physical review d 084021 (2009) self-force with (3þ1) codes:...

Self-force with (3þ 1) codes: A primer for numerical relativistsIan Vega,1 Peter Diener,2,3 Wolfgang Tichy,4 and Steven Detweiler1

1Institute for Fundamental Theory, Department of Physics, University of Florida, Gainesville, Florida 32611-8440, USA*2Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA†3Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA‡

4Department of Physics, Florida Atlantic University, Boca Raton, Florida 33431, USA(Received 29 August 2009; published 15 October 2009)

Prescriptions for numerical self-force calculations have traditionally been designed for frequency-

domain or (1þ 1) time-domain codes which employ a mode decomposition to facilitate in carrying out adelicate regularization scheme. This has prevented self-force analyses from benefiting from the powerful

suite of tools developed and used by numerical relativists for simulations of the evolution of comparable-

mass black hole binaries. In this work, we revisit a previously-introduced (3þ 1) method for self-forcecalculations and demonstrate its viability by applying it to the test case of a scalar charge moving in a

circular orbit around a Schwarzschild black hole. Two (3þ 1) codes originally developed for numericalrelativity applications were independently employed, and in each we were able to compute the two

independent components of the self-force and the energy flux correctly to within

instant affects the motion of the particle at that sameinstant. In curved spacetime, due to scattering with thecurvature, the motion of the particle is affected by fieldsit gave rise to in its causal past. This makes the resultingphysics much richer than of the flat spacetime case. Thesame scenario also applies to scalar and electric chargedparticles moving in curved spacetime. While often inter-esting in their own right, these also serve as useful andtechnically less-demanding toy problems for testing newmethods and techniques.

The effect of the self-force is a small acceleration on thepoint mass resulting in a secular deviation away from whatwould otherwise have been geodesic motion in the back-ground spacetime. This self-force will need to be calcu-lated and used to modify the motion of the point mass asoften as is practical throughout the course of the inspiral.Formal expressions for this self-force given in terms of anintegral over the particle’s entire past history have beenironed out in the literature [7–9], but these are hardlyconvenient for practical calculations. The challenge isthen to come up with an efficient way to compute self-forces based on these formal expressions. Several usefulprescriptions have been developed to address this issue[10,11], but most influential of these is the mode-sumprescription [12–14], which we shall describe below.

An alternative viewpoint of the self-force scenario is thatinstead of a backreacting ‘‘force’’ accelerating the pointmass in the background spacetime, the motion of the pointmass is really geodesic motion on the distorted backgroundgeometry [6,15]. In this framework, the task is to determinethe appropriate distorted geometry upon which to imposegeodesic motion (or equivalently, the correct smooth po-tentials governing the motion of scalar and electriccharges). The procedure thus involves first computing themetric perturbation hab induced by the point mass on thebackground spacetime metric gab (which would be diver-gent at the location of the point mass), and then appropri-ately regularizing this metric perturbation to give thecorrect smooth perturbation hRab. The motion of the pointmass will then be geodesic motion in the perturbed space-time gab þ hRab.

This perspective has been useful on both theoretical andpractical fronts. Most notably, it has reconciled the notionof a self-force with our understanding of the equivalenceprinciple [15]. It has also served as the basis of convenientvariants of the original mode-sum prescription [16–24],and it has thus advanced our understanding of this impor-tant calculational technique. The method presented in thismanuscript is another offshoot of this alternativeviewpoint.

Much progress has been made in the calculation of theself-force on a charge that moves momentarily along someprescribed geodesic of the background spacetime. In par-ticular, the mode-sum paradigm has contributed tremen-dously to our understanding of the elements of a self-force

calculation and continues to serve as the conceptual back-drop upon which all other calculational schemes are to beunderstood. It has now been employed to evaluate the self-force or self-force effects in a variety of contexts—scalar[16–22,25], electromagnetic [26], and gravitational[24,27,28]—for point sources moving along geodesics ina Schwarzschild background. Results for the Kerr space-time are rare. The first calculation of a self-force on a scalarcharge moving through this background has been achievedonly recently [29]. Despite this good progress, however,little has been done to achieve a dynamic calculation of theself-force that is then used to implement backreaction onmoving point sources.One of the reasons for this is that computing a self-force

is a complicated process. A typical mode-sum calculationfirst requires a decomposition of the problem (i.e. fieldsand sources) into modes with, say, spherical or spheroidalharmonics. This is done in order to avoid having to handlethe divergence in the physical retarded field numerically.Each mode component of the retarded field turns out to befinite at the location of the charge, and thus, numericallyaccessible. It is the sum of these modes that diverges.Buried in each mode is the piece that actually contributesto the self-force. The central insight of the mode-sumprescription is a way to access this relevant piece, basedon an asymptotic analysis of the divergence in the retardedfield. In calculating the self-force then, each computedmode is appropriately regularized (using an analytic ex-pression determined by the asymptotic analysis) to leavethe piece that contributes to the self-force. The regularizedpieces are then summed to get the full self-force.Convergence of this sum is typically slow, going as�1=ln, where l is the maximum mode number, and n isdetermined by the degree to which one analytically char-acterizes the asymptotic behavior of the divergent physicalfields.In [23] (henceforth referred to as Paper I), we introduced

a method for calculating the self-force designed principallyto obviate the apparent necessity of a decomposition intomodes in order to perform the regularization of the retardedfield. Through this alternative method, we proposed a (3þ1) approach to self-force calculations, which fits squarelywith the expertise and infrastructure found in the numericalrelativity community. Our technique is reviewed in betterdetail below. Its core idea, however, is straightforward:rather than regularizing the retarded field, one can insteadappropriately regularize the source term in the field equa-tion from the start, and thus have the evolution codes dealwith sufficiently-differentiable fields and sources that re-quire no further regularization. In other words, we dealwith regularization not as a post-processing step, but aspreliminary work that needs to be performed before anynumerical run. This work consists of appropriately replac-ing the �-function representation of a point mass sourcewith a regularized effective source. Designing this effec-

VEGA et al. PHYSICAL REVIEW D 80, 084021 (2009)

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tive source for the wave equation can be done in such a waythat a numerical evolution yields a differentiable fieldwhose gradient at the location of the particle automaticallygives the full self-force. In addition, this resulting regularfield is such that it becomes the physical retarded solutionin the wavezone, from which fluxes and the all-importantwaveforms can be extracted.

Two important features of this approach stand out. First,the evolution code never has to deal with divergent quan-tities (though the fields and sources are of finite differ-entiability); second, both self-force calculation andwaveform extraction are trivial, with accuracies limitedchiefly by the accuracy that can be provided by the evolu-tion code. It is these two features that call out to thenumerical relativity community at large, for in effect allthat is required for a self-force calculation is a (3þ 1) codethat can accurately evolve wave equations with sources oflimited differentiability. The ease with which one calcu-lates a self-force in this approach suggests no significantimpediment to implementing backreaction on the particle.

