4. MiSaTaQuWa force for radiation reaction

Transcription

1 4. MiSaTaQuWa force for radiation reaction [ ] g = πgt G 8 g = g ( 0 ) + h M>>μ v/c can be large + h ( ) M + BH μ Energy-momentum of a point particle 4 μ ν δ ( x z( τ)) μ dz T ( x) = μ dτ z z z = -g dτ μ

2 Linear perturbation in μ δ G h = 8π G T ( 1) geodesic on g 4 μ ν δ ( x z( τ)) μ dz T ( x) = μ dτ z z z = -g dτ Master variable ζ: () 1 ( 1) ζ = h ψ ζ or s ψ s ~ μ background metric ( a component of Weyl tensor) = φ lm( t, r) Y lm( Ω) lm : expanded in spherical (spheroidal) harmonics [ ] L ζ = S T Regge-Wheeler-Zerilli/Teukolsky eq.

3 From ζ, we can calculate: Waveform at infinity. de/dt GW, dl z /dt GW, etc. ( ) ~ O Gμ the orbit deviates from a geodesic on g How can we incorporate this deviation? Use de/dt & dl z /dt to determine the evolution of the orbital parameters (adiabatic approximation). But, this cannot predict the phase shift in orbit It cannot deal with non-adiabatic case. Δφ? Evaluate self-force from h acting on the particle.

4 For point particle, Self-force problem δ G [ h] = 8 π G T h 1 x z( τ) x α α h diverges at x = z ( τ) self-force (back-reaction) in a curved background: α Dz 1 [ ] [ h] z z ( h ( x ) +... ) z =F α h μδγ = μ μ ν z dτ α μ ν α μ ν μ ; ~ geodesic eq. on g + h singular!

5 Breakdown of perturbation theory? Yes! & No! Yes, because a point particle is ill-defined in GR. Mass is non-renormalizable in GR Gm bare lim mbare has no well-defined limit. r0 0 r0 No, because regular exact solution (BH) in GR. Mass renormalization is unnecessary cf. EM theory: point particle exists mass is renormalizable m phys e = lim mbare + r0 0 r 0 : two parameters to tune the limit

6 Namely, in GR: Identify the point particle with a BH solution of mass μ X singular horizon Embed the BH geometry in the linearly perturbed metric g = g + h : matching at x-z(τ) >>Gμ Matched Asymptotic Expansion Thorne & Hartle ( 84)

7 Simplest example Consider a point particle in the flat background g = η h ( x) = η μα η νβ Gμ ( z z x z + η α β αβ α ; ( τ ) = 0 z z( τ ret ) In the rest frame { X a } of the particle: c d cd Gμ ( Z Z + η ) a hab ( X) = ηacη bd ; Z = ( 1000,,, ) X This is just the Newtonian part of the Schwarzschild metric. Thus a Schwarzschild BH of mass μ can be naturally matched to g = g + h at ) X >> Gμ background geodesic eq. EOM unchanged. No self-force correction to all orders in Gμ 0

8 In General Curved Background: Hadamard decomposition of G (ret) in harmonic (Lorenz) gauge 0 0 [ ( )] G (ret) x,z ( x z ) u x,z v αβ = θ αβ δ σ αβ θ σ x,z σ ( xz, ) : world interval between x and z ( ~ 1 x z ) μ τ ( τ) h x = d G x,z z z α β (ret) (ret) αβ x α ( - ) u : direct part αβ v : tail part αβ direct part z α ( τ ) tail part (ret) x = h h h + h (direct) v αβ (direct) (tail) contains divergence curvature scattering is a solution of source-free eq. but not h (tail)

9 Matched asymptotic expansion z α ( τ ) g α R [ g ] ~ βγδ 1 L matching region Gμ << X ~ Gμ L << L g ( x) = g + h ( 0) (external: valid at X >> Gμ ) coordinate transformation: σ σ g g ~ i (BH) g ab( T, X ) = Hab + δhab (internal: valid at X << L ) μ μ ( x, z( τ)) -( x - z ) = - f ( T) X + f ( T) X X +... α ( xz, ( τ)) g ( xz, ) z = 0 ; ; μ μ i μ i j i i j ; μ μα μ ν x x ( X ) = g ( x) X X ab a b identify ab with ab in the matching region. g gμα : parallel transport bi-tensor

10 external scheme internal scheme ~ (BH) g = g + h g ab = Hab + δhab ( 0 ab ab ab background Riemann ~ 1/L perturbation in Gμ 1 1 () gab = ηab + hab + hab +... L L 1 1 hab = G μ hab + hab + () hab +... L L () + ( G ) 1 μ (() hab + () hab + 1 () hab +...) L L background Riemann ~ Gμ / X 3 perturbation in 1/ L H = η + Gμ H + ( G μ) H +... (BH) ab ab ab () 1 δhab = Hab + Gμ Hab + ( G μ) () Hab +... L 1 () () () + Hab + Gμ Hab + ( G μ) () Hab +... L ab matching condition: (n) (m) Hab = (n) (m) h ab +O ((m+1) (n+ 1) Gμ /L ) (n) (m) H ~ ab ( Gμ) n L m X (n-m)

11 Asymptotic matching to O(Gμ) flat part background EOM regular (tail) part EOM with self-force m n 1 1/L 1/L 1 Gμ η ab Gμ X 0 0 X L Gμ X L external background linear perturbation singular (direct) part no effect on EOM BH (=point particle) quadrupolar deformation Center of mass gauge condition

12 Regularized Gravitational Self-force MiSaTaQuWa force: (named by Eric Poisson) α 1 α μ ν F [ h (tail) (x)] ( h (tail) ; ( x ) +... ) z z Tail part of the metric perturbation Mino, Sasaki and Tanaka ( 97), Quinn and Wald ( 99) τ ( x ) ( tail ) αβ ( ) αβ h x d τ ' v x,z τ' T z τ' Regularized self-force is determined by the tail part E.O.M. with self-force = geodesic on g + h tail But h x ( tail) is NOT a solution of Einstein equations. g + h tail meaning of the metric was unclear

13 Detweiler - Whiting s S-R decomposition (improved over direct-tail decomposition) PRD 67, 0405 (003) ret 0 0 sym G x,z = θ x z G x,z sym 1 G ( x,z) = u( x,z) δ ( σ) v( x, z) θ( σ) 8π S sym 1 1 G x,z = G x, z + v x, z = u x,z δ σ + v x,z θ σ 8π 8π 4 h S x = d x g G S ( x,x ) T ( x ) :satisfies pert eqs. R ret S ( ret adv ) 1 G x, z = G x,z G x,z = G ( x,z) G ( x,z) vx, ( z) 8π R ret S h x = h x h x :satisfies source-free pert eqs. ( ) h h = O x z R tail Both give the same force EOM = geodesic on g + h R solution of (linearized) vacuum Einstein eqs.