Yasushi Mino ( 蓑泰志) Center for Gravitational Wave Astronomy University of Texas at Brownsville

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Yasushi Mino ( 蓑泰志 )

Center for Gravitational Wave AstronomyUniversity of Texas at Brownsville

E-mail : [email protected]

Index 1: Introduction: LISA project 2: Self-force calculation 3: Radiation Reaction Formula 4: Gauge Problem 5: Adiabatic Expansion 6: Conclusion

Workshop on EFT Techniques in Grav. Wave Physics,Carnegie Mellon University, July 23-25, 2007

Recent issues of the self-force problem

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1: Introduction: LISA Project

See LISA project homepage http://lisa.jpl.nasa.gov/See also the upcoming file Intersteller by S.Spielberg.

• LISA is a project for GW detection in space, proposed by ESA and NASA, starting 201X.. • Because of the long baseline and because it is free from the seismic noise, it is sensitive to the low-frequency gravitational waves.



HzLc 2_linebasesensitive 10/

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Promising Targets of LISA: Massive objects… supermassive Black Hole, Primordial GWs, QSO,

Among them, GWs from the so-called EMRIs are the primary target of LISA.



GMcT //1 3dynGW ◎MgM 740 1010

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LISA primary target: EMRI (= Extreme-Mass Ratio Inspirals)the inspiralling binary of a supermassive BH (10^5-7M) and a compact object (1-10M)

For the GW detection by matched filtering and for the extraction of the astrophysical information, it is necessary to establish a theoretical method to predict waveforms.



•Event rate is expected to be more than one per year.•“Waveforms are calculable.”

•Strong evidence for the presence of black holes•Standard candles for Observational Cosmology (Observable area is almost the entire universe)

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Problem: How do we calculate the gravitational waveforms?

Extreme mass ratio 610/ Mm

Black hole perturbation formalism

M

m

)( 2BlackHole Ohgg

)(ˆ 2BlackHole

BlackHole

OhGgG

hgG

Metric:

Einstein Eq.

BlackHole

)4(clePointParti )(ˆ

g

zxdmThG

Perturbed Einstein eq.

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To predict the waveform during the LISA observation time, it is necessary to consider the radiation reaction effect to the orbit.



•Dynamical time

•Radiation reaction time

•Dephasing time

•LISA observation time

Due to the BH uniqueness theorem, the black hole must be a Kerr black hole.

Due to the radiation reaction, the orbital motion is dissipative.

minutes dyn fewT

years radrad few

T

E

ET

weeksraddephase several

TT

years )(LISA severalfewT

t rad

2rad t

dephaseLISA TT

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2: Self-force calculation



By the post-Newtonian calculation, we usually calculate the gravitational energy flux and calculate the orbital evolution by

GWorb EE

We cannot use the same approach for the orbit around the Kerr black hole.

Kerr geodesic; CLE z ,, GW GW , zz LLEE

GWCC

OK!

NO!

Balance formula

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2: Self-force calculation



By the post-Newtonian calculation, we usually calculate the gravitational energy flux and calculate the orbital evolution by

GWorb EE

We cannot use the same approach for the orbit around the Kerr black hole.

Kerr geodesic; CLE z ,, GW GW , zz LLEE

GWCC

OK!

NO!

We use the MiSaTaQuWa self-force.

)(lim Singularfull xhFhFFVd

dm

zxMiSaTaQuWa

Balance formula

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MiSaTaQuWa self-force;


dm

zxMiSaTaQuWa

A general formalism for the self-force acting on a particle:

•The particle moves in a curved vacuum background.•It is only for the leading order. (linear perturbation)

•If the particle is small enough, it can be represented by a “point” particle.•The metric perturbation induced by the particle is divergent along the orbit and the regularization to remove the divergence is prescribed within the classical framework.

•It is formulated under the harmonic gauge condition.•The singular part is derived only around the orbit.

•The tail effect of the wave propagation induces the force.

02

1

;

hgh

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MiSaTaQuWa self-force;


dm

zxMiSaTaQuWa

Schwarzschild BH; convenient in Regge-Wheeler gauge possible in the harmonic gauge, but, need numerics.

full metric perturbation …?

