Progress on orbiting particles in a Kerr background

Transcription

1 Progress on orbiting particles in a Kerr background John Friedman Capra 15 Abhay Shah, Toby Keidl

2 I. Intro II. Summary of EMRI results in a Kerr spacetime A. Dissipative ( adiabatic ) approximation (only dissipative part of self-force used) B. Full self force for scalar particle C. Point-mass in circular orbit III. Review of method computing self-force for Kerr in a radiation gauge A. Mode-sum renormalization B. Ω, u t for circular orbit C. Self-force (not yet completed)

3 I. Intro The departure from geodesic motion to order m/m has two parts: Dissipative part associated with the loss of energy to gravitational waves, odd under ingoing Conservative part even under ingoing outgoing outgoing

4 The dissipative part of self force plays the dominant role and is much easier to handle: The part of the field odd under ingoing ½( h retarded h advanced ). outgoing is Because h retarded and h advanced have the same source, the odd combination is sourcefree and regular at the particle. The conservative part of the force, is computed from ½( h retarded + h advanced ) a field singular at the particle. One must renormalize the field.

5 II. Summary of EMRI results in a Kerr spacetime A. Dissipative ( adiabatic ) approximation: only dissipative part of self-force used Method and discussion: Mino 05, Drasco, Flanagan, Hughes 05, Pound, Poisson, Nickel 05 Hinderer, Flanagan 08 Point-mass computations with only dissipative part of self force are well in hand: Kennefick, Ori 06 Drasco, Flanagan, Hughes, Franklin 05, 06 Ganz, Hikida, Nakano, Sago, Tanaka 06, 07 Burko, Khanna 07 Mino 08 Review: T. Tanaka, Prog. Theor. Phys. Suppl. 163, 120 (2006) [arxiv:grqc/ ].

6 Sundararajan, Khanna, Hughes, Drasco 08 Orbit constructed as set of short geodesics: Using black hole perturbation theory compute the evolution of three constants of geodesic motion, E(t), L z (t), and Q(t). Choose initial conditions and find the inspiral trajectory [r(t), θ(t), φ(t)]. From this trajectory, find EMRI waveform. Drasco movie: orbit with a/m = 0.9, initial eccentricity = 0.7, inclined at 60 o to equatorial plane

7 Sundararajan, Khanna, Hughes, Drasco 08 Orbit constructed as set of short geodesics: Using black hole perturbation theory compute the evolution of three constants of geodesic motion, E(t), L z (t), and Q(t). Choose initial conditions and find the inspiral trajectory [r(t), θ(t), φ(t)]. From this trajectory, find EMRI waveform. Drasco movie: orbit with a/m = 0.9, initial eccentricity = 0.7, inclined at 60 o to equatorial plane

8 B. Full self-force for scalar particles Computations in Kerr background that include conservative part of self-force for a particle with scalar charge: Static Ottewill, Taylor 12 Circular orbits Warburton, Barack 10 (frequency-domain) Dolan, Barack, Wardell 11 (time-domain) Eccentric orbits Warburton, Barack 11 (frequency-domain)

9 C. Massive particles in circular orbit Perturbed metric renormalized, quantities Ω and u t, invariant under helically symmetric gauge transformations computed. Shah, JF, Keidl Dolan Self-force in progress...

10 III. Review of method computing self-force for Kerr in a radiation gauge A single complex Weyl scalar, either ψ 0 or ψ 4, determines gravitational perturbations of a Kerr geometry (outside perturbative matter sources) up to changes in mass, angular momentum, and change in the center of mass. ψ 0 and ψ 4 are each a component of the perturbed Weyl tensor along a tetrad associated with the two principal null directions of the spacetime. Each satisfies a separable wave equation, the Teukolsky equation for that component.

