 in M. Favata, PRD 83, (2011), arxiv: Sam Dolan (Southampton) Self-force Calculations Lisbon 24 / 98

25 1. Comparisons: (III) The periastron advance 6πM [(1 e 2 )p] GR periastron advance δ = (e.g. 43 per century for Mercury). Conservative part of GSF δ O(µ) δ < 0 for all eccentric orbits δ is gauge-invariant (within restricted class of gauges) but its parameterization δ(p, e) is not. Numerical results in Barack & Sago, PRD 83, (2011), arxiv: Sam Dolan (Southampton) Self-force Calculations Lisbon 25 / 98

26 1. Comparisons: (III) The periastron advance Periastron advance was recently compared between NR, PN, EOB and GSF in comparable mass regime 1/8 µ/m 1. Le Tiec et al. PRL 107, (2011) [arxiv: ] Remarkably, the GSF prediction works well even in comparable mass regime if we replace µ/m with symmetric mass ratio: µ/m µm/(µ + M) 2 Plots on next slide show K = Ω φ /Ω r = 1 + δ/(2π). Sam Dolan (Southampton) Self-force Calculations Lisbon 26 / 98

27 1. Comparisons: (III) The periastron advance From Le Tiec, Mroué, Barack, Buonanno, Pfeiffer, Sago and Taracchini, PRL 107, (2011), arxiv: Sam Dolan (Southampton) Self-force Calculations Lisbon 27 / 98

28 1. Comparisons: (III) The periastron advance From Le Tiec, Mroué, Barack, Buonanno, Pfeiffer, Sago and Taracchini, PRL 107, (2011), arxiv: Sam Dolan (Southampton) Self-force Calculations Lisbon 28 / 98

29 2. Calibration of Effective One-Body theory Damour and collaborators have fed GSF results into the EOB model. Idea: Compare precession of small-eccentricity orbits at first-order in µ Ω 2 ( r µ ) = 1 6x + ρ(x) + O ( (µ/m) 2) M where Ω 2 φ x [(M + µ)ω φ ] 2/3. PN theory gives the (weak-field) expansion ρ P N (x) = ρ 2 x 2 + ρ 3 x 3 + (ρ c 4 + ρ log 4 ln x)x 4 + (ρ c 5 + ρ log 5 ln x)x 5 + O(x 6 ) ρ 2, ρ 3 are given by 3PN. logarithmic contributions at 4PN and 5PN (ρ log 4 and ρ log 5 ) have been derived by Damour ρ c 4 and ρc 5 are (presently) unknown in PN. Sam Dolan (Southampton) Self-force Calculations Lisbon 29 / 98

30 2. Calibration of Effective One-Body theory Using accurate GSF results, {ρ 2, ρ 3, ρ log 4, ρlog 5 } may be tested, and the unknown parameters ρ c 4 and ρc 5 may be constrained: ρ c 4 = , ρc 5 = , ρlog 6 < 0. Determination of ρ(x) in the range 0 x 1/6 gives first info on strong-field behaviour of a combination of EOB functions a(u) and d(u) [where u = G(M + µ)/(c 2 r EOB )]. Advantage of GSF calibration: Both GSF and EOB split naturally into conservative and dissipative effects. GSF data for ρ(x) may be fitted with simple 2-point Pade approximation that also makes use of PN information. Sam Dolan (Southampton) Self-force Calculations Lisbon 30 / 98

31 2. Calibration of EOB model From Barack, Damour and Sago, Phys. Rev. D 82, (2010) [arxiv: ]. Sam Dolan (Southampton) Self-force Calculations Lisbon 31 / 98

32 3. Self-forced evolutions Want to evolve self-forced orbits over 10 5 cycles! Pound & Poisson [PRD77, (2008)] described a method of osculating geodesics for self-forced evolutions. Requires a fit to GSF data over a range of (p, e) with analytic model. First evolutions recently performed by Warburton, Akcay, Barack, Gair & Sago [arxiv: ]. Animations follow, showing two simulations: (i) with full GSF; (ii) with only dissipative part of GSF. Sam Dolan (Southampton) Self-force Calculations Lisbon 32 / 98

33 3. Self-forced evolutions Rather than computing GSF w.r.t. geodesics of background, the aim is to evolve self-consistently in the time domain. Diener, Vega, Wardell and Detweiler [arxiv: ] have looked at scalar-field case: Sam Dolan (Southampton) Self-force Calculations Lisbon 33 / 98

