PHYSICAL REVIEW D 67, 104016 共2003兲
Self-force of a scalar field for circular orbits about a Schwarzschild black hole
Steven Detweiler, Eirini Messaritaki, and Bernard F. Whiting
Department of Physics, P.O. Box 118440, University of Florida, Gainesville, Florida 32611-8440
共Received 20 February 2003; published 21 May 2003兲
The foundations are laid for the numerical computation of the actual worldline for a particle orbiting a black
hole and emitting gravitational waves. The essential practicalities of this computation are illustrated here for a
scalar particle of infinitesimal size and small but finite scalar charge. This particle deviates from a geodesic
because it interacts with its own retarded field ␺ ret. A recently introduced Green’s function G S precisely
determines the singular part ␺ S of the retarded field. This part exerts no force on the particle. The remainder of
the field ␺ R⫽ ␺ ret⫺ ␺ S is a vacuum solution of the field equation and is entirely responsible for the self-force.
A particular, locally inertial coordinate system is used to determine an expansion of ␺ S in the vicinity of the
particle. For a particle in a circular orbit in the Schwarzschild geometry, the mode-sum decomposition of the
difference between ␺ ret and the dominant terms in the expansion of ␺ S provide a mode-sum decomposition of
an approximation for ␺ R from which the self-force is obtained. When more terms are included in the expansion, the approximation for ␺ R is increasingly differentiable, and the mode sum for the self-force converges
more rapidly.
DOI: 10.1103/PhysRevD.67.104016
PACS number共s兲: 04.25.Nx, 04.20.Cv, 04.30.Db, 04.70.Bw
I. INTRODUCTION
In general relativity, a particle of infinitesimal mass will
orbit a black hole of large mass along a worldline ⌫ which is
an exact geodesic in the background geometry determined by
the large mass alone. If the orbiting particle is not infinitesimal, having a small finite mass, its orbit will no longer be a
geodesic in the background of the larger mass, and gravitational waves will be emitted by the system — at infinity. In
the neighborhood of the small particle, local measurements
cannot separately distinguish the background of the large
mass from a smooth perturbation to it caused by the presence
of the smaller mass 关1兴. The actual orbit of the particle can
be analyzed in a linearization of the Einstein equations via a
perturbation expansion in the ratio of the masses. Through
first order in this ratio, ⌫ is known to be a geodesic of a
geometry perturbed from the background of the large mass
by the presence of the smaller one 关1兴. The difference of the
worldline from a geodesic in the background is said to arise
from the interaction of the orbiting particle with its own
gravitational field. It is said to result from a ‘‘self-force,’’
even though, in the perturbed geometry determined by both
the small and large masses, the orbit would be observed to be
geodesic.
In a strict sense, if the particle is of infinitesimal size, then
its own field is singular along its worldline, and there the
perturbation analysis fails. This difficulty can be avoided by
allowing the size of the particle to remain finite while invoking the conservation of the stress-energy tensor within a
world tube which surrounds the worldline, in a manner similar to Dirac’s 关2兴 classical analysis. The balance of energy
and momentum indicates how to calculate the self-force in a
way which is independent of the size of the particle. The
limit of vanishing size may then be taken without confusion.
In curved spacetime, analyses beginning with DeWitt and
Brehme 关3兴 and subsequently by Mino, Sasaki and Tanaka
关4兴 and by Quinn and Wald 关5,6兴 formally resolve the difficulty presented by the singularity in curved spacetime with a
0556-2821/2003/67共10兲/104016共18兲/$20.00
Hadamard expansion 关3兴 of the Green’s function near ⌫. The
retarded Green’s function G ret(p,p ⬘ ) incorporates physically
appropriate boundary conditions and describes the field ␺ ret
of a particle moving through a given spacetime. In most past
discussions of the self-force, G ret(p,p ⬘ ) is commonly divided into a ‘‘direct’’ part, which has support only on the past
null cone of the field point p, and a ‘‘tail’’ part, which has
support inside the past null cone and is a result of the curvature of spacetime. The analyses show that ␺ tail⫽ ␺ ret⫺ ␺ dir is
necessarily finite at the particle and suggest that it is the part
of ␺ which belongs on the right-hand side of an equation for
the self-force 关6兴
Fa ⫽qⵜa ␺ .
共1兲
In these approaches, the use of ␺ tail instead of ␺ ret constitutes
a form of regularization of the singular ␺ ret. Actually, while
finite, ␺ tail is generally not differentiable on the worldline 关3兴
if the Ricci scalar of the background is not zero. Similarly,
the electromagnetic potential A tail
a 共respectively, the gravitatail
) is not differentiable at the
tional metric perturbation h ab
particle if (R ab ⫺ 61 g ab R)u b 共respectively, R cadb u c u d ) is nonzero in the background. In all such cases, some version of
averaging must be invoked to make sense of the self-force.
Moreover, we find it instructive to observe that the tail part
of the field is necessarily associated with a nonphysical inhomogeneous source, i.e. ⵜa ⵜ a ␺ tail⫽0: cf. Eq. 共30兲.
In this paper an alternative regularization of the field ␺ ret
is used to compute the self-force where, in particular, by
regularization we mean not only controlling the singular behavior, but also the differentiability. We have recently given
a precise procedure for decomposing the retarded field in
neighborhood of ⌫ into two parts 关1兴:
␺ ret⫽ ␺ S⫹ ␺ R,
共2兲
where ␺ S is a solution of the inhomogeneous field equation
for the particle, and is determined in the neighborhood of the
67 104016-1
©2003 The American Physical Society
PHYSICAL REVIEW D 67, 104016 共2003兲
DETWEILER, MESSARITAKI, AND WHITING
worldline entirely by local analysis via Eq. 共36兲. As both ␺ ret
and ␺ S are inhomogeneous solutions of the same differential
equation, it follows that ␺ R, defined by Eq. 共2兲 is necessarily
a homogeneous solution and is therefore expected to be differentiable on ⌫. In Ref. 关1兴 we showed that ␺ R formally
gives the correct self-force when substituted on the righthand side of Eq. 共1兲 in place of ␺ tail. In this paper ␺ R is used
for an explicit computation of the self-force. We consider ␺ S
to be associated with the singular source, and ␺ R with the
regular remainder.
While the procedure which follows from Eq. 共2兲 is well
understood in principle, its application to physically interesting situations remains a challenge. In this paper we consider
a particle endowed with a scalar charge q in circular motion
about a Schwarzschild black hole. On a technical level, a
spherical harmonic decomposition of both ␺ ret and ␺ S provides the multipole components of each, and the mode by
mode sum of the difference of these components determines
␺ R and, thence, the self-force.
In Sec. II we give a brief overview of the relation between
our work and that of earlier authors. We also summarize our
analytical results and introduce the additional regularizing
parameters which allow us to obtain increased convergence
in our mode sum representation of the self-force.
A special set of coordinates is described in Sec. III; the
Thorne-Hartle-Zhang 共THZ兲 coordinates, introduced by
Thorne and Hartle 关7兴 and extended by Zhang 关8兴, are locally
inertial on a geodesic. These coordinates are convenient for
describing the scalar wave equation in the vicinity of the
geodesic, where the metric takes a particularly advantageous
form. In Sec. IV the Hadamard expansion for the Green’s
function, discussed in detail by DeWitt and Brehme 关3兴, is
described in terms of Synge’s 关9兴 ‘‘world function’’ ␴ (p,p ⬘ ),
which is defined as half of the square of the geodesic distance between two points p and p ⬘ . We obtain both ␴ (p,p ⬘ )
and the Hadamard expansion in terms of the THZ coordinates.
Section V outlines the determination of the regularization
parameters given below in Eqs. 共13兲 to 共15兲. These results
are in agreement with, but extend by going to higher order,
the work of Barack and Ori 关10–12兴 and Mino, Nakano and
Sasaki 关12,13兴.
In Sec. VI, with a concrete application of our method, we
examine a scalar charge in a circular orbit of the Schwarzschild geometry at a radius of 10M . It is in this section that
we see the practical advantage of using a higher order approximation in the regularization of ␺ S. The additional parameters we find enable us to increase dramatically the rate
of convergence in the self-force summation.
In several Appendixes we include details concerning the
THZ coordinates, the mathematical analyses which focus on
calculation of the regularization parameters and a brief summary of details concerning the integration of the scalar wave
equation in the Schwarzschild geometry.
Notation. a,b, . . . are four dimensional space-time indices. i, j, . . . are three dimensional spatial indices.
(t s ,r, ␪ , ␾ ) are the usual Schwarzschild coordinates. t,x,y,z
are locally-inertial THZ coordinates attached to the geodesic
⌫, and ␳ 2 ⬅x 2 ⫹y 2 ⫹z 2 . The geodesic ⌫ is given as x a
⫽za(␶), where ␶ is the proper time along ⌫. The flat spatial
metric in Cartesian coordinates is ␦ i j . The flat Minkowski
metric in Minkowski coordinates is ␩ ab ⫽(⫺1,1,1,1). The
points p and p ⬘ refer to a field point and a source point on
the world line of the particle, respectively. In the coincidence
limit p→p ⬘ . An expression such as O( ␳ n ) means of the
order of ␳ n as x i →0 in the THZ coordinates. But note that
the differentiability of such an order term is only necessarily
C n⫺1 at x i ⫽0.
II. OVERVIEW OF RELATION TO EARLIER WORK
Formally, although our approach differs from that of
Barack and Ori, our method of implementation is similar to
that in their pioneering analysis in Refs. 关10,11,14兴. In their
procedure, which Burko has implemented 关15,16兴 both for a
scalar field with radial and with circular orbits of Schwarzschild, the self-force may be thought of as being evaluated
from 关30兴
ret
dir
F self
a ⫽ lim 关 F a 共 p 兲 ⫺F a 共 p 兲兴 ,
共3兲
p→p ⬘
where p ⬘ is the event on ⌫ where the self-force is to be
determined, p is an event in the neighborhood of p ⬘ , and the
relationship between Fa (p) and ␺ (p) is as given in Eq. 共1兲.
dir
To make use of this equation, both F ret
a (p) and F a (p) are
ret
expanded into multipole ᐉ modes, with F ᐉa (p) determined
numerically. Typically the source is expanded in terms of
spherical harmonics, and then a similar expansion for ␺ ret is
used
ret
␺ ret⫽ 兺 ␺ ᐉm
共 r,t 兲 Y ᐉm 共 ␪ , ␾ 兲
ᐉm
共4兲
ret
where ␺ ᐉm
(r,t) is found numerically. The individual ᐉm
components of ␺ ret in this expansion are finite at the location
ret
is
of the particle even though their sum is singular. Then F ᐉa
ret
finite and results from summing qⵜa ( ␺ ᐉm Y ᐉm ) over m. The
ᐉ-mode expansion of F dir
a (p) was initially determined by a
local analysis of the Green’s function for an orbit at r o in
Schwarzschild coordinates in Ref. 关10兴,
冉 冊
dir
⫽ ᐉ⫹
lim F ᐉa
r→r o
1
Ca
⫹O 共 ᐉ ⫺2 兲 ,
A a ⫹B a ⫹
2
1
ᐉ⫹
2
共5兲
in which it was found that the O(ᐉ ⫺2 ) terms yield precisely
zero when summed over ᐉ. Moreover, for circular geodesics
in the equatorial plane of the Schwarzschild geometry, the
regularization parameter C a ⫽0 and A a and B a also vanish
except for their r components. The values of A r and B r , first
determined by Barack and Ori 关10–12兴, are given below in
Eqs. 共13兲 and 共14兲. A further term, which we shall denote as
D a⬘ , was also introduced in Refs. 关10–12兴 and shown there
to be zero. It refers to the sum of the O(ᐉ ⫺2 ) terms in Eq.
共5兲. We comment further about the contribution of D ⬘a towards the end of this section. The self-force is ultimately
calculated as
104016-2
PHYSICAL REVIEW D 67, 104016 共2003兲
SELF-FORCE OF A SCALAR FIELD FOR CIRCULAR . . .
⬁
F self
a ⫽
兺
ᐉ⫽0
冋
冉 冊
ret
lim F ᐉa
⫺ ᐉ⫹
p→ p ⬘
1
Ca
A ⫺B a ⫺
2 a
1
ᐉ⫹
2
册
⫹D a⬘ .
ret
S
R
R
F self
a ⫽F a ⫺F a ⫽F a ⬅qⵜa ␺
r→r o
共6兲
Burko 关15兴 notes in his numerical analysis that the terms in
this sum scale as 1/ᐉ 2 for large ᐉ, the sum converges as 1/ᐉ,
and it is evident from his results that he computes to at least
ᐉ⫽80 and finds improved convergence with Richardson extrapolation.
