Regularization parameters for the self-force of a scalar particle in a general orbit about a Schwarzschild black hole

0.34 MB·English

by  Dong-Hoon Kim

Regularization parameters for the self-force of a scalar particle in a general orbit
about a Schwarzschild black hole
Dong-Hoon Kim
Center for Quantum Spacetime, #310 Ricci Annex Hall,
Sogang University, Shinsu-dong Mapogu Seoul 121-742, Korea
The interaction of a charged particle with its own field results in the"self-force" on theparticle,
which includes but is more general than theradiation reaction force. In the vicinity of the particle
incurvedspacetime,onemayfollowDiracandsplittheretardedfieldoftheparticleintotwoparts,
(1) the singular source field, q/r, and (2) the regular remainder field. The singular source field
∼
exerts no force on the particle, and the self-force is entirely caused by the regular remainder. We
describe an elementary multipole decomposition of the singular source field which allows for the
calculation of theself-force on a scalar-charged particle orbiting a Schwarzschild black hole.
0
1
0 I. INTRODUCTION
2

n According to the equivalence principle in general relativity, a particle of infinitesimal mass orbits a black hole of
a
large mass along a geodesic worldline Γ in the background spacetime determined by the large mass alone. For a
J
particle of small but finite mass, the orbit is no longer a geodesic in the background of the large mass because the
5
particle perturbs the spacetime geometry. This perturbation due to the presence of the smaller mass modifies the
2
orbit of the particle from an original geodesic in the background. The difference of the actual orbit from a geodesic

] in the background is said to result from the interaction of the moving particle with its own gravitationalfield, which
c is called a self-force (author?) [1].
q
Historically, Dirac (author?) [2] first gave the analysis of the self-force for the electromagnetic field of a particle
-
r in flat spacetime. He was able to approach the problem in a perturbative scheme by allowing the particle’s size to
g
remain finite and invoking the conservation of the stress-energy tensor inside a narrow world tube surrounding the
[
particle’s worldline. Dewitt and Brehme (author?) [3]extended Dirac’s problemto curvedspacetime. Mino,Sasaki,

1 and Tanaka (author?) [4] generalized it for the gravitational field self-force. Quinn and Wald (author?) [5] and
v Quinn (author?) [6] workedout similar schemes for the gravitational,electromagnetic,andscalar field self-forces by
1
taking an axiomatic approach.
2
InDirac’s (author?) [2] flatspacetime problem,the retardedfield is decomposedinto twoparts: (i) The firstpart
3
is the “mean of the advanced and retarded fields” which is a solution of the inhomogeneous field equation resembling
4
. the Coulomb q/r piece of the scalar potential near the particle, (ii) The second part is a “radiation” field which is a
1
homogeneoussolutionofMaxwell’sequations. Diracdescribesthe self-forceasthe interactionofthe particlewiththe
0
radiation field, a well-defined vacuum field solution.
0
1 In the analyses of the self-force in curved spacetime, the Hadamard form of Green’s function (author?) [3] is
: employed to describe the retarded field of the particle. Traditionally, taking the scalar field case for example, the
v retarded Green’s function Gret(p,p′) is divided into “direct” and “tail” parts: (i) The first part has support only on
i
X the past null cone of the field point p, (ii) The second part has the support inside the past null cone due to the
r presenceofthe curvatureofspacetime. Accordingly,the self-forceonthe particlewouldconsistoftwopieces: (i) The
a first piece comes from the direct part of the field and the acceleration of the worldline in the background geometry;
this corresponds to Abraham-Lorentz-Dirac(ALD) force in flat spacetime, (ii) The second piece comes from the tail
partofthe fieldandis presentincurvedspacetime. Thus,the descriptionofthe self-forcein curvedspacetimeshould
reduce to Dirac’s result in the flat spacetime limit. In this approach, the self-force is considered to result via
=q ψ, (1)
a a
F ∇
from the interaction of the particle with the quantity (author?) [1]
ψself =ψret ψdir. (2)
−
Although this traditional approach provides adequate methods to compute the self-force, it does not share the
physicalsimplicityofDirac’sanalysiswheretheforceisdescribedentirelyintermsofanidentifiable,vacuumsolution
of the field equations: unlike Dirac’s radiation field, the quantity in Eq. (??) is not a homogeneous solution of the
field equation 2ψ = 4π̺. Moreover, the integral term in this quantity comes from the tail part of the Green’s
∇ −
functionandisgenerallynotdifferentiableontheworldlineiftheRicciscalarofthe backgroundisnotzero(similarly,
the electromagnetic potential Atail and the gravitational metric perturbation htail are not differentiable at the point
a ab
2
ofthe particleunless R 1g R ub andR ucud,respectivelyarezerointhe background(author?) [7]). Thus,
ab− 6 ab cadb
some version of averaging process must be invoked to make sense of the self-force.