Barack-Mino-Nakano-Ori-Sasaki method;

lmlmlm

zx

lmlmlm

zx

YrthFhF

YrthFhFVd

dm

),(),(lim

),(),(lim

Singular""full

Singular""full

Kerr BH; “convenient” in Radiation gauge in the harmonic gauge … in progress (2D numerics)

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We have a good progress in the self-force calculation, however, it will need too much computational power:

•Even possible, the calculation of the metric perturbation in the harmonic gauge is difficult. A present idea is to implement 2D code.

•One has to calculate the self-force at each orbital point.•The convergence is not good.

We may need as much as 10^18 templates!

m

tiim(m,ωm erthh ),(fullfull

l

lmlmlm

zx

ll

B

l

A

YrthFhFVd

dm

)2/3)(2/1()2/1(

),(),(lim

2222222

Singular""full

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3: Radiation Reaction Formalism

What is the relation b/w Balance formula & MiSaTaQuWa self-force?

radiativeGW

radiativeGW , hFLhFE L

zE Hint by Galt’sov :

advancedretardedradiative

2

1hhh radiative field

Geodesics around the Kerr black hole have symmetry.

,tt ),,(),,( CLECLE zz A geodesic is transformed into itself.

),,(),,( CLECLE zz

The self-force changes the sign.

2/

2/)( lim

T

TTFdF time average

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x

Past Light cone

x

FutureLight cone

)(,,retarded

CLE z

)(,,advanced

CLE z

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)(,,)(,,advancedretarded

CLECLE zz

)(,,)(,,

2

1

)(,,)(,,2

1)(,,

retardedretarded

advancedretardedradiative

CLECLE

CLECLECLE

zz

zzz

The radiative self-force is expressed with the two-point average of the retarded self-force.

r

)(,,retarded

CLE z

r

)(,,retarded

CLE z

)(,,radiative

CLE z

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Time averaged self-force is derived by the radiative field.

retardedradiative,,,, CLECLE zz

Easy to calculate:•regularization without regularization

•most efficient convergence

•gauge invariant

sing.adv.sing.ret.adv.ret.radiative

2

1

2

1GGGGGGG

)(,,

,,,,radiative

l

lmz

lmlmzz

vOCLE

CLECLE

exponential convergence

There is a convenient method even for a Kerr black hole.

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Is the time averaged self-force sufficient to predict the orbital evolution?

nn

nninnCLE

CLE

r

rrr eFFF

FFF

CLE

,

),(

forceSelf

,,

,,

,,

The evolution of (E,L,C) by the entire self-force under a reasonable gauge; E,L,C

t

Under the radiation reaction formula;

CLE FFFCLE ,,,, RRformula

t

E,L,C

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t t

“Radiation reaction approximation”

t

We fail the prediction!!

aaaa EEEFEd

d0)0(,)(),(;

aaaa EEEFE

d

d0)0(,)(

E,L,CE,L,C

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t t

Radiation reaction formalism

t

We succeed the prediction!!

aaaa EEEFEd

d0)0(,)(),(;

aaaaa EEEEFE

d

d00)0(,)(

E,L,CE,L,C

Radiation gauge condition; it exists only when 12

v

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4: Gauge Problem

We use the “Adiabatic Approximation” for the orbital prediction.

t

x

At each instant, the orbit is a almost background geodesic. Using this geodesic, we calculate the self-force at each instant.

But, at an instant, the self-force is entirely gauge dependent….

Can this problem be solved by higher order perturbation expansion?

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Fools’ derivation of the self-force;

The linear metric perturbation requires the geodesic as the source.

By the standard metric perturbation, we expand the metric and the orbit with the geodesic as a background.

The expansion is valid only when the orbital deviation is small.

0][ TThG

)3(3)2(2)1(

geodesic

)3(3)2(2)1(BlackHole

)( zzzzz

hhhgg

t

x

zzgeodesic

)(geodesic Ozz

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By the gauge transformation, one can always bring the full orbit into the background geodesic during the entire region where the perturbative expansion is valid.

By choosing this gauge condition, the self-force is vanish to the full order of the perturbative expansion.

t

x0ForceSelf

F

Note: Gauge is a freedom to assign the coordinates to a perturbed geometry. It has nothing to do with the causality or hyperbolicity of the Einstein equation.