34 4. Resonances (I) Two timescales: (i) orbital period M, (ii) radiation reaction µ 1. Hinderer & Flanagan (2010) describe two-timescale expansion for EMRIs, using action-angle variables. Action : constants of motion : J ν = ( E/µ, L z /µ, Q/µ 2) Angle : phase variables q α = (q t, q r, q θ, q φ ). q r q r + 2π as orbit goes r = r min r max r min with period τ r = 2π/ω r. Frequencies ω α (J) = (ω r, ω θ, ω φ ) Generic geodesic orbits on Kerr are ergodic (space-filling). Isometries of Kerr (q t, q φ ) irrelevant, (q r, q θ ) relevant params. Sam Dolan (Southampton) Self-force Calculations Lisbon 34 / 98

35 4. Resonances (II) 1. Geodesic approximation (η = 0): Solution : dq α dτ dj ν dτ = ω α (J) = 0 q α (τ, η = 0) = ω α τ (1) J ν (τ, η = 0) = const. (2) Timescale : unchanging Sam Dolan (Southampton) Self-force Calculations Lisbon 35 / 98

36 4. Resonances (III) 2. Adiabatic approximation: dq α dτ dj ν dτ = ω α (J) = η G (1) ν (q r, q θ, J) orbital average Solution : Timescale : τ rad.reac. η 1 q α (τ, η) = η 1ˆq(ητ) J ν (τ, η) = Ĵ(ητ) Sam Dolan (Southampton) Self-force Calculations Lisbon 36 / 98

37 4. Resonances (IV) 3. Post-adiabatic approximation: dq α dτ dj ν dτ = ω α (J) + ηg (1) α (q r, q θ, J) + O(η 2 ) = ηg (1) ν (q r, q θ, J) + η 2 G (2) ν (q r, q θ, J) + O(η 3 ). Two timescales : η 1 (secular) and 1 (oscillatory). Sam Dolan (Southampton) Self-force Calculations Lisbon 37 / 98

38 4. Resonances (V) Is adiabatic approximation justified? i.e. is it always OK to neglect fast-oscillating parts? Consider Fourier decomposition G (1) ν (q r, q θ, J) = k r,k θ G (1) νk r,k θ (J)e i(krqr+k θq θ ) and q r = ω r τ + ω r τ , q θ = ω θ τ + ω θ τ k r q r + k θ q θ = (k r ω r + k θ ω θ ) τ + (k r ω r + k θ ω θ ) τ Cannot neglect higher Fourier components if resonance condition k r ω r + k θ ω θ = 0 is satisfied! i.e. when ω r /ω θ passes through low-order integer ratio. Sam Dolan (Southampton) Self-force Calculations Lisbon 38 / 98

39 4. Resonances (VI) Duration of resonance set by (k r ω r + k θ ω θ ) τ 2 1, i.e. τ res 1/ pη where p k r + k θ, η = µ/m. Net change in constants of motion is J η/p Net change in phase is q 1/ ηp Need to know precise first-order SF and (possibly) dissipative part of 2nd-order SF to model resonance accurately. Without complete knowledge, a resonance effectively resets the phase and kicks the orbital parameters. Sam Dolan (Southampton) Self-force Calculations Lisbon 39 / 98

40 4. Resonances (VII) Credit: Hinderer & Flanagan, arxiv: Sam Dolan (Southampton) Self-force Calculations Lisbon 40 / 98

41 GSF on Kerr: The frontier Schw. separability l-mode regularization easy! decompose h ab in tensor spherical harmonics Y lm(i) ab use Lorenz gauge b h ab = 0 with gauge constraint damping solve 1+1D in time domain, or ODEs in freq. domain apply l-mode regularization: F self µ = l=0 [ F l,ret µ A(l + 1/2) B C/(l + 1/2) ] D Sam Dolan (Southampton) Self-force Calculations Lisbon 41 / 98

42 GSF on Kerr: The frontier Kerr hard choices... lack of separability... Teukolksy variables Ψ 0, Ψ 4... spin-weighted spheroidal harmonics... metric reconstruction in radiation gauge (Chrzanowski) Lorenz gauge? l = 0, 1 modes? Hertz potential approach under development by Friedman et al. tensor spheroidal harmonics... [don t exist?] Full 3+1D approach... expensive! m-mode + 2+1D evolution... practical compromise. Proof-of-principle for m-mode recently established with scalar-field toy model for circular orbits on Kerr Φ R = m= Φ m Re imϕ, F m r = q r Φ m R, F r = m= F m r Sam Dolan (Southampton) Self-force Calculations Lisbon 42 / 98