In this paper an a(cid:0)lternative met(cid:1)hod to split the retarded field ψret as suggested by Ref. (author?) [1] is used
such that the splits resemble those by Dirac: (i) The first part, the Singular Source field ψS, is an inhomogeneous
field similar to the Coulomb potential piece and exerts no force on the particle, (ii) The second part, the Regular
Remainder field ψR, being an homogeneous solution of the field equation, is analogous to Dirac’s radiation field and
entirely responsible for the self-force. This alternative method is reviewed briefly in Section II.
In Section III we give a brief overview of the mode-sum decomposition scheme to evaluate the self-force. We
consider a particle with a scalar charge q in general geodesic motion about a Schwarzschild black hole in this paper.
A spherical harmonic decomposition of both ψret and ψS having this condition should be performed to provide the
multipole components of each. Then, the mode by mode sum of the difference of these components determines ψR,
and, thence the self-force. The multipole components of ψret can be determined numerically while the multipole
components of ψS are derived analytically. In particular, the multipole moments of ψS are generically referred to as
theRegularization Parameters;theentirepaperfocusesontheanalyticaltasktofindtheseregularizationparameters.
We summarize our analytical results at the end of the Section.
The description of ψS becomes advantageously simple in a specially chosen co-moving frame: the THZ normal
coordinates of this frame, introduced by Thorne and Hartle (author?) [8] and extended by Zhang (author?) [9]
are locally inertial on a geodesic. In Section IV we obtain a simple form of ψS using the THZ coordinates and then
re-express it in terms of the background, i.e. Schwarzschild coordinates via the coordinate transformation between
the THZ and the backgroundcoordinates.
Section V outlines the derivation of the regularization parameters given below in Eqs. (11) to (18). These results
are in agreement with Barack and Ori (author?) [10] and Mino, Nakano, and Sasaki (author?) [11].
In Appendix A we give a detailed description of the THZ coordinates. Appendix B provides some mathematical
details concerning the hypergeometric functions and the different representations of the regularizationparameters in
connection with them.
Notation: (t, r, θ, φ) are the usual Schwarzschild coordinates. (T, X, Y, Z) are the initial static normal co-
ordinates, reshaped out of the Schwarzschild coordinates. ( , , , ) are the THZ normal coordinates attached
to the geodesic Γ, and ρ √ 2+ 2+ 2. The points p TandXp′YrefZer to a field point and a source point on the
≡ X Y Z ′
worldline of the particle, respectively. In the coincidence limit p p.
→
II. DECOMPOSITION OF THE RETARDED FIELD
The recent analysis of the Green’s function decomposition by Detweiler and Whiting (author?) [1] shows an
alternative way to split the retarded field into two parts
ψret =ψS+ψR, (3)
whereψS andψR arenamedtheSingularSourcefieldandtheRegularRemainderfield,respectively. Thesourcefunc-
tion for a point particle on the worldline Γ is ̺(p)=q ( g)−1/2δ4(p p′(τ′))dτ′. Like ψret, ψS is an inhomogeneous
− −
solution of the scalar field equation
R
2ψ = 4π̺ (4)
∇ −
in the neighborhood of the particle. And ψS is determined in the neighborhood of the particle’s worldline entirely
by local analysis. ψR, defined by Eq. (3), is then necessarily a homogeneous solution and is therefore expected to be
differentiable on Γ. According to Ref. (author?) [1], ψR will formally give the correctself-force when substituted on
the right hand side of Eq. (1) in place of ψtail. In this paper we adopt this decomposition, and try to determine an
analytical approximation to ψS, which is to be subtracted from ψret for an explicit computation of the self-force.
III. MODE-SUM DECOMPOSITION AND REGULARIZATION PARAMETERS
By Eq. (1) the self-force can be formally evaluated from
self = lim ret(p) S(p) = lim R(p)
Fa p→p′ Fa −Fa p→p′Fa
(cid:2) (cid:3)
3
q lim ψR =q lim (ψret ψS), (5)
a a
≡ p→p′∇ p→p′∇ −
′ ′
where p is the event on Γ where the self-force is to be determined and p is an event in the neighborhood of p. For
use of this equation, both ret(p) and S(p) would be expanded into multipole ℓ-modes, with ret(p) determined
Fa Fa Fℓa
numerically.
Typically,if the backgroundgeometry is Schwarzschildspacetime, the sourcefunction ̺(p) is expandedin terms of
spherical harmonics, and then similar expansion for ψret is made
ψret = ψret(r,t)Y (θ,φ), (6)
ℓm ℓm
ℓm
X
whereψret(r,t)isfoundnumerically. Theindividualℓmcomponentsofψret inthisexpansionarefiniteatthelocation
ℓm
of the particle even though their sum is singular. Then, ret is finite and can be obtained as
Fℓa
ret =q ψretY , (7)
Fℓa ∇a ℓm ℓm
m
X
where a represents each component of t, r, φ, θ in the Schwarzschild geometry.
