Gauge and averaging in gravitational self-force

Samuel E. Gralla

Enrico Fermi Institute and Department of Physics, University of Chicago, 5640 S. Ellis Avenue, Chicago, Illinois 60637, USA(Received 9 May 2011; published 25 October 2011)

A difficulty with previous treatments of the gravitational self-force is that an explicit formula for the

force is available only in a particular gauge (Lorenz gauge), where the force in other gauges must be found

through a transformation law once the Lorenz-gauge force is known. For a class of gauges satisfying

a ‘‘parity condition’’ ensuring that the Hamiltonian center of mass of the particle is well-defined, I show

that the gravitational self-force is always given by the angle average of the bare gravitational force. To

derive this result I replace the computational strategy of previous work with a new approach, wherein the

form of the force is first fixed up to a gauge-invariant piece by simple manipulations, and then that piece is

determined by working in a gauge designed specifically to simplify the computation. This offers

significant computational savings over the Lorenz gauge, since the Hadamard expansion is avoided

entirely and the metric perturbation takes a very simple form. I also show that the rest mass of the particle

does not evolve due to first-order self-force effects. Finally, I consider the ‘‘mode sum regularization’’

scheme for computing the self-force in black hole background spacetimes, and use the angle-average form

of the force to show that the same mode-by-mode subtraction may be performed in all parity-regular

gauges. It appears plausible that suitably modified versions of the Regge-Wheeler and radiation gauges

(convenient to Schwarzschild and Kerr, respectively) are in this class.

DOI: 10.1103/PhysRevD.84.084050 PACS numbers: 04.25.Nx

I. INTRODUCTION

The leading-order deviation from geodesic motion pro-portional to the mass of a body—interpreted as the forcedue to the body’s own gravitational field—is known as thegravitational self-force. A recurring source of difficulty inboth the theoretical treatment and practical computation ofthe self-force has been the choice of gauge in which themetric perturbation and force are expressed. In particular,early treatments [1–4] required a specific gauge choice—Lorenz gauge—even to define the perturbed trajectory (viaa point particle hypothesis coupled with ‘‘Lorenz-gaugerelaxation’’ [2,5] to allow nongeodesic motion), and aproposed extension of the results to other gauges [6] re-stricted to gauge vectors that are continuous at the particle,even though the metric perturbation is singular. At a theo-retical level, this elevates a particular gauge to fundamentalstatus, to the exclusion of other gauges that seem equallynice, such as any gauge where the point particle 1=rsingularity corresponds to linearized Schwarzschild inCartesian Schwarzschild coordinates, as opposed to theCartesian isotropic coordinates that correspond to Lorenzgauge.1 And at a practical level, one has excluded thestandard gauges of black hole perturbation theory [6].

Previous work [5] (hereafter paper I) eliminated thefundamental status of the Lorenz gauge by giving a defini-tion of perturbed motion holding for any gauge where theparticle is represented by a 1=r singularity. However, in this

work we still relied on the Lorenz gauge for our computa-tions and, more importantly, for the expression of the finalresult, as a formula holding in the Lorenz gauge togetherwith a generalized transformation law. At a theoreticallevel, we still have a preferred role for the Lorenz gauge;and at a practical level, the results suggest that the compu-tation of a self-force in an alternative gauge must alwaysproceed through Lorenz gauge, eliminating much of theappeal of working in alternative gauges in the first place.In this paper I will identify a class of gauges based on the

requirement that the center of mass as defined by Reggeand Teitelboim [7] is well-defined (in the ‘‘near zone’’),and show that the force in any such gauge is given by theangle average of the bare force in that gauge. To derive thisresult I adopt the assumptions of paper I but take a newcomputational approach, wherein the form of the force inany gauge is fixed up to a gauge-invariant piece by simplemanipulations, and then that piece is determined by work-ing in a gauge chosen specifically to make the computationas simple as possible. This approach avoids much of thecomputational complexity of previous work (eliminatingthe Hadamard expansion entirely and significantly reduc-ing the calculation needed thereafter), while organizing thecomputation so that the final equation automatically takes agauge-independent form.The precise results are as follows. We define the (lowest-

order) mass M, spin Sab, and center of mass deviation Za

of the particle as tensors on a timelike worldline � (fourvelocity ua) in a vacuum background metric gab.

2 Then we

1The (discontinuous) gauge vector that changes the singularityfrom isotropic to Schwarzschild type is given by �i ¼ ni ¼ xi=rin Fermi normal coordinates.

2The spin is antisymmetric and satisfies uaSab ¼ 0. Thedeviation satisfies Zaua ¼ 0.

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find that � is a geodesic, that the mass and spin are constant(parallel propagated), and that the deviation Za satisfies

ubrbðucrcZaÞ ¼ 1

4�limr!0

ZFad�þ Rbcd

aubZcud

þ 1

2MRbcd

aSbcud; (1)

where Fa is the ‘‘bare gravitational force,’’

Fa ¼ �ðgab þ uaubÞ�rdhbc � 1

2rbhcd

�ucud; (2)

and h�� is the metric perturbation of a point particle,

Gð1Þab½h� ¼ 8�M

Z��4ðx; zð�ÞÞuaubd�; (3)

which must be expressed in a ‘‘parity-regular’’ gauge,where the singular part of the spatial metric is even parityon the sphere, Cijðt;� ~nÞ ¼ Cijðt; ~nÞ in Eq. (6). The bare

force Fa is familiar from the perturbed geodesic equationand here is defined only off of � (and only locally, whereua is extended off � by parallel transport along spacelikegeodesics orthogonal to ua) since the metric perturbation isdivergent. The integral in (1) is defined by using theexponential map based on � to associate a flat metric, interms of which the integral is over a fixed two-sphere ofspatial distance r in the hyperplane orthogonal to ua, withna its unit normal and d� its area element, and the inte-gration is done componentwise under the exponential map.I also define the perturbed mass of the particle and showthat it is constant in time.

The first term on the right-hand side of Eq. (1) isproportional to the metric perturbation; I therefore referto this term as the gravitational self-force. We see that theforce in any parity-regular gauge is given by the angleaverage of the bare force in that gauge, so that the self-force may be viewed as the net gravitational force on theparticle. If the Lorenz gauge (which is parity-regular) isadopted and the Hadamard form for the metric perturbationis computed (choosing the retarded solution with no in-coming radiation) and plugged in, then this term reduces tothe standard ‘‘tail integral’’ expression for the self-force(e.g., [4]). The second term corresponds to the geodesicdeviation equation and reflects the particle’s desire to moveon a new geodesic once it has been displaced from theoriginal. The third term is the Papapetrou spin force. If theparity condition is violated, Eq. (1) does not hold, andthe equation of motion takes a complicated form involvingan explicit gauge transformation to a parity-regular gauge,Eq. (B2).