The idea of finding a good substitute for the �-functionsource in the context of self-force calculations is also beingpursued by others in similar ways [30–32]. Barack andGolbourn [30] introduced a technique consisting of anm-mode (azimuthal mode) regularization of the�-function source. Their regularization is also guided byanalysis of the singular behavior of the retarded field at thelocation of the particle. First, one solves (2þ 1) waveequations with regularized sources, then extracts the con-tribution to the self-force due to each azimuthal mode, andfinally sums these to get the full self-force. Their approachis similar to ours in that it provides a way of representingpoint sources on a grid, but in keeping with the generalstrategy of the original mode-sum procedure (which was anl-mode sum), it is likely to inherit some of the propertiesour approach seeks to overcome.

A new approach to evaluating the retarded field byCañizares and Sopuerta [32] splits the problem into innerand outer domains marked by the location of the pointcharge. They situate the point source along the boundaryshared by both domains, and then just impose appropriatejump conditions on the fields that cross the boundary. Withthe benefit of having to deal with only smooth fields, thismethod takes advantage of the exponential convergence ofa pseudospectral implementation for evaluating the re-tarded field. In calculating a self-force, however, they arestill restricted to performing a mode sum of what remainsafter regularizing the output of their evolution code. Achief advantage of our method is precisely the fact that itescapes the requirement of a mode decomposition.

In Paper I, we reported an implementation of our methodusing a time-domain (1þ 1) code to compute the self-force and retarded field in the wavezone. By first breakinginto modes, this implementation ran counter to the verymotivations underlying our technique. This was done,

however, mainly to provide a quick proof of principleand to establish a more direct connection with more famil-iar approaches to self-force calculation.In this work, we report for the first time on the feasibility

of our technique in its intended setting. As a result, we haveachieved the first calculation of a self-force in a (3þ 1)framework. Two different codes [33,34] were employed inthe implementation of our method, both originally in-tended for numerical relativity applications. As inPaper I, we focus on the simplest possible strong-fieldscenario involving a scalar point charge interacting withits own scalar field while in a circular orbit around aSchwarzschild black hole. In choosing to deal with thissimple case, we also intend for this document to be a self-force primer and an invitation to numerical relativists whoare on the lookout for new challenges. The insights gainedfrom self-force analyses should prove useful to thosewishing to tackle the extreme-mass-ratio regime of blackhole binaries.

Outline and notation

Section II describes our formalism for replacing a point-particle �-function source in a standard field equation witha particular abstract effective source. The effect is that notonly does the resulting field equal the usual retarded fieldin the wavezone, but also the field is finite and differen-tiable at the particle. This allows it to be used directly incalculating the self-force acting on the particle. Some de-tails regarding the effective source are given in Sec. III.We describe a practical test application for our approach

to self-force computations in Sec. IV. This test applicationhas been previously well studied by the self-force com-munity by using more traditional self-force techniqueswhich are not particularly adaptable to numerical relativity[16–19].Section V contains the implementation details of mod-

ifications, with an eye on applications to self-force prob-lems, of two different previously developed numericalrelativity projects.Details of the results following from the applications of

these two codes to our test are given in Sec. VI. Wecompare the time and the radial components of the self-force, calculated by our two independent codes, with eachother and with very accurate (but tediously obtained) well-known frequency-domain results [16–19]. The time com-ponent of the self-force removes energy from the particle,and we also check its consistency with the energy flux viaradiation down the black hole and out at infinity.Section VII gives a summary of the apparent strengths

and weaknesses of our effective source method for regu-larizing self-force problems. In very general terms wedescribe how currently available computer codes mightbe adapted specifically to self-force problems.We have three appendices. Appendix A gives a (1þ 1)

example of applying traditional finite differencing opera-

SELF-FORCE WITH (3þ 1) CODES: A PRIMER FOR . . . PHYSICAL REVIEW D 80, 084021 (2009)

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tors to a wave equation where the source is of limiteddifferentiability. This elucidates the discussion of conver-gence. In Appendix B we derive the relationship betweenthe time component of the self-force and the radiativeenergy flux into the black hole and out at infinity, in thecase of a circular orbit and a scalar field. And we describe avery elementary, illustrative flat-space toy problem inAppendix C, which demonstrates how a problem involvinga �-function point source can be transformed into one witha smooth source in a mathematically precise way.

For our tensor notation, we denote regular four-dimensional spacetime indices with letters taken from thefirst third of the alphabet a; b; . . . ; h, indices which arepurely spatial in character are taken from the middle third,i; j; . . . ; q, and indices from the last third r; s; . . . ; z and also� and � are associated with particular coordinate compo-nents. The operator ra is the covariant derivative operatorcompatible with the metric at hand. Partial derivatives withrespect to t are denoted @t and with respect to a genericspatial coordinate by @i.

For the Schwarzschild metric, we use a coordinate sys-tem introduced by Eddington [35] and commonly knownas Kerr-Schild coordinates to describe a Schwarzschildblack hole.

Our use of the (3þ 1) formalism follows York [36] in allaspects except his labels for tensor indices.

II. FIELD REGULARIZATION FOR A SCALARCHARGE

In this section, we review the discussion of our methodfound in Paper I. We shall discuss it for the case of a scalarpoint charge moving in curved spacetime. A typical self-force computation first involves solving the minimally-coupled scalar wave equation with a point charge q source,

rarac ret ¼ �4�qZ�

�ð4Þðx� zð�ÞÞffiffiffiffiffiffiffi�gp d�; (1)for the retarded field c ret. Here, ra is the derivativeoperator associated with the metric gab of the backgroundspacetime, and � is the world line of the charge defined byzað�Þ and parameterized by the proper time �. The physicalsolution of the resulting wave equation will be a retardedfield that is singular at the location of the point charge. Aformal expression for the self-force given by