The singular source field ψS is determined analytically in the neighborhood of the particle’s worldline via local
analysis (see Section IV). Then, ψS is evaluated and the mode-sum decomposition of this quantity is performed
a
∇
to provide
S =q ψS Y , (8)
Fℓa ∇a ℓm ℓm
m
X
which is also finite at the location of the particle.
Then, using Eqs. (5), (7), and (8) the self-force is finally
self = lim ret(p) S(p)
Fa p→p′ Fℓa −Fℓa
ℓ
X (cid:2) (cid:3)
= q lim (ψret ψS )Y (9)
p→p′∇a ℓm− ℓm ℓm
ℓ m
X X
evaluated at the location of the particle.
In Section V the regularization parameters are derived from the multipole components of ψS evaluated at the
a
∇
sourcepointandareusedtocontrolbothsingularbehavioranddifferentiability. WefollowBarackandOri(author?)
[12] in defining the regularizationcounter terms, except that the singular source field ψS is used in place of ψdir
1 C
lim S = ℓ+ A +B + a +O(ℓ−2), (10)
p→p′Fℓa 2 a a ℓ+ 1
(cid:18) (cid:19) 2
and show
q2 r˙
A =sgn(∆) , (11)
t r21+J2/r2
o o
−1
q2E 1− 2rMo
A = sgn(∆) , (12)
r − r2 (cid:16)1+J2/r(cid:17)2
o o
A =0, (13)
φ
q2 F 3F
3/2 5/2
B = Er˙ , (14)
t ro2 "(1+J2/ro2)3/2 − 2(1+J2/ro2)5/2#
4
−1 −1
q2 F1/2 [1−2 1− 2rMo r˙2]F3/2 3 1− 2rMo r˙2F5/2
B = + + , (15)
r ro2 −(1+J2/ro2)1/2 2(cid:16)(1+J2/(cid:17)ro2)3/2 (cid:16)2(1+J(cid:17)2/ro2)5/2 
 
q2 F F 3(F F )
1/2 3/2 5/2 3/2
B = r˙ − + − , (16)
φ J "(1+J2/r2)1/2 2(1+J2/r2)3/2#
o o
C =C =C =0, (17)
t r φ
A =B =C =0, (18)
θ θ θ
where ∆ r r , E u = (1 2M/r )(dt/dτ) (τ: proper time) and J u = r2(dφ/dτ) are the conserved
≡ − o ≡ − t − o o ≡ φ o o
energy and angular momentum, respectively, and r˙ ur = (dr/dτ) . Also, shorthand notations are used for the
≡ o
hypergeometric functions, F F p,1;1; J2 (see Appendix B for more details about the hypergeometric
p ≡ 2 1 2 ro2+J2
functions and the representations of th(cid:16)e regularizati(cid:17)onparameters in connection with them).
IV. DETERMINATION OF ψS VIA THE THZ NORMAL COORDINATES
ItwasmentionedearlierinSectionsII andIII thatψS is determinedinthe neighborhoodofthe particle’sworldline
entirelyby localanalysis. When analyzedbysome speciallocalcoordinatesysteminwhichthe backgroundgeometry
looks as flat as possible, the scalarwaveequationmight take a simple formand ψS might look like a simple Coulomb
potential piece. Detweiler, Messaritaki, and Whiting (author?) [7] (cited henceforth as Paper I) show
ψS =q/ρ+O(ρ2/ 3), (19)
R
whereρ=√ 2+ 2+ 2 with , , beingspatialcomponentsinthatlocalinertialcoordinatesystemand rep-
X Y Z X Y Z R
resentsalengthscaleofthebackgroundgeometry(thesmallestoftheradiusofcurvature,thescaleofinhomogeneities,
and time scale for changes in curvature along Γ).
Describing this special coordinate system more precisely, first, it must be a normal coordinate system where on
Γ, the metric and its first derivatives match the Minkowski metric, and the coordinate measures the proper time.
T
Normal coordinates for a geodesic, however, are not unique, and we use particular ones that were introduced by
Thorne and Hartle (author?) [8] and extended by Zhang (author?) [9] to describe the external multipole moments
of a vacuum solution of the Einstein equations: namely, the THZ NORMAL COORDINATES.
However, in order to derive the regularization parameters from the multipole components of ψS, ρ in Eq. (19)
a
∇
mustbe expressedin terms ofthe coordinatesofthe originalbackground,which is the Schwarzschildgeometryin our
problem. Then, this requires us to find the expressions of , , (THZ) in terms of t, r, φ, θ (Schwarzschild): the
X Y Z
task here is simply to find the coordinate transformation between two different geometries.