A practical technique for computing self-forces in blackhole background spacetimes is known as mode-sum regu-larization ([8] and many other references). In this approachone numerically solves for the spherical harmonic modesof the metric perturbation (the sum over which diverges atthe particle) and performs a mode-by-mode subtraction

that regularizes the sum in such a way that the correctself-force is computed. Extensive work has determined theform of the subtraction in the Lorenz gauge, which auto-matically holds for gauges that are smoothly related.Taking advantage of a connection between mode decom-positions and averaging (and hence self-force), I show thatthe same subtraction may in fact be performed in all parity-regular gauges. It appears plausible that suitably modifiedRegge-Wheeler and radiation gauges (convenient to per-turbations of Schwarzschild and Kerr, respectively) will bein this class, with especially strong evidence in the case ofradiation gauge (see discussion at the end of Sec. V). If so,then the results of this paper would provide a theoreticaland practical foundation for the computation of self-forceeffects in gauges convenient to black hole spacetimes.I use the conventions of Wald [9]. Greek indices label

tensor components, while early-alphabet Latin indicesa; b; . . . are abstract indices. When coordinates t, xi areused, the time and space components are denoted by 0 andmidalphabet Latin indices i; j; . . . , respectively.

II. REVIEW OF FORMALISM

The central idea of paper I is to introduce a mathe-matically precise formulation of the notion of the nearzone of a body and to use the requirement of a sensiblenear zone to demand a sensible perturbation family. Givena family of metrics gabð�Þ in coordinates ðt; xiÞ, we definea scaled metric �gab � ��2gab and scaled coordinates �t ���1ðt� t0Þ and �xi � ��1xi. Denoting the scaled metric inscaled coordinates by �g �� ��, consider the � ! 0 limit,

�gð0Þ�� �� � �g �� ��j�¼0. This limit effectively ‘‘zooms in’’ on the

spacetime point (t ¼ t0, xi¼0), and will recover the near

zone of a body if the one-parameter family contains a bodywhose radius and mass shrink down linearly with � to theworldline xi ¼ 0 (denoted by �). By demanding itsexistence (and associated conditions), we automaticallyconsider bodies of small size and mass. Note that thecomponents of the original and scaled metrics are relatedsimply by ‘‘plugging in the new coordinates,’’

�g �� ��ð�; t0; �t; �xiÞ ¼ g��ð�; t ¼ t0 þ ��t; xi ¼ � �xiÞ: (4)

This equation relates components of the scaled metric inscaled coordinates, �g �� ��, to corresponding components of

the original metric in the original coordinates, g��.

One may perform perturbation theory in either the origi-nal (‘‘far-zone’’) picture or the scaled (‘‘near-zone’’) pic-ture. The far-zone background and perturbations will be

denoted by gðnÞ�� � ð1=n!Þð@�Þng��j�¼0 (I also write h�� ¼gð1Þ��), while the near-zone background and perturbations

will be denoted by �gðnÞ�� �� � ð1=n!Þð@�Þn �g �� ��j�¼0. Our as-

sumptions together with the choice of Fermi coordinates(e.g., [4]) about � for the far-zone background metricconstrain the far-zone quantities to take the form

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gð0Þ�� ¼ �� þ B�i�jðtÞxixj þOðr3Þ (5)

h�� ¼ gð1Þ�� ¼ C��ðt; ~nÞr

þD��ðt; ~nÞ þ rE��ðt; ~nÞ þOðr2Þ(6)

gð2Þ�� ¼ F��ðt; ~nÞr2

þH��ðt; ~nÞr

þ K��ðt; ~nÞ þOðrÞ; (7)

where we have defined r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ijx

ixjq

and ni ¼ xi=r. Here

B�i�j is related to the Riemann curvature on the back-

ground worldline xi ¼ 0 (see e.g. [4] for the exact expres-sion), whereas C��; . . . ; K�� are unspecified smooth

functions on R� S2 (denoted by arguments ðt; ~nÞ). Thelack of a linear term in Eq. (5) is a consequence of � beinggeodesic, as shown in paper I. We also showed that themetric perturbation has effective point particle source,Eq. (3). In light of Eq. (4) the near-zone perturbation seriestakes the form

�g ð0Þ�� �� ¼ �� þ

C��ðt0; ~nÞ�r

þ F��ðt0; ~nÞ�r2

þO

�1

�r3

�(8)

�g ð1Þ�� �� ¼ D�� þ

H��

�rþO

�1

�r2

�þ �t

� _C��

�rþ

_F��

�r2þO

�1

�r3

��

(9)

�gð2Þ�� �� ¼ B�i�j �xi �xj þ �rE�� þ K�� þO

�1

�r

�

þ �t

�_D�� þ

_H��

�rþO

�1

�r2

��

þ 1

2�t2� €C��

�rþ

€F��

�r2þO

�1

�r3

��; (10)

where an overdot denotes a t derivative, and the ordersymbols Oð1=�rnÞ refer to �t-independent functions. Thedependence on ðt0; ~nÞ is suppressed in Eqs. (9) and (10)for readability. Note that the lack of growing-in-�r terms inEq. (9) is inherited from the absence of a linear term in (5),which is a consequence of � being geodesic (and choosingFermi coordinates). The (stationary and asymptotically

flat) near-zone background metric �gð0Þ�� �� represents the local

state of the body at time t0, and as such its propertiesshould characterize those of the body. In particular, the

multipole moments of �gð0Þ�� �� should correspond to those of

the body, at lowest nontrivial perturbative order. To the(far-zone) perturbative order considered in this paper, onlythe monopole and dipole moments can play a role. Wetherefore define3

Mðt0Þ � �1

8�lim�r!1

Zni@i �g

ð0Þ00 �r

2d� (11)

Diðt0Þ � 3

8�lim�r!1

Z�gð0Þ00 n

i �r2d� (12)

Sijðt0Þ � 3

8�lim�r!1

Z�gð0Þ0½inj� �r

2d� (13)

and refer to M, Di, and Sij as the (lowest-order) mass,

mass dipole, and spin (current dipole) of the body.4 InEqs. (11)–(13), the bars on the indices of coordinate com-ponents of the near-zone metric have been dropped, and theintegrals are taken on fixed coordinate two-spheres withrespect to the ‘‘flat’’ volume element d�. It is well known(and easily checked) that the mass dipole may be set to zeroby translating the coordinates, �xi ! �xi � Zi, with

Ziðt0Þ � Diðt0Þ=Mðt0Þ: (14)