Fað�Þ ¼ qðgab þ uaubÞrbc retðzð�ÞÞ (2)would thus be undefined without a proper regularizationprescription. Early analyses [7,8,37] were based upon aHadamard expansion of the Green function and showedthat for a particle moving along a geodesic, the self-forcecould be described in terms of the particle interacting onlywith the ‘‘tail’’ part of c ret, which is finite at the particleitself. The mode-sum prescription is effectively a way ofregularizing the right-hand side of Eq. (2) to retrieve the

force due to this tail part. Later [15], it was realized that asingular part of the field c S, which exerts no force on theparticle itself, could be identified as an actual solution toEq. (1) in a neighborhood of the particle. A formal de-scription of c S in terms of parts of the retarded Green’sfunction is possible, but generally there is no exact func-tional description for c S in a neighborhood of the particle.Fortunately, as is shown in [16], an intuitively satisfyingdescription for c S results from a careful expansion aboutthe location of the particle:

c S ¼ q=�þOð�3=R4Þ as � ! 0; (3)where R is a constant length scale of the backgroundgeometry, and � is a scalar field which simply satisfies�2 ¼ x2 þ y2 þ z2 in a very special Minkowski-like lo-cally inertial coordinate system centered on the particle,first described by Thorne, Hartle, and Zhang [38,39] andapplied to self-force problems in Refs. [6,16,40]. A de-tailed discussion of these coordinates can be found in[6,16] and Appendix A of Paper I. Not surprisingly, thesingular part of the field, which exerts no force on theparticle itself, appears as approximately the Coulomb po-tential to a local observer moving with the particle.Our proposal for solving Eq. (1) and determining the

self-force acting back on the particle now appears elemen-tary. First, we define

~c S � q=� (4)as a specific approximation to c S. By construction, weknow that ~c S is singular at the particle, and is C1 else-where. Also, within a neighborhood of the world line of theparticle

rara ~c S ¼ �4�qZ�

�ð4Þðx� zð�ÞÞffiffiffiffiffiffiffi�gp d�þOð�=R4Þ;as � ! 0: (5)

It must be pointed out that for Eqs. (3) and (5) to bevalid, the Thorne-Hartle-Zhang (THZ) coordinates must beknown correctly to Oð�4=R3Þ. Knowing the THZ coordi-nates only to Oð�3=R2Þ would spoil the remainder inEq. (5), which would then have a direction dependentdiscontinuity in the limit as � ! 0. The local coordinateframe must be known precisely enough in terms of theglobal coordinates for the Coulomb-like potential to be agood representation of the local singular field.Next, we introduce a window function W which is a C1

scalar field with

W ¼ 1þOð�4=R4Þ as � ! 0; (6)and W ! 0 sufficiently far from the particle, in particular,in the wavezone and at the black hole horizon. The require-ment thatW approaches 1 this way, i.e.Oð�4Þ, is explainedbelow.


084021-4

Finally, we define a regular remainder field

c R � c ret �W ~c S; (7)which is a solution of

rarac R ¼ �raraðW ~c SÞ � 4�qZ�

�ð4Þðx� zð�ÞÞffiffiffiffiffiffiffi�gp d�(8)

from Eq. (1). Because of our use of ~c S as an approxima-tion of the full singular field, c R will be contaminated withOð�3=R4Þ pieces that are only C2 at the location of thecharge but do not affect the self-force.

The effective source of this equation

Seff � �raraðW ~c SÞ � 4�qZ�

�ð4Þðx� zð�ÞÞffiffiffiffiffiffiffi�gp d� (9)is straightforward to evaluate analytically, and the twoterms on the right-hand side have �-function pieces thatprecisely cancel at the location of the charge, leaving asource which behaves as

Seff ¼ Oð�=R4Þ as � ! 0: (10)Thus the effective source Seff is continuous but not neces-sarily differentiable, C0, at the particle while being C1elsewhere.1

A solution c R of

rarac R ¼ Seff (11)is necessarily C2 at the particle. Its derivative

rac R ¼ raðc ret �W ~c SÞ � ~c SraW¼ raðc ret � c SÞ þOð�2=R4Þ� ! 0 (12)

provides the approximate self-force acting on the particlewhen evaluated at the location of the charge. It should beclear why the behavior of W is chosen as in Eq. (6): Awindow function with this behavior would not spoil theOð�2=R4Þ error already incurred by using the q=� ap-proximation for the singular field. Also, in the wavezone,W effectively vanishes and c R is then identically c ret andprovides both the waveform as well as any desired fluxmeasured at a large distance.

General covariance dictates that the behavior of Seff inEq. (9) may be analyzed in any coordinate system. But,only in the specific coordinates of Refs. [38,39], or theTHZ coordinates, is it so easily shown [16] that the simpleexpression for c S in Eq. (3) leads to the Oð�=R4Þ behav-ior in Eq. (10) and then to the C2 nature of the solution c R

of Eq. (11).Self-consistent dynamics of a scalar charge requires that

the self-force act instantaneously. Thus a simultaneous

solution of the coupled equations

rarac R ¼ Seffðxð�Þ; uð�ÞÞ; (13)

mdub

d�¼ qðgbc þ ubucÞrcc R (14)

evolves c R while self-consistently moving the charge viathe self-force of Eq. (14).Our method effectively regularizes the field itself rather

than the gradient of the field, and this regularization isimplicit in the construction of the effective source, asopposed to most existing self-force calculations in whichthe divergent pieces of the individual modes are explicitlysubtracted out. Once c R is determined, our method has noneed for any further regularization. The derivatives of c R

determine the self-force, providing instantaneous access;while c R is identical to c ret in the wavezone, allowingdirect access to fluxes and waveforms.The tedious aspects of our method reside primarily in the

construction of the effective source. This is mainly due tothe need for the transformation from THZ coordinates tothe background coordinates. This transformation is a func-tion of the location and four velocity of the particle at anygiven instant. For fully consistent dynamics where theparticle location and four velocity are constantly beingmodified, this transformation will itself be changing.Thus, this coordinate transformation will unavoidablyhave to be determined and then applied numerically.Once the effective source is appropriately constructed,

the only remaining requirement is a code capable of evolv-ing the wave equation with a C0 source.

III. EFFECTIVE SOURCE

At the heart of our approach is the use of a convenientregular representation of a point-particle source. We referto this as the effective source. The two main elementswhich enter this are (1) the approximate singular field~c S ¼ q=� whose explicit form in terms of the chosenbackground coordinates depends on the position and fourvelocity of the particle, and (2) the window function Wwhose main purpose is to localize the support of theapproximate singular field to within the vicinity of theparticle.In tackling the same physical test application as in

Paper I, no modifications of ~c S were needed for our (3þ1) runs, apart from a trivial replacement of the backgroundcoordinates in which to express the effective source. (Here,we use ingoing Kerr-Schild coordinates as opposed to theSchwarzschild coordinates of Paper I).However, for the current implementation, we did seek

out a more adaptable window function. In (1þ 1), itproved sufficient to use a simple window function havinga Gaussian-like profile in r:

1With �2 � x2 þ y2 þ z2, a function which is Oð�nÞ as � !0, is at least Cn�1 where � ¼ 0.