From our perspective, the self-force at a point p ⬘ on ⌫ is
formally given by
⫹
B r ⫽⫺
⫽ⵜa
兺
ᐉm
S/ret
⫽ⵜa
F ᐉa
兺m
S/ret
␺ ᐉm
Y ᐉm ,
共10兲
the self-force is
F self
a ⫽
兺ᐉ 共 F ᐉaret ⫺F ᐉaS 兲
r o 共 r o⫺2M 兲
冋
2r o2 共 r o⫺2M 兲
r o⫺3M
4
共11兲
evaluated at r o . In the above expressions the difference in
multipole moments must be taken before the summation
over ᐉ.
In our approach the regularization parameters are derived
from the multipole components of ⵜa ␺ S evaluated at the
source point and are used to control both singular behavior
and differentiability. In Sec. V we consider circular orbits of
the Schwarzschild geometry at radius r o and show that
共13兲
r o2 共 r o⫺2M 兲
冋
D r⫽
⫺
⫹
⫺
at the source point p ⬘ . Further, with the definitions
关 r o共 r o⫺3M 兲兴 1/2
册冋
1/2
r o⫺3M
共8兲
共9兲
⫹O 共 ᐉ ⫺6 兲
F 1/2⫺
册
共 r o⫺3M 兲 F 3/2
,
2 共 r o⫺2M 兲
共14兲
and
R
␺ ᐉm
Y ᐉm
兺
ᐉm
ret
S
⫺ ␺ ᐉm
兲 Y ᐉm
共 ␺ ᐉm
共 2ᐉ⫺3 兲共 2ᐉ⫺1 兲共 2ᐉ⫹3 兲共 2ᐉ⫹5 兲
A r ⫽⫺ sgn共 ⌬ 兲
and the self-force can be determined by evaluating the vector
field
F self
a ⫽ⵜa
E r1 P3/2
where the regularization parameters are independent of ᐉ and
given by
共7兲
evaluated at the source point p ⬘ . Formally, the function ␺ is
defined only in a neighborhood of p ⬘ ; however for calculational purposes, the function may be extended in any smooth
manner throughout the spacetime. While the spherical harmonic components of this extended function in the
Schwarzschild geometry are not uniquely determined, they
still provide a convergent expression for ␺ S for events near
p ⬘ . Thus, in the Schwarzschild geometry the spherical harmonic expansions of ␺ S and ␺ ret yield
1
2 冑2D r
A r ⫹B r ⫺
2
共 2ᐉ⫺1 兲共 2ᐉ⫹3 兲
共12兲
S
R
ret
S
␺ ᐉm
共 r,t 兲 ⫽ ␺ ᐉm
共 r,t 兲 ⫺ ␺ ᐉm
共 r,t 兲 ,
冉 冊
S
lim F ᐉr
⫽ ᐉ⫹
册冋
1/2
⫺
M 共 r o⫺2M 兲 F ⫺1/2
2r o4 共 r o⫺3M 兲
共 r o⫺M 兲共 r o⫺4M 兲 F 1/2
8r o4 共 r o⫺2M 兲
共 r o⫺3M 兲共 5r o2 ⫺7r oM ⫺14M 2 兲 F 3/2
16r o4 共 r o⫺2M 兲 2
3 共 r o⫺3M 兲 2 共 r o⫹M 兲 F 5/2
16r o4 共 r o⫺2M 兲 2
册
.
共15兲
E r1 has not yet been determined analytically, but the constant
P3/2 in Eq. 共12兲 is independent of ᐉ and is given in Eq.
共D23兲; the ᐉ dependence of the E r1 term, and of higher order
parameters (E rk , k⬎1), is discussed in Sec. V. In these expressions F q refers to the hypergeometric function
1
2 F 1 关 q, 2 ;1;M /(r o⫺2M ) 兴 . The A r and B r terms agree with
the results of Refs. 关10–13兴 restricted to circular orbits.
When summed over all ᐉ, the D r and E rk terms individually
give no contribution to the self-force. This is consistent with
the results in 关10–13兴, but note the different definition of D r
there, which we have referred to above as D ⬘a . Our results
thus yield the identical self-force to that of Barack and Ori.
As we shall show in Sec. IV, in general ␺ S can be known
only approximately. If we ignore the D r and E rk terms in the
approximation for ␺ S, then the approximation for ␺ R is only
C 1 . Hence, the D r and E rk terms must be included for ␺ R to
be a homogeneous solution of Eq. 共30兲 as discussed above.
Although we have just indicated above that the D r and E rk
terms give no overall contribution to the self-force, we find
that understanding the nature of these additional terms can be
used to speed up dramatically the convergence of the sum in
Eq. 共11兲. We have used this understanding in obtaining the
results of Sec. VI.
104016-3
PHYSICAL REVIEW D 67, 104016 共2003兲
DETWEILER, MESSARITAKI, AND WHITING
III. THZ NORMAL COORDINATES
The scalar wave equation takes a simple form when written in a particular coordinate system in which the background geometry looks as flat as possible. Consider a geodesic ⌫ through a background vacuum spacetime geometry
g ab . Let R be a representative length scale of the background geometry—the smallest of the radius of curvature,
the scale of inhomogeneities, and the time scale for changes
in curvature along ⌫. A normal coordinate system can always be found 关17兴 where, on ⌫, the metric and its first
derivatives match the Minkowski metric, and the coordinate
t measures the proper time. Normal coordinates for a geodesic are not unique, and we use particular coordinates which
were introduced by Thorne and Hartle 关7兴 and extended by
Zhang 关8兴 to describe the external multipole moments of a
vacuum solution of the Einstein equations. In Appendix A,
we give a constructive algorithm for finding these THZ coordinates for any particular geodesic in a vacuum spacetime.
In THZ coordinates
g ab ⫽ ␩ ab ⫹H ab
⫽ ␩ ab ⫹ 2 H ab ⫹ 3 H ab ⫹O 共 ␳ 4 /R 4 兲 , ␳ /R→0, 共16兲
with
2 H ab dx
a
dx b ⫽⫺E i j x i x j 共 dt 2 ⫹ ␦ kl dx k dx l 兲
4
⫹ ⑀ kpq B q i x p x i dtdx k
3
冋
冋
册
20
2
⫺
Ėi j x i x j x k ⫺ ␳ 2 Ėik x i dtdx k
21
5
⫹
ternal multipole moments are spatial, symmetric, tracefree
tensors and are related to the Riemann tensor evaluated on ⌫
by
Bi j ⫽ ⑀ i pq R pq jt /2,
共20兲
Ei jk ⫽ 关 ⵜk R tit j 兴 STF
共21兲
3
Bi jk ⫽ 关 ⑀ i pq ⵜk R pq jt 兴 STF
8
共22兲
where STF means to take the symmetric, tracefree part with
respect to the spatial indices i, j and k. Ei j and Bi j are
O(1/R 2 ), and Ei jk and Bi jk are O(1/R 3 ). The dot denotes
differentiation of the multipole moment with respect to t
along ⌫. That all of the above external multipole moments
are tracefree follows from the assumption that the background geometry is a vacuum solution of the Einstein equations.
The THZ coordinates are a specialization of harmonic coordinates, and it is useful to define the ‘‘Gothic’’ form of the
metric
gab ⬅ 冑⫺gg ab
共23兲
H̄ ab ⬅ ␩ ab ⫺gab .
共24兲
as well as
册
5
1
x ⑀ Ḃq x p x k ⫺ ␳ 2 ⑀ pqi Ḃ j q x p dx i dx j
21 i j pq k
5
and
a
共19兲
and
A coordinate system is harmonic if and only if
共17兲
3 H ab dx
Ei j ⫽R tit j ,
⳵ a H̄ ab ⫽0.
共25兲
1
dx b ⫽⫺ E i jk x i x j x k 共 dt 2 ⫹ ␦ kl dx k dx l 兲
3
Zhang 关8兴 gives an expansion of gab for an arbitrary solution
of the vacuum Einstein equations in THZ coordinates, his
Eq. 共3.26兲. The lower order terms of H̄ ab in this expansion
are
2
⫹ ⑀ kpq B q i j x p x i x j dtdx k
3
H̄ ab ⫽ 2 H̄ ab ⫹ 3 H̄ ab ⫹O 共 ␳ 4 /R 4 兲
⫹O 共 ␳ 4 /R 4 兲 i j dx i dx j ,
共18兲
where ␩ ab is the flat Minkowski metric in the THZ coordinates (t,x,y,z), ⑀ i jk is the flat space Levi-Civita tensor, ␳ 2
⫽x 2 ⫹y 2 ⫹z 2 and the indices i, j, k, l, p and q are spatial
and raised and lowered with the three dimensional flat space
metric ␦ i j . Note that a term of O( ␳ 4 /R 4 ) is only known to
be C 3 in the limit. We call coordinates where H ab matches
only Eq. 共17兲 second order THZ coordinates; these coordinates are well defined up to the addition of arbitrary functions of O( ␳ 4 /R 3 ). Third order THZ coordinates match Eq.
共16兲 through the terms in Eq. 共18兲; these are well defined up
to the addition of arbitrary functions of O( ␳ 5 /R 4 ). The ex-
共26兲
where
tt
⫽⫺2E i j x i x j
tk
2
10
2
⫽⫺ ⑀ kpq Bqi x p x i ⫹
Ėi j x i x j x k ⫺ Ėi k x i ␳ 2
3
21
5
2 H̄
2 H̄
2 H̄
and
104016-4
ij
冋
⫽
冋
5 (i j)pq
1
x ⑀ Ḃqk x p x k ⫺ ⑀ pq(i Ḃ
21
5
j)
qx p␳
2
册
册
共27兲
PHYSICAL REVIEW D 67, 104016 共2003兲
SELF-FORCE OF A SCALAR FIELD FOR CIRCULAR . . .
G ret共 p,p ⬘ 兲 ⫽2⌰ 关 ⌺ 共 p 兲 ,p ⬘ 兴 G sym共 p,p ⬘ 兲
2
tt
i j k
3 H̄ ⫽⫺ E i jk x x x
3
3 H̄
tk
1
⫽⫺ ⑀ kpq Bqi j x p x i x j
3
3 H̄
ij
⫽O 共 ␳ 4 /R 4 兲 .
G adv共 p,p ⬘ 兲 ⫽2⌰ 关 p ⬘ ,⌺ 共 p 兲兴 G sym共 p,p ⬘ 兲
共28兲
At linear order in H̄ ab , the metric perturbation H ab is the
trace reversed version of H̄ ab ,
1
H ab ⫽H̄ ab ⫺ g ab H̄ c c ,
2
共29兲
共35兲
where ⌰ 关 ⌺(p),p ⬘ 兴 ⫽1⫺⌰ 关 p ⬘ ,⌺(p) 兴 equals 1 if p ⬘ is in
the past of a spacelike hypersurface ⌺(p) that intersects p,
and is zero otherwise. The terms in a Green’s function containing u and v are commonly referred to as the ‘‘direct’’ and
‘‘tail’’ parts, respectively.
A second symmetric Green’s function 关1兴
G S共 p,p ⬘ 兲 ⫽
1
关 u 共 p,p ⬘ 兲 ␦ 共 ␴ 兲 ⫹ v共 p,p ⬘ 兲 ⌰ 共 ␴ 兲兴 共36兲
8␲
precisely identifies the part of ␺ which is not responsible for
the self-force. In particular, the regular remainder
and Eqs. 共16兲–共18兲 are precisely the terms up to O( ␳ 4 /R 4 )
which correspond to Zhang’s 关8兴 expansion.
␺ R⫽ ␺ ret⫺ ␺ S
IV. GREEN’S FUNCTIONS FOR A SCALAR FIELD
is a homogeneous solution of the field equation 共30兲 and
completely provides the self-force when put on the righthand side of Eq. 共1兲 关1兴.
The scalar field equation
ⵜ 2 ␺ ⫽⫺4 ␲ %
ⵜ 2 G 共 p,p ⬘ 兲 ⫽⫺ 共 ⫺g 兲 ⫺1/2␦ 4 共 x ap ⫺x p ⬘ 兲 ,
a
共31兲
where p ⬘ represents a source point on ⌫, and p a nearby field
point. The source function for a point charge moving along a
worldline ⌫, described by p ⬘ ( ␶ ), is
% 共 p 兲 ⫽q
冕
共 ⫺g 兲
⫺1/2
A. Approximation for ␺ S
共30兲
is formally solved in terms of a Green’s function,
␦ „p⫺p ⬘ 共 ␶ 兲 … d ␶ ,
4
In this section approximate expansions are derived for G S
and ␺ S. For a vacuum spacetime (R ab ⫽0) which is nearly
flat Thorne and Kovács 关18兴 show that
u 共 p,p ⬘ 兲 ⫽1⫹O 共 ␳ 4 /R 4 兲 ,
冕
G 关 p, p ⬘ 共 ␶ 兲兴 d ␶ .
