A New Approach to Equation of Motion 　

for a fast-moving particle 　 　

T. FutamaseAstronomical Institute,

Tohoku University, Sendai, Japan22th Sept. 10 @Yukawa. Kyoto

Collaborators: P. Hogan(Dublin), Y. Itoh(Tohoku)

Why we need Fast motion approximation

• Equation of Motion(EOM) plays an important role in Gravitational Wave (GW) Astronomy

• Slow-motion (post-Newtonian ) approximation has been developed up to 3.5 pN order

• 4 pN approximation seems impossible at the moment and even if we have it, PN series converges very slowly

• There are interesting GW sources with high velocities

Fast-motion approximation will be necessary

There has been development in this field such as Mino, Sasaki and Tanaka, and others

However actual calculation of self and radiation reaction force needs special treatment (separation of divergent self-field) and is difficult for arbitrary motion

New method of fast motion approximation without any divergence and easy to calculate radiation reaction will be very useful

The problem we want to study

• A small charged particle (BH) moving in an arbitrary external gravitational and electromagnetic fields.

We have learned that infinities appears by considering delta function type source, and we have also learned that no infinities appear by considering point particle limit in post-Newtonian case.

We would like to consider point particle limit, but the limit will be taken along null direction on this case

We will use a (future) null coordinate with vertex at the particle

We do not want to have divergences anywhere in the derivation and EOM.

Point particle limit )(ux

We want to consider the following situation and taking a limit r -> 0 along the null direction

:)(ux a timelike world-line with 4-vel. v and 4-accelaration a

We also consider the situation where a small particle produces a small perturbation on a background except very small region around the particle(body zone)

02

This may be possible by shrinking the boundary of the body zone as m and assuming the following scaling

~~, 2 rasem

v

X

kr

r

The boundary always stays in the far zone viewed from the body, thus far zone expansion is possible at the boundary

22

22 2))(( ducdudrbdudychedxsheadudyshedxche

p

rds

,

,0

udr

dxk

kkg

We consider the following form of the metric in the coordinate (x,y, r, u)

Background space-time

Without a massive particle, the spacetime is regular on a timelike (r=0) world-line and in a nbd. of the world-line

)( 2rOg

)(ux

u=const.k

r

),( yx

A solution of Einstein-Maxwell equations

)(log

)(4

11,,),(

4

11

00

0

2222

Pu

kah

vP

yxyxyx

20

2220

2

2

)21(2)( durhdudrdydxPr

dXdXds

Therefore we assume our metric approaches to the flat metric at a timelike curve X=x(u) written in the following form(Newman-Unti,1962)

:

:

du

dva

du

dxv

where

4-velocity of the worlfdline r=0

Derivation 0:)(

uxX

v

X

kr

r0 vr

)(ux

)()(0 uvXxX

k

rX

uu

1

,

kah

khvku

kkk

r

k

0

0 ,,,0,

u

PPhyxuPk

0100

10 ),()(

)(

4

11,,),(

4

11 2222 yxyxyx

Coordinate transformation relating the Cartesian (X) and curvilinear (u,r,x,y)

10)( rPuxX

dyy

dxx

rPdrkduhrPv

drPdrkdPrdxdX

100

10

10

10

)(

20

2220

2

2

)21(2)( durhdudrdydxPr

dXdXds

Then

Small r expansion around r=0

.....1

......,

......,

......,

......,

......)1(

220

221

221

33

22

33

22

33

220

rcrhc

rbrbb

raraa

rr

rr

rqrqPp

22

22 2))(( ducdudrbdudychedxsheadudyshedxche

p

rds

0)21(2)( 20

2220

2 rdurhdudrdydxPr

Similary background Maxwell field KduMdyLdxA

......,

......,

......,

22

1

33

22

33

22

KrKrK

MrMrM

LrLrL

By solving Einstein-Maxwell equation order by order

.....1

......,

......,

......,

......,

......)1(

220

221

221

33

22

33

22

33

220

rcrhc

rbrbb

raraa

rr

rr

rqrqPp

.....ˆˆ)ˆ2(2

......,ˆˆˆˆ

......,ˆˆˆˆ

......,ˆˆ

......,ˆˆ

......)ˆˆ1(ˆ

2210

12

2

2210

1

2210

1

33

22

33

22

33

220

rcrccr

fm

r

ec

rbrbbr

bb

raraar

aa

rr

rr

rqrqPp

Introduce a small charged BH on the background

BH perturbation of Background

dxdxcpbagds ),,,,,(2

21101

101

ˆ,ˆˆ

,),(ˆˆ

OfObb

OemOaa

1ˆ OAA

)1(ˆ32100 OQQPP

1O 2O

All of these coefficients are chosen in order to satisfy the Reisnner-Nordstrom black hole limit, and higher-order terms are determined by solving Einstein-Maxwell field equations by order by order in r

