Progress on orbiting particles in a Kerr background

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

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)

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

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.

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/0508114].

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. http://gmunu.mit.edu/viz/emri_viz/emri_viz.html Drasco movie: orbit with a/m = 0.9, initial eccentricity = 0.7, inclined at 60 o to equatorial plane

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. http://gmunu.mit.edu/viz/emri_viz/emri_viz.html Drasco movie: orbit with a/m = 0.9, initial eccentricity = 0.7, inclined at 60 o to equatorial plane

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)

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...

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.

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

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 2 2 2 2 2 2 2 ( 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 1 + + s θ s 2 2 sin θ φ Source function S = T T, 2 nd -order differential operator

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

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.

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

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

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

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.

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 2 2 2 + 144M ω ωψ

Equivalent alternative involves radial derivatives along principal null geodesics: 4 4 0 ( 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, ). 1 2 3 4 0 4, r r r r 1 2 3 dr ( Kerr coordinates : ) r r

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

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 1 2 3 4 0lm r r r r 1 2 3 r = r max 0lm r = r min lm

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

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

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)

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

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

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

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)

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

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

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

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).

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.

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

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

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 0 0 0 0 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.

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] )

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.