1
u 共 p,p ⬘ 兲 ␦ ret关 ␴ 共 p,p ⬘ 兲兴
4␲
共32兲
共33兲
DeWitt and Brehme 关3兴 analyze scalar-field self-force effects by using the Hadamard expansion of the Green’s function. An important quantity is Synge’s 关9兴 ‘‘world function’’
␴ ( p, p ⬘ ) which is half of the square of the distance along a
geodesic between two nearby points p and p ⬘ . The usual
symmetric scalar field Green’s function is derived from the
Hadamard form to be 关3兴
G sym共 p,p ⬘ 兲 ⫽
1
关 u 共 p, p ⬘ 兲 ␦ 共 ␴ 兲 ⫺ v共 p, p ⬘ 兲 ⌰ 共 ⫺ ␴ 兲兴
8␲
共34兲
where u(p, p ⬘ ) and v (p, p ⬘ ) are bi-scalars described by DeWitt and Brehme. The ⌰(⫺ ␴ ) guarantees that only when p
and p ⬘ are timelike related is there a contribution from
v ( p, p ⬘ ).
The retarded and advanced Green’s functions are
共38兲
their Eqs. 共39兲 and 共40兲, and they evaluate the direct part of
the retarded Green’s function to be
where ␶ is the proper time along the worldline of the particle
with scalar charge q. The scalar field of this particle is
␺ 共 p 兲 ⫽4 ␲ q
共37兲
⫽
冉
1⫹O 共 ␳ 4 /R 4 兲
4 ␲␴˙
冊
␦ 共 t p ⫺t ret兲 ,
共39兲
␶ ret
where the dot denotes a derivative with respect to t p ⬘ .
We now express ␴˙ ret in terms of the THZ coordinates to
obtain Eq. 共47兲 below. When the source point p ⬘ is on
⌫, ␴ (p,p ⬘ ) is particularly easy to evaluate in THZ coordinates for p close to p ⬘ . Synge’s 关9兴 ‘‘world function’’
␴ (p,p ⬘ ) is shown by Thorne and Kovács 关18兴 to be
冉 冕
1
␴ 共 p,p ⬘ 兲 ⫽ x a x b ␩ ab ⫹
2
C
冊
H ab d␭ ⫹O 共 ␳ 6 /R 4 兲 ,
共40兲
their Eqs. 共37兲 and 共38兲, where the THZ coordinates of p ⬘
are (t p ⬘ ,0,0,0), x a is the coordinate of the field point p, and
the coordinates of the path of integration C are given by
␨ a (␭)⫽␭(x a ⫺t p ⬘ ␦ at ) with ␭ running from 0 to 1. We closely
follow the analysis in 关18兴, while using THZ coordinates,
and only work through lower orders in ␳ /R.
Given H ab ⫽ 2 H ab ⫹ 3 H ab from Eqs. 共17兲 and 共18兲, the
integral of a component of H ab along C is straightforward.
For example,
104016-5
PHYSICAL REVIEW D 67, 104016 共2003兲
DETWEILER, MESSARITAKI, AND WHITING
冕
C
H tt d␭⫽⫺
冕冉
C
冊
1
E i j ␨ i ␨ j ⫹ E i jk ␨ i ␨ j ␨ k d␭⫹O 共 ␳ 4 /R 4 兲
3
1
1
⫽⫺ E i j x i x j ⫺ E i jk x i x j x k ⫹O 共 ␳ 4 /R 4 兲 .
3
12
冋
d ␴ 共 p,p ⬘ 兲
冕
C
1
⫽⫺ 共 1⫺Htt 兲关 t p ⬘ ⫺t p ⫹x i Hit ⫺ ␳ 共 1⫹Htt 兲兴
2
ret
⫹O 共 ␳ 6 /R 5 兲
共41兲
1
⫽⫺ 共 1⫺Htt 兲关 ⫺2 ␳ 共 1⫹Htt 兲 ⫹O 共 ␳ 5 /R 4 兲兴
2
The other components give similar results. If we define
Hab ⬅
dt p ⬘
册
⫽ ␳ 关 1⫹O 共 ␳ 4 /R 4 兲兴 ;
the first equality follows from taking the derivative of the
second term in square brackets in Eq. 共46兲 with respect to
t p ⬘ , the second equality from evaluating at the retarded time,
and the third equality follows from Eqs. 共41兲 and 共42兲.
The direct part of the retarded Green’s function in Eq.
共39兲 is now
then Synge’s world function is
1
1
␴ 共 p,p ⬘ 兲 ⫽ x a x b ␩ ab ⫹ 共 t p ⫺t p ⬘ 兲 2 Htt
2
2
1
⫹ 共 t p ⫺t p ⬘ 兲 x i Hit ⫹ x i x j Hi j ⫹O 共 ␳ 6 /R 4 兲 ,
2
1
u 共 p,p ⬘ 兲 ␦ ret关 ␴ 共 p,p ⬘ 兲兴
4␲
1
⫽⫺ 共 1⫺Htt 兲关共 t p ⬘ ⫺t p ⫹x i Hit 兲 2
2
⫽ 共 4 ␲␳ 兲 ⫺1 ␦ 关 t p ⬘ ⫺t p ⫹x i Hit ⫹ ␳ 共 1⫹Htt 兲
⫹O 共 ␳ 5 /R 4 兲兴关 1⫹O 共 ␳ 4 /R 4 兲兴 .
⫺x x 共 ␩ i j ⫹Hi j 兲 / 共 1⫺Htt 兲兴 ⫹O 共 ␳ /R 兲 .
i j
共47兲
共42兲
H ab d␭,
6
4
共43兲
DeWitt and Brehme show that in general
The second equality depends upon the facts that Hab
⫽O( ␳ 2 /R 2 ) and that 兩 t p ⬘ ⫺t p 兩 ⫽O( ␳ ) near the null cone.
With the source point on ⌫,
x i x j 共 ␩ i j ⫹Hi j 兲 ⫽ ␳ 2 共 1⫹Htt 兲 ⫹O 共 ␳ 6 /R 4 兲
共44兲
v共 p,p ⬘ 兲 ⫽⫺
1
R 共 p ⬘ 兲 ⫹O 共 ␳ /R 3 兲 ,
12
v共 p,p ⬘ 兲 ⫽O 共 ␳ 2 /R 4 兲 .
共45兲
The result in Eq. 共44兲 depends upon the detailed nature of
2 H i j and 3 H i j in Eqs. 共17兲 and 共18兲 as well as upon the
definition of Hi j in Eq. 共42兲.
After the substitution of Eq. 共44兲 into Eq. 共43兲, factorization of ␴ yields
⫻ 关 t p ⬘ ⫺t p ⫹x i Hit ⫹ ␳ 共 1⫹Htt 兲兴 ⫹O 共 ␳ 6 /R 4 兲 .
共46兲
At the retarded time, p ⬘ is on the past null cone emanating
from p, where ␴ (p, p ⬘ )⫽0, and it follows that the first of the
factors in square brackets is ⬃ ␳ and the second must be
␳ ⫺1 ⫻O( ␳ 6 /R 4 )⫽O( ␳ 5 /R 4 ) to cancel the order term in Eq.
共46兲 and have ␴ (p, p ⬘ ) vanish precisely. Thus, differentiation
of Eq. 共46兲 with respect to t p ⬘ and evaluation at the retarded
time yields an expression which is dominated by the part
which results from the differentiation of the second term in
square brackets,
共49兲
共50兲
This follows from Eq. 共2.9兲 with substitutions from Eqs.
共2.14兲, 共2.15兲, 共1.76兲 and 共1.10兲 of Ref. 关3兴. The dominant
contribution to ␺ S from the v (p,p ⬘ ) term is O( ␳ 3 /R 4 ) in
the coincidence limit, p→⌫.
All together then, with Eqs. 共48兲 and 共49兲, substituted into
Eq. 共36兲 and an integration over the worldline,
␺ S⫽q/ ␳ ⫹O 共 ␳ 3 /R 4 兲 .
1
␴ 共 p,p ⬘ 兲 ⫽⫺ 共 1⫺Htt 兲关 t p ⬘ ⫺t p ⫹x i Hit ⫺ ␳ 共 1⫹Htt 兲兴
2
p→⌫,
but in vacuum, where R⫽0 from the Einstein equations,
where
␳ 2 ⫽x i x j ␩ i j .
共48兲
共51兲
We note that the third order THZ coordinates are only well
defined up to the addition of a term of O( ␳ 5 /R 4 ). Such an
addition would change 1/␳ by the sum of a term that is
O( ␳ 3 /R 4 ), and would be consistent with the order term of
Eq. 共51兲. The differentiability of the order term is of interest,
and a term of O( ␳ 3 /R 4 ) is C 2 in the limit that ␳ →0. In
light of the fact that ␺ R is a homogeneous solution, Eq. 共51兲
clarifies the relationship between the accuracy of an approximation for ␺ S and the differentiability of the subsequent approximation for ␺ R⫽ ␺ ret⫺ ␺ S, and the self-force ⳵ a ␺ R. Specifically, if the approximation for ␺ S is in error by a C n
function, then the approximation for ␺ R is no more differentiable than C n and the approximation for ⳵ a ␺ R is no more
differentiable than C n⫺1 .
104016-6
PHYSICAL REVIEW D 67, 104016 共2003兲
SELF-FORCE OF A SCALAR FIELD FOR CIRCULAR . . .
B. Intuitive understanding for ␺ S
Before continuing, it is instructive to provide an elementary, direct explanation of Eq. 共51兲 by taking full advantage
of the features of the THZ coordinates. The scalar wave operator in THZ coordinates, is
冑⫺gⵜ a ⵜa ␺ ⫽ ⳵ a 共 ␩ ab ⳵ b ␺ 兲 ⫺ ⳵ a 共 H̄ ab ⳵ b ␺ 兲
共52兲
or
冑⫺gⵜ a ⵜa ␺ ⫽ ␩ ab ⳵ a ⳵ b ␺ ⫺H̄ i j ⳵ i ⳵ j ␺ ⫺2H̄ it ⳵ (i ⳵ t) ␺
⫺H̄ tt ⳵ t ⳵ t ␺ .
共53兲
Direct substitution into Eq. 共53兲 shows how well q/ ␳ approximates ␺ S. If ␺ is replaced by q/ ␳ on the right hand
side, then the first term gives a ␦ function, the third and
fourth terms vanish because ␳ is independent of t, and in the
second term 2 H̄ i j has no contribution because of the details
given in Eq. 共27兲, and the remainder of H̄ i j yields a term that
scales as O( ␳ /R 4 ). Thus,
A coordinate rotation maps each Y ᐉm ( ␪ , ␾ ) into a linear
combination of the Y ᐉm ⬘ (⌰,⌽) which preserves the index ᐉ,
while m ⬘ runs over ⫺ᐉ . . . ᐉ. Thus, the ᐉ component of the
self-force, after summation over m, is invariant under the
coordinate rotation.
To obtain the regularization parameters: first we expand
⳵ r (q/ ␳ ) into a sum of spherical harmonic components whose
amplitudes depend upon r. Then we take the limit r→r o .
Finally only the m⫽0 components contribute to the selfforce at ⌰⫽0 because Y ᐉm (0,⌽)⫽0 for m⫽0. Thus, the
regularization parameters of Eq. 共12兲 are the (ᐉ,m⫽0)
spherical harmonic components of ⳵ r (q/ ␳ ) evaluated at r o .
In this section ⑀ is a formal parameter which is to be set to
unity at the end of a calculation; a term containing a factor of
⑀ n is O( ␳ n ). We use it to help identify the behavior of certain terms in the coincidence limit, x i →0. We have used
MAPLE and GRTENSOR extensively to obtain the results reported below.
A lengthy expression for ␳ 2 , for a circular orbit in the
Schwarzschild geometry, may be derived from the analysis
of Appendix B. The O( ⑀ 2 ) part of ␳ 2 is
冑⫺gⵜ a ⵜa 共 q/ ␳ 兲 ⫽⫺4 ␲ q ␦ 3 共 x i 兲 ⫹O 共 ␳ /R 4 兲 , ␳ /R→0.
˜␳ 2 ⬅
共54兲
From consideration of solutions of Laplace’s equation in flat
spacetime, it follows that a C 2 correction to q/ ␳ , of
O( ␳ 3 /R 4 ), would remove the remainder on the right-hand
side. We conclude that ␺ S⫽q/ ␳ ⫹O( ␳ 3 /R 4 ) is an inhomogeneous solution of the scalar field wave equation. And the
error in the approximation of ␺ S by q/ ␳ is C 2 .