Based on the idea from Weinberg (author?) [13], one can achieve this coordinate transformation to the level of
accuracywedesireforthisparticularproblemofmode-sumregularization,bytakingthefollowingtwostepsbasically:
(i) FindaninertialCartesiancoordinatesXAtoredirecttheSchwarzschildcoordinatesxa bytheTaylor’sexpansion
around the location of the particle, xa;
o
1
XA =XA+MA (xa xa)+ MA Γa (xb xb)(xc xc)+O[(x x )3], (20)
o a − o 2 a bc|o − o − o − o
where we may choose XA = 0 and MA = diag MT , MX , MY , MZ for convenience (this choice will
o a t r φ θ
redirect and rescale the Schwarzschild coordinates as T =MT (t t ), X =MX (r r ), Y =MY (φ φ ),
(cid:2) t − o (cid:3) r − o φ − o
Z =MZ (θ θ ).
θ o
−
5
(ii) Boost XA with uA, the particle’s four-velocity at p′ as measured in this Cartesian frame, to obtain the final
coordinates A′;
X
A′ = ΛA′ XA
A
X
= ΛA′ MA (xa xa)+ 1MA Γa (xb xb)(xc xc) +O[(x x )3], (21)
A a − o 2 a bc|o − o − o − o
(cid:20) (cid:21)
where
uT uX uY uZ
− − −
ΛA′A = 1+(uT −SY1M)(uX)2/u2 1+(u(Tu−T 1)u1)X(uuYY)/2u/2u2 ((uuTT −11))uuXYuuZZ//uu22  (22)
− −
 1+(uT 1)(uZ)2/u2 
 − 
 
with u2 (uX)2+(uY)2+(uZ)2 (author?) [14].
≡
According to Ref. (author?) [13], one can show out of Eq. (21)
gA′B′ = gab∂XA′ ∂XB′
∂xa ∂xb
= ηA′B′ +O[(x x )2], xa xa, (23)
− o → o
so that
∂gA′B′
=O[(x x )], xa xa, (24)
∂ C′ − o → o
X
1/2 −1/2
with the choice of MT = 1 2M , MX = 1 2M , MY = r sinθ , MZ = r . Eqs. (23) and (24)
t − ro r − ro φ o o θ − o
are the local inertial feature(cid:16)s as exp(cid:17)ected for norm(cid:16)al coord(cid:17)inates.
To simplify the calculations, we may confine the particle’s orbits to the equatorial plane θ =π/2. Then, we have
o
MA =diag f1/2, f−1/2, r , r , (25)
a o o
−
h i
where f 1 2M . Also, this constraint of the equatorialplane makes uZ =0. We can rewrite uA in terms of the
≡ − ro
Schwarzsch(cid:16)ildcoord(cid:17)inates and the constants of motion,
J
uA uT, uX, uY, uZ = f−1/2E, f−1/2r˙, , 0 , (26)
≡ r
(cid:18) o (cid:19)
(cid:0) (cid:1)
where E u = f(dt/dτ) (τ: proper time) and J u = r2(dφ/dτ) are the conserved energy and angular
momentum≡,−resptectively, andor˙ ur =(dr/dτ) . From t≡his iφt folloows that uo2 =f−1E2 1 and we have
≡ o −
f−1/2E f−1/2r˙ J/r 0
o
− −
ΛA′A = 1+r˙2/S(fY1M/2E+f) 1+JJr˙2//[[rro2((Ef−+1/f21E/2+)]1)] 00. (27)
o
 1
 
 
Now we are finally able to express ρ2 in terms of the Schwarzschildcoordinates. Using Eq. (21) one may write
ρ2 = I = δ ΛI ΛJ MC MD (xc xc)(xd xd)+ Γc (xa xa)(xb xb)(xd xd)
X XI IJ C D c d − o − o ab|o − o − o − o
+O[(x xo)4], (cid:2) (cid:3) (28)
−
where I, J =1, 2, 3. Then, using Eqs. (25) and (27), Eq. (28) can be eventually expressed as
2Er˙
ρ2 = (E2 f)(t t )2 (t t )(r r ) 2EJ(t t )(φ φ )
o o o o o
− − − f − − − − −
6
1 r˙2 2Jr˙ π 2
+ 1+ (r r )2+ (r r )(φ φ )+(r2+J2)(φ φ )2+r2 θ
f f − o f − o − o o − o o − 2
(cid:18) (cid:19) (cid:16) (cid:17)
MEr˙ M 2E2 r˙2 MJr˙
(t t )3+ 1+ + (t t )2(r r )+ (t t )2(φ φ )
− r2 − o r2 − f f − o − o r2 − o − o
o o (cid:18) (cid:19) o
MEr˙ 2(r M)EJ
(t t )(r r )2 o− (t t )(r r )(φ φ )
− f2r2 − o − o − fr2 − o − o − o
o o
π 2
+r Er˙(t t )(φ φ )2+r Er˙(t t ) θ
o o o o o
− − − − 2
M r˙2 (2r 5M(cid:16))Jr˙ (cid:17)
1+ (r r )3+ o− (r r )2(φ φ )
−f2r2 f − o f2r2 − o − o
o (cid:18) (cid:19) o
r˙2 2J2 r˙2 π 2
+r 1 + (r r )(φ φ )2+r 1 (r r ) θ
o − f r2 − o − o o − f − o − 2
(cid:18) o (cid:19) (cid:18) (cid:19) (cid:16) (cid:17)
π 2
r Jr˙(φ φ )3 r Jr˙(φ φ ) θ +O[(x x )4]. (29)
o o o o o
− − − − − 2 −
(cid:16) (cid:17)
Eq. (29) is substituted into Eq. (19) to determine ψS in terms of the Schwarzschild coordinates and will serve
significantly to derive the regularizationparameters in the next section.