This gives the new coordinates the interpretation of beingmass centered, so that Zi represents the center of massposition of the body in the original coordinates. Since theabove translation corresponds to a first-order gauge trans-formation in the far zone (recall �xi ¼ xi=�), we identify Zi

with the first-order deviation of the center of mass positionfrom its background position �. While the mass, spin, anddeviation are defined in the near zone ‘‘at spatial infinity’’(as a function of the coordinate time t0 along � at which thenear-zone limit is taken), they may equally well be viewedas tensors defined in the far-zone ‘‘at the spatial origin,’’i.e., as tensor fields on �. It was shown in paper I that themass and spin do not evolve with time t0 (i.e., that they areparallel propagated tensor fields on �), while a Lorenz-gauge equation of motion (together with a gauge trans-formation law) was derived for the deviation vector.The analysis of this paper will require transforma-

tion properties of Zi not only under ordinary translations,but also under transformations of the form � �xi ¼ið ~nÞ þOð1=�rÞ, i.e., under angle-dependent translations,or supertranslations. Using the well-known fact that�g00 ¼ �1þ 2M=�rþOð1=�r2Þ for all vacuum solutions ofthe form (8), we have by direct computation that � �g00 ¼2Mini= �r

2 þOð1=�r3Þ, so that Zi transforms as

�Zi ¼ 3

4�

Zjnjn

id�: (15)

Since the angle-average of a vector picks out an ‘ ¼ 1,electric parity part, we may restrict consideration to i ofthe form i ¼ Bjnjn

i þ Ci for constants Bi and Ci. The

associated change in Zi is then simply Bi þ Ci. Thus Zi

3Equation (13) for the spin holds only in coordinates where Ci0

vanishes. A formula for the spin holding in general coordinateswould take a more complicated form.

4The mass dipole may also be thought of as the time-spacecomponent of the spin tensor, S0i. However, we will work with aspin tensor that is orthogonal to �, S0i ¼ 0, defining a separatemass dipole.

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changes under supertranslations with Bi � 0, in addition toits change under ordinary translations i ¼ Ci.

III. COORDINATE CHOICESAND PERTURBED MASS

The mass, spin, and deviation of the body were definedat lowest order in the near zone, so that perturbativecorrections to these quantities would naturally be definedat first and higher orders in the near zone. However, whilethe background metric is stationary and asymptotically flat(so that its multipole moments are well-defined), the nth-order perturbation may have growth in n combined powersof �t and �r [cf. paper I and Eqs. (8) and (9)]. It turns out,however, that at first order the situation is more undercontrol. As already noted, the choice of Fermi coordinatesin the far-zone background—together with the fact that �is geodesic—eliminates growing-in-�r terms from the firstnear-zone perturbation, so that the perturbation is asymp-totically flat. Furthermore, as I now show, the constancyof the mass and spin similarly allows one to eliminate all�t=�r and �t=�r2 terms, so that the perturbation is stationary to

Oð1=�r2Þ. To see this, recall that �gð0Þ�� �� is a stationary and

asymptotically flat metric, so that, introducing ��� ¼diagð1; 1; 1; 1Þ and t ¼ ð1; 0; 0; 0Þ, it may (at each t0) beput in the standard form [10],

�gð0Þ�� �� ¼ �� þ 2M

�r��� þM2

2�r2ð3�� � t�t�Þ

� 4nitð�S�Þi�r2

þO

�1

�r3

�; (16)

where Si0 ¼ 0 for all t0 (implying Di ¼ 0, so that thecoordinates are mass centered and ‘‘track’’ the motion ofthe body), while a priori M and Sij may depend on t0,

reflecting evolution of the mass and spin. However, sincetheM and Sij in (16) do correspond to those defined in (11)

and (13), we know that these quantities are in fact inde-pendent of time t0, as shown in paper I and remarkedabove. Thus the near-zone background metric is identical(through orderOð1=�r2Þ) at all t0, which implies by the formof (9) that the near-zone perturbation is independent of �tthrough Oð1=�r2Þ, becoming simply

�g ð1Þ�� �� ¼ D��ð ~nÞ þH��ð ~nÞ 1�r þO

�1

�r2

�þ �tO

�1

�r3

�: (17)

Further simplification can now be made. The perturbation

�gð1Þ�� �� must satisfy the (vacuum) linearized Einstein equation

about the near-zone background �gð0Þ�� ��. The quantity D��

appears at order Oð1=�r2Þ (and higher) in the linearizedEinstein tensor, while H�� appears first at Oð1=�r3Þ.However, at these orders only the first two terms in thebackground (16) will appear, so that the background iseffectively Schwarzschild. Therefore, when consideringonly the explicitly displayed terms in the perturbation

(17), we may use well-known results [11,12] for perturba-tions of the Schwarzschild spacetime. In particular, sincestationary perturbations scaling as �r0 and �r�1 [as in (17)]are known to be ‘ ¼ 0 (spherically symmetric) up togauge, the perturbation in (17) is simply a change inmass, and a gauge may be chosen so that

�g ð1Þ�� �� ¼ 2�Mðt0Þ

�r��� þO

�1

�r2

�þ �tO

�1

�r3

�; (18)

where �M is an arbitrary constant (which here may a prioridepend on t0), which I refer to as the perturbed mass ofthe body. An explicit formula may be given in analogywith (11),

�Mðt0Þ ¼ �1

8�lim�r!1

Zni@i �g

ð1Þ00 �r

2d�; (19)

with the caveat that this equation holds only in coordinateswhereD�� ¼ _C�� ¼ 0. If one wishes to compute the mass

in other coordinates (such as when the Lorenz gauge isused and D�� is given by a tail integral), a more compli-

cated expression (which must implicitly involve transform-ing to appropriate coordinates) must be derived. However,since the mass is an intrinsic property of a spacetime, thereis no need to consider such coordinates in defining �M anddetermining its evolution.5 Note that since the perturbation(17) is still stationary at Oð1= �r2Þ, it may be possible todefine perturbed spin and deviation by an analogous pro-cedure. However, these quantities appear at one order in �higher than pursued in this paper and are not consideredhere.I now show that the perturbed mass does not evolve with

time. With our previous coordinate choices, the second-order near-zone perturbation takes the form

�gð2Þ�� �� ¼ B�i�j �xi �xj þ �rE�� þ K�� þ

�t

�r2 _�M���

þOð1=�rÞ þ �tOð1=�r2Þ þ �t2Oð1=�r3Þ: (20)

This perturbation satisfies the (vacuum) linearized Einstein

equation about �gð0Þ�� �� [Eq. (16)] with effective sources con-

structed from �gð1Þ�� �� [Eq. (18)]. However, it is easy to see that

these effective sources are Oð1=�r4Þ, �tOð1=�r6Þ, and�t2Oð1=�r8Þ, while the error in the linearized Einstein tensorintroduced by including only the explicitly displayed termsin the second-order perturbation (10) isOð1=�r3Þ, �tOð1=�r4Þ,and �t2Oð1=�r5Þ. Thus the effective source terms may be

ignored, and denoting the linearized Einstein tensor of �gð2Þ�� ��

5A definition of perturbed mass was given in [13], whichappears to correspond to Eq. (19) applied in the Lorenz gauge.This definition would not be sensible within our framework. Themass defined in [13] was found to evolve with time. Theconclusion that the perturbed mass evolves with time appearsto be at odds with the analysis of [14], where it was found thatenergy conservation is satisfied under the assumption of nochange in mass.