43 Puncture scheme : Scalar field implementation Local approximation Φ P for Detweiler-Whiting S field Φ S Covariant expansion power series approximation in coordinate differences δx a = x a x a, where x is field point, x is worldline point Classification: nth order expansion iff Φ [n] P Φ S O( δx δx n 2 ) 4th-order expansions are available [arxiv: , arxiv: ]. From local expansion Φ [n] P to global puncture field Φ[n] Let x become a function of x e.g. set same BL time coordinate, t = t Periodic continuation: e.g. δϕ 2 2(1 cos δϕ) = δϕ 2 + O(δϕ 4 ) Sam Dolan (Southampton) Self-force Calculations Lisbon 43 / 98 P :

44 Residual field + modal decomposition Introduce residual field: Φ [n] R = Φ Φ[n] P Residual field obeys wave equation, Φ R = S eff with effective source S eff = γ δ(x x(τ))dτ Φn R. Regularity: S eff O ( δx δx n 4) Sam Dolan (Southampton) Self-force Calculations Lisbon 44 / 98

45 Residual field + modal decomposition Introduce residual field: Φ [n] R = Φ Φ[n] P Residual field obeys wave equation, Φ R = S eff with effective source S eff = γ δ(x x(τ))dτ Φn R. Regularity: S eff O ( δx δx n 4) Decomposition in m modes: Φ R = m= 2+1D wave equations: Φ m Re imϕ, {Φ m P, Seff m } = 1 π {Φ P, S eff } e imϕ dϕ 2π π m Φ m R = S m eff Sam Dolan (Southampton) Self-force Calculations Lisbon 44 / 98

46 Mode sums and convergence Field is real Φ m = Φ m SF from mode sums, e.g. F r = q r Φm R m=0 where Φ m R = { Φ m R, m = 0 2 Re ( Φ m R eim ϕ(t)), m 0 Sam Dolan (Southampton) Self-force Calculations Lisbon 45 / 98

47 Mode sums and convergence Field is real Φ m = Φ m SF from mode sums, e.g. F r = q r Φm R m=0 where Φ m R = { Φ m R, m = 0 2 Re ( Φ m R eim ϕ(t)), m 0 Power law convergence Fr m m ζ in large-m regime Convergence rate ζ depends on order n of puncture. ζ = n for n even, and ζ = n 1 for n odd. Sam Dolan (Southampton) Self-force Calculations Lisbon 45 / 98

48 Puncture order and m-mode convergence For circular orbits, F r is conservative and F ϕ is dissipative. punc. order Φ R C S eff Φ m R Fr m Fϕ m 1 δx/ δx C 1 1/δx 2 m 2 2 δx C 0 1/ δx m 2 m 2 e λm 3 δx δx C 1 δx/ δx m 4 m 2 e λm 4 δx δx 2 C 2 δx m 4 m 4 e λm Sam Dolan (Southampton) Self-force Calculations Lisbon 46 / 98

49 World-tube construction Worldtube T of fixed dimensions δr, δθ Outside: m Φ m = 0 Inside: m Φ m R = Sm eff Across boundary δt : Φ m R = Φm Φ m P Sam Dolan (Southampton) Self-force Calculations Lisbon 47 / 98

50 Finite difference method in 2+1D t r * θ t=0 Sam Dolan (Southampton) Self-force Calculations Lisbon 48 / 98

51 Spatial profile of modes: r Sam Dolan (Southampton) Self-force Calculations Lisbon 49 / 98

52 Spatial profile of modes: θ Sam Dolan (Southampton) Self-force Calculations Lisbon 50 / 98

53 Spatial profiles: r and θ (m = 0) q 1 Ψm R/ret m = θ r * / M Sam Dolan (Southampton) Self-force Calculations Lisbon 51 / 98

54 Spatial profiles: r and θ (m = 1) q 1 Ψm R/ret m = θ r * / M Sam Dolan (Southampton) Self-force Calculations Lisbon 52 / 98

55 Spatial profiles: r and θ (m = 5) q 1 Ψm R/ret m = θ r * / M Sam Dolan (Southampton) Self-force Calculations Lisbon 53 / 98

56 Spatial profiles: r and θ (m = 10) q 1 Ψm R/ret m = θ r * / M Sam Dolan (Southampton) Self-force Calculations Lisbon 54 / 98

