1 Yasushi Mino Theoretical AstroPhysics Including Relativity (TAPIR), CalTech E-mail :...

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Yasushi MinoTheoretical AstroPhysics Including Relativity (TAPIR),

CalTech E-mail : [email protected]

Index 1: Introduction: LISA project 2: MiSaTaQuWa Self-force? 3: Adiabatic Metric Perturbation 4: Radiation Reaction Formula 5: Gauge and Validity 6: Conclusion

Astrophysics Seminar at University of Florida, Gainesville, Sept. 24, 2004

Self-Force in Radiation Reaction Formula

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Everyday we experience gravity. Why should we discuss the gravitational physics?

Newtonian Gravity :

Gravity as a potentialNo dynamical Freedom

= 4GEinstein Gravity :

Gravity as a geometryDynamical Freedom in Gravity

G = 16GT

We want to know the dynamics of the gravitational physics!



1: Introduction: LISA Project

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The dynamical degree of freedom in Einstein’s Gravity propagates the space-time as gravity waves. When linearized, we have a wave equation of the metric perturbation.

g = gbackground + h + O(h2)

Detection of gravitational waves is a strong evidence of the dynamical nature of the gravitational theory.

– 12;

; – R = 8GT

= h – 12 gh



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Einstein Gravity predicts a very strongly self-gravitating object, called “Black hole”.

Newtonian Gravity :

It makes a singularity if gravitationally collapsed.

Einstein Gravity :

The singularity is hidden by the horizon.

r

We want to know the nature of the strong gravity!

r rH

5

Projects of Gravity Wave detection : Experimental test of the relativistic gravitational theory New observational window to distant astrophysical objects Observation of highly relativistic gravitational phenomenaPromising Target: NS/BH Binary system, SuperNova, Primordial GWs, Pulser, GammaRayBurst, …..



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Space Project: LISA (joint project by NASA & ESA)

See LISA project homepage http://lisa.jpl.nasa.gov/



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8





9

LISA primary target

A compact object (~10M) is inspiralling into a super-massive blackhole (~10^6M).

*extreme mass ratio*eccentric orbit*relativistic motion

We need to know what gravitational waves are expected to be detected.



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Theory is challenged by Experiment.

Unlike other theoretical physics, we do not (did not?) have a theory to predict the observation until recently!

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2: MiSaTaQuWa Self-force?It is a good approximation to consider it as a two-body problem in GR.

The central black hole is considered to be a Kerr black hole.

For its extreme mass-ratio, a linear perturbation might be a good approximation. 510/ Mm

M

m

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Suppose we could use a linear perturbation: We approximate the supermassive black hole by a Kerr black hole, and consider the linear metric perturbation induced by an inspiralling compact object.

One can calculate the gravitational waveform by a linear perturbation, being given a orbit.

We need to solve the orbital equation.

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MiSaTaQuWaFVd

D

A binary system is known to emit gravitational waves, and the gravitational waves carry away the binding energy of the system. As a result, the orbit deviates from the background geodesic.

MiSaTaQuWa self-force was derived by a linear perturbation. It is considered to include the radiation reaction effect to the orbit.

(MiSaTaQuWa=Mino,Sasaki,Tanaka;Quinn,Wald)

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We come to have a regularization method to evaluate MiSaTaQuWa self-force. … Radiation Reaction Formula (Mino) Mode-decomposition method

Barack,Ori; Mino,Nakano,Sasaki;Detweiler,Messaritaki,Whiting,Kim

Power-expansion method (Mino,Nakano)

but … something weird …

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Do we have to consider the self-force in a certain gauge? Is the orbital evolution gauge-dependent?

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One may choose a gauge condition such that the self-force vanishes, and it is consistent with the linear perturbation!

1) The linear perturbation is derived by solving the linearized Einstein equation.

ThG ][

0 T

2) The linearized Einstein equation requires the conservation law in the background, i.e. a geodesic.

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3) The linear approximation is valid only when the orbital deviation from a geodesic is small.

t

x

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.

4) Because the orbital deviation from a geodesic is small, one can bring the orbit’s coordinates back to those of the geodesic.

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Conclusion:There is no self-force.

Problem:We have to extend “the linear perturbation formalism” so that it can describe the metric perturbation induced by a non-geodesic orbit.

Conclusion: MiSaTaQuWa self-force makes no physically meaningful prediction of the orbital evolution by itself.

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3: Adiabatic Metric Perturbation

We consider a quasi non-linear extension of the linear metric perturbation so that one can describe the metric perturbation induced by a non-geodesic orbit.

For this, we use 1) a physically reasonable class of gauge conditions,2) the picture of adiabatic approximation.

The adiabatic approximation is well known in classical mechanics, but, the application to a classical gauge field is not well known.

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Kerr black hole is a stationary solution of the Einstein equation, thus, the linearized Einstein equation of a Kerr background is time-independent.

)'()',(')( '''' xTxxGdxxh

)'(

'' )',()',( ttieGdxxG xx

We call this a physically reasonable class of gauge conditions.•One can easily extract the physical information of gravitational waves.•Technically feasible metric perturbation formalisms belong to this class.

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Step 2: We approximate the orbit by a geodesic on each foliation hypersurface.

x

t

Step 1: We consider the spacelike foliation.