V. REGULARIZATION PARAMETERS FOR A CIRCULAR
ORBIT OF THE SCHWARZSCHILD GEOMETRY
In Appendix B we give the detailed functional relationship between the THZ coordinates (t,x,y,z) and the
Schwarzschild coordinates (t s ,r, ␪ , ␾ ) for an orbit described
by r⫽r o with ␪ ⫽ ␲ /2 and ␾ ⫽⍀t s .
As seen in Sec. IV, an approximation to ␺ S is
␺ ⫽q/ ␳ ⫹O 共 ␳ /R 兲 .
S
3
4
⌬⬅ 共 r⫺r o兲 ,
␹ ⬅1⫺
M sin2 ⌽
.
r o⫺2M
共59兲
The dependence of ␳ 2 on the Schwarzschild coordinates
may be written solely in terms of ⌬, ␹ and ˜␳ by use of Eq.
共57兲 to remove cos ⌰, then Eqs. 共58兲 and 共59兲 remove r and
sin ⌽ respectively. We formally expand ⳵ r (1/␳ ) in powers of
⑀ to obtain
⳵ r 共 1/␳ 兲 ⫽⫺
⑀ ⫺2 r o⌬
˜␳ 3 共 r o⫺2M 兲
⫺
冋
⑀ ⫺1
r o⫺3M
1⫺
2 ␹ 共 r o⫺2M 兲
˜␳ r o
册
⫹O 共 ⌬ 2 /˜␳ 2 ,⌬ 4 /˜␳ 4 兲 ⫹ ⑀ 0 O 共 ⌬/˜␳ ,⌬ 2 /˜␳ 2 兲
⫹
⫹
⫺
sin ␪ cos 共 ␾ ⫺⍀t s兲 ⫽ cos ⌰
cos ␪ ⫽ sin ⌰ sin ⌽.
共58兲
and
共55兲
共56兲
共57兲
where
The regularization parameters result from evaluating the
multipole components of q/ ␳ at the location of the source.
The use in this manner of q/ ␳ , in lieu of ␺ S itself, is justified
because the error in the approximation to ␺ S, being
O( ␳ 3 /R 4 ), gives no contribution to ⵜa ␺ S as x→0.
To aid in the multipole expansion we rotate the usual
Schwarzschild coordinates to move the coordinate location
of the particle from the equatorial plane to a location where
sin ⌰⫽0 for a specific t s , following the approach of Barack
and Ori as described in 关11兴. Thus, we define new angles ⌰
and ⌽ in terms of the usual Schwarzschild angles by
sin ␪ sin 共 ␾ ⫺⍀t s兲 ⫽ sin ⌰ cos ⌽
r o⫺2M
r o⌬ 2
⫹2r o2
␹ 共 1⫺ cos ⌰ 兲
r o⫺2M
r o⫺3M
⑀ 1˜␳
ro
4
冋
⫺
M 共 r o⫺2M 兲 共 r o⫺M 兲共 r o⫺4M 兲
⫺
2 共 r o⫺3M 兲
8 ␹ 共 r o⫺2M 兲
共 r o⫺3M 兲共 5r o2 ⫺7r oM ⫺14M 2 兲
16␹ 2 共 r o⫺2M 兲 2
3 共 r o⫺3M 兲 2 共 r o⫹M 兲
16␹ 3 共 r o⫺2M 兲 2
⫹O 共 ␳ 2 兲 .
⫹O 共 ⌬ 2 /r o ,⌬ 4 /˜␳ 2 r o兲
册
共60兲
We consider the multipole expansion, in the limit that ⌬
→0, of the m⫽0 part of each coefficient of ⑀ n for n⫽⫺2 to
104016-7
PHYSICAL REVIEW D 67, 104016 共2003兲
DETWEILER, MESSARITAKI, AND WHITING
1. A convenient method to find the m⫽0 component in one
of the following terms involves integrating over the angle ⌽
using details described in Appendix C. The expansion of the
⌰ dependence in terms of Legendre polynomials is described in Appendix D.
First term. The ⌬→0 limit of the expansion of the ⑀ ⫺2
term in Eq. 共60兲 is
lim ⫺
⌬→0
冋
⑀ ⫺1
r o⫺3M
1⫺
2 ␹ 共 r o⫺2M 兲
˜␳ r o
⫽⫺
冋
r o⫺3M
冋
2r o4 共 r o⫺2M ⫺M sin2 ⌽ 兲共 1⫺ cos ⌰ 兲
⫻ 1⫺
⫺2
lim ⫺
⌬→0
⑀ r o⌬
˜␳ 3 共 r o⫺2M 兲
⫽ lim ⫺
⌬→0
冋
冉
⫻
⫽⫺
冉
冊冉
r o⌬
r o⫺2M
2
冊
⫽⫺
1/2
r o⌬
2r o 共 r o⫺2M 兲
␹ 共 1⫺ cos ⌰ 兲
⫹
r o⫺2M
r o⫺3M
2
2
ro
r o⫺2M
冊
兺
ᐉ⫹
ᐉ⫽0
册
冋
2r o4 共 r o⫺2M 兲
冓 冋
冋
lim ⫺
⌬→0
共61兲
where ⑀ has been set equal to 1 on the right-hand side here
and below, and where the second equality follows from Eq.
共D12兲 with the substitution
⌬→0
⑀ ⫺2 r o⌬
˜␳ 3 共 r o⫺2M 兲
⫽⫺sgn 共 ⌬ 兲
冔
关 r o共 r o⫺3M 兲兴 1/2
r o2 共 r o⫺2M 兲
⬁
兺
ᐉ⫽0
共62兲
冉 冊
ᐉ⫹
1
P 共 cos ⌰ 兲 .
2 ᐉ
In the coincidence limit P ᐉ ( cos ⌰)⫽1 and a term in this sum
is then
冉 冊
r o⫺3M
2 ␹ 共 r o⫺2M 兲
3/2
册
共65兲
.
册冋
册冔
2r o 共 r o⫺2M 兲
4
F 1/2⫺
共 r o⫺3M 兲 F 3/2
2 共 r o⫺2M 兲
册
⬁
兺
ᐉ⫽0
P ᐉ 共 cos ⌰ 兲 .
共66兲
In the coincidence limit P ᐉ ( cos ⌰)⫽1 and a term in this sum
is then
B r ⫽⫺
冋
r o 共 r o⫺2M 兲
4
册冋
1/2
r o⫺3M
F 1/2⫺
共 r o⫺3M 兲 F 3/2
2 共 r o⫺2M 兲
册
共67兲
which is the B r term in Eq. 共12兲 as given in Eq. 共14兲.
Third term. The O( ⑀ 0 ) term in Eq. 共60兲 is zero in the limit
that ⌬→0 for nonzero ⌰, and gives no contribution to the
sum in Eq. 共12兲 as follows from Eq. 共D7兲.
Last term. For the last, ⑀ 1 , term in Eq. 共60兲 we consider
lim ˜␳ ⫽
⌬→0
共63兲
1
关 r o共 r o⫺3M 兲兴 1/2
sgn 共 ⌬ 兲
⫺ ᐉ⫹
2
r o2 共 r o⫺2M 兲
⫺
1/2
r o⫺3M
⫻ 冑2
Integrating over ⌽ and dividing by 2 ␲ 共denoted by the angle
brackets 具 典 here and in Appendix C兲 via Eq. 共C7兲, to find the
m⫽0 contribution, results in
冓
␹
1/2
⑀ ⫺1
r o⫺3M
1⫺
␹ 共 r o⫺2M 兲
˜␳ r o
⫽⫺
lim ⫺
1
Integrating over ⌽ results in hypergeometric functions as
shown in Eq. 共C3兲, and the expansion of (1⫺ cos ⌰)⫺1/2 in
terms of the P ᐉ ( cos ⌰) is given in Eq. 共D7兲 and results in
r o 共 r o⫺2M ⫺M sin2 ⌽ 兲
␦ 2 ⫽⌬ 2 共 r o⫺3M 兲 / 关 2r o共 r o⫺2M 兲 2 ␹ 兴 .
册
冋
册
册
1/2
1/2
r o⫺3M
⫺3/2
2
1
P 共 cos ⌰ 兲 ,
2 ᐉ
r o⫺3M
2 ␹ 共 r o⫺2M 兲
⫻ 共 1⫺ cos ⌰ 兲 ⫺1/2
sgn 共 ⌬ 兲共 r o⫺3M 兲
1/2
冉 冊
⬁
⫻
ro
r o⫺2M
1/2
册
冋
2r o2 共 r o⫺2M 兲
r o⫺3M
册
1/2
␹ 1/2共 1⫺ cos ⌰ 兲 1/2.
共68兲
After the expansion of (1⫺ cos ⌰)1/2 described in Eq. 共D16兲
and the integration over ⌽ with Eq. 共C3兲, the multipole expansion of the ˜␳ terms in Eq. 共60兲 gives
共64兲
which determines the A r term in Eq. 共12兲 as given in Eq.
共13兲.
Second term. The ⌬→0 limit of the ⑀ ⫺1 term in Eq. 共60兲
is
104016-8
冋
2r o2 共 r o⫺2M 兲
r o⫺3M
⫺
⫹
册冋
1/2
⫺
M 共 r o⫺2M 兲 F ⫺1/2
2r o4 共 r o⫺3M 兲
共 r o⫺M 兲共 r o⫺4M 兲 F 1/2
8r o4 共 r o⫺2M 兲
共 r o⫺3M 兲共 5r o2 ⫺7r oM ⫺14M 2 兲 F 3/2
16r o4 共 r o⫺2M 兲 2
PHYSICAL REVIEW D 67, 104016 共2003兲
SELF-FORCE OF A SCALAR FIELD FOR CIRCULAR . . .
⫺
3 共 r o⫺3M 兲 2 共 r o⫹M 兲 F 5/2
16r o4 共 r o⫺2M 兲 2
⬁
⫻
兺
ᐉ⫽0
⫺2 冑2 P ᐉ 共 cos ⌰ 兲
.
共 2ᐉ⫺1 兲共 2ᐉ⫹3 兲
册
冋
⳵r ⫺
共69兲
In the coincidence limit P ᐉ ( cos ⌰)⫽1, and a term in this
sum is then
Dr
⫺2 冑2
,
共 2ᐉ⫺1 兲共 2ᐉ⫹3 兲
共70兲
which defines D r , as in Eq. 共15兲, and is a term in Eq. 共12兲.
The remainder. These derivations of the regularization parameters reveal a pattern for the ᐉ-dependence of higher order parameters, even if the overall scale of the parameter
remains unknown.
The successive terms in Eq. 共60兲 provide increasingly accurate approximations of ⳵ r ␺ S and also increasingly accurate
approximations of ⳵ r ␺ R⫽ ⳵ r ␺ ret⫺ ⳵ r ␺ S. In principle the
terms through O( ⑀ 0 ) are sufficient to calculate the radial
component of the self-force; this is effectively the level of
approximation described in Refs. 关10–13兴 and implemented
in Ref. 关15兴. With the inclusion of O( ⑀ 0 ) terms the approximation for ⳵ r ␺ R is C 0 , the remainder terms scale as ᐉ ⫺2 for
large ᐉ and their sum converges. But when the approximation of ⳵ r ␺ S is improved by the addition of the O( ⑀ ) terms,
the resulting approximation of ⳵ r ␺ R is then C 1 , and we see
below that the remainder terms scale as ᐉ ⫺4 resulting in a
more rapid convergence of the sum for the self-force.
As the approximation of ⳵ r ␺ S is improved by successive
terms of greater differentiability, the resulting approximation
of ⳵ r ␺ R is not only more differentiable but also leads to
increasingly rapid convergence of the sum for the self-force.
We can anticipate the details of how this occurs. From the
descriptions of the THZ coordinates in Appendix B and of
␺ S in terms of THZ coordinates in Sec. IV, we expect that the
O( ⑀ 2 ) term in Eq. 共60兲 is C 1 . A more accurate approximation to ␺ S could be provided by the modification
␳ 2 → ␳ 2 ⫹␭ N X N ,
共71兲
where the components of ␭ N ⫽O(1/R n⫺2 ) are not functions
of the coordinates but depend only upon the orbit and the
coordinate location of the particle. Here we borrow the notation of the analysis of STF tensors 关19兴 where N is a multiindex that represents n spatial indices i 1 i 2 . . . i n , however
while ␭ N is symmetric it is not necessarily tracefree. Also
X N represents X i 1 X i 2 . . . X i n where the X i represents one
of X, Y or Z defined in Appendix B. Now, ␳ 2 is already
determined through O( ⑀ 5 ), so that n is necessarily greater
than or equal to 6 for an improved approximation. And while
we do not know the actual value of ␭ N , we assume that such
a ␭ N exists that provides an improved approximation to ␺ S.