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about �gð0Þ�� �� by G �� ��, we have G �� �� ¼ 0 to the appropriate

order, i.e.,

G �� �� ¼ Oð1=�r3Þ þ �tOð1=�r4Þ þ �t2Oð1=�r5Þ: (21)

To determine the mass evolution _�M we focus on the

‘ ¼ 0 part of G �� ��. While the background �gð0Þ�� �� is not

spherically symmetric due to the presence of the spinterm, it is easy to see that to the relevant order in G �� ��

the spin term only contributes in product with the ‘‘B’’term in the perturbation (20). However, since the far-zonebackground spacetime is assumed to be vacuum, itsRiemann tensor may be decomposed into two rank-twosymmetric, trace-free spatial tensors (the ‘‘electric’’ and‘‘magnetic’’ parts, e.g. [4]), showing that the B term is pure‘ ¼ 2. Since the spin term is ‘ ¼ 1, the combination cantherefore make no contribution to the ‘ ¼ 0 part of G �� �� at

the relevant order. Instead, the ‘ ¼ 0 part is completelydetermined by the remaining terms in the perturbation,which ‘‘see’’ only the Schwarzschild metric. In particularonly the ‘ ¼ 0 parts of these terms may contribute to the‘ ¼ 0 part of G �� ��, and we conclude that the ‘ ¼ 0 part of

�gð2Þ�� �� must be a perturbation of the Schwarzschild spacetime

to the relevant order. We may now use Zerilli’s result [11]that ‘ ¼ 0 perturbations of Schwarzschild simply shift themass, being writable as a 1=�r term plus a gauge trans-formation. Since Oð1=�rÞ is higher than considered, we

have that the ‘ ¼ 0 part of �gð2Þ�� �� is pure gauge to the relevant

order,

�rE�� þ K�� þ

�t

�r2 _�M���

�‘¼0

¼ rM

ð���Þ þOð1=�rÞ þ �tOð1=�r2Þ þ �t2Oð1=�r3Þ; (22)

where rM

is the derivative operator compatible withSchwarzschild and �� is a vector field. We may now takea time derivative to find

@0rM

ð���Þ ¼ rM

ð�@0��Þ

¼ 2 _�M

�r��� þOð1=�r2Þ þ �tOð1=�r3Þ; (23)

which immediately implies _�M ¼ 0, since the mass per-

turbation 2 _�M=�r��� is not pure gauge.

Incorporating this result, we may now summarize thecoordinate choices made in this section as

�gð0Þ�� �� ¼ �� þ 2M

�r��� þM2

2�r2ð3�� � t�t�Þ

� 4nitð�S�Þi�r2

þO

�1

�r3

�(24)

�g ð1Þ�� �� ¼ 2�M

�r��� þO

�1

�r2

�þ �tO

�1

�r3

�(25)

�gð2Þ�� �� ¼ B�i�j �xi �xj þ �rE�� þ K��

þO

�1

�r

�þ �tO

�1

�r2

�þ �t2O

�1

�r3

�; (26)

with S0i ¼ 0. By fixing the mass and spin terms to astandard form at all t0 and choosing the mass dipole tovanish for all t0, the metric form has been made verysimple, and all nonstationarity has been eliminated fromthe relevant orders in �r. These properties make this gaugemuch simpler to use than the Lorenz gauge used in [5] andelsewhere. Rewritten in the far-zone, the perturbation se-ries in these coordinates becomes

h�� ¼ gð1Þ�� ¼ 2M���

rþ E��rþOðr2Þ (27)

gð2Þ�� ¼ M2

2r2ð3�� � t�t�Þ �

4nitð�S�Þir2

þ 2�M

r��� þ K�� þOðrÞ; (28)

with the background gð0Þ�� still given by (5).

IV. PARITY CONDITION

The definition of center of mass adopted in Sec. II isbased on the dipole moment of the time-time component ofa stationary, asysmptotically flat metric. An alternativedefinition of center of mass is given by equation (5.13) ofRegge and Teitelboim [7], derived as the conserved quan-tity canonically conjugate to the asymptotic boost symme-try of asymptotically flat general relativity.6 Like theHamiltonian notion of mass (normally referred to as theADM mass), this ‘‘Hamiltonian center of mass’’ involvesonly the spatial metric and is more general in that it may beapplied to time-dependent spacetimes in addition to thestationary spacetimes we consider. However, unlike theHamiltonian notion of mass, the center of mass comeswith an additional restriction: In order to ensure the exis-tence of the integral defining the center of mass, Regge andTeitelboim impose a ‘‘parity condition’’ that the monopoleð1=rÞ part of the spatial metric be even parity, Cijð ~nÞ ¼Cijð� ~nÞ in Eq. (8).7 This restriction is not necessary to

define the mass dipole, which is finite for any metric of theform (8).Since the general metric form (16) satisfies the parity

condition, we see that the parity condition is simply

6We note that a later formula due to Beig and O’Murchadha[15] is equivalent given the parity condition.

7Regge and Teitelboim also impose a parity condition on theextrinsic curvature. However, this condition is not needed for thecenter of mass and plays no role in our analysis.

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a coordinate condition in the context of stationary, asymp-totically flat spacetimes. Rotations and translations willautomatically preserve the parity condition, while a super-translation �xi ¼ ið ~nÞ þOð1=�rÞ must now satisfy i ¼ci þ �ið ~nÞ, with ci constant and �i odd parity, �ið� ~nÞ ¼��ið ~nÞ. This form provides a natural split between the‘‘pure translation’’ part ci and a ‘‘pure supertranslation’’part �i, which is odd parity. It is easily checked fromEq. (15) that the mass dipole center of mass changes byci under this transformation, so that the parity conditionremoves its supertranslation dependence. In this case thetransformation properties of the mass dipole center of massagree with those established by Regge and Teitelboimfor their center of mass; and since both notions give zeroon the metric (16), the two definitions are equivalent in ourcontext. Therefore, the question of which definition to useis simply the question of whether to impose the paritycondition.

