Regularization of the the second-order gravitational perturbations produced by a compact object Eran...
16
Regularization of the Regularization of the the second-order the second-order gravitational gravitational perturbations produced by perturbations produced by a compact object a compact object Eran Rosenthal Eran Rosenthal University of Guelph - Canada University of Guelph - Canada Amos Ori Amos Ori Technion – Israel Institute of Technology Technion – Israel Institute of Technology
-
Upload
warren-tirrell -
Category
Documents
-
view
216 -
download
0
Transcript of Regularization of the the second-order gravitational perturbations produced by a compact object Eran...
Regularization of the Regularization of the the second-order the second-order
gravitational perturbations gravitational perturbations produced by a compact produced by a compact
objectobject
Eran RosenthalEran RosenthalUniversity of Guelph - CanadaUniversity of Guelph - Canada
Amos OriAmos OriTechnion – Israel Institute of TechnologyTechnion – Israel Institute of Technology
PresentationPresentation of the problemof the problem
Problem: calculation of at the limit
0)2(h
such that .
1R/
Consider a Schwarzschild black-hole Consider a Schwarzschild black-hole with a mass moving in a vacuum with a mass moving in a vacuum background spacetime with radius of background spacetime with radius of curvaturecurvature
R
)( 3)2(2)1( Ohhg g
Background metric
Full metric
Practical motivationPractical motivationAccurate calculation of the accumulating phase for long gravitational wave trains emitted from an extreme mass-ratio binary system (Lior Burko, Eric Poisson). These calculations can be used to construct accurate waveforms for LISA.
This requires accurate calculation of the orbit (of the smaller mass object) in the background spacetime induced by the more massive object.
Geodesic in the background spacetime
Perturbative approach to the calculation of the orbit
First order self-force corrections
Second order self-force corrections
require
s
require
s
)1(
h
)2(
h
Gravitational Gravitational perturbationsperturbations
0R at the limit
0
Solution in the external zone
)(z world-line
Produced by a Schwarzschild black-hole with a Produced by a Schwarzschild black-hole with a mass moving in a vacuum backgroundmass moving in a vacuum background spacetime
0 g Background metric
1
duuzxGh ))](|[4)1(
(Lorenz gauge))1(
21)1()1( hghh
The world line is a geodesic with respect to the background
)(z
)(, zx
At the leading order is a geodesic with respect to the background spacetime. Higher order corrections will be discussed later.
)(z
)(, zx
(General gauge)
ShD ][ )2(
Second-order gravitational Second-order gravitational perturbationsperturbations
Calculation of - main Calculation of - main difficultiesdifficulties
)2(h
Consider the linear differential equations for obtained from
Einstein equations. Schematically written as )2(
h
)( 233
44
Obb
Naive construction of the 2nd order retarded solution diverges at every point in spacetime.
Non-integrable
source terms
!
''][)'|(4)( 41''
'')2( xdghSxxGxh ret
(Lorenz gauge)
)()()()()( &][ 11111 hhhhhS
][][ )1()2( hShD
Distance from the “world-line”
)( 011
)1( Oah
(Lorenz gauge)
)(][ 234)2( ObaShD
)(][ 34 OaD
Require
)2()2( hh
)(][][ 3)2( ODShD
Consider in Fermi coordinates.
In the vicinity of the world-line:
g
Regularization of the singularity inRegularization of the singularity in 4 S
)()( 11 hhA
)1()1(B hh
ghD 2)1( )(
ghhC )1()1(
""2)1(
hFinding a Finding a causalcausal
)2()2( hh
Will be discussed now
1 BA cc1 DC cc
)()( 7 2641
DCBA cccc A B
C D
1x
t )(x
)(z
)( z
)(][][ 3)2( ODShD
)1()1()1()1( & hhhh ][DS
Schematically written:
RS hhh )1()1()1(
)( 1O )( 0O
Regularization of the singularity inRegularization of the singularity in3 ][ DS
Sz
R hh )1()]([
)1( )( 3O
New 1st order gauge
01 )]([)(
zRh
)( 2O
)(][][ 2)2( ODShD
Particular solution in (1Particular solution in (1stst order) Fermi order) Fermi gaugegauge
)2()2( hh
)(, zx
)2(h Retarded solution
)(z
Consider the following 1st order gauge: Fermi gauge
01 )]([)(
zRh
01 )]([)(
zRh
0selff Geodesic: No
corrections of order
The world-lineCorrections to a geodesic world-line which come from the first order self-force induce a 2nd order corrections to the gravitational perturbations.
selff
SHhhh )2()2(
General solution in (1General solution in (1stst order) Fermi order) Fermi gaugegauge
1. Boundary conditions at infinity: No incoming waves (requires additional regularization at infinity!)
Requirements on which fix
3. Divergent boundary conditions as determined from matched asymptotic expansions
0
0][ SHhD )(, zx
)(2h
SHh
2. Causality
)shomogeneou()gauge pure(
RetardedSHh
Kirchhoff representation (assuming boundary conditions at infinity)
Only divergent boundary conditions as are required
0
'
'''''''''
''4
1)(
dSGhhGxh g
SHSHSH
)( 2O
)(zxThe semi-homogeneous part satisfies
Required boundary conditions asRequired boundary conditions as 0
Fermi coordinates
0][ SHhD
00 n
nn
n
nnSH BAh )()(
)()()( 3221 Ohhg g
Expansions in the buffer zone
R
(Thorne and Hartle 1985)
22 R
1R
...
...
...
............
12 R
2R 22 R
22 1
11 R
SHhhh )2()2(
Schg
][11gR
][22gR
)( 3RO
||
if )( 0 Oh SH 0)( xh SH
00 n
nn
n
nnSH BAh )()(
2nd order solution in (1st order ) Fermi gauge
is formally given by:)()( 22
hh
][][ )( DShD 2
)2(h Retarded
solution
ResultResultss
in 2nd order Lorenz gauge
)(, 2h
""2)1(
h )1()1()1()1( & hhhhS
Schematically:
Considering the equation away from Considering the equation away from the world-linethe world-line
Introducing Introducing
Choosing (1Choosing (1stst order) Fermi gauge order) Fermi gauge
Determining from boundary Determining from boundary conditions as conditions as
Summary of the methodSummary of the method
""2)1(
h
SHh0