Such a correction to ␳ 2 ultimately results in the addition
of
␭ NX N
⫹O 共 ␳ 4 /R 5 兲
2˜␳ 3
册
共72兲
to ⳵ r (1/␳ ) in Eq. 共60兲, which is of O( ⑀ 2 ) and C 1 if n⫽6.
This result is consistent with Eq. 共51兲 and with the discussion
following Eq. 共54兲 in Sec. IV above. With this improved
approximation the resulting error term in Eq. 共60兲 would be
O( ⑀ 3 ) and C 2 .
Finding higher order THZ coordinates might not be the
best way to correct the approximation for ␺ S, but any such
correction involving an expansion about ⌫ would necessarily
take the generic form of a homogeneous polynomial in X, Y
and Z 共or equivalently in x i ) divided by ˜␳ raised to some
integral power. Thus to find higher order corrections to ␺ S,
we are led to consider the multipole expansion of a term such
as in Eq. 共72兲, for n⭓6 and to determine the nature of the
regularization parameters that would result.
First X, Y and Z are replaced by their definitions in terms
of the usual Schwarzschild coordinates in Eqs. 共B1兲–共B3兲.
Then, the angles are changed to ⌰ and ⌽ via Eq. 共56兲. And
finally all coordinate dependence is written in terms of ⌬, ␹
and ˜␳ as described above in Eq. 共60兲. The result is a sum of
terms each of which is of O( ⑀ n⫺4 ), for n⭓6 and whose
coordinate dependence is contained in the functions ⌬
共which depends only upon r), ␹ 共which depends only upon
⌽) and ˜␳ . As x→0 both ⌬ and ˜␳ are O( ⑀ ) while ␹
⫽O(1), so that each term must include a factor ⌬ q˜␳ p for
integers q and p where q⫹ p⫽n⫺4⭓2. In fact, careful
analysis of the above substitutions shows that q⭓0. All of
the ⌰ dependence resides in ˜␳ p .
The Legendre polynomial expansion of ˜␳ p , for odd p, is
discussed at length in Appendix D. There we show that when
p⫽⫺1 共respectively, p⬍⫺1), a consequence of Eq. 共D7兲
关respectively, Eq. 共D11兲兴 is that the expansion coefficients
scale as a constant 共respectively, diverge as (r⫺r o) p ) when
r→r o . The coefficients of the expansion ⌬ q˜␳ p then scale as
(r⫺r o) q 关respectively, (r⫺r o) q⫹p ]. In both cases the power
is an integer ⭓2. In the limit that r→r o these terms approach zero and give no contribution to the regularization
parameters. The case of p even and negative gives a similar
result, but is not discussed in the Appendix. We conclude that
if p⬍0 then no contribution to the regularization parameters
results.
If p⭓0 and q⬎0 then the Legendre polynomial expansion of ˜␳ p is well behaved and finite as r→r o , but the product ⌬ q˜␳ p vanishes in the limit r→r o and gives no contribution to the regularization parameters.
If q⫽0 and p⭓2 and is even, then the ⌰ dependence is
in the form of a polynomial in cos ⌰ which has an expansion
in terms of the Legendre polynomials only up to P p (⌰), and
because ˜␳ ⫽0 when r⫽r o and ⌰⫽0 the sum of this finite
number of terms is zero and gives no contribution to the
regularization parameters. This case always results when the
improvement to ⳵ r ␺ S is O( ⑀ n ) for n being even.
The only remaining case is q⫽0 and p⬎2 being a positive odd integer. In the limit that r→0, ˜␳ p ⬀(1⫺ cos ⌰)p/2.
104016-9
PHYSICAL REVIEW D 67, 104016 共2003兲
DETWEILER, MESSARITAKI, AND WHITING
The Legendre polynomial expansion of this function is discussed in detail in Appendix D. We see in Eq. 共D22兲 that, for
k a positive integer
⬁
共 1⫺ cos ⌰ 兲 k⫹1/2⫽
兺
ᐉ⫽0
A k⫹1/2
P ᐉ 共 cos ⌰ 兲
ᐉ
共73兲
where
A k⫹1/2
⫽ 共 2ᐉ⫹1 兲 Pk⫹1/2 / 关共 2ᐉ⫺2k⫺1 兲共 2ᐉ⫺2k⫹1 兲 •••
ᐉ
⫻ 共 2ᐉ⫹2k⫹1 兲共 2ᐉ⫹2k⫹3 兲兴 ,
共74兲
for a constant Pk⫹1/2 given in Eq. 共D23兲. When ⌰
⫽0, P ᐉ ( cos ⌰)⫽1 and such terms do contribute additional
regularization parameters in the mode sum representation of
the self-force. This case always results when the improvement to ⳵ r ␺ S is O( ⑀ n ) for n being odd.
We now see that every other higher order correction to ␺ S
provides an additional regularization parameter. The ᐉ dependence is necessarily of the form
E ka A k⫹1/2
ᐉ
共75兲
where E ka is independent of ᐉ but still undetermined. The
first term of this sort, for k⫽1 is included in Eq. 共12兲. It is
important to note that for each value of k the sum of these
terms from ᐉ⫽0 to infinity is necessarily zero, and need not
be included in the self-force analysis. However we see in the
next section that including these additional coefficients dramatically speeds up the convergence of the self-force sum
Eq. 共11兲.
VI. APPLICATION
In this section we apply the formalism developed above to
determine the self-force F rR for a scalar charge in orbit at r
⫽10M about a Schwarzschild black hole. In the numerical
work we use units where M ⫽1 and q⫽1. Appendix E describes the practical details for numerically integrating the
ret
. To compute the
scalar wave equation to determine the F ᐉr
self-force, the A r , B r and D r terms must first be removed
ret
as in Eqs. 共11兲 and 共12兲—a process which deterfrom F ᐉr
mines residuals which fall off as ᐉ ⫺4 for large ᐉ. Removing
the contribution of each successive E rk improves the falloff of
the residuals by an additional two powers of ᐉ.
ret
for all ᐉ, summing the residuals after
If we have F ᐉr
removing A r , B r and D r would give us the self-force. With
ret
evaluated only for finite ᐉ, we must make some attempt
F ᐉr
to obtain the higher ᐉ contributions. To do this we will numerically determine the E rk coefficients by fitting the residuals, considered as a function of ᐉ, with a linear combination
of terms whose ᐉ dependence is given by the E r1 term in Eq.
共12兲 for k⫽1 and by Eq. 共75兲 for integers k⬎1.
ret
A comparison of the integration results, F ᐉr
, for different
values of an accuracy parameter in the numerical routine,
revealed that a systematic effect remained in our best data for
ret
F ᐉr
. In order to avoid fitting to that systematic effect, we
chose to add to our data a small random component which
ret
FIG. 1. The upper portion of the figure displays F ᐉr
as a function of ᐉ, along with the result of it being regularized by A r , B r ,
and D r . The lower portion displays the residual after a numerical fit
of from 1 to 5 additional parameters, E r1 . . . E r5 . A point where the
data on a particular curve changes sign from being negative to
positive is labeled with ⫹, from positive to negative by 䊊.
was capable of swamping any trace of the systematic effect,
and which allowed us to have precise control of the error in
ret
the F ᐉr
. This random component also provided the opportunity to use Monte Carlo analysis to determine the statistical
significance of our result of the self-force.
In fitting for the E rk we avoided small values of ᐉ, which
may contain significant physical information not associated
ret
. Thus we fitted the residues
with the large ᐉ falloff of F ᐉr
for ᐉ from 13 to 40 while determining from 1 to 5 of the E rk
coefficients. Figure 1 summarizes the results of this numeriret
as a function of ᐉ.
cal analysis. The curve labeled F ᐉ is F ᐉr
ret
S
S
The curves A, B and D show F ᐉr ⫺F ᐉr where F ᐉr
successively includes the contribution from the regularization parameters A r , B r and D r . The E 1 to E 5 curves show the
residuals after numerically fitting from 1 to 5 of the E rk coefficients and removing their contributions successively.
We actually used a singular value decomposition from
Ref. 关20兴 to fit the residuals. It provided an independent estimate of the uncertainty of the E rk ’s, which is entirely compatible with the Monte Carlo results. This represented a valuable, overall consistency check of our analysis. The E rk
coefficients which result from a fit of four coefficients are
given in Table I along with their uncertainties in brackets.
After fitting four coefficients and removing their contribution, we obtained an rms residual of 2.8⫻10⫺14 over the
104016-10
ret
TABLE I. The fitted parameters of F ᐉr
.
k
E rk
E rk part of F rR
1
2
3
4
1.80504(4)⫻10⫺4
⫺1.000(3)⫻10⫺4
4.3(3)⫻10⫺5
⫺5.6(5)⫻10⫺5
2.78201(7)⫻10⫺9
6.90(2)⫻10⫺12
3.1(2)⫻10⫺14
7.7(6)⫻10⫺16
PHYSICAL REVIEW D 67, 104016 共2003兲
SELF-FORCE OF A SCALAR FIELD FOR CIRCULAR . . .
fitting range, which is completely determined by the size of
ret
the random component we had introduced to the original F ᐉr
to swamp the systematic effect. Four coefficients evidently
fit the data down to the noise. It was clear that fitting a fifth
coefficient, or more, did not improve the quality of the fit.
The self-force F rR⫽1.37844828(2)⫻10⫺5 was obtained
ret
by summing, over the range of our data, F ᐉr
with the A r , B r
and D r terms removed as in Eqs. 共11兲 and 共12兲. The remainder of the sum to ᐉ⫽⬁ was approximated by the contributions of the E r1 , E r2 , . . . sums from 41 to ⬁, once the E rk
coefficients had been determined. The uncertainty was obtained from the Monte Carlo simulation. The table also
shows the individual contribution of each E rk to the self-force
F rR as well as the amount of the uncertainty in F rR which is
attributable to that E rk . Without including the effects of the
E rk tails, we would have found the sum out to ᐉ⫽40 to be
1.37817⫻10⫺5 . Fitting the higher order terms has allowed
us to increase dramatically the effective convergence to our
final result.
Our result is consistent with Fig. 共4A兲 of Burko’s analysis
ret
, he ef关15兴. With the A r and B r terms removed from F ᐉr
fectively calculates the total self-force by summing data
points on the equivalent of our curve B out to a large enough
value of ᐉ that convergence is obtained while using Richardson extrapolation.
A future manuscript will apply our methods to the investigation of physical questions.
VII. DISCUSSION
In a previous paper 关1兴, we outlined our method for computing the self-force. This hinged on realizing that ␺ R⫽ ␺ ret
⫺ ␺ S is a homogeneous solution of the field equation and we
proved that it gave the same result as methods based on
using the tail part of the retarded Green’s function. As a
consequence of this we have obtained the same regularization parameters as all previous authors 关10–13兴 using a regularization procedure based on mode sum expansions. Exact
computation of ␺ R would yield a homogeneous solution of
the field equation. Under interesting physical circumstances,
we anticipate that ␺ R should consequently have a high level
of differentiability 关31兴. The level of differentiability of an
approximation for ␺ R is limited by the accuracy of the approximation for ␺ S. To improve the level of differentiability
of our approximation for ␺ R beyond that of ␺ tail, we have
thus been led to explore higher order approximations to ␺ S.
Following earlier work 关21兴, we have used THZ coordinates to obtain the simple approximation ␺ S⬇q/ ␳ . Our key
analytical result is the expansion in Eq. 共60兲 which is based
on this approximation. The regularization parameters are derived from the mode sum representation of each term in Eq.
共60兲. The parameters A r and B r come from the first two
terms. The parameter C r is seen to be zero, directly from the
third term. All these results are consistent with previous
work by others 关10–13兴. Our D r parameter comes from the
fourth term in Eq. 共60兲, which we compute analytically and
for which we find the specific ᐉ dependence of 1/(2ᐉ
⫺1)(2ᐉ⫹3). Direct inspection of this term shows that the
sum goes to zero in the coincidence limit, so it does not
contribute to the self-force. Nevertheless, it is recognition of
the large ᐉ behavior of the mode sum expansion for this, and
similar higher order terms characterized by the E rk , which
leads to dramatically improved convergence in the mode
summation. Understanding of the specific nature of the ᐉ
dependence in the mode sum representation of the higher
order terms was obtained by an analysis of general methods
for improving the approximation to ␺ S. Our numerical application of this scheme amply illustrates the benefits of estimating the E rk parameters in order to accelerate convergence.