57 Time evolution of m-modes on worldline Sam Dolan (Southampton) Self-force Calculations Lisbon 55 / 98

58 Low-m modes and power law relaxation Low-m modes take longest to relax Fit power-law decay model e.g. for m = 0, Φ m R (t) = Φ m R ( ) + c 2t η +... Local power law index η for Ψ m=0 R t / M r 0 = 6M r 0 = 7M r 0 = 8M r 0 = 10M r 0 = 14M r 0 = 20M Sam Dolan (Southampton) Self-force Calculations Lisbon 56 / 98

59 Richardson extrapolation (I) Extrapolation to infinite resolution Results depends on grid resolution x, e.g. : t = xm, r = xm, θ = πx/6 Second-order-accurate FD method error O(x 2 ) Ψ m (x) = Ψ m (x = 0) + c 2 x 2 + c 3 x Fit results of runs at various resolutions to this model, and extrapolate to x = 0 Sam Dolan (Southampton) Self-force Calculations Lisbon 57 / 98

60 Richardson extrapolation (II) r -1! m h = 1/ h = 1/24 h = 1/ h = 1/48 h = 1/56 h = 1/ t / M Sam Dolan (Southampton) Self-force Calculations Lisbon 58 / 98

61 Richardson extrapolation (III) Sam Dolan (Southampton) Self-force Calculations Lisbon 59 / 98

62 Modal convergence: F m φ Exponential convergence of dissipative component Sam Dolan (Southampton) Self-force Calculations Lisbon 60 / 98

63 Modal convergence: F m r Power-law convergence of conservative component Puncture orders n = 2, 3 and 4 (M/q) 2 F m r m 2nd-order 3rd-order 4th-order (M/q) 2 F m r e-05 1e-06 m 2 1e-07 2nd-order 3rd-order 4th-order 1e m m 4 Sam Dolan (Southampton) Self-force Calculations Lisbon 61 / 98

64 Modal convergence: F m r 4th-order puncture... m 4 convergence Sam Dolan (Southampton) Self-force Calculations Lisbon 62 / 98

65 Modal convergence: F m r rescaled variable m 4 F m r Sam Dolan (Southampton) Self-force Calculations Lisbon 63 / 98

66 Results: scalar-field SF on Kerr F r for circular orbits in equatorial plane Radial component of SF, (M 2 /q 2 )Fr self r 0 = 6M r 0 = 10M r 0 = r isco a = 0.9M 4.941(1) (7) a = 0.7M 4.102(1) (2) a = 0.5M 3.290(1) (2) a = 0M (2) (1) (2) a = +0.5M 2.423(4) (9) (5) a = +0.7M 9.530(3) (9) (4) a = +0.9M (5) (1) (9) Sam Dolan (Southampton) Self-force Calculations Lisbon 64 / 98

67 Results: scalar-field SF on Kerr F φ for circular orbits in equatorial plane a = 0.9M a = 0.7M a = 0.5M a = 0M Angular component of SF, (M/q 2 )Fφ self r 0 = 6M r 0 = 10M r 0 = r isco (1) (1) (1) (1) (1) (1) (3) (1) (1) a = +0.5M (3) (1) (4) a = +0.7M (3) (1) (1) a = +0.9M (8) (1) (2) Sam Dolan (Southampton) Self-force Calculations Lisbon 65 / 98

68 GSF on Kerr So much for the scalar field... what about the interesting case? Einstein equations : G ab R ab 1 2 g abr = 8πT ab Vacuum background + stress-energy T ab small parameter µ = m/m Metric split : background + perturbation : Trace-reversed perturbation h ab : g ab = ĝ ab + µh ab h ab = h ab 1 2 g abh Sam Dolan (Southampton) Self-force Calculations Lisbon 66 / 98

69 Gravitational SF on Kerr Linearized equations: L hab c c hab + 2 R c a d b h cd + g ab Z c ;c Z a;b Z b;a = 16πT ab where Z b a hab Mixed hyperbolic-elliptic type equations. Impose Lorenz gauge conditions Z a = 0 Z a = 0. How to enforce gauge conditions? Gauge-constraint damping [Gundlach et al. 05] c c hab + 2 R c a d b h cd + n a Z b + n b Z a = 16πT ab. Sam Dolan (Southampton) Self-force Calculations Lisbon 67 / 98

70 Gravitational SF on Kerr m-mode decomposition: h ab = α ab (r, θ)u ab (r, θ, t)e imφ, (no sum) 10 wave equations: sc u ab + M ab (u cd,t, u cd,r, u cd,θ, u cd ) = S ab Sam Dolan (Southampton) Self-force Calculations Lisbon 68 / 98