In the above analysis we have ignored the term O[(x x )3] in Eq. (21) and its contribution to ρ2, which is
o
O[(x x )4] in Eqs. (28) and (29). To the level of acc−uracy we desire for the mode-sum regularization in this
o
−
paper, that is to say, to the determination of C-terms, it is not necessary to specify the term O[(x x )4] in ρ2
o
(hence not necessary to specify O[(x x )3] in the spatial THZ coordinates , , ). Even witho−ut specifying
o
− X Y Z
the term O[(x x )4] in ρ2, one can prove that C -terms in Eq. (10) always vanish (see Subsection VC). In fact,
o a
the clarification−of O[(x x )3] for the THZ coordinates in Eq. (21) requires more involved analyses of coordinate
o
−
transformations,whichwouldbe beyondthe scope ofthis paper. Readersmay referto Appendix A for moredetailed
description of the THZ coordinates, specified up to the quartic order.
V. REGULARIZATION PARAMETERS FOR A GENERAL ORBIT OF THE SCHWARZSCHILD
GEOMETRY
In Section IV, we have seen that an approximation to ψS is
ψS =q/ρ+O(ρ2/ 3). (30)
R
Following Paper I (author?) [7], the regularization parameters can be determined from evaluating the multipole
components of ∂ (q/ρ) (a = t, r, θ, φ for the Schwarzschild background) at the location of the source. The error,
a
O(ρ2/ 3)intheaboveapproximationisdisregardedsinceitgivesnocontributionto ψS aswetakethe“coincidence
a
R ∇
limit”, x x , where x denotes a point in the vicinity of the particle and x the location of the particle in the
o o
→
Schwarzschildgeometry.
In evaluating the multipole components of ∂ (q/ρ), singularities are expected with certain terms. To help identify
a
thosesingularities,weintroduceanorderparameterǫwhichistobesettounityattheendofacalculation: weattach
ǫn to each O[(x x )n] part of ρ2 in Eq. (29) and may re-express ρ2 as
o
−
ρ2 =ǫ2 +ǫ3 +ǫ4 +O(ǫ5), (31)
II III IV
P P P
where , , and represent the quadratic, cubic, and quartic order parts of ρ2, respectively. Here we pretend
II III IV
P P P
that the quartic part is also specified: this will help us to perform the structure analysis for C -terms later in
IV a
P
Subsection VC when we prove that these regularization parameters always vanish.
Whenweexpress∂ (1/ρ)intheLaurentseriesexpansiontoidentifythetermsaccordingtotheirsingularpatterns,
a
every denominator of this expansion should take the form of n/2 (n = 3, 5, 7). Due to this special position, which
PII
becomes singular in the coincidence limit, would play a significant role in inducing the multipole decomposition.
II
P
However,thequadraticpart ,directlytakenfromEq.(29),maynotbefullyreadyforthistaskyet. First,φ φ
II o
P −
needs to be decoupledfrom r r so that we have independent complete square forms of each,which is necessary for
o
−
inducing the Legendre polynomial expansions later. Coupling between t t and φ φ is not significant because
o o
− −
upon fixing t=t all terms having t t will vanish. Thus, we reshape our quadratic term in Eq. (29) into
o o
−
2Er˙r2
= (E2 f)(t t )2 o (t t )∆ 2EJ(t t )(φ φ′)
PII − − o − f(r2+J2) − o − − o −
o
7
E2r2 π 2
+ o ∆2+ r2+J2 (φ φ′)2+r2 θ (32)
f2(r2+J2) o − o − 2
o
(cid:0) (cid:1) (cid:16) (cid:17)
with
Jr˙∆
′
φ φ , (33)
≡ o− f(r2+J2)
o
where ∆ r r , and an identity r˙2 =E2 f 1+J2/r2 is used for simplifying the coefficient of ∆2. Here, taking
≡ − o ′ − o
the coincidence limit ∆ 0, we have φ φ (the same idea is found in Mino, Nakano,and Sasaki (author?) [11]).
