Date post: | 03-Jan-2016 |
Category: |
Documents |
Upload: | riley-mcdonald |
View: | 17 times |
Download: | 0 times |
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 ?