On finding fields and self force in a gauge appropriate to separable wave equations

(gr-qc/0611072)

2006 Midwest Relativity Meeting

Tobias KeidlUniversity of Wisconsin--Milwaukee

In collaboration with:John Friedman, Eirini Messaritaki,

Alan Wiseman

MotivationLaser Interferometer Space Antenna (LISA)

•Dedicated space-based gravitational wave observatory •Launch date ~2014-2020•5 year expected lifetime

Motivation• Look for Extreme Mass

Ratio Inspirals

• Estimate LISA will see ~10-1000 events per year (J.R. Gair et al 2004)

• Develop waveforms templates suitable for LISA to detect gravitational waves

• Inspiral can be modeled within perturbation theory

• Can treat captured object as a small point perturbation on the background spacetime

Graphic stolen from www.srl.caltech.edu

1<<Mm

Point Particle Regularization

• Regularized gravitational self-force MiSaTaQuWa

Mino ， Sasaki and Tanaka (‘97)Quinn and Wald (‘97)

• Detweiler and Whiting (‘03)particle follows geodesic of hrenormalized

hrenormalized =hretarded −hsingular

Gauge dependent Known only for Harmonic gauge

Need to solve 10 coupled PDEs

Teukolsky Formalism

r8

Δ(r)∂2Ψ4

∂t2+ 4r5 Mr

Δ(r)−1

⎛

⎝⎜⎞

⎠⎟∂Ψ4

∂t−Δ(r)2

∂∂r

1Δ(r)

∂ r4Ψ4( )∂r

⎛

⎝⎜⎜

⎞

⎠⎟⎟−

r4

sinθ∂∂θ

sinθ∂Ψ4

∂θ⎛

⎝⎜⎞

⎠⎟

−r4

sin2θ∂2Ψ4

∂φ2+

4ir4

sin2θ∂Ψ4

∂φ+ r4 4cotθ + 2( )Ψ4 =4πr2T(vx)

€

Δ(r) = r2 − 2Mr T(v x ) = Source Function

•For a background Kerr black hole, there are two complex projections of the perturbed Weyl tensor are gauge independent: Ψ0,Ψ4

•Solvable by a use of the Teukolsky equation (written below for Ψ4 in Schwarzschild)

Ψ4 = Oh•Related to the metric by a 2nd order differential operator

Point Particle Regularization

Ohrenormalized =Ohretarded −Ohsingular

Ψ4renormalized = Ψ 4

retarded − Ψ 4singular

Gauge Independent Calculate from Harmonic gauge or directly

Solve Teukolsky equation numerically

But…

•This method gives us Ψ0 or Ψ4, not the metric

•In vacuum, there is a prescription for reconstructing the metric from Ψ0 or Ψ4

Ψ0 or Ψ4 do not determine the s=0,1 piece

•Use jump condition across particle across spin 0 and 1 projections of the Einstein equations to fix remaining metric pieces (Price, Shankar and Whiting)

hμυ =L 3Ψ + L 3ΨL3 is a second order derivative operator

Ψ4 = L1Ψ + L2ΨL1,L2 are fourth order derivative operators

(additional equations for remaining Weyl components)

Work by Chrzanowski, Kegeles & Cohen, Wald, Lousto & Whiting, Ori

•Can use a formalism by Kegeles and Cohen to construct a scalar potential from Ψ0 or Ψ4

•“Ingoing Radiation Gauge” metric

Radiation Gauge Metric

hμνlμ =0 h=0

Outline of Calculation

1. Compute Ψ4ret from Teukolsky Equation

2. Use hsingular in harmonic gauge to compute Ψ4sing

3. Ψ4ren = Ψ4

ret - Ψ4sing is a sourcefree solution to the

Teukolsky equation

4. Calculate renormalized metric from Ψ4ren

5. Use jump condition on the Einstein equations to find s=0,1 piece of metric

6. Calculate self force from perturbed geodesic equation