084021-5

WðrÞ ¼ exp��ðr� roÞ

N

N

�; (15)

where ro is the radius of the circular orbit in Schwarzschildcoordinates, while N and are parameters to be chosenaccording to the requirements described in Sec. II. It iseasily verified that all of these required conditions can bemet for a sufficiently largeN. In principle, these conditionsmake it a reasonable choice regardless of the numericalimplementation. In practice, however, this original effec-tive source had some properties that could potentiallyburden certain (3þ 1) codes. (Some results from our earlyruns with the original source were, in fact, what motivatedthe construction of a new one.) Specifically, the choice of aGaussian-like window leads to a significant large-amplitude, short-scale (� ) structure away from theparticle. This was not an issue in the (1þ 1) case, wherehigh r resolution (�r�M=25) and high angular resolution(with a spherical harmonic decomposition going to as high

as L ¼ 39) were practical. Of course, the extra structureaway from the particle need not necessarily be a problemfor all (3þ 1) codes. One can maintain the Gaussian-likewindow and simply adjust its width to lessen the artifi-cial short-scale structure. Some of the runs presented be-low were performed with this original window, usingN ¼ 8, ro ¼ 10M, and ¼ 5:5M. The width was chosenin order to make the profile as wide as possible while stilleffectively vanishing before the horizon is reached. Theseruns show sufficiently good results as well.Nevertheless, there is merit in using a more flexible

window function, for instance, one with more adjustableparameters that can be tuned to the needs of any (3þ 1)code. A convenient choice makes use of the smooth tran-sition functions introduced in [41]. Like in Paper I, we havechosen to apply a window function only along the r direc-tion, in keeping with the spherical symmetry of the back-ground spacetime. Consider the smooth transition function

fðxjx0; w; q; sÞ ¼

8>><>>:0; x � x012 þ 12 tanh

�s�

�tan½ �2w ðx� x0Þ� � q

2

tan½ �2wðx�x0Þ�

��; x0 < x< x0 þ w

1; x � x0 þ w:(16)

This is a function that smoothly transits from zero to one inthe region x0 < x< x0 þ w. It comes with four adjustableparameters fx0; w; q; sg:

(1) x0: defines where the transition begins.(2) w: gives the width of the transition region.(3) q: determines the point x1=2 ¼ x0 þ ð2w=�Þ�

arctan q where the transition function fðxÞ ¼ 1=2.(4) s: influences the slope sð1þ q2Þ=ð2wÞ at x1=2 afterw

and q are chosen.Using this transition function, a window function for a

particle at r ¼ R could be

WðrÞ ¼�fðrjðR� �1 � w1Þ; w1; q1; s1Þ r � R1� fðrjðRþ �2Þ; w2; q2; s2Þ r > R; (17)

and WðrÞ ¼ 1 in the region R� �1 < r < Rþ �2.This satisfies all of the key requirements for a window

function (and more):(a) WðRÞ ¼ 1;(b) dnW=drnjr¼R ¼ 0, for all n;(c) W ¼ 0 if r 2 ½0; R� �1 � w1� [ ½Rþ �2 þ

w2;1Þ (thus making it truly of compact support);(d) and W ¼ 1 if r 2 ½ðR� �1Þ; ðRþ �2Þ�.For the actual runs that used this window function, we

settled on the following choices for these parameters:f�1 ¼ �2 ¼ 0M; q1 ¼ 0:6, q2 ¼ 1:2; s1 ¼ 3:6, s2 ¼ 1:9;w1 ¼ 7:9M, w2 ¼ 20Mg. The inner width w1 was chosenso that the window and effective source go to exactly zerojust outside the event horizon. The rest were picked afterextensively looking at many parameter combinations. Theprimary criteria were simply that the effective source

would be sufficiently small everywhere and that it didnot possess structure at extremely small scales. A system-atic search for the optimal set of parameters vis-a-vis itseffect on self-force accuracy was not conducted in thisstudy and is left for future work.One important attribute of the new window function is

that, for a wide range of parameter choices, it leads to aneffective source whose overall structure away from theparticle is significantly less pronounced than that produced

-0.01-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

0

2

4

6

8

10

12

-10-8

-6-4

-2 0

2 4

6 8

10

0 0.01 0.02 0.03 0.04

Sef

f

xy

Sef

f

FIG. 1 (color online). Equatorial profile of the new effectivesource, Seff , at ~t ¼ 0. The axes are defined simply by x ¼r sin� cos� and y ¼ r sin� sin�, where r, �, � are just theSchwarzschild coordinates (or the Kerr-Schild coordinates ofSec. VA). The charge in this plot is located at X ¼ 10M andY ¼ 0, where the C0 behavior of the source is not apparent onthis scale. Note that much of the structure induced by the newwindow function is between the charge and the event horizon.


084021-6

by the original Gaussian-like window. Comparing the neweffective source in Fig. 1 with the original source used inPaper I (shown as Fig. 1 of that paper), one notes imme-diately that the artificial structure resulting from the newwindow is almost 2 orders of magnitude smaller.Moreover, this structure is mainly located at r < R (whereR ¼ 10M).

It is instructive to look at the structure of Seff at thelocation of the particle. The effective source, Seff , is C

0 atthe particle due to the level of the approximation used forthe singular field c S. This C0 behavior is sufficient forcalculating the self-force. In our approach, this yields anevolved regular field c R that is C2 at the location of thecharge, from which derivatives can be computed to give theself-force. In Fig. 2, the C0 nature of the effective source isrevealed. The effective source is certainly a nonsingularrepresentation of a point charge source which is amenableto the (3þ 1) codes we have used for calculating the self-force.

IV. TEST APPLICATION

The physical scenario that is analyzed in this paperinvolves a particle with mass m and scalar charge q in aperpetual circular orbit around a Schwarzschild black holewhile emitting scalar radiation. The effects of the self-force(which include the effects of the emission of radiation)would lead to the gradual decay of the circular orbit. Forsimplicity in this analysis, we keep the charge in a circularorbit and compute the external force needed to counteractthe scalar self-force.

