Transcript Tobias Keidl - Midwest Relativity Meeting

On finding fields and self force in a gauge appropriate
to separable wave equations
(gr-qc/0611072)
2006 Midwest Relativity Meeting
Tobias Keidl
University of Wisconsin--Milwaukee
In collaboration with:
John Friedman, Eirini Messaritaki,
Alan Wiseman
Motivation
Laser Interferometer Space Antenna (LISA)
•Dedicated space-based gravitational wave observatory
•Launch date ~2014-2020
•5 year expected lifetime
Motivation
m
• Look for Extreme Mass
 1
M
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
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
Need to solve 10
coupled PDEs
Known only for
Harmonic gauge
Teukolsky Formalism
•For a background Kerr black hole, there are two complex projections of
the perturbed Weyl tensor are gauge independent: 0,4
•Solvable by a use of the Teukolsky equation
(written below for 4 in Schwarzschild)

2
 1  r 4



r8  4
Mr

4
5
2
4

4r

1

(r)

 (r)  t
(r) t 2
r  (r) r
 

 4 
r4  
sin 
sin   
 
2
r 4   4 4ir 4  4
4
2
 2


r
4cot


2


4

r
T (x)
4
2
2


sin  
sin 

(r)  r 2  2Mr

T(x)  Source Function
•Related to the metric by a 2nd order differential operator

 4  Oh
Point Particle Regularization
Ohrenormalized  Ohretarded  Ohsingular

renormalized
4
Gauge Independent

retarded
4

Solve Teukolsky
equation numerically
singular
4
Calculate from
Harmonic gauge or
directly
But…
•This method gives us 0 or 4, not the metric
•In vacuum, there is a prescription for
reconstructing the metric from 0 or 4
•0 or 4 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)
Radiation Gauge Metric
Work by Chrzanowski, Kegeles & Cohen, Wald, Lousto & Whiting, Ori
•Can use a formalism by Kegeles and Cohen to construct a scalar
potential from 0 or 4
 4  L1  L2 
L1 ,L2 are fourth order derivative operators
(additional equations for remaining Weyl components)
h  L3  L3
L3 is a second order derivative operator
•“Ingoing Radiation Gauge” metric
h l   0
h0
Outline of Calculation
1. Compute 4ret from Teukolsky Equation
2. Use hsingular in harmonic gauge to compute 4sing
3. 4ren = 4ret - 4sing is a sourcefree solution to
the Teukolsky equation
4. Calculate renormalized metric from 4ren
5. Use jump condition on the Einstein equations to
find s=0,1 piece of metric
6. Calculate self force from perturbed geodesic
equation