One may take one of two alternative viewpoints on thismatter. First, since the formal Hamiltonian analysis yieldsa center of mass formula that diverges (in general) incoordinates that violate the parity condition, one may arguethat such coordinates are ‘‘too irregular’’ to admit a notionof center of mass, even if the mass dipole formula is finite.Alternatively, one may view the mass dipole formula asproviding an extension of the Regge Teitelboim center ofmass to a larger class of coordinates within the stationarycase. In any case, the parity condition adds a number ofsimplifying properties in the context of the present work:It eliminates the supertranslation dependence of the massdipole, it allows the equations of motion to be expressedpurely in terms of the local spacetime metric (see discus-sion in Appendix B), and it makes the mode-sum regulari-zation scheme gauge invariant.

V. EQUATION OF MOTION

In the gauge of Sec. III, the equation of motion for thedeviation is simply ZiðtÞ ¼ 0. In principle, therefore,giving the change in Zi under a change in gauge providesthe complete description of motion. However, the moreuseful description of motion in other gauges, Eq. (1), maybe derived as follows. Begin with gauge transformationsx0� ¼ x� þ ��� of the form8

�� ¼ �ðt; ~nÞ þOðrÞ; (29)

where �� is assumed smooth in r at fixed ðt; ~nÞ, so that inparticular � is a smooth function of its arguments. It iseasy to check that such transformations preserve theform of Eqs. (5) and (6), and thus are allowed under ourassumptions. As described in [16], we furthermore believe

(but have not proven) that expressibility in the form ofEq. (29) is a necessary condition on an allowed trans-formation, except in the case of certain trivial one-parameter families, where additional logr terms areallowed. Thus we believe (but have not proven) that suchtransformations correspond precisely to the coordinatechoices allowed by our formalism (not including the paritycondition) at first order in �, for nontrivial families ofsolutions. In order to preserve the parity conditionCijðt;� ~nÞ ¼ Cijðt; ~nÞ, we must restrict the form of � so

that

�i ¼ ciðtÞ þ �iðt; ~nÞ þOðrÞ; (30)

with �i odd parity, �iðt;� ~nÞ ¼ ��iðt; ~nÞ. I will refer togauge transformations of the above form as parity-regulartransformations, and I will define parity-regular gauges asthose that are related to the gauge of Sec. III by a parity-regular transformation. We believe that parity-regulargauges are the general class allowed by our assumptionsplus the parity condition, except possibly in trivial cases.Thus we believe that one may check if a given gauge isparity-regular by checking that the metric perturbationtakes the form (6) with Cijðt;� ~nÞ ¼ Cijðt; ~nÞ. However,absent a complete proof of the assertions in [16], one mustinstead check that the gauge vector relating to a known-parity-regular gauge is of the form (30). At the end of thissection, I discuss the parity-regularity of some commongauge choices.Under a change of gauge of the form (30), the near-zone

background coordinates change by �x0i ¼ �xi þ ciðt0Þ þ�iðt0; ~nÞ þOð1=�rÞ. Using Eq. (15), we see that the devia-tion Ziðt0Þ changes by ciðt0Þ, as noted in the previoussection. We may express this in terms of the gauge vector�� by taking an angle average over a small constant-rsphere,9

�Zi ¼ h�iir!0 � 1

4�limr!0

Z�id�: (31)

Equation (31) gives the change in deviation due to a parity-regular transformation made on any perturbation of theassumed form (6).The key manipulation now follows. Consider the second

time derivative, � €Zi ¼ @0@0�Zi. We have

8In the Appendix of paper I an opposite sign convention, x0� ¼x� � ���, was used for the definition of the gauge vector.

9We could equivalently express �Zi as an average over a circleor over two antipodal points, since these averages all agree for a‘‘constant plus odd parity’’ function. The entire derivation of theequation of motion could then proceed unchanged, so that theself-force in fact may equivalently be written as the average ofthe bare force over a (constant geodesic distance) sphere, circle,or pair of points.

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� €Zi ¼ h@0@0�iir!0

¼ hr0r0�i þ R0j0kxk@j�i

þr0ri�0 �rir0�0 þ Ri00j�jir!0

¼���r0�h0i � 1

2ri�h00

��r!0

� R0i0jh�jir!0

¼ �hFiir!0 � R0i0j�Zj: (32)

In the second line we have rewritten in terms of covariantderivatives (R0k0lx

l@k�i is a Christoffel term), as well asadded zero in the form of the Ricci identity. However, since@j�i is even parity to leading order and the Riemann tensor

is smooth, the first Riemann term vanishes by virtue of theparity condition. Noting that the remaining derivatives of�i appear only in symmetrized form, in the third line wereexpress in terms of the change in the metric perturbation,�h�� ¼ �2rð���Þ, finding precisely the gravitational

force form of Eq. (2). We also separate off the remainingRiemann term, which takes a geodesic deviation form. Inthe last line we use Eqs. (2) and (31), where the finiteness10

of hFiir!0 allows us to pull the � out of the angle average.We may now ‘‘drop the �s’’ to obtain

€Z i ¼ hFiir!0 � R0j0iZj þAi; (33)

where Ai is the constant of integration—an unknowngauge-invariant acceleration. Thus by simple manipula-tions we have fixed the form of the equation of motionup to a gauge-invariant piece and may now work in anyconvenient (parity-regular) gauge to determine Ai.

Since the gauge of Sec. III has ZiðtÞ ¼ 0, it immediatelyeliminates two terms in (33), giving simply

A i ¼ �hFiir!0 (34)

in this gauge. The interpretation of the gauge is that thegravitational self-force is exactly opposite to the gauge-invariant force, so that the total force is zero, and there isno deviation from geodesic motion (ZiðtÞ ¼ 0). Since theangle average of a three-vector picks out an ‘ ¼ 1, electricparity part, we need consider only the ‘ ¼ 1, electric paritypart of the bare force Fi in order to compute Ai fromEq. (34). Since only the ‘‘E’’ term in the perturbation (27)contributes, we may focus on the ‘ ¼ 1, electric parity partof E��. To do so, we return to second-order near-zone

perturbation theory.As remarked above in the derivation of the constancy of

the perturbed mass, the relevant terms in the second-order

metric perturbation �gð2Þ�� �� satisfy the linearized Einstein

equation off of the �gð0Þ�� �� to the relevant order, Eq. (23).