o
→ → (cid:0) (cid:1)
Also, in order to be multipole-decomposed globally, the quadratic part needs to be completely analytic and smooth
over the entire two-sphere. For this purpose we rewrite it as
2Er˙r2
= (E2 f)(t t )2 o (t t )∆ 2EJ(t t )sinθsin(φ φ′)
PII − − o − f(r2+J2) − o − − o −
o
E2r2
+ o ∆2+(r2+J2)sin2θsin2(φ φ′)+r2cos2θ
f2(r2+J2) o − o
o
+O[(x x )4], (34)
o
−
where one should notice that by replacing φ φ′ =sin(φ φ′)+O[(φ φ′)3] and 1=sinθ+O[(θ π/2)2] we create
− − − −
the O[(x x )4] terms.
o
−
To aid in the multipole decomposition we rotate the usual Schwarzschild coordinates by following the approach of
Barackand Ori (author?) [10] and Paper I (author?) [7] such that the coordinate location of the particle is moved
from the equatorial plane θ = π to a location where sinΘ=0 (Θ: new polar angle). We define new angles Θ and Φ
2
in terms of the usual Schwarzschildangles by
′
sinθcos(φ φ) = cosΘ
−
′
sinθsin(φ φ) = sinΘcosΦ
−
cosθ = sinΘsinΦ. (35)
Also, under this coordinate rotation, a spherical harmonic Y (θ,φ) becomes
ℓm
ℓ
Yℓm(θ,φ)= αℓmm′Yℓm′(Θ,Φ), (36)
m′=−ℓ
X
where the coefficients αℓ depend on the rotation (θ,φ) (Θ,Φ) as well as on ℓ, m, and m′, and the index ℓ
mm′ →
is preserved under the rotation (author?) [15]. As recognized already in Paper I (author?) [7], there is a great
advantage of using the rotated angles (Θ,Φ): after expanding ∂ (q/ρ) into a sum of spherical harmonic components,
a
we take the coincidence limit ∆ 0, Θ 0. Then, finally only the m=0 components contribute to the self-force at
→ →
Θ = 0 since Y (0,Φ) = 0 for m = 0. Thus, the regularization parameters of Eq. (10) are just (ℓ, m = 0) spherical
ℓm
harmonic components of ∂ (q/ρ) e6 valuated at xa.
a o
Now, using these rotated angles, we may re-express Eq. (34) as
2Er˙r2
= (E2 f)(t t )2 o (t t )∆ 2EJ(t t )sinΘcosΦ
PII − − o − f(r2+J2) − o − − o
o
J2sin2Φ r2E2∆2
+2 r2+J2 1 o +1 cosΘ
(cid:0) o (cid:1)(cid:18) − ro2+J2 (cid:19)2f2(ro2+J2)2 1− Jr2o2s+inJ22Φ − 
+O[(x x )4],  (cid:16) (cid:17)  (37)
o
−
whereanapproximationsin2Θ=2(1 cosΘ)+O(Θ4)isused,theerrorfromwhichisessentiallyO(Θ4)=O[(x x )4]
o
− −
and can be absorbed into . Here one should note that through a series of modifications of the quadratic part of
IV
P
Eq. (29) we have created additional quartic order terms apart from the desired form. Then, we may remove these
additional terms from the quadratic part and incorporate them into the quartic part in Eq. (31). It is not
IV
P
necessary, however, to specify this quartic part: as already mentioned above, later in Subsection VC we will show
that the quartic part starts appearing from the ǫ0-term and verify that the regularization parameters of the
IV
ǫ0-term always vanish bPy analyzing the generic structure of the ǫ0-term.
