Capra V - Penn State
May-June 2002
Mode-sum approach:
a year’s progress
Leor Barack - AEI
with
Carlos Lousto, Yasushi Mino, Amos Ori,
Hiroyuki Nakano, Misao Sasaki
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Capra V - Penn State
May-June 2002
Adiabatic orbital evolution
via self-force
Adiabatic inspiral ( tradiation  tdynamic) described as a sequence of geodesics:
reaction
z ( )  z geod [ ; E ( ), Lz ( ), Q( )].
Need to obtain
this!
Local rate of change of E, Lz , Q given by
1 C

C
F
u
w/

(E.g., Ori 96)

C (u , x )  E (u , x ), Lz (u , x ), Q(u , x )
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Capra V - Penn State
May-June 2002
Talk Plan
 Review of the mode sum approach
 Derivation of the RP: “direct force” approach
 Derivation of the RP: l-mode Green’s function approach
 Analytic approximation
 Gauge problem - and resolution strategy
 Implementation for radial trajectories
 Implementation for circular orbits (in progress)
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Capra V - Penn State
May-June 2002
Mode-sum prescription
Fself  lim x z Ftail ( x)


 lim x z F full ( x )  Fdir ( x )



l
l
  lim x  z F full
( x )  Fdir ( x )
x
z

l 0


 F
l 0
l
full ( x
 z )  ( AL  B  C




l
L)   Fdir
( x  z )   AL  B  C L 
l 0
l 1/ 2

D

l
Fself   F full
( x  z )  AL  B  C L  D
l
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Capra V - Penn State
May-June 2002
Derivation of the RP : Scalar field
(based on the analytic form of the direct force)
 1  O (x 2 ) 
dir
dir
dir

F ( x )  q  , ,
 ( x )  q





[Mino, Nakano, & Sasaki, 2001]
Introducing


z
x
x
   0  O(x 2 ), P( n )  x n ,
one obtaines (Barack, Mino, Nakano, Ori, & Sasaki, 2002)
Fdir   03 P(1)   05 P( 4)   07 P( 7 )  O(x )
Now expand in Ylm, sum over m, and take xz:
Fdir
l ( x  z )  A L  B
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Capra V - Penn State
May-June 2002
Scalar RP in Schwarzschild: summary
(Coordinates chosen such that the orbit is equatorial)
q2 
A r   2
,
r fV
q 2 r
At   2 ,
r V
A  0
q 2 (r 2  22 ) Kˆ ( w)  (r 2  2 ) Eˆ ( w)
Br  2
r
f (1  2 r 2 )3 / 2

q 2 r Kˆ ( w)  2 Eˆ ( w)
Bt  2
r
V 3 / 2

q 2 r Kˆ ( w)  Eˆ ( w)
B 
r  (  r )V 1/ 2
C  D  0


w

 ut
  u
r  u r
f  1  2M r
w  2 ( 2  r 2 )
V  1  2 r 2
Eˆ , Kˆ  elliptic intgs .
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Capra V - Penn State
May-June 2002
Derivation of the RP: gravitational case
(based on the analytic form of the direct force)



Fself  Ftail ( x  z )   k
k



tail
 ; x  z
h
In harmonic
gauge!
1
1
1
1
 u  u  g   g  g   u g  u   g  u  u   u u  u  u 
2
4
4
2
Similarly:
û

z
x
x


full, dir
F full
(
x
)


k
h
, dir
 ;
h  4 uˆ  uˆ
dir
1  O(x 2 )

[Mino, Sasaki, & Tanaka, 1997]
Again, we obtain Fdir ( x )   03 P(1)   05 P( 4 )   07 P( 7 )  O(x )
Note: the P(n)’s, hence the RP values, depend on
our choice of the off-worldline extension of k
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Capra V - Penn State
May-June 2002
Considerations in choosing the k-extension
 In our (non-elegant, yet convenient) scheme, the l-mode contributions
are defined through decomposing each vectorial -component in scalar Ylm:

F full
   k  ( x )h (i )lm ( r , t ) Y(i )lm ( ,  ) ;  
l , m ,i

lml 'm ' l 'm '
l '
F full
Y ( ,  )    F full
lm,l 'm '
l'
Wish to design k(,) such that decomposing this in Ylm turns out simple!
 Derivation of the gravitational RP better be simple
 Numerical computation of the full modes better be simple
 Calculation of the “gauge difference” (see below) better be simple
Note: Final result (Fself) does not depend on the k-extension; one just needs
to be sure to apply the same extension to both the RP and the full modes.
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Capra V - Penn State
May-June 2002
RP values in Schwarzschild : Grav. case
k-extension
“fixed contravariant
I components”
(in Schwarzcshild
coordinates)
“revised I” (some k
II components multiplied by
certain functions of )
g(x): “same as”;
III u(x): parallelly-propagated
along a normal geodesic
IV “revised III” (under study)
RP values
Rgrav 



(  u u ) R
sca
Same as I
grav
R
 R
scalar
Same as III
especially convenient for…
Numerical calculation of the full
modes
Calculation of the scalar-harmonic
l-modes (assures finite mode
coupling)
Calculating the “gauge difference”
Calculating the full modes &
gauge difference, still with a finite
mode-coupling
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Capra V - Penn State
May-June 2002
Deriving the RP via local analysis of the
(l-mode) Green’s function
 This method relies only on MST/QW’s original tail formula,

tail 
 

  full
 Fdir ( )  lim  0   (G )d '  (G)d '   lim  0  (G)d ',
 


 


with the Green’s function equation,
 Gl expanded asG
l
G  RG   4 ...
 G0  G1 L1  G2 L2  , where Gn  Gn ( Lx); z 
 Analytic solutions for the Gn’s (in terms of Bessel functions)
 G0, G1, G2 suffice for obtaining A, B, C, & also D
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Capra V - Penn State
May-June 2002
Green’s function method - cont.
 Method applied so far for radial & circular orbits in
Schwarzschild, for both scalar [LB & Ori, 2000] and gravitational [LB
2001,2] cases
 Agreement in all RP values for all cases examined. Especially
important is the independent verification of D=0, which cannot be
easily checked numerically.
 Green’s function method involes tedious calculations (though
by now mostely automated) and difficult to apply to general orbits.
 It’s main advantage: quite easily extendible to higher
orders in the 1/L expansion! (Direct force method requires
knowledge of higher-order Hadamard terms.)
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Capra V - Penn State
May-June 2002
Gauge problem
 Self force calculations require h (or it’s tensor-harmonic modes
h(i )lmY(i )lm ) in the harmonic gauge.
 BH perturbations are not simple in that gauge, even in
Schwarzschild: separable with respect to l,m but various elements of the
tensor-harmonic basis remain coupled. Unclear how this set of coupled
equations will behave in numerical integrations.
 Two possible strategies:
Either tackle the perturbation equations in the harmonic gauge ; or
Formalize and calculate the self force in a different gauge: e.g.,
Regge-Wheeler gauge (for Schwarzschild), or the radiation gauge
(for Schawrzschild or Kerr)
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Capra V - Penn State
May-June 2002
Gauge transformation of the self force
[LB & Ori, 2001]
Radiation or RW
H R
h
  ;    ;
gauges
 Then,
H R

:

given

H R

  

  
(Fself
)



[(
g

u
u
)


R
u
 u ]
, full


 If F attains a well defined limit xz, then (RP)R=(RP)H; however  F not necessarily well defined at z. Examples:
H Radiation
is direction-dependent even for a static particle
Fself
in flat space
H RW
Fself
is direction-dependent for all orbits in
Schwarzschild, except strictly radial ones
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Capra V - Penn State
May-June 2002
Dealing with the gauge problem
 For radial trajectories in Schwarzschild - no gauge problem! One
l
implements the mode-sum scheme with the full modes Ffull calculated
in the RW gauge, and with the same RP as in the H gauge.
 If F is discontinuous (direction-dependent)
but finite as xz: average over spatial directions?
 If F admits at least one direction from
which xz is finite: define a “directional”
force?
In both cases, a useful sense
of the resulting quantities
must be made by prescribing
the construction of desired
gauge- invariant quantities
out of them. [Ori 02]
A general strategy: Calculating the force in an “intermediate” gauge,
obtained from the R-gauge by modifying merely its “direct” part (such
that it resembles the one of the H-gauge). This is how it’s done:
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Capra V - Penn State
May-June 2002
Self force in the “intermediate” gauge
ˆ
(H )
( R)
 h
  ;     ; for R  Hˆ,
 Solve h
ˆ
(H )
(H )
where h
is the direct part of h
ˆ
ˆ
 ConstructF RH ( RH )
 Now:

H
H
H
Fself
 Ffull
 Fdir
ˆ
ˆ
ˆ
H H
H H
H R
RH
Fself
 F full
 F full
 F full
Hˆ
self
F
F
R
full
 F
R  Hˆ
full
H
 Fdir
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Capra V - Penn State
May-June 2002
Mode-sum scheme in the Ĥ gauge


(H )
l ( R)
l ( RH )
Fself
  Ffull
 Ffull
 AL  B  C L  D
ˆ
ˆ
l
R-gauge modes l-modes of F (to be
(easy to obtain) obtained analytically)
H-gauge RP
 Note: Must apply the same k-extension for Fl, Fl, and the RP!
 Preliminary Results for radiation gauge in Schwarzschild:
Fl has no contribution to A, B, or C !
Calculation of contribution to D (zero?) under way
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Capra V - Penn State
May-June 2002
Implementing the mode-sum method:
radial trajectories (RW gauge)
 We worked entirely within the RW gauge:
l-mode MP derived (via twice differentiating) from Moncrief’s
scalar function yl, satisfying y ,ltt y ,lr*r*  V (r )y l  Source
Then, l-mode full force derived through
l
Ffull
  k  hl ;
Mode-sum formula applied with H-gauge RP - see Lousto’s talk.
 Parameters A, B, C for the RW modes obtained independently by
applying the Green’s function technique to Moncrief’s equation.
We found (RP)RW=(RP)H.
Explicit demonstration of the RP’s “gauge-invariance” property
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Capra V - Penn State
May-June 2002
Large l analytic approximation

l
l
Ftail   F full
 Fdir

l
Conjecture: l-mode expansion of Ftail converges faster than any power of l.
Flfull and Fldir have the same 1/L expansion
In particular:
O(L-2) term of Flfull can be inferred from that of Fldir
This, in turn, is obtained by extending the Green’s
function method one further order
(conjecturing here is not risky, since the result is tested numerically)
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Capra V - Penn State
May-June 2002
analytic approximation - cont.
 For radial trajectories in Schwarzschild (in RW gauge) we found
(LB & Lousto, 02)
rl
rl
Freg
 Fbare
 Ar L  B r  C r / L
15  2 2 2

   4M / r  1 L2  O( L 4 )
2
16 r

Ftlreg  



15 2  d 2
  r r L 2  O( L 4 )
16
d
In perfect
agreement with
numerical results!
(see Lousto’s talk)
 Integrated “energy loss” by l-mode at large l (for r0  ,   1):
 ( EH )
 ( EH )
 d  

l


F
tl
 d 

15
(  / M )  L 2
32
but gauge-dependent!
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Capra V - Penn State
May-June 2002
Implementing the mode-sum method:
circular orbits (Ĥ gauge)
 Current status:


(H )
l ( RW )
l ( RW  H )
Fself
  Ffull
 Ffull
 AL  B  C L  D
ˆ
ˆ
l
Full modes in RW gauge:
Numerical calculation of
the tensor-harmonic MP
modes (via Moncrief).
Still need to prescribe &
carry out the construction
of the “scalar-harmonic”
force mode.
H-gauge RP
Gauge difference: being
calculated (by solving for ,
obtaining F, and decomposing
into modes.)
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Capra V - Penn State
May-June 2002
What’s next?
 Circular case will provide a first opportunity for comparing
self-force and energy balance approaches.
 Generic orbits in Schwarzschild: already have the RP and
numerical code; soon will have Fl.
 Orbits in Kerr: key point - be able to obtain MP. Progress relies on this.
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