6

4

3

2

1

5

Step 3: We patch the linear metric perturbations of geodesics on each foliation surface.

h6

h5

h4

h3

h2

h1

Adiabatic extension of the linear perturbation:

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Linear perturbation:

)(][ ThG : a geodesic

)(][ )()( tt ThGAdiabatic metric perturbation:

(t) : time-evolving geodesic

Adiabatic metric perturbation is valid as long as the extra term is sufficiently small. (around a year)

Linear perturbation is valid as long as the orbit does not deviate from a geodesic. (around a week)

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4: Radiation Reaction Formula

Radiation Reaction Formula was originally formulated by a linear perturbation in the physically reasonable gauge.(Strictly speaking, all the formula so far is based on the linear perturbation, and we cannot make any physical interpretation.)

For a meaningful discussion of the orbital evolution, we have to consider the adiabatic extension of Radiation reaction formula in a manner consistent with the adiabatic metric perturbation.

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4-a: “Original” Radiation Reaction Formula

We consider a geodesic. Geodesics are characterized by 6 parameters;

},,{

},,{ cc

CLEtr

Primary constants ;Secondary constants ;

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2

2222

22222

sin22

1

21

cossin

sin

LMraMrE

d

d

aMrLEd

dt

KaL

aEd

d

KraLEard

dr

222222

22

2222

sin2

2

cos

Mraar

aMrr

ar

2

d

d

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n

inn

rrr

n

inn

e

eRr r

)(2,)(

)(2,)(

)(

)(

cee

ceTeTTt

n

inninnr

t

n

inninnr

r

r

)()(

)()(

)(

)(

r/-motion t/-motion

Asymptotic gravitational field in the physically reasonable class of gauge conditions;

TTTr

,, : three principal frequencies, functions of (E,L,C)

Tm

Tn

Tntihrh r

rmnnmnn

mnnmnn

r

r

r

r

),,(

,,),,(

),,( ,)exp()(

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nnrr

nnaa

r

r ininCLEFCLEF,

),( exp),,();,,(

We consider the self-force acting on (E,L,C).

V

d

DVFV

d

DFV

d

DF CCLLEE ,,

)2

1,,(

VVCVLVE CLE

In the physically reasonable class of gauge conditions, the linear metric perturbation gives the force as



It is proven that the zero-mode is gauge-invariant and can be obtained by the radiative part of the field.

)( )0,0()0,0()0,0( advancedaradiativearetardeda FFF

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nm

nminmtnmttt

nm

nminmrnmrrr

nm

nminm

r

r

r

ecccccccc

e

eKLECLECLE

,

),(),(2

)0,0(

,

),(),(2

)0,0(

,

),()0,0(

,,2

,,

,,2

,,

,,,,,,

Perturbative evolution of the orbital constants becomes;

One can see that the perturbative evolution of the orbit and the linear metric perturbation becomes in valid at the dephasing time scale;

)( 2/1 OTt dephase (around a week)

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4-b: “Adiabatic” Radiation Reaction FormulaNow the “constants” could evolve non-perturbatively;

},,,,,{ ccCLE tr

},,,,,{ )()()()()()()(

ccCLE tr

nnrr

nnaa

r

r ininCLEFCLEF,

)()()(),(

)()()(~~exp),,();,,(

In the physically reasonable class of gauge conditions, the adiabatic metric perturbation gives the force as

)(2~)(

/)(//

r

rr (We ignore O(^2) terms here.)

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We have the adiabatic evolution equation of the orbit. (See the upcoming paper in detail.)

ccbbaa Fcd

dF

d

dFΕ

d

d

,,

The behavior of these evolution equations;

)(),(),( tOFtOFOF cba

)(][ )()( tt ThG)( 2tO

Einstein Equation for the adiabatic metric perturbation;

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Validity of the adiabatic metric perturbation;

)( 1 OTmetric

Validity of the adiabatic orbital evolution;

Beyond this time scale, we do not solve the Einstein equation to the accuracy O().

)( 1 OTorbit

Beyond this time scale, the second order effect will change the orbital evolution.

This time scale is around several months for LISA.

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5: Gauge and Validity

nnrr

nnaa

r

r ininFCLEF,

),()()()(

~~exp);,,(

)0,0(aF is proven to be gauge invariant in the physically reasonable class. What about, non-zero components?

We found that the non-zero modes are totally gauge dependent. By a special choice of gauge, one can eliminate the non-zero mode.

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)0,0()()()( );,,( aa FCLEF

We call this gauge condition the radiation reaction gauge.

In this gauge, the self-force has only the dissipative term. This DOES NOT mean that the self-force does not have conservative terms. The conservative effect of the self-force is renormalized into the initial conditions.

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General gauge in the physically reasonable class;

)();,,( )()()( OCLEF a

)();,,( 5)()()( vOCLEF a

Radiation reaction gauge;

(v) is a typical velocity of the system, and is 0.3 at most.

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Validity of the adiabatic metric perturbation;

)( 51 vOTmetric

Validity of the adiabatic orbital evolution;

Beyond this time scale, we do not solve the Einstein equation to the accuracy O().

)( 2/51 vOTorbit Beyond this time scale, the second order effect will change the orbital evolution.

This time scale is around several years for LISA.

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6: Conclusion

A method for an orbital prediction using a linear metric perturbation is established for the first time.

The method can predict the orbital evolution for long enough for LISA project.

The calculation technique of the method proposed here is already established and the required computational power is minimum.

Coding to calculate the waveform is in progress.

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