8
After removing O[(x x )4] from Eq. (37) we may define
o
−
2Er˙r2
ρ˜2 (E2 f)(t t )2 o (t t )∆ 2EJ(t t )sinΘcosΦ
≡ − − o − f(r2+J2) − o − − o
o
J2sin2Φ r2E2∆2
+2 r2+J2 1 o +1 cosΘ . (38)
(cid:0) o (cid:1)(cid:18) − ro2+J2 (cid:19)2f2(ro2+J2)2 1− Jr2o2s+inJ22Φ − 
 (cid:16) (cid:17) 
In particular, when fixing t=t , we define
o
ρ˜2 ρ˜2 =2 r2+J2 χ δ2+1 cosΘ (39)
o ≡ t=to o −
with (cid:12)(cid:12) (cid:0) (cid:1) (cid:0) (cid:1)
J2sin2Φ
χ 1 (40)
≡ − r2+J2
o
and
r2E2∆2
δ2 o . (41)
≡ 2f2(r2+J2)2χ
o
Now we rewrite Eq. (31) by replacing the original quadratic part with the modified form ρ˜2 above,
II
P
ρ2 =ǫ2ρ˜2+ǫ3 +ǫ4 +O(ǫ5), (42)
III IV
P P
where the new quartic part includes the additional quartic order terms that result from modification of the
IV
P
quadratic part . Then, based on this redefined ρ2, we have the following expression of ∂ (1/ρ) in a Laurent
PII a |t=to
series expansion
∂ 1 = 1 ∂a ρ˜2 t=toǫ−2+ 1 ∂aPIII|t=to + 3 ∂a ρ˜2 PIII t=to ǫ−1+O(ǫ0). (43)
a(cid:18)ρ(cid:19)(cid:12)(cid:12)t=to −2 (cid:0) ρ˜3o(cid:1)(cid:12)(cid:12) (−2 ρ˜3o 4 (cid:2) (cid:0) (cid:1)ρ˜(cid:3)5o (cid:12)(cid:12) )
(cid:12)
Eq. (39), when ins(cid:12)erted into Eq. (43), plays a very important role in calculating the regularization parameters for
the rest of the section: out of Eq. (39), we induce Legendre polynomial expansions in terms of cosΘ. Sometimes we
may have the dependence on ρ˜2 not only in the denominators but also in the numerators on the right hand side of
o
Eq. (43). In the numerators the dependence can be found from the terms containing sinnΘ or cosnΘ since Eq. (39)
canbesolvedforcosΘ. Afterfindingalloftheρ˜2 dependence, therestofthe calculationsinvolveintegratingoverthe
o
angle Φ . The techniques involved in Legendre polynomial expansions and integration over Φ are described in detail
in Appendices C and D of Paper I (author?) [7].
Below in Subsections VA and VB, we present the key steps of calculating the regularizationparameters.
A. Aa-terms
We take the ǫ−2 term from Eq. (43) and define
Q [ǫ−2] q2 ∂a ρ˜2 t=to (44)
a ≡− 2 (cid:0) ρ˜3o(cid:1)(cid:12)
(cid:12)
Then, we proceed with our calculations of the regularizationparameters one component at a time.
1. At-term:
First we complete the expression for Q [ǫ−2] by recalling Eqs. (38) and (39)
t
q2
Q [ǫ−2] = ρ˜−3 ∂ ρ˜2
t − 2 o t t=to
(cid:0) (cid:1)(cid:12)
(cid:12)
9
= q2 2 r2+J2 χ δ2+1 cosΘ −3/2 2Er˙ro2∆ +2EJsinΘcosΦ
2 o − f(r2+J2)
(cid:18) o (cid:19)
= q2(cid:2)E(cid:0)r˙ro2∆χ−3(cid:1)/2 (cid:0) δ2+1 cosΘ(cid:1)(cid:3) −3/2
2√2f(r2+J2)5/2 −
o
q2EJχ−3/2cosΦ(cid:0) ∂ δ2+1 (cid:1) cosΘ −1/2, (45)
− √2(r2+J2)3/2 ∂Θ −
o (cid:12)∆
(cid:12) (cid:0) (cid:1)
(cid:12)
where ∂ means that ∆ is held constant while th(cid:12)e differntiation is performed with respect to Θ.
∂Θ ∆
According to Appendix D of Paper I (author?) [7], for p 1
(cid:12) ≥
(cid:12)
∞
δ2+1 cosΘ −p−1/2 = 2ℓ+1 [1+O(ℓδ)]P (cosΘ), δ 0, (46)
− δ2p−1(2p 1) ℓ →
ℓ=0 −
(cid:0) (cid:1) X
and for p=0
∞
δ2+1 cosΘ −1/2 = √2+O(ℓδ) P (cosΘ), δ 0. (47)
ℓ
− →
(cid:0) (cid:1) Xℓ=0h i
Then, by Eqs. (46) for p=1, (47), and (41), in the limit δ 0 (equivalently ∆ 0) Eq. (45) becomes
→ →
q2r˙r χ−1 ∞ 1
lim Q [ǫ−2] = sgn(∆) o ℓ+ P (cosΘ)
∆→0 t (r2+J2)3/2 2 ℓ
o ℓ=0(cid:18) (cid:19)
X
q2EJχ−3/2cosΦ ∞ ∂
P (cosΘ). (48)