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Fools’ derivation of the self-force casts two questions:

Q-1) Should we use some specific gauge condition for the self-force?

Q-2) The standard perturbation expansion is valid only in the dephasing time. Can we define a consistent perturbation expansion valid long enough for the waveform prediction?

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Q-1) Should we use some specific gauge condition for the self-force? …

Basically NO! The waveform is observable and gauge invariant.But, the orbit is not directly observable, therefore, it can be gauge dependent. The modulation of the waveform due to radiation reaction also comes from the non-linear term.

)2(2

)3(3

)1(geodesic

)2(2)1(BlackHole

)( z

h

zz

hh

z

gg

][],[2],,[][

][],[][

][][

)2()1()2((2))1()1()1((3))3(Linear

)1()1()1((2))2(Linear

geodesic)1(Linear

zThhGhhhGhG

zThhGhG

zThG

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There is no physical reason to use some specific gauge condition for the self-force. MiSaTaQuWa self-force has no direct physical meaning by itself.

However,

Practically YES! By using a specific gauge condition, the self-force can have a physically meaningful information, which is the phase modulation.

Paper in progress...

If the metric perturbation is obtained in the form,

the self-force carries the physical information of the phase modulation.

)',;',;';'( )',(

)|'()',( ')(

''''

geodesic'''')1(

rrttGxxG

zxTxxGdxxh

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5: Adiabatic Expansion

A new metric perturbation scheme for the waveform calculation;

• It must not expand the orbit from the background geodesic.• It must have the picture of “adiabatic approximation” to the leading order.

One can formulate a new metric perturbation scheme by two-scale expansion.

1)(~onrad.reacti

dyn. OT

T

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))(|()( tCxTxT

1st step … the source term We use the point source. Instead of the orbital coordinates, we consider the evolution of the orbital elements.

2nd step … the foliationWe introduce the foliation and use it for the orbital parameter.

3rd step … the metric perturbationThe metric perturbation is defined to the function of the orbital elements.

)|( CxT ( ; Source term for a geodesic)

)()),(|()( xfffCxTxT

))(|())(|( )2(2)1(BlackHole fCxhfCxhgg

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Ansatz of the two-scale (derivative) expansion;

For example, a partial derivative becomes

)()( nn

n

OfCdf

d

fdf

dC

dC

dCxfCx fCC

)()|())(|(

C is effectively regarded as constant.Higher order term

0th order equation becomes

which is the linear perturbation in the picture of the adiabatic approximation

)|()]|([ )1(Linear CxTCxhG ))(|()1( fCxh

28x

t

C6

h=h(C1)



C5

C4

C3

C2

C1

h=h(C2)

h=h(C3)

h=h(C4)

h=h(C5)

h=h(C6)Adiabatic Approximation;

•We assign the orbital elements, C(t), at each instant.•On the foliation, f, we use the linear metric perturbation induced by the geodesic of C(f).

For the purpose of the self-force, the choice of the foliation function is not relevant.

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1st post-adiabatic equation becomes schematically

fdf

dCCxh

dC

d

fdf

dCCxh

dC

d

CxhCxhGCxhG

2)1(

)1(

)1()1((2))2(Linear

)|(

)|(

)]|(),|([)]|([

Effect of the orbital deviation from the geodesic

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6: Conclusion



Self-force calculation Possible, but, the technique is not yet established not an effective calculation

Radiation reaction formalism well established, most effective One can use a semi-analytic method, but, not yet applied.

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Theoretical question: Is the adiabatic expansion convergent? What is the “right” choice of the foliation? What is the “right” constraint on the gauge? Astrophysical Question: What is the validity in using the adiabatic approximation? … PN estimation says YES for EMRI (circular orbit) Can we have the evidence of the BH uniqueness? How about Intermediate-mass-ratio inspirals?

Practical Question: How efficient can we calculate the waveforms? What is the best data analysis strategy?



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Question to EFT people Adiabatic expansion must be taken into account in NRGR method because it is systematic formulation of the particles & metric. … What foliation do you use?

Documents

Yasushi Mino ( 蓑 泰志) Center for Gravitational Wave Astronomy University of Texas at Brownsville

Yasushi Mino ( 蓑泰志) Center for Gravitational Wave Astronomy University of Texas at Brownsville