In principle, neither the use of ␺ R instead of ␺ tail, nor the
specific use of THZ coordinates are intrinsically necessary in
the computation of the self-force. Indeed, many authors have
used neither, and have yet obtained analytical results for the
regularization parameters, and/or numerical results for the
self-force 关10–14,22–24兴. It is clear that other methods
might also be used to calculate D r as well as the general ᐉ
dependence of our E rk terms. What we believe is important is
that the form of these higher order terms has been determined, and that the inclusion of these terms has a dramatic
impact on the effectiveness and accuracy of numerical work.
Unquestionably, the intensity of work in this area is paying off, and as efficient computational techniques are recognized and implemented, a greater volume of results will become available. This will be especially true in relation to the
long awaited detection of gravitational waves from binary
inspiral sources, represented by a small compact object in
orbit about a comparatively large black hole.
ACKNOWLEDGMENTS
The analysis presented in Appendix A was performed by
Dong Hoon Kim, and we are grateful to Lior Burko for providing us with some unpublished details of his numerical
analysis. This research has been supported in part by the
Institute of Fundamental Theory at the University of Florida
共E.M兲, NSF Grant No. PHY-9800977 共B.F.W.兲 and NASA
Grant No. NAGW-4864 共S.D.兲 with the University of
Florida. One of us 共B.F.W.兲 is also grateful to Petr Hájı́ček
and the Swiss National Fonds for the opportunity to visit and
work at the University of Bern, Switzerland.
APPENDIX A: THE DETERMINATION
OF THZ COORDINATES
A particular THZ coordinate system (t,x,y,z) is associated with any given geodesic ⌫( ␶ ) of a vacuum spacetime
and is a harmonic coordinate system as well as being ‘‘locally inertial and Cartesian.’’ By Zhang’s 关8兴 definition a ‘‘locally inertial and Cartesian’’ 共LIC兲 coordinate system has the
spatial origin x i ⫽0 on the worldline ⌫, and has the metric
being expandable about ⌫ in powers of ␳ in a particular form
which we describe as g ab ⫽ ␩ ab ⫹ ␳ p ⫻ homogeneous polynomials in x i of degree q, for non-negative integers p and q
with p⫹q⭓2.
The defining features of nth order THZ coordinates are
that
104016-11
PHYSICAL REVIEW D 67, 104016 共2003兲
DETWEILER, MESSARITAKI, AND WHITING
共i兲 On ⌫: t measures the proper time along the geodesic,
the spatial coordinates x, y and z are all zero, g ab 兩 ⌫ ⫽ ␩ ab
and all of the first derivatives of g ab vanish.
共ii兲 At linear, stationary order 共cf. Ref. 关8兴兲 H̄ i j
⫽O(x n⫹1 ).
共iii兲 The coordinates satisfy the harmonic gauge condition
⳵ a gab ⫽O(x n ).
To find THZ coordinates associated with a particular geodesic, it is easiest to satisfy these conditions in order.
Given a general set of coordinates Y A and a particular
point p a new set of coordinates X a may be defined by
␨ a ⫽ ␨ a i jk X i X j X k ,
where the ␨ ai jk are functions only of t to be determined below.
Thus
a
⫽X a ⫹ ␨ a i jk X i X j X k ,
x (new)
⫹O 关共 Y ⫺Y p 兲 兴 Y
A
→Y Ap ,
ab
ab
⫽H̄ old
⫹ ⳵ a ␨ b ⫹ ⳵ b ␨ a ⫺ ␩ ab ⳵ c ␨ c ⫹O 共 ␨ 2 兲
H̄ new
g ab ⫽g AB
⳵Xa ⳵Xb
Y A →Y Ap ,
ij
⳵ i ␨ j ⫹ ⳵ j ␨ i ⫺ ␩ i j ⳵ c ␨ c ⫽⫺H̄ old
⫹O 共 X 3 兲 ,
共A11兲
1
␨ i jkl ⫹ ␨ jikl ⫺ ␩ i j ␨ m mkl ⫽⫺ H̄ old
.
3 i jkl
共A12兲
We use the decomposition of symmetric trace-free 共STF兲 tensors of Ref. 关26兴 and let
⫽O 关共 Y ⫺Y p 兲兴 ,
Y A →Y Ap .
共A3兲
A specific choice for the A a and B a A result in the coordinates
of p being X ap ⫽ ␶ p ␦ at . And condition 共i兲 is satisfied by repeating this construction along ⌫ while parallel propagating the
coordinate basis.
Thus, the coordinates that satisfy condition 共i兲 are denoted
X a . And
3
5
1
␨ i jkl ⫽A 具 i jkl 典 ⫹ ⑀ ip 具 j B p kl 典 ⫹ ␦ i 具 j C kl 典 ⫺ H̄ i p p( j ␦ kl) ,
4
7
5
共A13兲
where 具 ••• 典 implies taking the STF part of the enclosed
indices and A i jkl , B pkl and C kl are STF tensors on all of
their indices. The coefficient ⫺1/5 in the last term is chosen
to make the trace of Eq. 共A13兲 agree with the trace of Eq.
共A12兲. The remainder of the solution for ␨ i jkl is
gab ⫽ ␩ ab ⫺H̄ ab
⫽ ␩ ⫺H̄
ab
共A10兲
We find the necessary gauge transformation in two steps.
To satisfy condition 共ii兲 we require that the gauge transformation obey
共A2兲
so that
⳵X
共A9兲
which implies that
⫽ ␩ ab ⫹O 关共 Y ⫺Y p 兲 2 兴 ,
c
1 abi j 1 abi j abi j bai j
H̄ ⫽ H̄ ⫹ ␨ ⫹ ␨ ⫺ ␩ ab ␨ k ki j .
3 new 3 old
⳵Y A ⳵Y B
⳵ g ab
共A8兲
or
共A1兲
where ⌫ A BC are the usual Christoffel symbols, the A a are
arbitrary constants, and the B a A are also arbitrary constants
restricted by the condition that B a A , considered a matrix, be
invertible. Weinberg 关25兴 shows that
X i →0
changes H̄ ab to
1
X a ⫽A a ⫹B a A 共 Y A ⫺Y Ap 兲 ⫹ B Aa ⌫ A BC 共 Y B ⫺Y Bp 兲共 Y C ⫺Y Cp 兲
2
3
共A7兲
ab
i jX
i
X ⫹O 共 X /R 兲 ,
j
3
3
X →0,
i
1
1
1
A i jkl ⫽⫺ H̄ 具 i jkl 典 ⫹ ␦ (i j H̄ p kl)p ⫹ H̄ p p(i j ␦ kl)
6
7
42
共A4兲
here X in the order term can refer to any of the spatial coordinates, and the
H̄ ab i j ⬅
1 ⳵ 2 H̄ ab
2 ⳵ X i⳵ X j
冏
⫺
1
1
B pkl ⫽⫺ ⑀ i j( p H̄ i kl) j ⫹ ␦ 共 kl ⑀ p) i j H̄ iq q j
3
15
共A5兲
⌫
are functions only of t.
To satisfy conditions 共ii兲 and 共iii兲 we use a gauge transformation of the form
a
a
x (new)
⫽X (old)
⫹␨a
共A14兲
共A15兲
and
1
1
C kl ⫽ H̄ i ikl ⫹ 共 H̄ p jkp ⫹H̄ p k j p ⫹ ␦ jk H̄ pq pq 兲 , 共A16兲
3
15
共A6兲
where changes in geometrical objects are calculated only
through linear order in ␨ a . In this application, we are interested in the vicinity of ⌫ and accordingly let
1
␦ ␦ H̄ pq pq ,
35 (i j jk)
but only if H̄ i jkl satisfies the auxiliary condition that
104016-12
H̄ kli i ⫺H̄ i kli ⫺H̄ i lki ⫹ ␦ kl H̄ pq pq ⫽0.
共A17兲
PHYSICAL REVIEW D 67, 104016 共2003兲
SELF-FORCE OF A SCALAR FIELD FOR CIRCULAR . . .
This condition is automatically satisfied when H̄ ab is derived
from a vacuum solution of the Einstein equations, cf. Eq.
共35.64兲 of Ref. 关17兴.
The spatial component of condition 共iii兲 is now satisfied
through O(x) because H̄ i j vanishes. The time component of
condition 共iii兲 will be satisfied through O(x) also if
⳵ ⳵ b␨
b
t
bt
⫽⫺ ⳵ b H̄ old
,
⳵ b ⳵ b ␨ t ⫽⫺ ␨¨ t i jk X i X j X k ⫹6 ␨ ti ik X k
X i →0,
共A19兲
or
1
␨ ti ik ⫽⫺ H̄ ti ik .
3
共A20兲
An elementary solution for ␨ t i jk is
1
␨ t i jk ⫽⫺ H̄ tp p(i ␦ jk) .
5
To find the THZ coordinates for a circular orbit in the
Schwarzschild geometry it is convenient to take advantage of
the spherical symmetry of the background geometry rather
than to follow the general procedure described in the preceding section. The orbit ⌫, given by ␾ ⫽⍀t where ⍀
⫽ 冑M /r o3 is the orbital frequency at Schwarzschild radius
r o , is tangent to a Killing vector field ␰ a ⬅ ⳵ / ⳵ t s ⫹⍀ ⳵ / ⳵ ␾ .
We choose three helping functions X, Y and Z which are Lie
derived by the Killing vector, L␰ X⫽L␰ Y⫽L␰ Z⫽0, and
their gradients are spatial and orthogonal to the 4-velocity of
the geodesic and to each other when evaluated on ⌫. These
helping functions are
共A21兲
The combination of Eqs. 共A13兲 and 共A21兲 provides the
necessary gauge transformation that results in satisfaction of
conditions 共ii兲 and 共iii兲 for second order THZ coordinates.
The third order coordinates may be found following a
similar procedure where the gauge transformation is of the
form
a
a
⫽x (old)
⫹ ␨ a i jkl X i X j X k X l ,
x (new)
APPENDIX B: THZ COORDINATES FOR A CIRCULAR
ORBIT OF SCHWARZSCHILD
共A18兲
thus
⫽⫺2H̄ ti ik X k ⫹O 共 X 2 兲
are of the same order as terms quadratic in the Riemann
tensor, the presence of nonlinearities complicate the construction of THZ coordinates. We have not needed detailed
knowledge about these higher order THZ coordinates
throughout this paper.
X i →0.
X⬅
关 r sin ␪ cos 共 ␾ ⫺⍀t s兲 ⫺r o兴
共 1⫺2M /r o兲 1/2
⫹
⫹
⫹
MX
2r o 共 r o⫺2M 兲共 r o⫺3M 兲
3
共 1⫺2M /r o兲 1/2
共B1兲
,
冉
r o⫺2M
Y⬅r o sin ␪ sin 共 ␾ ⫺⍀t s兲
r o⫺3M
冊
1/2
共B2兲
and
共A22兲
Z⬅r o cos 共 ␪ 兲 .
At the fourth and higher orders, where terms in the metric
expansion involving two derivatives of the Riemann tensor
x̃⫽
r⫺r o
共B3兲
Two more useful functions are
M
r o2 共 1⫺2M /r o兲 1/2
冋
⫺
冉
冊 册
r o⫺3M
X2
⫹Z 2
⫹Y 2
2
r o⫺2M
关 ⫺M 2 X 2 ⫹Y 2 共 r o⫺3M 兲共 3r o⫺8M 兲 ⫹3Z 2 共 r o⫺2M 兲 2 兴
M
r o5 共 1⫺2M /r o兲 1/2共 r o⫺3M 兲
冋
MX 4
共 r o2 ⫺r oM ⫹3M 2 兲 X 2 Y 2
⫹
共 28r o2 ⫺114r oM ⫹123M 2 兲
8 共 r o⫺2M 兲
28
⫹
X 2Z 2
MY 4
共 14r o2 ⫺48r oM ⫹33M 2 兲 ⫹
共 3r o3 ⫺74r o2 M ⫹337r oM 2 ⫺430M 3 兲
14
56共 r o⫺2M 兲 2
⫺
M 2 Y 2 Z 2 共 7r o⫺18M 兲 M Z 4
⫺
共 3r o⫹22M 兲
4 共 r o⫺2M 兲
56
册
and
104016-13
共B4兲
PHYSICAL REVIEW D 67, 104016 共2003兲
DETWEILER, MESSARITAKI, AND WHITING
ỹ⫽r sin ␪ sin 共 ␾ ⫺⍀t s兲
冉
r o⫺2M
r o⫺3M
冊
1/2
⫹
MY
2r o
3
冋
⫺2X 2 ⫹Y 2
冉
冊 册
r o⫺3M
M XY
⫹Z 2 ⫹
5
r o⫺2M
14r o 共 1⫺2M /r o兲 1/2共 r o⫺3M 兲
⫻ 关 2M X 2 共 4r o⫺15M 兲 ⫹Y 2 共 14r o2 ⫺69M r o⫹89M 2 兲 ⫹2Z 2 共 r o⫺2M 兲共 7r o⫺24M 兲兴 .