As further remarked, the ‘ ¼ 1 spin term in the back-ground appears only in combination with the ‘ ¼ 2

‘‘B’’ term in the perturbation. While there is no contribu-tion to the ‘ ¼ 0 mode relevant for the perturbed mass,there can be a contribution to the ‘ ¼ 1 mode relevant forthe deviation, which significantly complicates the analysis(see Appendix A). However, if the spin is assumed to bezero from the outset, then, as in the mass evolution case,the metric perturbation ‘‘sees’’ only Schwarzschild to therelevant order, and we can make use of Zerilli’s [11]analysis of perturbations of Schwarzschild. In particular,Zerilli showed that ‘ ¼ 1, electric parity perturbations arepure gauge so that by a (second-order near-zone) gaugetransformation we may eliminate the ‘ ¼ 1, electric paritypart of E�� entirely; whence it immediately follows from

(34) that Ai ¼ 0. Thus the equation of motion in thespinless case may be derived with very little effort, involv-ing only the few lines of algebra of Eq. (32). If the spin isnot assumed zero, more algebra is required (though stillsignificantly less than needed when the Lorenz gauge isused). This case is treated in Appendix A, leading to

MAi ¼ 1

2SklRkl0i; (35)

showing that Ai is simply the acceleration due tothe Papapetrou spin force. We have thus justified the finalequation of motion, which appears in covariant form in (1).I now discuss the parity-regularity of some common

gauge choices. The Lorenz gauge is convenient for localseries expansions about the particle. From Eq. (27), thegauge of Sec. III already satisfies the Lorenz conditionat leading and subleading order, r�ðh�� � ð1=2Þhg��Þ ¼Oð1Þ. It is then easy to check that the gauge vector to a fullLorenz gauge must take the form �i ¼ CiðtÞ þOðrÞ, sothat the Lorenz gauge is parity-regular.The Regge-Wheeler gauge is convenient for perturba-

tions of Schwarzschild. Barack and Ori [6] show that ina few specific cases the gauge vector takes the form (29)(bounded but direction dependent), and their formulae alsoimply that in these cases one can choose the vector to satisfythe required parity property, Eq. (30). However, Hopper andEvans [17] have shown that in general the Regge-Wheelergauge metric perturbations contain a delta-function in theSchwarzschild radial coordinate at the position of the par-ticle, so that in general the Regge-Wheeler gauge is toosingular to define the motion by our procedure. However,using the explicit formulae for the coefficient of the deltafunction found in [17], it should be possible to simplyeliminate the delta function by a gauge transformationduring the process of reconstructing the metric perturba-tions. If the resulting gauge is parity-regular (a suggestionconsistent with Barack and Ori’s restricted results), then theresults of this paper would enable self-force computationsto be made within the Regge-Wheeler formalism, usingmode-sum regularization if desired.The radiation gauge is convenient for perturbations of

Kerr. In [18] it was checked that a radiation gauge may be

10The angle average of Fi is manifestly finite in the gauge ofSec. III, and, by reversing the calculations of (32), may be easilyseen to transform finitely (see also (40)).

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chosen so that the metric perturbations near the particletake the form ð1=rÞC��ðt; ~nÞ, with Cijðt; ~nÞ even parity on

the sphere, so that the singularity is properly 1=r and theparity condition is satisfied.11 However, it was not checkedthat C��ðt; ~nÞ is smooth (as assumed in this paper), and in

fact it can be seen from the analysis of [19] that C��ðt; ~nÞcontains a jump discontinuity along a great circle, which isinherited from a discontinuity in the metric perturbationlocated at the radial coordinate of the particle. Such adiscontinuity seems unlikely to threaten the validity ofthe results, since all of the relevant formulae remain de-fined. However, in order to be certain that the results of thispaper may be applied to the gauge of [18], a carefulanalysis of the gauge vector relating to some parity-regulargauge must be performed. Armed with the explicit form ofsuch a vector, it should be straightforward to check if thearguments of this paper still hold.

VI. MODE-SUM REGULARIZATION

The computation of gravitational self-forces on blackhole background spacetimes is an important problem forgravitational-wave astronomy of extreme mass-ratio inspi-rals (e.g., [20]). The angle-average formula suggests astraightforward way of proceeding: first, numerically com-pute the metric perturbations of a point particle in anyparity-regular gauge and then perform an average to de-termine the force in that gauge. However, while simple inprinciple, such a procedure may be difficult to carry out inpractice, due to the singular nature of the quantity beingaveraged. Instead, to achieve an accurate result it is likelypreferable to use an alternative technique, such as that ofmode-sum regularization, first introduced in [8] and widelyemployed thereafter. This method takes advantage of thefact that numerical calculations in black hole spacetimesusually employ a spherical (or spheroidal) harmonic de-composition, which, in particular, has the property that theindividual modes of the bare force are finite at the particle.One then performs a finite subtraction on each mode,which is designed so that the resulting sum converges tothe correct self-force. Extensive work has determined theform of this subtraction (in terms of ‘‘regularization pa-rameters’’) for arbitrary orbits of Schwarzschild and Kerrin the Lorenz gauge. In this section, I show that the mode-sum regularization scheme is gauge invariant under theparity condition, in the sense that the same subtractionmay be employed to determine the force in any parity-regular gauge. Combined with the Lorenz-gauge results of[21], this provides a complete regularization prescriptionfor Kerr in parity-regular gauges.

Let ð~t; ~r; �; �Þ be Boyer-Lindquist coordinates for theKerr spacetime. For a given point along the worldline �

where we wish to compute the self-force, we may rotateand time translate the coordinates so that the particle is

located at ~t ¼ ~� ¼ 0, while taking the remaining coordi-

nate positions to be ~r ¼ r0 and � ¼ �0. Despite the lack ofa full rotational symmetry we (following [8]) nevertheless

also perform an additional rotation in the � direction, sothat the particle is located at the pole of the new coordi-nates. More precisely, define new angular coordinates

ð~�; ~�Þ bycos� ¼ � sin~� cos ~� cos�0 þ cos~� cos�0 (36)

tan� ¼ sin~� sin ~�

sin~� cos ~� cos�0 þ cos~� sin�0(37)

to obtain ‘‘rotated Boyer-Lindquist’’ coordinates ~t; ~r; ~�; ~�in which the particle position is given by ~t ¼ 0, ~r ¼ r0,~� ¼ 0. In these coordinates the metric components aresmooth everywhere except for the pole � ¼ 0, wherethey acquire nontrivial direction-dependent limits. Belowwe will need the lowest-order relationship between thespatial Fermi coordinates xi and the rotated Boyer-Lindquist coordinates, restricted to the sphere. A straight-forward computation gives this to be

xij~t¼0;~r¼r0 ¼ i ~� cos ~�þ i ~� sin ~�þOð~�2Þ; (38)

where i and i are constants independent of ~� and ~�(dependent on the Boyer-Lindquist position �0, the three-velocity, and the mass and spin parameters of the Kerr

metric), and where the Oð~�2Þ term may depend on ~�.The advantage of placing the particle at the pole is the

simplification of the spherical harmonic description by theelimination of modes with nonzero m when the series isevaluated at the particle. Let a subscript ‘ denote the ‘th

term in the expansion evaluated at the pole/particle, A‘ ¼RAY‘0d ~� for integrable functions Að~�; ~�Þ, and suitably

generalized for distributions. When viewed in light of ourangle-average result, the mode-sum regularization pre-scription amounts to finding some Si‘ such that

hFiir!0 ¼X1‘¼0

ðFi‘ � Si‘Þ: (39)

This formula relates an average over an infinitesimalsphere surrounding the particle to a spherical harmonicdecomposition on a finite sphere surrounding the blackhole, evaluated at the particle (see Fig. 1). Such a connec-tion between mode sums and local averaging is familiarfrom ordinary Fourier series, where, if a function is ofbounded variation, its series converges to the two-sidedaverage at a discontinuity. For spherical harmonic expan-sions, an analogous result (e.g. Sec. III22b of [22]) statesthat if the average of a function over latitude lines is ofbounded variation (as a function of latitude), then itsspherical harmonic series evaluated at the pole (‘‘Laplace

11At the time of the writing of [18], I believed (and communi-cated to the authors of [18]) that only this check was required forthe results of the present paper to hold.