With the charge in perpetual circular motion, and in theabsence of other external sources which may violate thissymmetry, the system is helically symmetric. For any fieldG, there must then exist a helical Killing vector a, suchthat

L G ¼ 0: (18)In Schwarzschild coordinates, this Killing vector is simply

a@

@xa¼ @

@tþ� @

@�; (19)

and

L c R ¼ arac R ¼ 0: (20)For the circular orbit problem we have chosen, the fourvelocity ua is tangent to the Killing field at the location ofthe particle. Thus,ra� is already orthogonal to ua, and theself-force is given directly by

Fa ¼ qrac R (21)withno need for the projection operator,present in Eq. (14),for our test application. There are then only two indepen-dent components of the self-force, Ft and Fr, with F� ¼�Ft=� from Eq. (20), and F� ¼ 0, by virtue of the systemhaving a reflection symmetry about the equator.For circular orbits, there exists a useful relation between

the scalar energy flux and Ft. In terms of the Kerr-Schildcoordinates described next, this appears as

dE

dt

��r¼2MþdE

dt

��r¼R¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 3M

ro

sFt; (22)

where

dE

dt

��r¼2M¼ �4M2I

_c 2d�; (23)

and

dE

dt

��r¼R1 ¼ R2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R

R� 2M

s IR

�2M

R_c 2 þ

�1� 2M

R

�_c @rc

�� d�: (24)

Here, ro is the radius of the circular orbit, and R is the finiteouter extraction radius. The field c is actually the retardedfield, but one can instead use c R, as long as the surfaceintegrals are evaluated outside the support of the windowfunction, where (by design) c ret ¼ c R. This simple rela-tion is proved explicitly in Appendix B. We use this as aconsistency check on our self-force results.

A. Coordinates for a Schwarzschild black hole

We describe the Schwarzschild metric as

gab ¼ �ab þHkakb (25)using a coordinate system first identified by Eddington2

and commonly known as Kerr-Schild ingoing coordinates(t; x; y; z), where �ab is the flat Minkowskii metric with

-2e-07 0 2e-07 4e-07 6e-07 8e-07 1e-06 1.2e-06 1.4e-06

9.9 9.925

9.95 9.975

10 10.025

10.05 10.075

10.1 -0.1-0.075

-0.05-0.025

0 0.025

0.05 0.075

0.1

0

4e-07

8e-07

1.2e-06

1.6e-06

Sef

f

x y

Sef

f

FIG. 2 (color online). Seff zoomed in at the location of thecharge.

2Actually Eddington [35] and Finkelstein [42] wrote down theSchwarzschild metric using the precise coordinates of Eq. (26).But, somehow the Eddington-Finkelstein duo are associated witha coordinate system that contains an ingoing or outgoing nullcoordinate, although neither explicitly introduced or used such anull coordinate. Kerr and Schild (nearly 40 years afterEddington) described the Kerr metric in a form that reduces toEq. (26) in the Schwarzschild a ! 0 limit. Bowing to currentconventions of the numerical relativity community rather than tohistorical accuracy, we label the coordinate system in use as‘‘Kerr-Schild.’’


084021-7

(� 1; 1; 1; 1) along the diagonal,

kadxa ¼ �dt�

�x

rdxþ y

rdyþ z

rdz

�¼ �dt� dr;

(26)

which is the ingoing principle null vector, and

H ¼ 2Mr

(27)

with r2 ¼ x2 þ y2 þ z2. This is equivalent to the usualSchwarzschild form of the metric

ds2 ¼ ��1� 2M

r

�d~t2 þ dr

2

1� 2M=rþ r2ðd�2 þ sin2�d�2Þ (28)

with the Schwarzschild time coordinate ~t related to theKerr-Schild coordinates by

~t ¼ t� 2M lnðr=2M� 1Þ (29)and the usual flat-space relationships between (r; �; �) and(x; y; z).

The Kerr-Schild form of the metric is popular in thenumerical relativity community, because a constant t hy-persurface is nonsingular and horizon penetrating whichallows for convenient imposition of boundary conditions orfor excision.

However, it can be confusing to compare the compo-nents of the self-force as evaluated in these Kerr-Schildcoordinates with the components as evaluated inSchwarzschild coordinates. The Kerr-Schild radial coordi-nate rKS equals the Schwarzschild radial coordinate rSch,but constant t and constant ~t surfaces are not the same. Therelationships between the components of the self-force forthese two coordinates systems are

FSch~t ¼ FKSt ; (30)

FSchr ¼�

2M

ro � 2M�FKSt þ FKSr : (31)

B. The (3þ 1) version of the Schwarzschild metric inKerr-Schild coordinates

For the Schwarzschild metric in Kerr-Schild coordi-nates, the contravariant form of the metric (25) is

gab ¼ �ab �Hkakb: (32)With the (3þ 1) formalism [36], the contravariant com-

ponents of the metric are closely related to the lapsefunction �, shift vector i, and spatial metric �ij of afoliation of spacetime by

gab ¼ ��2 j=�2

i=�2 �ij � ij=�2� �

: (33)

This relationship gives

� gtt ¼ 1þH ¼ ��2; (34)

git ¼ Hxi=r ¼ i=�2; (35)and

gij ¼ �ij � ij=�2; (36)which implies that

�ij ¼ �ij � H1þH

xixj

r2: (37)

Also the determinants of the metrics are related byffiffiffiffiffiffiffi�gp ¼ � ffiffiffiffi�p : (38)C. The wave equation in Kerr-Schild coordinates

A form of the wave operator convenient for computationis

rarac ¼ 1ffiffiffiffiffiffiffi�gp @að ffiffiffiffiffiffiffi�gp gab@bc Þ: (39)In the (3þ 1) formalism, after substitution for the contra-variant components of the metric from Eq. (32) and withthe time-independence of gab, we obtain

�2rarac ¼ �@t@tc þ i@t@ic þ �ffiffiffiffi�p @i� ffiffiffiffi

�p�

i@tc

�

þ �ffiffiffiffi�

p @i��

ffiffiffiffi�

p ��ij �

ij

�2

�@jc

�(40)

for the wave equation in the Schwarzschild geometry withKerr-Schild coordinates and �, i, and �ij given inEqs. (34)–(37).

V. (3þ 1) IMPLEMENTATIONSIn the following, we describe the finite differencing and

pseudospectral codes used in the numerical experiments.

A. Three-dimensional multiblock finite difference code

We solve the wave Eq. (13) for c on a fixedSchwarzschild background with a source over a multiblockdomain using high order finite differencing. The code isdescribed in more detail in [33]; here, we will just sum-marize its properties. We use touching blocks, where thefinite differencing operators on each block satisfies a sum-mation by parts (SBP) property and where characteristicinformation is passed across the block boundaries usingpenalty boundary conditions. Both the SBP operators andthe penalty boundary conditions are described in moredetail in [43]. The code has been extensively tested andwas used in [44] to perform simulations of a scalar fieldinteracting with Kerr black holes and was used to extractvery accurate quasinormal mode frequencies.