共B5兲
In terms of the functions defined above, the THZ coordinates (t,x,y,z) are
x⫽x̃ cos 共 ⍀ † t s兲 ⫺ỹ sin 共 ⍀ † t s兲
共B6兲
y⫽x̃ sin 共 ⍀ † t s兲 ⫹ỹ cos 共 ⍀ † t s兲
共B7兲
and
where ⍀ † ⫽⍀ 冑1⫺3M /r o, along with
z⫽r cos 共 ␪ 兲 ⫹
MZ
2r o 共 r o⫺3M 兲
3
关 ⫺X 2 共 2r o⫺3M 兲 ⫹Y 2 共 r o⫺3M 兲 ⫹Z 2 共 r o⫺2M 兲兴 ⫹
M XZ
14r o 共 1⫺2M /r o兲 1/2共 r o⫺3M 兲
5
⫻ 关 M X 2 共 13r o⫺19M 兲 ⫹Y 2 共 14r o2 ⫺36r oM ⫹9M 2 兲 ⫹Z 2 共 r o⫺2M 兲共 14r o⫺15M 兲兴
共B8兲
and
t⫽t s共 1⫺3M /r o兲 1/2⫺
⫹
r⍀Y
共 1⫺2M /r o兲
⍀M XY
14r o 共 r o⫺2M 兲共 r o⫺3M 兲
3
⫹
1/2
⍀M Y
r o2 共 1⫺2M /r o兲 1/2共 r o⫺3M 兲
冋
⫺
X2
r o⫺3M
⫹M Z 2
共 r ⫺M 兲 ⫹M Y 2
2 o
3 共 r o⫺2M 兲
册
关 ⫺X 2 共 r o2 ⫺11r oM ⫹11M 2 兲 ⫹Y 2 共 13r o2 ⫺45r oM ⫹31M 2 兲 ⫹Z 2 共 13r o⫺5M 兲共 r o⫺2M 兲兴 .
共B9兲
The set of functions (t,x̃,ỹ,z) forms a noninertial coordinate system that corotates with the particle in the sense that
the x̃ axis always lines up the center of the black hole and the
center of the particle, the ỹ axis is always tangent to the
spatially circular orbit, and the z axis is always orthogonal to
the orbital plane. The spatial coordinates are all Lie derived
by the Killing vector, L␰ x̃⫽L␰ ỹ⫽L␰ z⫽0.
The x a coordinates (t,x,y,z) are locally inertial and nonrotating in the vicinity of ⌫, but these same coordinates appear to be rotating when viewed far from ⌫ as a consequence
of Thomas precession as revealed in the ⍀ † t s dependence in
Eqs. 共B6兲 and 共B7兲 above.
The determination of the THZ coordinates was tedious
but not difficult. We looked for the relationship between the
THZ coordinates and the usual Schwarzschild coordinates
X A ⫽(t s ,r, ␪ , ␾ ) by using the usual rule for the change in
components of a tensor under a coordinate transformation,
g ab ⫽g AB
⳵xa ⳵xb
⳵XA ⳵XB
共B7兲 were chosen so that t measured the proper time on the
orbit, g ab 兩 ⌫ ⫽ ␩ ab and ⳵ c g ab 兩 ⌫ ⫽0. This much could be done
easily by hand and resulted in coordinates that satisfied condition 共i兲 in Appendix A.
The O(X 3 ) and O(X 4 ) terms were found by use of
GRTENSOR running under MAPLE following a procedure similar to that in Appendix A except that we used homogeneous
polynomials of the form ␨ a i jk X i X j X k and ␨ a i jkl X i X j X k X l
along with Eq. 共B10兲 to determine ␨ a i j . . . which resulted in
the satisfaction of conditions 共ii兲 and 共iii兲 in Appendix A.
Note that ultimately the THZ coordinates (t,x,y,z) are
linear combinations of products of C ⬁ functions of the
Schwarzschild coordinates, and so are C ⬁ functions themselves.
It is convenient to note that the natural Minkowski metric
that goes with the Thorne-Hartle coordinates is ␩ ab dx a dx b
⬅⫺dt 2 ⫹dx 2 ⫹dy 2 ⫹dz 2 , while its components in the original Schwarzschild coordinates are
␩ AB ⫽ ␩ ab
共B10兲
where g AB is the Schwarzschild geometry in the Schwarzschild coordinates. The terms through O(X 2 ) 共the X in the
order term represents any of X, Y, or Z) in the definitions of
t, x̃, ỹ and z and the rotation represented in Eqs. 共B6兲 and
⳵xa ⳵xb
⳵XA ⳵XB
.
共B11兲
Another form for ␩ ab is
␩ ab ⫽⫺ⵜa tⵜb t⫹ⵜa xⵜb x⫹ⵜa yⵜb y⫹ⵜa zⵜb z
or
104016-14
共B12兲
PHYSICAL REVIEW D 67, 104016 共2003兲
SELF-FORCE OF A SCALAR FIELD FOR CIRCULAR . . .
resented in terms of complete elliptic integrals of the first
and second kinds respectively,
␩ ab ⫽⫺ⵜa tⵜb t⫹ⵜa x̃ⵜb x̃⫹ⵜa ỹⵜb ỹ⫹ⵜa z̃ⵜb z̃
⫹ 共 x̃ 2 ⫹ỹ 2 兲 ⵜa 共 ⍀ † t s 兲 ⵜb 共 ⍀ † t s 兲
⫹2 关 x̃ⵜ(a ỹ⫺ỹⵜ(a x̃ 兴 ⵜb) 共 ⍀ t s 兲 .
冉
And from the above definitions it readily follows that L ␰ ␳ 2
⫽L␰ z⫽L␰ ⵜa t⫽0, while L␰ x, L␰ y, and L␰ t are nonzero.
冊
共C8兲
冊
共C9兲
2
1 1
K 共 ␣ 兲 ⫽ 2 F 1 , ,1; ␣ ⫽F 1/2
␲
2 2
共B13兲
†
and
冉
APPENDIX C: INTEGRALS OVER ⌽
1 1
2
E 共 ␣ 兲 ⫽ 2 F 1 ⫺ , ,1; ␣ ⫽F ⫺1/2 .
␲
2 2
The approach in this and the following Appendix is similar to that of Appendixes C and D of Ref. 关13兴.
In Sec. V we define
APPENDIX D: LEGENDRE POLYNOMIAL EXPANSIONS
␹ ⬅1⫺ ␣ sin ⌽
We require the coefficients A ᐉp/2( ␦ ) in the expansion
共C1兲
2
⬁
共 ␦ ⫹1⫺u 兲
2
where
␣⬅
M
.
r o⫺2M
冉
冊
2
␲
冕
␲ /2
0
共 1⫺ ␣ sin2 ⌽ 兲 ⫺p d⌽
1
⫽ 2 F 1 p, ;1; ␣ ⬅F p .
2
⫽
冕
冕
␲
2
␲
1
␲ /2
0
共C3兲
for ␦ →0,
共D1兲
⬁
⫽
共 1⫺2tu⫹t 兲
共 1⫺ ␣ sin2 ⌽ 兲 ⫺p d⌽
兺
ᐉ⫽0
t ᐉ P ᐉ共 u 兲 ,
兩 t 兩 ⬍1.
t
⫺1/2
共 1⫺t 兲
0
⫺1/2
共 1⫺ ␣ t 兲
⌫共 c 兲
⌫ 共 b 兲 ⌫ 共 c⫺b 兲
⫺a
冕
dt,
1
0
⫺p
dt,
共C4兲
⬁
共 e ⫹e
T
⫺T
⫺2u 兲
⫺1/2
⫽
兺
ᐉ⫽0
e ⫺(ᐉ⫹1/2)T P ᐉ 共 u 兲 ,
The expansion
Re 共 c 兲 ⬎Re 共 b 兲 ⬎0.
冊
1
1
⫺1, ;1; ␣ ⫽1⫺ ␣ ⫽F ⫺1
2
2
e T ⫹e ⫺T ⫽2⫹T 2 ⫹O 共 T 4 兲 ,
T⫽ ␦ 冑2
共C6兲
冊
共D5兲
具 共 1⫺ ␣ sin2 ⌽ 兲 ⫺1 典 ⫽ 2 F 1 1, ;1; ␣ ⫽ 共 1⫺ ␣ 兲 ⫺1/2⫽F 1 .
共C7兲
The latter is used in Eq. 共63兲 and leads to the A r term in Eq.
共12兲. The special cases p⫽ 21 and p⫽⫺ 12 are also easily rep-
共D6兲
in Eq. 共D4兲 provides
⫽ 冑2⫹O 共 ᐉ ␦ 兲 ,
A ⫺1/2
ᐉ
1
2
T→0,
followed by the substitution
and
冉
T⬎0.
共D4兲
t b⫺1 共 1⫺t 兲 c⫺b⫺1
Two elementary special cases of Eq. 共C3兲 are
冉
共D3兲
Eq. 共D2兲 implies
共C5兲
具 1⫺ ␣ sin ⌽ 典 ⫽ 2 F 1
共D2兲
With T defined from
t⫽e ⫺T ,
1
⫻ 共 1⫺tz 兲
2
A ᐉp/2共 ␦ 兲 P ᐉ 共 u 兲 ,
2 ⫺1/2
where t⫽ sin2 ⌽, and from the integral representation of the
hypergeometric function, Eq. 共15.3.1兲 of Ref. 关27兴
2 F 1 共 a,b;c;z 兲 ⫽
兺
ᐉ⫽0
for both positive and negative odd-integral values of p. Note
that if p is a positive even integer then the left-hand side is a
p/2 degree polynomial in u and the sum terminates with
ᐉ⫽ p.
First we analyze the negative odd-integral values of p via
induction. The generating function for Legendre polynomials
is
This result follows almost immediately from
具 ␹ ⫺p 典 ⫽
⫽
共C2兲
And we use
具 ␹ ⫺p 典 ⫽ 具 共 1⫺ ␣ sin2 ⌽ 兲 ⫺p 典 ⫽
p/2
␦ →0,
共D7兲
which is used in the second term of Eq. 共60兲 to arrive at Eq.
共66兲 and leads to the B r term of Eq. 共12兲 and to the absence
of a term in Eq. 共12兲 which might have resulted from the
third term of Eq. 共60兲.
Differentiation of both sides of Eq. 共D4兲 with respect to T
yields
104016-15
PHYSICAL REVIEW D 67, 104016 共2003兲
DETWEILER, MESSARITAKI, AND WHITING
1
⫺ 共 e T ⫹e ⫺T ⫺2u 兲 ⫺3/2共 e T ⫺e ⫺T 兲
2
⬁
⫽
兺⫺
ᐉ⫽0
A 1/2
ᐉ ⫽
冉 冊
1 ⫺(ᐉ⫹1/2)T
e
P ᐉ共 u 兲 ,
2
ᐉ⫹
T⬎0.
共D8兲
Simplification and expansion about T⫽0 gives
⫽
兺
ᐉ⫽0
⬁
共 1⫺u 兲 k⫹1/2⫽
共 2ᐉ⫹1 兲
P ᐉ 共 u 兲关 1⫹O 共 ᐉT 兲兴 ,
2T
T→0,
共D9兲
and repeated differentiation extends this result to
共 e T ⫹e ⫺T ⫺2u 兲 ⫺k⫺1/2
⬁
⫽
共 2ᐉ⫹1 兲
P ᐉ 共 u 兲关 1⫹O 共 ᐉT 兲兴 ,
兺
ᐉ⫽0 2 共 2k⫺1 兲 T 2k⫺1
Finally, for k⭓1 the expansion and substitution of Eqs. 共D5兲
and 共D6兲 result in
2ᐉ⫹1
␦
共 2k⫺1 兲
⫽
A k⫹1/2
0
兺
ᐉ⫽0
␦ →0.