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series’’) converges to the average on an infinitesimal lati-tude line surrounding the pole. (This result is easily under-stood at a formal level, by noting that Y‘0 is independent of~�, so that the formula for A‘ takes an average over ~�.) Thetheorem does not apply to the bare force Fi (which isdivergent), but it does apply to the change in bare forceunder a change in gauge, �Fi. In particular, a simplecalculation [which reverses the calculations internal tothe average in (32)] gives

�Fi ¼ @0@0�i þ R0

i0j�

j � R0k0lx

l@k�i þOðrÞ (40)

for any transformation of the form (29). If the transforma-tion is parity-regular, Eq. (30), we see that �Fi has a Fermicoordinate expansion of the form

�Fi ¼ CiðtÞ þ Siðt; ~nÞ þOðrÞ; (41)

where Si is smooth and odd parity,Siðt;� ~nÞ ¼ �Siðt; ~nÞ.12 Restricting to the background co-ordinate sphere ~t ¼ 0, ~r ¼ r0 and expanding in ~� at fixed~�, we have

�Fij~t¼0;~r¼~r0ð~�; ~�Þ¼Cið0ÞþSið0; ~nj~t¼0;~r¼~r0ð~�; ~�ÞÞþOð~�Þ(42)

¼ Cið0Þ þ Sið0; ~nj~t¼0;~r¼~r0ð~� ¼ 0; ~�ÞÞþOð~�Þ (43)

¼ Cið0Þ þ Sið0; ~n0ð ~�ÞÞ þOð~�Þ; (44)

where the Oð~�Þ terms may depend on ~�, and we havedefined

~n 0ð ~�Þ ¼ lim~�!0

~nð~t ¼ 0; ~r ¼ ~r0; ~�; ~�Þ: (45)

In moving from Eq. (42) to Eq. (43) we have used the factthat the restriction of ~n to the background coordinate

sphere is smooth in ~� at fixed ~� [see Eq. (38) and recallni ¼ xi=r], as well as the fact that Si is smooth in ~n.Equation (44) shows that the restriction of �Fi to thebackground coordinate sphere is continuous (in fact

smooth, by our assumptions) in ~� at fixed ~�. In particular

its average over ~� is of bounded variation, and by thetheorem we haveX

‘

ð�FiÞ‘ ¼ lim~�!0

1

2�

Z�Fij~t¼0;~r¼r0ð~�; ~�Þd ~� (46)

¼ Cið0Þ þ 1

2�

ZSið0; ~n0ð ~�ÞÞd ~�; (47)

where in the second line we have plugged in the form ofEq. (44). Since Si is odd parity, the second term on the

right-hand side will vanish if ~n0 is odd under ~� ! ~�þ �.This property is expected from the geometry of the setup(Fig. 1) and is easily confirmed from Eqs. (45) and (38).Thus the term involving Si vanishes, so that the Laplaceseries for �Fi in fact converges to Ci. However, the angle-average that computes the self-force also returns Ci on theform (41). Therefore, when the parity condition is satisfiedthe averages agree, and we have simplyX

‘

ð�FiÞ‘ ¼ h�Fiir!0; (48)

showing that the Laplace series for the change in bare force�Fi in fact converges to its local inertial angle-average,i.e., to the change in self-force it effects. This means inparticular that no extra Si‘ must be subtracted in the new

gauge, since merely the process of decomposing �Fi intomodes and resumming returns its contribution to the self-force. To see this explicitly, let Fi

old denote the bare force in

a gauge that satisfies the parity condition and writeX‘

ðFi‘ � Si‘Þ ¼

X‘

½ðFioldÞ‘ þ ð�FiÞ‘ � Si‘�

¼ X‘

½ðFioldÞ‘ � Si‘� þ h�Fiir!0

¼ hFiold þ �Fiir!0: (49)

FIG. 1. A diagram illustrating the geometrical setup of themode-sum regularization argument. The particle is at the poleof rotated Boyer-Lindquist coordinates. The mode decomposi-tion is taken relative to the background coordinate sphere, whilethe self-force is given by an average over the local inertialsphere. For the change in bare force, a general theorem relatesthe mode sum at the particle/pole to the average over the polarcircle, which agrees with the average over the local inertialsphere when the parity condition is satisfied.

12The parity condition is not required to show the applicabilityof the theorem but will be necessary for the later analysis of thissection.

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In writing the second line we have used (48), and in writingthe third line we have used Eq. (39). This shows thatEq. (39) holds in the new gauge if it held in the old, i.e.,that Si‘ is a correct piece to subtract in any parity-regular

gauge.We note that previous work has organized the subtrac-

tion so that one first subtracts an Si‘ of the form Si‘ ¼Aið‘þ 1=2Þ þ Bi þ Ci=ð‘þ 1=2Þ, then sums over modes(the result is now finite), and finally adds in a finite residualDi to get the correct self-force. (In terms of our Si‘, D

i is a

‘‘finite piece’’ Di � P‘ðSi‘ � Si‘Þ.13) The (‘-independent)

A, B, C, D are the ‘‘regularization parameters’’ for theparticular orbit and spacetime, which by the results of thissection do not depend on the choice of parity-regulargauge.

ACKNOWLEDGMENTS

I gratefully acknowledge Robert Wald and JohnFriedman for helpful conversations, Laszlo Szabados forhelpful correspondence, and an anonymous referee for ahelpful comment. This research was supported in part byNSF Grant No. PHY08-54807 to the University ofChicago.