084021-8

After the standard (3þ 1) split, the wave equation iswritten in first order in time, first order in space form interms of the variables � � @tc and�i � @ic . The systemof equations being integrated is then

@t� ¼ i@i�þ �ffiffiffiffi�p @i��

ffiffiffiffi�

p �gij�j þ

i�

�2

�� 2Seff ;

(41)

@t�i ¼ @i�; (42)@tc ¼ �; (43)

where Eq. (41) follows from Eq. (40) with gij � �ij ��2ij as in Eq. (36). Equation (42) is an elementaryconsequence of the definition of �i. The primary dynami-cal variables are u ¼ ð�;�iÞ, while c is evolved via anordinary differential equation (no spatial derivatives).Across a boundary with unit normal vector i, the charac-teristic modes are

w0i ¼ �i � j�ji; (44)

w� ¼ ðii �Þ�þ giji�j; (45)where the speeds of the two transverse modes in Eq. (44)are �0 ¼ 0, and the speeds of the two normal modes inEq. (45) are �� ¼ �ii � �.

The only necessary modifications to the code describedin [33], in order to apply it to the problem at hand, were theaddition of the source term in Eq. (41) and the addition ofthe code to interpolate the time derivative � and the spatialderivatives �i of the scalar field to the location of theparticle.

In addition, some optimizations were performed.OpenMP pragmas and directives were added to allow forsimultaneous OpenMP and MPI parallelization for betterperformance on modern multicore machines. Also, a loadbalancing issue arose that could potentially lead to verypoor scaling, because Seff is expensive to calculate only inthe spherical shell where it is nonzero. This issue wassolved by adding data structures that were distributedevenly among all MPI processes, with just the right sizeand shape to cover the spherical shell. The source is thenevaluated first (all processors working simultaneously) onthis distributed data structure and then copied into the mainthree-dimensional grid functions.

1. Boundary conditions and initial data

The simulations below were all performed using the 6-block system, providing a spherical outer boundary andspherical inner excision boundary without any coordinatesingularities. We use the Schwarzschild solution in Kerr-Schild coordinates as the background metric for the scalarfield evolution. The inner radius was chosen to be Rin ¼1:8M, and the outer boundary was chosen to be at Rout ¼400M in most cases (it was placed at Rout ¼ 600M in a few

runs for more accurate extraction of the fluxes). Since weare using SBP finite differencing operators, we can evalu-ate the right-hand sides for the evolution equations, i.e.@tu ¼ ð@t�; @t�iÞ, even as we approach the outer boundary(using more and more one-sided stencils). At the outerboundary, we then convert both u and @tu to characteristicvariables using Eqs. (44) and (45), i.e. we obtain (w0i ; w

�)and (@tw

0i ; @tw

�). We only have to apply a boundarycondition to w�, since this is the only incoming mode.We do this by adding a suitable penalty term (only at theouter boundary) to @tw

� of the form Tðg� w�Þ, where Tis a penalty parameter that has to be chosen to be consistentwith the SBP operator and the speed of the mode in order toachieve stability (see more details in [33]), and g is thedesired incoming characteristic mode. In this case, we useg ¼ 0, i.e. zero incoming mode. We then transform(@tw

0i ; @tw

þ; @tw� þ Tðg� w�Þ) back to a new @tu thatis used by the time integrator to update the primary vari-ables. At the inner boundary, the geometry ensures that allcharacteristics leave the computational domain; i.e. thereare no incoming modes, and therefore we do not apply anyboundary condition there.We do not, a priori, know the correct field configuration

and start the simulation with zero scalar field c ðt ¼ 0Þ ¼0, zero time derivative �ðt ¼ 0Þ ¼ 0, and zero spatialderivatives �iðt ¼ 0Þ ¼ 0, as if the scalar charge suddenlymaterializes at t ¼ 0. After the system is evolved for a feworbits, the initial transient has decayed and the systemapproaches a helically symmetric end state. We used the8-4 diagonal norm SBP operators and added some compat-ible explicit Kreiss-Oliger dissipation to all evolvedvariables.With this code, we have performed runs for a scalar

charge on circular orbits of radius ro ¼ 10M with both thewide Gaussian profile window (N ¼ 8 and ¼ 5:5M) andthe smooth transition function window described inSec. III. We find that the extracted self-force is independentof the window function (as it should be) and that the onlydifference between the runs is in the shape and amplitudeof the initial scalar wave pulse.

2. Convergence

The convergence of the code has been extensively testedin [43], where the evolution of a plane wave moving acrossa spherical grid was used as a test problem (i.e. no source).It was shown that for each implemented finite differencingorder, the code was converging at the expected order. Forexample, for the 8-4 SBP operators used here, we found theexpected fifth order global convergence.As the source is only C0 at the world line of the particle,

it is to be expected that the scalar field will be C2 there,while the main evolution variables � ¼ @c =@t and �i ¼@c =@xi should be C1 on the world line of the particle andC1 everywhere else. With the finite differencing code,there will be some stencils which are penetrated by the


084021-9

world line at a particular event. For those stencils, the finitedifferencing errors will be affected by the limited differ-entiability of both the source and the field at the particle.We would expect that any traditional centered finite differ-encing operator applied to a C1 field (regardless of order)should then only be first-order accurate: the second deriva-tive is discontinuous across the world line, and so thesecond order terms in the Taylor expansion of the operatorwill not cancel.

Naively, one would then expect that the solution for �and �i at the particle and thus the extracted self-forcewould only converge to first order. However, as shown inAppendix A for the wave equation in (1þ 1) dimensions,the errors in � in fact converge at second order in the L2norm for a C0 source. In the Appendix, it is also shown thatthe error is of high frequency with the frequency increasingwith resolution. Thus, for our test application we cannotdemonstrate pointwise convergence for the quantities �and �i. But we expect that the amplitude of any noisegenerated near the particle location will converge at secondorder. We find below that the extracted self-force compo-nents at the location of the particle are indeed noisy, butthat the noise converges to zero at second order.

B. The pseudospectral code

We solve the wave Eq. (13) for c on a fixedSchwarzschild background with a source using pseudo-spectral techniques.

We use the SGRID code [34,45,46] to numerically evolvec . This code uses a pseudospectral method in which allevolved fields are represented by their values at certaincollocation points. From the field values at these points, itis also possible to obtain the coefficients of a spectralexpansion. As in [34,45], we use standard spherical coor-dinates with Chebyshev polynomials in the radial directionand Fourier expansions in both angles. Within this method,it is straightforward to compute spatial derivatives. Toobtain the results described below, the SGRID code uses atmost 3� 53þ 2� 161 ¼ 481 collocation points in theradial and only 64� 48 in the angular directions. Thissmall number of points makes it so efficient that it canrun on a single PC or laptop.