2ᐉ⫹1
共D11兲
␦
关 1⫹O 共 ᐉ ␦ 兲兴 ,
␦ →0,
兺ᐉ
t ᐉ A 1/2
ᐉ 2/共 2ᐉ⫹1 兲
1
⫽ k⫹
2
⫻
冕
兺
⬁
共 1⫺u 兲 1/2⫽
and
⫺2 冑2
P ᐉ共 u 兲
兺
ᐉ⫽0 共 2ᐉ⫺1 兲共 2ᐉ⫹3 兲
A k⫺1/2
Pᐉ
ᐉ
ᐉ⫽0
⬁
ᐉ⫽0
A k⫺1/2
ᐉ
⬘ ⫺ P ᐉ⫹1
⬘
P ᐉ⫺1
2ᐉ⫹1
共D19兲
,
A k⫹1/2
P ᐉ⬘
ᐉ
冉 冊兺 冋
1
⫽ k⫹
2
共D14兲
Now, expand the left-hand side in powers of t to determine
the A 1/2
ᐉ . This results in
⬁
⬁
共D13兲
from the normalization of the Legendre polynomials,
2 ␦ ᐉᐉ ⬘
.
P ᐉ 共 u 兲 P ᐉ ⬘ 共 u 兲 du⫽
2ᐉ⫹1
⫺1
1
共 1⫺u 兲 k⫺1/2
2
where the prime denotes differentiation with respect to u, and
the last equality follows from Eq. 共12.23兲 of Ref. 关28兴. A
resummation of this last expression yields
ᐉ⫽1
1
共D18兲
冉 冊
冉 冊兺
冉 冊兺
1
⫽⫺ k⫹
2
共D12兲
is used in the first term of Eq. 共60兲 to obtain Eq. 共63兲 and,
subsequently, the A r term of Eq. 共12兲.
Next, for positive odd-integral values of p in Eq. 共D1兲,
first let p⫽1 and multiply the left-hand side of Eq. 共D2兲 by
(1⫺u) 1/2 and the right-hand side by 兺 ᐉ A 1/2
ᐉ P ᐉ (u). Then,
integrate over u from ⫺1 to 1; the right-hand side is
共D17兲
2 k⫹1/2
.
3
k⫹
2
A k⫹1/2
P ᐉ⬘ ⫽⫺ k⫹
ᐉ
For k⫽1
⫽
A ⫺3/2
ᐉ
A k⫹1/2
P ᐉ共 u 兲
ᐉ
for ᐉ⭓1 are obtained from Eq.
The coefficients A k⫹1/2
l
共D15兲 by induction on k. The derivative of Eq. 共D17兲 provides
⬁
关 1⫹O 共 ᐉ ␦ 兲兴 ,
兺
ᐉ⫽0
which defines the expansion coefficients A k⫹1/2
. The first
ᐉ
is
obtained
by
multiplying
both
sides of
coefficient A k⫹1/2
0
Eq. 共D17兲 by 1⫽ P 0 (u), integrating over u from ⫺1 to 1
and using the orthogonality of the Legendre polynomials to
yield
T→0.
共D10兲
⫽ 2k⫺1
A⫺k⫺1/2
ᐉ
共D16兲
The latter is used in the ⑀ 1 term of Eq. 共60兲 to obtain Eq. 共69兲
and, subsequently, the D r term of Eq. 共12兲.
For other positive odd-integral values of p⬎1 in Eq.
共D1兲, consider the Legendre polynomial representation of
(1⫺u) k⫹1/2, with k a positive integer,
共 e T ⫹e ⫺T ⫺2u 兲 ⫺3/2
⬁
⫺2 冑2
.
共 2ᐉ⫺1 兲共 2ᐉ⫹3 兲
⬁
k⫺1/2
A ᐉ⫹1
ᐉ⫽1
2ᐉ⫹3
⫺
k⫺1/2
A ᐉ⫺1
2ᐉ⫺1
册
P ᐉ⬘ .
共D20兲
For ᐉ⭓1
冉 冊冋
⫽ k⫹
A k⫹1/2
ᐉ
共D15兲
1
2
k⫺1/2
A ᐉ⫹1
2ᐉ⫹3
⫺
k⫺1/2
A ᐉ⫺1
2ᐉ⫺1
册
共D21兲
in terms of A k⫺1/2
with the help of Eq.
provides A k⫹1/2
ᐉ
ᐉ
共D18兲. The final result is
104016-16
PHYSICAL REVIEW D 67, 104016 共2003兲
SELF-FORCE OF A SCALAR FIELD FOR CIRCULAR . . .
A k⫹1/2
⫽Pk⫹1/2共 2ᐉ⫹1 兲 / 关共 2ᐉ⫺2k⫺1 兲
ᐉ
␺ ⫽ 兺 ␺ ᐉm 共 r 兲 e i ␻ m t Y ᐉm 共 ␪ , ␾ 兲 ,
共D22兲
where
Pk⫹1/2⫽ 共 ⫺1 兲
关共 2k⫹1 兲 !! 兴
k⫹1 k⫹3/2
2
2
共D23兲
共E7兲
ᐉ,m
⫻ 共 2ᐉ⫺2k⫹1 兲 ••• 共 2ᐉ⫹2k⫹1 兲共 2ᐉ⫹2k⫹3 兲兴 ,
and the ᐉm component of the scalar wave equation becomes
d 2 ␺ ᐉm
dr 2
for k a positive integer or zero. Equation 共D22兲 is used to
furnish the ᐉ dependence in the E r1 term of Eq. 共12兲.
The significant conclusions of this appendix are summarized in Eqs. 共D7兲, 共D11兲 and 共D22兲.
冋
⫽⫺
册
2 共 r⫺M 兲 d ␺ ᐉm
ᐉ 共 ᐉ⫹1 兲
␻ 2r 2
⫺
␺ ᐉm
⫹
r 共 r⫺2M 兲 dr
共 r⫺2M 兲 2 r 共 r⫺2M 兲
⫹
q ᐉm
␦ 共 r⫺r o兲 .
r o⫺2M
共E8兲
We know that
APPENDIX E: INTEGRATION OF THE SCALAR
WAVE EQUATION
* ,
Y ᐉ,⫺m ⫽ 共 ⫺1 兲 m Y ᐉ,m
The scalar field resulting from a charge q moving in a
circular orbit of the Schwarzschild geometry is most easily
found following an approach similar to that of Breuer et al.
关29兴 or, more recently, Burko 关15兴. The wave equation for the
scalar field is
and the reality of % and of the final solution for ␺ (t,r, ␪ , ␾ )
requires similar expressions for q ᐉ,⫺m and ␺ ᐉ,⫺m .
The boundary conditions of interest require only ingoing
waves at the event horizon
ⵜ 2 ␺ ⫽⫺4 ␲ %,
where the scalar field source %, being distinct from ␳ , represents a point charge q moving through spacetime along a
worldline ⌫( ␶ ), described by coordinates z a ( ␶ ). This source
is
% 共 x 兲 ⫽q
冕
␺ ᐉm ⫽e ⫺i ␻ r * /r,
*
共E2兲
共 ⫺g 兲 ⫺1/2 ␦ 共 r⫺r o兲 ␦ 共 ␪ ⫺ ␲ /2兲 ␦ 共 ␾ ⫺⍀t 兲
⫽r
⫽
q
mt
Y ᐉm 共 ␪ , ␾ 兲 ,
␺ ᐉm 共 r 兲 ⫽
a n⫽
共E3兲
␻ m ⬅⫺m⍀,
共E4兲
* 共 ␲ /2,0 兲
4 ␲ q Y ᐉm
,
ro
dt/d ␶
共E5兲
and
dt
1
⫽
.
d ␶ 冑1⫺3M /r o
Also, decomposing ␺ provides
共E12兲
an
e ⫺i ␻ r *
r n⫽0 r n
兺
共E13兲
and, with Eq. 共E8兲, obtain a recursion relation for a n :
where
q ᐉm ⫽
共E11兲
An expansion of ␺ ᐉm starts the numerical integration at
large r. We assume that
q ␦ 共 r⫺r o兲 ␦ 共 ␪ ⫺ ␲ /2兲 ␦ 共 ␾ ⫺⍀t 兲 dt/d ␶
ᐉm
␦ 共 r⫺r o兲 e i ␻
兺
4
␲
ro
ᐉm
共E10兲
where
⫻ ␦ „t⫺t 共 ␶ 兲 …d ␶
⫺2
r→⬁,
r ⫽r⫹2M log共 r/2M ⫺1 兲 .
with ␶ the proper time along the worldline. For a circular
orbit at radius r o , expanding % in terms of spherical harmonic components provides
冕
r→2M ,
and only outgoing waves at infinity
共 ⫺g 兲 ⫺1/2 ␦ 4 „x a ⫺z a 共 ␶ 兲 …d ␶
⫽q 共 ⫺g 兲 ⫺1/2共 dt/d ␶ 兲 ⫺1 ␦ 3 „x i ⫺z i 共 t 兲 …,
%⫽q
␺ ᐉm ⫽e i ␻ r * /r,
共E1兲
共E9兲
n 共 n⫺1 兲 ⫺ᐉ 共 ᐉ⫹1 兲
M 共 n⫺1 兲 2
a n⫺1 ⫺
a n⫺2 ,
2i ␻ n
i␻n
共E14兲
with the starting values of a 0 ⫽1 and a n⬍0 ⫽0. This is an
asymptotic expansion, and we begin the integration of Eq.
共E8兲 at a value of r which is just big enough that the sum in
Eq. 共E13兲 reaches machine accuracy before beginning to diverge. We numerically integrate Eq. 共E8兲 in to the radius of
the orbit r o : this provides us with a homogeneous solution
␺ ⬁ with proper boundary conditions at large r.
Similarly an expansion of ␺ ᐉm for small r⫺2M starts the
numerical integration near the event horizon. We assume that
␺ ᐉm 共 r 兲 ⫽
共E6兲
e i␻r*
b 共 r⫺2M 兲 n
r n⫽0 n
兺
and, with Eq. 共E8兲, obtain a recursion relation for b n :
104016-17
共E15兲
PHYSICAL REVIEW D 67, 104016 共2003兲
DETWEILER, MESSARITAKI, AND WHITING
b n ⫽⫺
⫺
⫺
12i ␻ M 共 n⫺1 兲 ⫹ 共 2n⫺3 兲共 n⫺1 兲 ⫺ 共 ᐉ 2 ⫹ᐉ⫹1 兲
2M 共 4in ␻ M ⫹n 兲
2
12i ␻ M 共 n⫺2 兲 ⫹ 共 n⫺2 兲共 n⫺3 兲 ⫺ᐉ 共 ᐉ⫹1 兲
4M 共 4in ␻ M ⫹n 兲
2
2
i ␻ 共 n⫺3 兲
2M 共 4in ␻ M ⫹n 兲
2
2
with the starting values of b 0 ⫽1 and b n⬍0 ⫽0. We begin the
integration of Eq. 共E8兲 at a value of r⫺2M which is just
small enough that the sum in Eq. 共E15兲 reaches machine
accuracy within a reasonable number of terms. We numerically integrate Eq. 共E8兲 out to the radius of the orbit r o : this
provides us with a homogeneous solution ␺ H with proper
boundary conditions near the event horizon.
The retarded field is
ret
␺ ᐉm
⫽
再
r⬍r o
r⬎r o ,
B
dr
⫺A
H
d ␺ ᐉm
dr
冊
⫽⫺
ro
q ᐉm
r o⫺2M
共E18兲
which yields
b n⫺2
冉
H
A ␺ ᐉm
共E16兲
b n⫺3 ,
H
A ␺ ᐉm
,
⬁
B ␺ ᐉm ,
冉
b n⫺1
⬁
d ␺ ᐉm
⬁
d ␺ ᐉm
dr
⬁
⫺ ␺ ᐉm
H
d ␺ ᐉm
dr
冊
⬁
⫽⫺ ␺ ᐉm
q ᐉm
r o⫺2M
共E19兲
and
冉
H
B ␺ ᐉm
⬁
d ␺ ᐉm
dr
⬁
⫺ ␺ ᐉm
H
d ␺ ᐉm
dr
冊
H
⫽⫺ ␺ ᐉm
q ᐉm
. 共E20兲
r o⫺2M
The ᐉ component of the radial self-force for ␺ ret in Eq. 共11兲,
is then given by
ᐉ
ret
⫽
F ᐉr
共E17兲
兺
m⫽⫺ᐉ
ret
d ␺ ᐉm
dr
冏
.
共E21兲
ro
with the match at r o determined by the ␦ -function source of
Eq. 共E8兲,
ret
in the determiSection VI describes the efficient use of F ᐉr
nation of the self-force.
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关30兴 We use ␺ ret throughout in places where other authors have
used ␺ full or ␺ total to denote the ‘‘actual’’ field 关2兴.
关31兴 The differentiability of ␺ R is controlled by boundary conditions and initial data. We consider nondifferentiable initial data
or shock waves coming in from boundaries to be physically
unreasonable.
104016-18