11 Null tetrad Newman-Penrose Formalism l α, n α, m α, m α θˆ ˆ θ ˆ θ n l n l n l r e.g.,kinnersley tetrad for Schwarzschild 2 α r α α α 1 α α l = t + r n = t r 2 2 r α 1 ( ˆα ˆα) α 1 m m ( ˆα ˆ α = θ + iφ = θ iφ ) 2 2 α β γ δ ψ = δc lmlm = δc ψ = δc 0 γδ 2 lmlm 4 = r 2Mr nmnm

12 ψ = δc l m l m ψ = δc n m n m α β γ δ α β γ δ 0 γδ 4 γδ T O Teukolsky equation: operator s 2 2 ( cot ) O s ψ = S separable ( r + a ) 2 2 M ( r a ) 4Mar s s+ 1 = a sin θ 2s r iacosθ 2 t + t t φ r r 1 ar ( M) icosθ sinθ 2s + 2 sinθ θ θ sin θ φ 2 2 a s θ s 2 2 sin θ φ Source function S = T T, 2 nd -order differential operator

13 Solution: ψ 0 is a sum over angular and time harmonics of the form ψ = R ( r) S ( θ) e 0mω 2 mω 2 mω i( mφ ωt) spin-weighted spheroidal harmonic

14 ψ 0 involves 2 derivatives of the metric perturbation h To recover the metric from ψ 0 involves 2 net integrations. The method is due to Chrzanowski and Cohen & Kegeles, with a clear and concise derivation by Wald. First integrate 4 times to obtain a potential Ψ, the Hertz potential. Then take two derivatives of Ψ to find h The resulting metric is in a radiation gauge.

15 Outgoing Radiation Gauge (ORG) β h n = 0 h= 0 5 constraints, similar to those for ingoing waves in flat space with a transverse-tracefree gauge. The metric perturbation satisfying these conditions is given by h = L Ψ where L is a 2 nd -order differential operator involving only (angular derivative operator) and ð t

16 Weyl scalar ψ 0 Hertz potential Ψ metric perturbation h and expression for self-force a α renormalization coefficients renormalized a α (radiative part)

17 Weyl scalar ψ 0 ret Compute 0 from the Teukolsky equation as a mode sum over l,m,ω.

18 Weyl scalar ψ 0 Hertz potential Ψ For vacuum: ret ret ret Find the Hertz potential from 0 or 4 either algebraically from angular equation or as a 4 radial integrals from the radial equation. ret ret The angular harmonics of 0 and 4 are defined for r>r 0 or r<r 0, with r 0 the radial coordinate of the particle.

19 Explicitly, 1 ð ia sin θ Ψ + 12 M Ψ = ψ 8 Integrate 4 times with respect to θ ( ) 4 t t 0 Algebraic solution for vacuum, valid for circular orbit: For each frequency and angular harmonic Ψ mω = 8 m ( 1) Dψ + 12iM D 0 m ω 0mω M ω ωψ

20 Equivalent alternative involves radial derivatives along principal null geodesics: ( l) r (,,, ur) For each angular harmonic of ψ 0, this gives a unique solution satisfying the Teukolsky equation: e.g., for r>r 0, 4 4 ( l ) (,,, ur ) 0 dr dr dr dr (, u r, ) , r r r r dr ( Kerr coordinates : ) r r

21 When the orbit is not circular, one cannot use the algebraic method to find Ψ near the particle. Inside the spherical shell between r min and r max, ψ 0lmω has a nonzero source and the vacuum algebraic relation fails: r = r max r = r min 0 0lm 0lm has a source lm sourcefree algebraic

22 Radial integration commutes with decomposition into spherical harmonics: Can use Ψ lmω near the particle if computed by radial integration: lm dr dr dr dr lm r r r r r = r max 0lm r = r min lm

23 Weyl scalar ψ 0 Hertz potential Ψ metric perturbation h and expression for self-force a α Find, in a radiation gauge, the components of and its derivatives that occur in the expression for a α ret by taking derivatives of. e.g.: h mm ( n )( n ) ret h

24 Weyl scalar ψ 0 Hertz potential Ψ metric perturbation h and expression for self-force a α renormalization coefficients

25 renormalization coefficients ret Compute a from the perturbed geodesic equation as a mode sum truncated at max. Compute the renormalization vectors A a and B a (and C a?), numerically matching a power series in ret to the values of a. (Shah et al)