71 2+1D Wave Equations (Schw.) f sc u ab + M ab ( u cd,t, u cd,r, u cd,θ, u cd ) = 0 M 00 = 2 `2r 2 ( u 01 u 00 ) + u 00 u 11 M 01 = 2f 2 (cos θu 02 + imu 03 ) r 2 sin θ M 02 = f(u im cos θu 03 ) r 2 sin 2 θ M 03 = f(u 03 2im cos θu 02 ) r 2 sin 2 θ M 11 = 4f 2 (cos θu 12 + imu 13 ) r 2 sin θ r 4 + 4f (u 00 u 11 ) r 3 + 2f 2 (u 22 + u 33 ) r 3 + 2( u 00 + u 11 2u 01 ) r 2 2f 2 (u 01 + θ u 02 ) r 2 + 2( u 12 u 02 ) r 2 + f[(4 + r)u r θ u 01 ] r 3 f 2 u 02 r 2 + 2fimu 01 r 2 sin θ + 2( u 13 u 03 ) r 2 + f(4 + r)u 03 r 3 f 2 u 03 r 2 + 2[2r2 ( u 01 u 11 ) + u 11 u 00 ] r 4 4f(u 00 u 11 ) r 3 2f 2 (2ru 11 + u 22 + u r θ u 12 ) r 3 + 2f 3 (u 22 + u 33 ) r 2 M 12 = f(u im cos θu 13 ) r 2 sin 2 θ 2f 2 [cos θ(u 22 u 33 ) + imu 23 ] r 2 sin θ + f[(4 + r)u r θ u 11 ] r 3 f 2 (5u θ u 22 ) r 2 + 2( u 02 u 12 ) r 2 Sam Dolan (Southampton) Self-force Calculations Lisbon 69 / 98

72 2+1D Wave Equations (Schw.) f sc u ab + M ab ( u cd,t, u cd,r, u cd,θ, u cd ) = 0 M 13 = f(u 13 2im cos θu 12 ) r 2 sin 2 θ + f(4 + r)u 13 r 3 f 2 (5u θ u 23 ) r 2 M 22 = 2f[u 22 u im cos θu 23 ] r 2 sin 2 θ 2f 2 (u 22 + u 33 ) r 2 M 23 = 2f[2u 23 im cos θ(u 22 u 33 )] r 2 sin 2 θ M 33 = 2f(u 22 u im cos θu 23 ) r 2 sin 2 θ + 2f(u 11 + u 33 ) r 2 2f 2 (u 22 + u 33 ) r 2. 2f[2f cos θu 23 + im(fu 33 u 11 )] r 2 sin θ + 2( u 03 u 13 ) r 2 + 2(u 00 u 11 ) r 3 + 2f(u 11 + u θ u 12 ) r 2 2f(cos θu 13 imu 12 ) r 2 sin θ + 4f(cos θu 12 + imu 13 ) r 2 sin θ + 2f(u 23 + θ u 13 ) r 2 + 2(u 00 u 11 ) r 3 Sam Dolan (Southampton) Self-force Calculations Lisbon 70 / 98

73 Gauge constraint damping Imperfect, gauge-violating initial data Z a b hab 0. Gauge-violation itself obeys a wave equation: Z a = 0. How to drive system towards Lorenz gauge solution Z a = 0? Gauge Constraint Damping: add extra term to wave equations featuring gauge violation vector Z a, i.e. h ab + 2R c a d b h cd + (n a Z b + n b Z a ) = 0. so that Z a obeys a damped wave equation Sam Dolan (Southampton) Self-force Calculations Lisbon 71 / 98

74 2nd-order puncture scheme Barack, Golbourn & Sago (2007) give a 2nd-order puncture formulation: h P ab (x) = µ [ χ ab, χ ab = u a u b + (Γ c ad u b + Γ c bd u a)u c δx d] x= x ɛ [2] P For circular orbits in equatorial plane, this reduces to χ 00 = C 00 + D 00 δr χ 01 = D 01 sin δφ χ 03 = C 03 + D 03 δr χ 13 = D 13 sin δφ χ 33 = C 33 + D 33 δr Sam Dolan (Southampton) Self-force Calculations Lisbon 72 / 98