APPENDIX A: DERIVATION IN THE CASEOF NON-ZERO SPIN

When the spin is nonzero, we may not rely on Zerilli’sresults for Schwarzschild [11], and the analysis of second-order Einstein equation, Eq. (23), becomes more compli-cated. In this case it pays to systematically consider thecontributions to the linearized Einstein tensor G �� �� from

the various terms in the background (24) and perturbation(26). Since all terms are stationary to the relevant orders,no �t dependence will appear, and we may count orders in1=�r. At leading order Oð1Þ in G �� ��, the only contributions

are from the B term in the perturbation (26) and the flat‘‘’’ term in the background. Since the B term is (by theFermi coordinate construction) a perturbation of flat space-time, the linearized Einstein equation is automaticallysatisfied and we learn no new information. At next orderOð1=�rÞ in G �� ��, both the E term in the perturbation and the

mass term 2M���= �r in the background can now contrib-

ute. Expanding the background metric in powers of 1= �r,the linearized Einstein equation at Oð1=�rÞ may be written

Gð1Þ ½E�� �r� þ 2Gð2Þ

½2M= �r���; B�i�j �xi �xj� ¼ 0; (A1)

where Gð1Þ and Gð2Þ

are the first and second-order Einsteintensors (respectively) off of flat spacetime. However, sincethe B term has no ‘ ¼ 1, electric parity part while the Mterm is spherically symmetric, the second term in (A1) hasno ‘ ¼ 1, electric parity part. Therefore the ‘ ¼ 1, electricparity part of the E term satisfies the vacuum linearizedEinstein equation about flat spacetime,

Gð1Þ ½ðE�� �rÞ‘¼1;þ� ¼ 0; (A2)

where the subscript ‘‘‘ ¼ 1, þ’’ indicates the ‘ ¼ 1, elec-tric parity part. Since ‘ ¼ 1, electric parity perturbations offlat spacetime that scale linearly with �r are pure gauge(e.g., [23]), all solutions to Eq. (A2) may be writtenðE�� �rÞ‘¼1;þ ¼ @ð�E�Þ, where E� is ‘ ¼ 1 and electric

parity. If we make a (second-order near-zone) gauge trans-formation generated by E, then we may set the ‘ ¼ 1,electric parity part of E�� to zero. However, one can check

that in general the required gauge vector E� is �t dependent

(despite the E term being stationary), so that this gaugetransformation would introduce �t dependence at higherorders, in violation of our previous choices that eliminatedsuch dependence. [In particular, terms of order �t=�r and

�t2= �r2 would appear in �gð2Þ�� ��, contradicting the previous

choices _H�� ¼ €C�� ¼ 0—see Eqs. (10) and (26).] With-

out invalidating these choices we may only make gaugetransformations whose gauge vector is �t independent. Onemay check that this allows us to put E�� in the form

ðE�� �rÞ‘¼1;þ ¼ �2ai �xit�t�; (A3)

where ai is an arbitrary spatial vector, named since in thisform ðE�� �rÞ‘¼1;þ is an ‘‘acceleration perturbation’’ famil-

iar from Fermi coordinates about an accelerated worldline.Making such a gauge transformation (and ‘‘absorbing’’ itseffects at Oð1Þ into the arbitrary tensor K��), Eq. (34)

becomes simply

A i ¼ ai: (A4)

We have now made coordinate choices that eliminate all�t dependence and reduce the relevant ‘ ¼ 1, electric paritypart of E�� into a simple form with one unknown, the

gauge-invariant acceleration Ai ¼ ai. It remains to usethe linearized Einstein equation at order Oð1=�r2Þ to deter-mine Ai. Again expanding the background in powers of1=�r, at this order the ‘ ¼ 1, electric parity parts of thesecond-order Einstein equation may be written as

�Gð1Þ

½K��� þ 2Gð2Þ

�2M���

�r;�2Ai �x

it�t�

�

þ 2Gð2Þ

��4nitð�S�Þi�r2

; B�i�j �xi �xj

��‘¼1;þ

¼ 0; (A5)

13Remarkably, it has been found (by lengthy computation in theLorenz gauge) that Di ¼ 0 in every circumstance, so that thesubtraction of Aið‘þ 1=2Þ þ Bi þ Ci=ð‘þ 1=2Þ in fact returnsthe correct force. This surprising relationship between thelarge-‘ expansion of a point particle metric perturbation (whichuniquely determines A, B, C) and the physical self-force has thusfar defied a more fundamental explanation.

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where terms that can give no ‘ ¼ 1, electric part have notbeen displayed. This equation gives relationships betweenK��, Ai, M, Sij, and R�i�j. To determine a relationship

not involving the unknown tensor K��, first write out the

linearized Einstein tensor about flat spacetime for station-ary perturbations, and note that there is a particular ‘ ¼ 1part14 that vanishes for all K��. Then computing this

particular ‘ ¼ 1 part of the remaining two terms willdetermine Ai in terms of the mass, spin, and Riemanntensor. Performing this calculation yields MAi ¼12 S

klRkl0i, as claimed in Eq. (35).

APPENDIX B: EQUATION OF MOTIONIN PARITY-IRREGULAR GAUGES

I now consider the form of the equations of motion inparity-irregular gauges, adopting the mass dipole definitionof center of mass. Under the general gauge transformations(29), we have from Eq. (15) that15

�Zi ¼ 3

4�limr!0

Z�jnjn

i ¼ 3h�jnjniir!0: (B1)

Beginning with the equation of motion in a parity-regulargauge, we may now derive an equation for Zi in a newgauge,

€Z i � hFiir!0 þ R0j0iZj �M�1SklRkl0i

¼ hð�@0@0�jð�ij � 3ninjÞ þ R0j0kx

k@j�i

� R0i0j�kð�k

j � 3nknjÞÞir!0; (B2)

where Eq. (40) has been used. If the parity condition issatisfied, and �i ¼ ci þ �ið ~nÞ þOðrÞ with �i odd parity,then the right-hand side vanishes and the equation ofmotion retains the original form, depending only on thelocal spacetime metric (at zeroth and first order in pertur-bation theory). However, if the parity condition is notsatisfied, then the right-hand side does not in generalvanish,16 and the equation of motion for Zi takes a com-plicated form involving the gauge transformation to somereference gauge. Another way to see the difficulty is torepeat the calculations of (32) for a general gauge trans-formation, giving

� €Zi ¼ 3hð�Fj þ R0k0lxl@k�j � R0j0k�

kÞnjniir!0: (B3)

Without the parity condition the Riemann terms do notsimplify into the geodesic deviation form R0i0j�Z

j. It

appears that no expression in terms of just �Zi and �h��

is possible so that the equation for Zi in parity-irregulargauges must involve a gauge vector explicitly. In particu-lar, there appears to be no natural separation of the termscontributing to €Zi into ‘‘self’’ and other forces in the caseof a parity-irregular gauge.

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14In the notation of [5], this part is ð2=3ÞRDi þ RE

i , involvingonly spatial trace-free components of the linearized Ricci tensor.15As already noted in Sec. V, the sign convention for �� used inthis paper differs from that used in the Appendix of paper I.

16For example, with �i ¼ Bjnjni with Bi constant, the right-

hand side of (B2) evaluates to ð16=15ÞR0i0jBj.

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