As in [34], we introduce an extra variable

� � � 1�ð@tc � i@ic Þ (46)

in order to obtain a system of equations that is first order intime

@tc ¼ i@ic � ��; (47)

@t� ¼ � 1ffiffiffiffi�p @i½ ffiffiffiffi�p ði�þ ��ij@jc Þ� þ �Seff ; (48)which results from Eqs. (40) and (46). For the time inte-gration, we use a fourth order accurate Runge-Kutta

scheme. We implement Eq. (48) in the code using theequivalent, specific form

@t� ¼ i@i�� gij@i@jc þ ��i@ic � gijð@ic Þ@j�þ �K�þ �Seff ; (49)

where K is the trace of the extrinsic curvature of a constantt hypersurface, and �i is given in terms of the Christoffelsymbols of the 3 metric as �i ¼ �jk�ijk. In ingoing Kerr-Schild coordinates, K and �i are given by

K ¼ 1ð1þHÞ3=2H

r

�1þ 3M

r

�; (50)

�i ¼ 1ð1þHÞ2H

r

�3

2þ 4M

r

�xi

r: (51)

For the time integration, we use a fourth order accurateRunge-Kutta scheme. As in [34], we find that it is possibleto evolve this system in a stable manner if we use a singlespherical domain, which extends from some inner radiusRin (chosen to be within the black hole horizon) to amaximum radius Rout. In this case, one needs no boundaryconditions at Rin since all modes are going into the holethere and are thus leaving the numerical domain. At Rout,we have both ingoing and outgoing modes. We imposeconditions only on ingoing modes and demand that theyvanish. However, since we need more resolution near theparticle, it is advantageous to introduce several adjacentspherical domains. In that case, one also needs boundaryconditions to transfer modes between adjacent domains.We were not able to find interdomain boundary conditionswith which we could stably evolve the system (49). For thisreason, we introduce the three additional fields

�i ¼ @ic ; (52)

and we evolve the system

@tc ¼ i@ic � ��@t� ¼ i@i�� gij@i�j þ ��i�i � gij�i@j�

þ �K�þ �Seff@t�i ¼ j@j�i þ�j@ij � �@i��@i�: (53)

Note that this system is now first order in both space andtime, and it can be stably evolved using the methods de-tailed below. Also notice that we evolve the Cartesiancomponents of all fields. Because of the introduction ofthe additional fields� and�i, our evolution system is nowsubject to the constraints

@tc ¼ i@ic � ��; �i ¼ @ic : (54)


084021-10

1. Characteristic modes

The characteristic modes of the system (53) are [47]

w� ¼ �� i�i; w0i ¼ �i � j�ji; wc ¼ c :(55)

For our shell boundaries, i is a spatial outward-pointingunit vector. The fieldsw0i and w

c have velocity�i, whilew� have velocity �i � �i.

2. Domain setup, boundary conditions, and initial data

We typically use 4 adjacent spherical shells as ournumerical domains. The innermost shell extends fromRin ¼ 1:9M to ro ¼ 10M. The next two interdomainboundaries are at 18:1M and 27:5M. The outermost shellextends from 27:5M to Rout ¼ 210M. The outermost shellalways has 161 collocation points in the radial direction.The inner shells all have the same number of points. Wevary their number between 29 and 53. For simulations thatlast longer than about 390M, we have observed that re-flections from the outer boundary can reach the particle andintroduce errors in the self-force. For this reason, we havealso performed simulations where we add an additionalouter shell with 161 radial points that extends from 210Mto R0out ¼ 400M. As we can see in Fig. 3, we can nowevolve to at least 600M without spurious boundary effects.For our simulations, we have used the Window function inEq. (17).

As mentioned above, we do not impose any boundaryconditions at Rin. At Rout, we impose boundary conditionsin the following way. First, we compute @tw

þ from thefields @tc , @t�, and @t�i at the boundary. Then, weimpose the conditions

@tw� ¼ ��=r; @tw0i ¼ ð�ki � kiÞ@k@tc ;

@twc ¼ i�i � ��

(56)

on the ingoing modes. Finally, we recompute @tc , @t�,and @t�i from @tw

�, @tw0i , and @twc . The motivation forthe outer boundary conditions in Eq. (56) is as follows. Thefirst equation is equivalent to assuming the Sommerfeldcondition c ¼ fðt� rÞ=r for some unknown function f.The other two conditions are derived from the constraintsin Eq. (54) and can thus be considered constraintpreserving.For the interdomain boundaries, we simply compute

@tw�, @tw0i , and @twc from @tc , @t�, and @t�i at the

boundary in each domain. On the left side of the boundary,we then set the values of the left going modes @tw

�, @tw0i ,and @tw

c equal to the values just computed on the rightside of the boundary. On the right side of the boundary, weset @tw

þ equal to the value computed on the left side. Thisalgorithm simply transfers all modes in the direction inwhich they propagate.As initial data, we simply use c ¼ � ¼ �i ¼ 0.

3. Spectral filters

In order to obtain a stable evolution, we apply a filteralgorithm in the angular directions after each evolutionstep. As in [45], we project our double Fourier expansiononto spherical harmonics. After setting the highest l modein c and � to zero, we recompute all fields at the collo-cation points. This filter algorithm removes all unphysicalmodes and also ensures that c and� always have one lessmode than �i.

4. Noise reduction

If we compute the coefficients in a Fourier series expan-sion of the effective source for a particle moving along acircular orbit, we expect them to be of the form

hmðtÞ ¼ hmð0Þeim�t; (57)where hmð0Þ are the coefficients at time t ¼ 0, m is themode number, and � is the orbital angular velocity.However, in the SGRID code, we use discrete Fourier trans-forms instead of Fourier series, so the resulting coefficientshave a more complicated time dependence for any finiteresolution.The collocation points in the SGRID code are fixed. This

means that the moving particle periodically approachesgrid points. Thus for any given resolution, the discreteFourier coefficients of the effective source will show amodulation (in addition to the expected phase factor) onthe time scale it takes to move from one grid point to thenext. This modulation is a source of extra noise. In oursimulations, we have removed this extra noise by thefollowing procedure. We simply compute the coefficientshmð0Þ once and for all at t ¼ 0. For any later time, we

0 100 200 300 400 500 600t/M

3.40×10-5

3.45×10-5

3.50×10-5

3.55×10-5

3.60×10-5

3.65×10-5

3.70×10-5

3.75×10-5

3.80×10-5

Ft

Rout

=210M

R’out

=400M

FIG. 3 (color online). The broken line shows Ft for an outerboundary located at Rout ¼ 210M. We can see reflections arriv-ing at the location of the particle at around 390M. If the outerboundary is moved out to R0out ¼ 400M (solid line), no sucheffects can be observed during an evolution time of 600M.


084021-11

evaluate the source by taking the inverse discrete Fouriertransform of hmð0Þeim�t, so we avoid any extra modulationor noise.