75 2nd-order puncture scheme Effective source: S eff ab = h P ab + 2Rc a d b h P cd m-mode decomposition: h P (m) ab and S eff(m) ab Puncture and source found in terms of symmetric elliptic integrals I m 1,..., Im and antisymmetric integrals J m 1,..., J m 5... Z π Z π π Z π π Z π π π Z π π ɛ 3 P sin δφ e imδφ d(δφ) = ɛ 3 P sin δφ cos δφ e imδφ d(δφ) = ɛ 5 P sin δφ cos2 (δφ/2) e imδφ d(δφ) = ɛ 5 P sin δφ sin2 (δφ) e imδφ d(δφ) = ɛ 5 P sin δφ sin2 (δφ/2) e imδφ d(δφ) = Wardell and co. developing a 4th-order scheme i ˆqm B 3/2 ρ 1K K(i/ρ) + ρ 2 q1e m E(i/ρ) iγ B 3/2 [qm 2K K(γ) + qm 2E E(γ)] iγ ˆqm B 5/2 3K K(γ) + ρ 2 q3e m E(γ) i ˆqm B 5/2 ρ 4K K(i/ρ) + ρ 2 q4e m E(i/ρ) iγ 2 ˆqm B 5/2 ρ 5K K(i/ρ) + ρ 2 q5e m E(i/ρ) Sam Dolan (Southampton) Self-force Calculations Lisbon 73 / 98

76 Metric Perturbations : Time evolution Regularized field at particle in circular orbit: r 0 = 7M, m = Regularized perturbation on worldline as a function of time r! "" tt tr t! rr!! t / M Sam Dolan (Southampton) Self-force Calculations Lisbon 74 / 98

77 Gauge Violation : Time evolution r 0 = 7M, m = Gauge violation on worldline as a function of time 10-1 Gauge violation Z a Z t Z r Z! t / M Sam Dolan (Southampton) Self-force Calculations Lisbon 75 / 98

78 Metric Perturbations : Angular Profile r 0 = 7M, m = "# Metric perturbations: angular profile r# h ab ## "" tt tr t# rr t" r" !/4!/2 3!/4! " Sam Dolan (Southampton) Self-force Calculations Lisbon 76 / 98

79 Metric Perturbations : Angular Profile r 0 = 7M, m = 2 h ab !/4!/2 3!/4! " tt h ab t# !/4!/2 3!/4! " h ab tr h ab ## !/4!/2 3!/4! " 0 0!/4!/2 3!/4! " Sam Dolan (Southampton) Self-force Calculations Lisbon 77 / 98

80 l-mode and m-modes Project out: m-modes u m ab (t, r, θ) onto lm modes h(i) lm (t, r) of Barack/Lousto/Sago. Use tensor spherical harmonics i = , Z π h (1) lm (r, t) = 2π 0 Z π h (2) lm (r, t) = 2π 0 Z π h (3) lm (r, t) = 2π 0 Z π h (4) lm (r, t) = 4π 0 Z π h (5) lm (r, t) = 4π 0 Z π h (6) lm (r, t) = 2π 0 Z π h (7) lm (r, t) = 2π 0 sin x (u 00 + u 11 ) Ylm (x)dx (3) sin x 2u 01 Ylm (x)dx (4) sin x (u 00 u 11 ) Ylm (x)dx (5) [sin x u 02 x imu 03 ] Ylm dx (6) [sin x u 12 x imu 13 ] Ylm dx (7) sin x (u 22 + u 33 ) Ylm dx (8) [sin x(u 22 u 33 )D 2 + 2u 23 D 1 ] Ylm dx (9) Sam Dolan (Southampton) Self-force Calculations Lisbon 78 / 98

81 Comparison with l-modes Projection from m modes onto lm modes of Barack/Lousto/Sago i = 1 i = 2 i = 3 i = 4 i = 5 i = 6 i = 7 l = 2, m = i i i i i i i i i i i i i i Sam Dolan (Southampton) Self-force Calculations Lisbon 79 / 98

82 Problem: Time Evolution of m = 0 mode Regularized metric perturbations Metric perturbations on the worldline : m = 0 tt tr t! rr ""!! t / M Sam Dolan (Southampton) Self-force Calculations Lisbon 80 / 98

83 Radial Profile : m = 0 mode Metric perturbations Radial profile of metric perturbations at t = 50M tt tr t! rr ""!! t = 50M r * / M Sam Dolan (Southampton) Self-force Calculations Lisbon 81 / 98

84 Radial Profile : m = 0 mode Metric perturbations Radial profile of metric perturbations at t = 100M tt tr t! rr ""!! t = 100M r * / M Sam Dolan (Southampton) Self-force Calculations Lisbon 82 / 98

85 Radial Profile : m = 0 mode Metric perturbations Radial profile of metric perturbations at t = 150M tt tr t! rr ""!! t = 150M r * / M Sam Dolan (Southampton) Self-force Calculations Lisbon 83 / 98