where

KduMdyLdxA

......,

......,

......,

22

1

33

22

33

22

KrKrK

MrMrM

LrLrL

......,ˆˆˆ)ˆ(

......,ˆˆˆ

......,ˆˆˆ

22

101

33

22

0

33

22

0

KrKrKr

KeK

MrMrMM

LrLrLL

RN BH as r->0

1021

100

ˆ,ˆ

,ˆˆ

OKOK

OML

duhr

eduhr

rder

dXevdXAA

00

11)(log

durhdrdXv

krhvruxXvr

)1(

)1(,))((

0

0

We assume that A is predominantly the Lienard-Wiechert form near r=0

200200ˆ,ˆˆ OehKOML

Derivation method of Equation of Motion

)1(ˆ)(ˆ32100

2220

220 OQQPPdydxPrds

Necessary conditions for the 2-surfaces with the above line-elements to be smooth, non-trivial deformations of 2-spheres will be the equations of motion of the black hole.

Neglecting O(r^4)-terms, the line element induced on these null hypersurfaces are given by

The null-hypersurfaces u=const. in the perturbed spacetime are approximately future null-cones for small r

The wave fronts can be approximated 2-spheres near the black hole

Solve Einstein-Maxwell equation (R=T_EM) order by order in m and e

EMababab TR ˆ

)(

)(

)(

)(

)(

)(

)(

)(

13

23

344

02

12

23

34

434

133

02

12

22

34

121

32

3

121

32

12

121

32

2211

02

13

23

4

rOOrOr

rOOrOrOrOr

rOO

rOOrOrOr

rOOOrOr

rOOOrOr

rOOOrOr

rOOrOrOr

A

A

AA

baba FM |

)(

)(

)(

22

4

1313

121

32

rOOrM

rOOOrM

rOOOrOrM A

2110

21

21

202

202121111

12

012

020

2200400)2(

821

2

1122844

34log2

OvkeFQQc

baPqecMbLaeKKeKe

bPy

aPx

Py

L

x

M

y

L

x

MPPAA

11222

01

2222121

201

20

4)1( 242

OvkFOy

M

x

LPK

y

M

x

LqeMbLaPKPM

21144)2( 66)2(2

1OvpeFpmaQQ

vvhkhp ,

02 pp

211 )(662 OuAvpeFpmaQQ

211 33)2(2

1OpveFpmaQQ

pOpveFma ,)( 2

2220 yxP

Same equation is obtained by taking angular average over a small sphere around r=0

Equation governing the perturbation of the 2-sphere around BH

where Laplacian on the 2-sphere

Q_1 is l=0 or l=1 trivial perturbation, thus we can take Q_1=0

In the next order

32102244)2( )2(2

1OAAAQQ lll

l=1 spherical harmonic

l=2spherical harmonic

l=0 spherical harmonic

kkuA

Py

umYuVePx

umXuUevpFuKe

pGappFFhevpFmepahevpeFpmaA

vvFFeGA

l

l

l

)(

)(log))(2)((3)(log))(2)((3)(12

1288466

3

164

2

02

022

221

20

)1(ˆ)(ˆ32100

2220

220 OQQPPdydxPrds

Induced metric on the two-sphere around BH

32122 3

12 OAAQQ

32122 6

1

2

1)2(

2

1OAAQQ ll

Provided A_0=O_3, this equation can be integrated without introduction of directional singularities

Or, taking angular average over a small sphere around r=0

vvFFeG 2

3

4

Eqn. for Q_2

Smooth non-trivial l=2 perturbations of the wave fronts near the black hole

322

3

4

3

2OTvFFheaheveFma

31 OA For Q2 to be free of directional singularities we must havewhich gives us the following EOM

06 22 QQ

223222 12

1

18

12 AQOAQQ

where

vGFFF

m

eT 2

)()(2 2 uFuKe

vvFFeG 2

3

4

with

Remaining perturbation

Problem?

Does not coincide with DeWitt-Breme( a charged particle on a fixed curved background and no EM background) . Ours solves Einstein-Maxwell equation consistently up to O_2. We may have to go O_3

Kerr solution with mass m and angular momentum (a_0,b_0,c_0) can be put in the form

We consider a general metric which approaches to the above form

EOM with spin

22220

222 2)()()( dcdudbddyaddxpFrdsKerr

dyFdxFdud xy with

22

2220

22

22

20

22

20

22000

10

220

)(21

,,

)](4/11[

)(4/11

Fr

FFp

Fr

mrc

Fr

Fpb

Fr

Fpa

yxcybxapF

yxp

yx

xy

Perturbed Space-time

Expansion near r=0

22

22 2))(( dcdrddbdychedxshedadyshedxche

p

rds

Perturbed space-time as approximate vacuum space-time

where

F is an l=1 spherical harmonics and can be written as

334)3( 2ˆ OFFR u

2Odu

ds i

44)2( R̂ uuRsma i

Conclusions

• We have developed a new method for fast moving small self-gravitating particle without any divergence

• Does not coincide with DeWitt-Breme(testEM on fixed curved background) . Ours solve Einstein-Maxwell equation consistently up to O_2.

• Gravitational radiation reaction in O_4 ?