− (r2+J2)3/2 ∂Θ ℓ
o ℓ=0 (cid:12)∆
X (cid:12)
Then, we integrate lim∆→0Qt[ǫ−2] over Φ and divide it by 2π (we den(cid:12)(cid:12)ote this process by the angle brackets “ ”)
hi
q2r˙r χ−1 ∞ 1
lim Q [ǫ−2] =sgn(∆) o ℓ+ P (cosΘ), (49)
D∆→0 t E (ro2+J(cid:10)2)3/(cid:11)2 Xℓ=0(cid:18) 2(cid:19) ℓ
wherewe exploitthe factthat χ−3/2cosΦ =0to getridofthe secondpartinEq. (48)[18]. Appendix C ofPaperI
(author?) [7] provides χ−1 = F 1,1;1;α F =(1 α)−1/2, where α J2/ r2+J2 . Substituting this into
(cid:10) 2 1 2 (cid:11) ≡ 1 − ≡ o
Eq. (49), the regularization parameter A can be finally determined when we take the coefficient of the sum on the
t
(cid:10) (cid:11) (cid:0) (cid:1) (cid:0) (cid:1)
right hand side in the coincidence limit Θ 0
→
q2 r˙
A =sgn(∆) . (50)
t r21+J2/r2
o o
2. Ar-term:
Similarly, we have
q2
Q [ǫ−2]= ρ˜−3 ∂ ρ˜2 . (51)
r − 2 o r t=to
(cid:0) (cid:1)(cid:12)
Here, before computing ∂ ρ˜2 we reverse the process via Eqs. ((cid:12)32), (34), (37), and (38) to obtain the relation
r t=to
(cid:0) (cid:1)(cid:12)
(cid:12)
ρ˜2 = +O[(x x )4]
II o
P −
2Er˙r2
= (E2 f)(t t )2 o (t t )∆ 2EJ(t t )(φ φ′) (52)
− − o − f(r2+J2) − o − − o −
o
E2r2 π 2
+ o ∆2+ r2+J2 (φ φ′)2+r2 θ
f2(r2+J2) o − o − 2
o
(cid:16) (cid:17)
+O[(x x )4]. (cid:0) (cid:1) (53)
o
−
10
Differentiating this with respectto r andgoingthroughthe processvia Eqs. (34) and(35), Eq.(51) canbe expressed
with the help of Eq. (39) as
Q [ǫ−2]= q2 2 r2+J2 χ δ2+1 cosΘ −3/2 ro2E2∆ +fJr˙sinΘcosΦ (54)
r −f2 o − r2+J2
(cid:20) o (cid:21)
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
[19]. Then, the rest of the calculation is carried out in the same fashion as for the case of A -term above. We obtain
t
q2 E
A = sgn(∆) . (55)
r − r2f(1+J2/r2)
o o
3. A -term:
φ
First we have
q2
Q [ǫ−2] = ρ˜−3 ∂ ρ˜2 . (56)
φ − 2 o φ t=to
(cid:0) (cid:1)(cid:12)
Taking the same steps as used for A -term above via Eqs. (53), (34),(cid:12)and (35) in order, we obtain
r
∂ ρ˜2 =2 r2+J2 sinΘcosΦ+O[(x x )3]. (57)
φ t=to o − o
(cid:0) (cid:1)(cid:12) (cid:0) (cid:1)
Then, in a similar manner to that emplo(cid:12)yed in the previous cases, in the limit ∆ 0 Eq. (56) becomes
→
q2χ−3/2cosΦ ∞ ∂
lim Q [ǫ−2]= P (cosΘ) (58)
∆→0 φ − (r2+J2)1/2 ∂Θ ℓ
o ℓ=0 (cid:12)∆
X (cid:12)
(cid:12)
[20]. The right hand side vanishes through “ ” process because χ−3/2c(cid:12)osΦ =0. Hence,
hi
A =0. (cid:10) (cid:11) (59)
φ
4. A -term:
θ
It is evident from the particle’s motion, which is confined to the equatorial plane θ = π, that no self force is
o 2
acting onthe particle inthe directionperpendicular to this plane. This is due to the fact that both the derivativesof
retarded field and the singular source field with respect to θ tend to zero in the coincidence limit. Our calculation of
A should support this. Through the same process as employed before, we have
θ
q2
Q [ǫ−2]= ρ˜−3 ∂ ρ˜2 (60)
θ − 2 o θ t=to
(cid:0) (cid:1)(cid:12)
with (cid:12)
∂ ρ˜2 =2r2sinΘsinΦ+O[(x x )3]. (61)
θ t=to o − o
(cid:0) (cid:1)(cid:12)
Then, similarly as in the case of Aφ-term ab(cid:12)ove
q2r2χ−3/2sinΦ ∞ ∂
lim Q [ǫ−2]= o P (cosΘ). (62)
∆→0 θ − (r2+J2)3/2 ∂Θ ℓ
o ℓ=0 (cid:12)∆
X (cid:12)
(cid:12)
Again, via “ ” process, the right hand side vanishes because χ−3/2sinΦ(cid:12) =0. Thus,
hi
A =0.(cid:10) (cid:11) (63)
θ