86 The low multipoles stability problem The growing solutions arise even for vacuum perturbations. The growing solutions are (locally) Lorenz-gauge They are homogeneous and pure-gauge: h ab = ξ a;b + ξ b;a They are scalar gauge modes: ξ a = Φ ;a. The growing solutions satisfy ingoing conditions at horizon: ( u t + 2 ln(1 2/r) t ) u = 0 r The growing solutions are traceless h = h a a = 0. The problem is entirely in l = m = 0 and l = m = 1 modes. Q. Why has no-one evolved Schw. l = 0 and l = 1 modes in time-domain? A. Negative potentials (r < 3M), unstable evolutions. Sam Dolan (Southampton) Self-force Calculations Lisbon 84 / 98

87 Lorenz-Gauge Monopole Modes Pure-gauge modes generated by gauge vectors ξ a Lorenz-gauge h ab = ξ (a;b) h ab = ξ (a;b) 1 2 g abξ c ;c h ;b ab = 0 ξ a;b b = 0 Two scalar monopole gauge modes ξ a = Φ ;a ( Φ) ;a = 0 Φ = {0, const.} Trace : h = ξ a ;a = Φ = {0, const} Trace-free, static scalar gauge mode Φ 0 = 1 2 ln f Pseudo-static mode Φ = t Φ 0 = t 2 ln f, Φ = 0, u 00, u 11, u 22 t, u Sam Dolan (Southampton) Self-force Calculations Lisbon 85 / 98

88 Pseudo-static modes Pseudo-static (i.e. linearly-growing) locally Lorenz-gauge modes in monopole How do they arise in time domain? To see, use conservation laws (due to symmetries of Ricci-flat background) to reduce degrees of freedom. Monopole: Four coupled 2nd order equations + two gauge constraints + one conservation equation. After reducing degrees of freedom, find wave equation with negative potential. Sam Dolan (Southampton) Self-force Calculations Lisbon 86 / 98

89 Conserved quantities in non-radiative multipoles (I) Symmetries: Background spacetime has Killing vectors X a : a X b + b X a = 0 Stress-energy is conserved, a T ab = 0, so we can construct a conserved vector: j a T ab X b a j a = 0. The vector j a = ( 16π) 1 W ab X b can be written j a = b F ab, where F ab = F ba i.e. the divergence of an antisymmetric tensor F ab where ( 16π)F ab = h ac;b X c h bc;a X c h ac X c ;b + h bc X c ;a Apply Stokes theorem Conserved integrals on two-surfaces Sam Dolan (Southampton) Self-force Calculations Lisbon 87 / 98

90 Conserved quantities in non-radiative multipoles (II) Gauss s theorem: Stokes theorem (j a = F ab ;b): j a dσ a = j a dσ a Σ 1 Σ 2 Σ F ab ;bdσ a = 1 2 γ Σ F ab ds ab Σ Σ 1 2 Σ Sam Dolan (Southampton) Self-force Calculations Lisbon 88 / 98

91 Conservation Law (III) Integrate on constant-t hypersurfaces, on concentric spheres: X a (t) Energy E, X a (φ) Ang. Mom. L z in perturbation Ang. mom. in l = 1 odd-parity sector, energy is in monopole (l = 0), { [ ] 4π r 2 F (t) r2 E u t, r 1 < r 0 < r 2, 01 = r 1 0, otherwise. Locally conserved quantity in monopole (l = m = 0) equations: { r 2 ( htt,r h ) 4E, r > r 0, tr,t 2f 1 htt + 2f h rr = 0, r < r 0. Sam Dolan (Southampton) Self-force Calculations Lisbon 89 / 98

92 Monopole equations Monopole has four equations (u 00, u 01, u 11, u 22 = u 33 ) + two gauge constraints. Trace equation evolve stably Use conserved quantity C = { 4E r < r 0 0 r > r 0 Hierarchical system of equations for {H, X, Y } [ D 2 2f ( r 2 1 4M r where D 2 = 2 t + 2 r 2fM/r3 D 2 H = 0 D 2 X = 2f r 4 H 3fC )] r 3 Y = 4f r 2 H + 2f r C H = r h a a, X = (2rf) 1 [u 11 (2r 3)u 00 ], Y = rf 1 (u 00 u 11 ). Sam Dolan (Southampton) Self-force Calculations Lisbon 90 / 98