26 Weyl scalar ψ 0 Hertz potential Ψ metric perturbation h and expression for self-force a α renormalization coefficients renormalized a α (radiative part)

27 renormalized a α (radiative part) Subtract singular part of expression mode-by-mode renα retα α α a = a AL+ B a renα Shah uses l max = 75 = lim max max = 0 α C ( + ) L a renα

28 The missing pieces ψ 0 and ψ 4 do not determine the full perturbation: Spin-weight 0 and 1 pieces undetermined. There are algebraically special perturbations of Kerr, perturbations for which ψ 0 and ψ 4 vanish: changing mass δm changing angular momentum δj (and singular perturbations to C-metric and to Kerr-NUT). And gauge transformations

29 ret h [ ψ ] via CCK procedure 0 h ret [ δ m] from the conserved current associated with the background Killing vector t α. h ret [ δ J] from the conserved current associated with the background Killing vector φ α, for the part of δj along background J. (L. Price)

30 h [ δm], h [ δj] α = δ(2 α δ α ) β α = δ α φ β () t β β ( φ) β j T Tt j T ackground δ T α β = 0 j α j α α () t α ( φ) = 0, = 0 α δ α () t α ( φ) α m = j ds J = j ds 1 = m(2 u α t ) = mu α ut α α α φ α

31 This is enough to compute the self-force-induced change in two related quantities, a change invariant under gauge transformations generated by helically symmetric gauge vectors: U = u t at fixed Ω Ω (at fixed u t ) Each computable in terms of h ren u α u β

32 Ω for circular orbits in a Kerr background a < 0 counter-rotating a > 0 corotating

33 U for circular orbits in a Kerr background a < 0 counter-rotating a > 0 corotating r 0 /M a/m = 0.9 M U a/m = -0.9 M Comparisons underway with Alexandre Le Tiec (PN) and Sam Dolan a/m = (time-domain 0 calculation).

34 To find the self-force itself, one needs two final pieces: h ret [ J ] δ the part of δj orthogonal to the background J h ret [ CM ] the change in the center of mass Each is pure gauge outside the source, but the gauge transformation is discontinuous across the source.

35 o 2 h [ J ], h [ CM ] δ If they are pure gauge, how can they have a source? g ξ h = g Θ( r r ) is not pure gauge at r=r 0 ( h = g is pure gauge) g ξ Θ( r r ) 0 0

36 For Schwarzschild these are l=1 perturbations, with axial and polar parity, respectively. How do we identify them in Kerr? The idea is to find the part of the source that has not contributed to h ret [ ψ] + h [ δm] + h [ δj] One could in principle simply subtract from δt the contribution from these three terms. Writing E h : = δg

37 we have h ret = 8 πδt, remaining ret 8πδT = 8 πδt E ( h [ ψ ] + h[ δ m] + h[ δ J]) Find ξ at r 0 from the jump condition r r + ε r + ε gauge ( E h ) = 8 ε r ε πδt remaining For h gauge gauge continuous, the jump in E h involves only the few terms in E with second derivatives in the radial direction orthogonal to u α.

38 But Now we re back to the old difficulty of handling terms that are singular at the particle. Instead of trying directly to evaluate E + + ret 8 πδt ( h [ ψ ] h[ δ m] h[ δ J]) use the fact that h sing has source δt : r r gauge sing ret h E h E h h m + ε r + ε gauge ren E h E h m ε r ε 0 0 E 0 0 = ( ) ( [ ψ] + [ δ ] + h[ δj] ) = ( h [ ψ] + [ δ ] + h[ δj] )

39 Future problems: Key problems involving conservative part of self-force are not yet done Self-force on particle in circular orbit in Kerr (underway in modified radiation gauge and Lorenz gauge) and orbital evolution. Self-force on particle in generic orbit in Kerr and orbital evolutions. Identify and include relevant 2 nd -order corrections. Include particle spin (some calculations already done). In our (Abhay Shah s) mode-sum computation, form of singular field agrees with Lorenz. Why? (What happens to a logarithmic divergence of the gauge vector at the position of the particle?) Analytically find renormalization coeffs in radiation gauge.

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