93 Monopole equations H and X equations evolve stably. Y equation does not. Y equation resembles a Regge-Wheeler equation ] [ 2 t r 2 V 12 (r) Y =... where V ls (r) = f i.e. here l = 1, s = 2. ( l(l + 1) r 2 2M(1 ) s2 ) r 3 Potential turns negative within r < 3M growing modes. In principle, Y can be recovered from H, X by integrating conservation law on spatial slices: (ry ) = r[c 2X fh]. r Sam Dolan (Southampton) Self-force Calculations Lisbon 91 / 98

94 Challenges for time-domain Lorenz gauge formulation How do we evolve l = 0, l = 1 modes in time domain in 1+1D? e.g. how do we eliminate trace-free, massless, locally-lorenz gauge modes? (in monopole and dipole) How do we enforce the physical boundary condition at the horizon? Ideas... : 1 Use of generalized Lorenz gauge to promote stability, h ;ν µν = H µ ( h αβ ; r). 2 Horizon-penetrating coordinates? (e.g. ingoing Eddington-Fink.). Hyperboloidal slicing? 3 Restricted set of variables, with reconstruction of metric by integrating first-order conservation equations? Sam Dolan (Southampton) Self-force Calculations Lisbon 92 / 98

95 Generalized Lorenz gauge I have tried a generalized Lorenz gauge (GLG) of the form h ;b ab = H a(h tr ) For circular orbits, we want the monopole part of h tr to be zero. I can achieve stable evolutions if I make H a proportional to an ingoing null vector. With analytically-known l = 0, m = 0 monopole as initial data, the 2+1D scheme evolves stably with h tr 0 as grid spacing 0. I have not yet found a GLG which stabilizes the m = 1, l = 1 even-parity dipole but since the undesirable mode grows linearly (whereas physical part exp(imωt)), I can eliminate it: h 1 Ω 2 2 t 2 h Sam Dolan (Southampton) Self-force Calculations Lisbon 93 / 98

96 Recent progress These tricks now make it possible to compute Lorenz-gauge GSF for circular orbits, using 2+1D approach. With a second-order puncture, and max. resolution r = M/16, I can compute F r to accuracy greater than 0.05%, for r 0 = 6M on Schw. I have written the first Lorenz-gauge 2+1D Kerr code. To validate, may compare F t against the energy fluxes computed via the Teukolsky formalism Sam Dolan (Southampton) Self-force Calculations Lisbon 94 / 98

97 Recent progress: Dissipative GSF in Kerr Comparing Energy Flux (Finn Thorne 2000) with F t component of Self-Force n = 2 n = 4 n = 6 n = 8 de/d! F t a = 0.5M, r isco = 4.233M t / M m = 2 mode (radiative) Showing results of various grid resolutions x M/n. Sam Dolan (Southampton) Self-force Calculations Lisbon 95 / 98

98 Recent progress: Dissipative GSF in Kerr Comparing Energy Flux (Finn Thorne 2000) with F t component of Self-Forc F t n = n = 4 n = 6 a = 0.5M, r isco = 4.233M n = 8 de/d! extrapolated value nd-order puncture S ln( r r 0 ) x 2 ln x convergence. 0.4% here error still unexplained... t / M Sam Dolan (Southampton) Self-force Calculations Lisbon 96 / 98

99 Summary of progress: m-mode 2 + 1D method Scalar field, first-order puncture: Barack & Golbourn [arxiv: ]. Second-order GSF formulation: Barack, Golbourn & Sago [arxiv: ]. Scalar-field, fourth-order punc, Schw.: Dolan & Barack [arxiv: ] Scalar-field, Kerr, circ orbits: Dolan, Wardell & Barack [arxiv: ] Scalar-field, Kerr, eccentric orbits: Thornburg (in progress) GSF, Schw, circ. orbits, 2nd order: Dolan & Barack (in progress) GSF, Kerr, circ. orbits, 4th order: coming soon (I hope!). Sam Dolan (Southampton) Self-force Calculations Lisbon 97 / 98

100 Summary GSF programme for black hole inspirals recently came-of-age (in 2009) with comparison of physically-meaningful numerical results on Schwarzschild with other methodologies. First comparisons of GSF with PN, EOB and NR have been successful. First self-forced orbits and waveforms produced recently (Gair et al.) Very interesting resonance phenomenon (Hinderer & Flanagan) expected for EMRIs on Kerr. Details require Kerr GSF. First GSF calculations on Kerr underway (Dolan 2011; Friedman 2011). Second-order formalism is under discussion; numerical work someway off. Lots of interesting calculations still to do! Sam Dolan (Southampton) Self-force Calculations Lisbon 98 / 98