5. Convergence

As the source Seff is C0 at the particle, we expect that c

is C2 and�i is C1 there. This implies that with our spectral

code c is expected to be fourth order convergent at theparticle. This expectation is confirmed by the results pre-sented in Fig. 4. The solid line shows the difference in cbetween a low and medium angular resolution run, whilethe broken line shows the difference between the mediumand high resolution run scaled by a factor of s ¼ 3:21chosen such that the two lines coincide. This factor isrelated to the order of convergence O by

s ¼ ð1=NlowÞO � ð1=NmedÞO

ð1=NmedÞO � ð1=NhiÞO: (58)

For an order of convergence of O ¼ 4, we would have

s ¼ ð1=64Þ4 � ð1=80Þ4

ð1=80Þ4 � ð1=96Þ4 ¼ 2:78: (59)

The scale factor of s ¼ 3:21 thus corresponds to conver-gence of an order between 4 and 5.

VI. SELF-FORCE AND ENERGY FLUX

In this section, we present and then comment on ourresults. We compute Ft and Fr, the two nontrivial compo-nents of the self-force for a scalar charge in a circular orbitaround Schwarzschild, and we show consistency betweenthe results from the two codes.

Using Eqs. (23) and (24), we also compute the scalarenergy flux across the event horizon and some finite outerboundary, referring to this outer boundary as the extractionradius. For ease of comparison, these fluxes are expressedas the t component of the self-force, based on the relationgiven by Eq. (22).A representative summary of the accuracies we achieved

is presented in Table I below.

A. The dissipative piece, Ft

The mode sum of the t component of the self-force @tcR

for the case of a scalar charge in a circular orbit inSchwarzschild is known to converge exponentially in l,and is thus typically calculated extremely accurately.Despite the divergence in c ret at the location of the scalarcharge, @tc

ret is smooth there and requires no regulariza-tion. This arises, because the retarded and advanced fieldsfor a charge in a circular orbit are related by

@tcret ¼ �@tc adv: (60)

Writing c ret as

c ret ¼ 12ðc ret � c advÞ þ 12ðc ret þ c advÞ; (61)we see that the time derivative of the second term vanishes.The first term is clearly smooth at the location of thecharge, because it is a solution of the homogeneous waveequation. For generic orbits, Eq. (60) will not be true, andall components of @ac

ret will need to be regularized.By instantaneously switching on our source at t ¼ 0, the

early part of the evolution will be contaminated by initial

0 50 100 150 200 250 300 350t/M

0.0

2.0×10-7

4.0×10-7

6.0×10-7

8.0×10-7

1.0×10-6

1.2×10-6

1.4×10-6

(ψNθ=64

− ψNθ=80

)(ψ

Nθ=80 − ψ

Nθ=96)∗3.21

FIG. 4 (color online). Differences in c at the particle locationfor runs with different angular resolutions given by N�. All threecases are for Nr ¼ 29 and N� ¼ 3N�=4. If we scale the differ-ence between medium and high resolutions by 3.21, the twocurves coincide. This corresponds to convergence of an orderbetween 4 and 5.

TABLE I. Summary of (3þ 1) results. The top half of thetable reports the computed components of the self-force for acharge in a circular orbit ro ¼ 10M. These were extractedaround time, t ¼ 600 M. The error is determined by a compari-son with an accurate frequency-domain calculation [16], whichreports Ft ¼ 3:750227� 10�5 and Fr ¼ 1:378448� 10�5. Thebottom half of the table reports the computed energy fluxesthrough the event horizon and the two sphere defined by outerextraction radius R. The R ¼ 1 case is an extrapolation to aninfinite outer extraction radius that was performed on results ofthe multiblock code (as explained in Sec. VI C). For ease ofcomparison with the local self-force, all energy fluxes are ex-pressed as Ft according to Eq. (22). MB stands for multiblockcode.

Code Result Error

Ft MB ð3:728–3:748Þ � 10�5 0.05%–0.6%Ft SGRID ð3:7481–3:7487Þ � 10�5 0.05%Fr MB ð1:384–1:389Þ � 10�5 0.4%–0.8%Fr SGRID ð1:384–1:386Þ � 10�5 0.4%–0.5%_EðR ¼ 150Þ MB 3:773� 10�5 0.6%_EðR ¼ 150Þ SGRID 3:771� 10�5 0.6%_EðR ¼ 300Þ MB 3:761� 10�5 0.2%_EðR ¼ 1Þ MB 3:7502� 10�5 0.0005%


084021-12

data effects. After some time, these transient effects propa-gate out of the numerical domain and the system settlesdown to its helically symmetric end state.

The results of Fig. 5 are from the multiblock code. Itplots Ft evaluated at the location of the charge as a functionof time t. Helical symmetry would correspond to a hori-zontal line, and we see that the plot gradually approachesthis while also getting to the correct self-force (based onhighly-accurate frequency-domain results in the literature).

The particle makes about two full orbits (Torb ¼2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiR3=M

p 200M) before Ft is reached to within 1%.The result improves as initial data effects further diminish.

On top of the evolution toward the correct self-force,there appears to be some sort of modulated noise. Thisbehavior is the result of two factors. The high-frequencycomponent is due to the fact that the source is only C0 (andhence derivatives of the scalar field are only C1) at thelocation of the particle. The finite differencing schemeemployed here uses stencils near the particle locationthat enclose this nonsmoothness, and this is expected tointroduce some noise. The frequency of this noise corre-sponds exactly to the particle travel time from one gridpoint to the next. The low-frequency modulation that en-velopes the noise has a period of about 50M, which isexactly the time between crossings of interpatch bounda-ries. This is due to the inflated sphere coordinates usedwithin the individual blocks. The angular resolution isslightly higher near the edges than in the middle of theblock.

We observe that the amplitude of this noise decreaseswith increased angular resolution. At 40� 40 angularresolution, we obtain values for the self-force between3:7� 10�5 and 3:75� 10�5 i.e. within 1.3% of thefrequency-domain value (which we will consider in thefollowing to be exact). At an angular resolution of 60� 60,the amplitude of the noise is smaller by a factor of 2.25corresponding to second order convergence and an error ofabout 0.6% of the exact value.Both results correspond to a radial resolution of �r ¼

M=10. The inner (excision) boundary was placed at Rin ¼1:8M and the outer boundary at Rout ¼ 400M. Modifyingthe radial resolution (to�r ¼ M=15) does not significantlyimpact the amplitude of the noise.In Fig. 6, we compare the results from the two codes.

There is good agreement between the two, except that theSGRID result has noticeably less noise than the multiblock

result. This is due to the extra noise reduction performed bythe SGRID code, as described in Sec. VB 4. For the SGRIDresult shown here, the number of collocation points in the �and� directions wereN� ¼ 64 andN� ¼ 48, respectively.In the r direction, Ninr ¼ 53 collocations points were usedin each of the three inner shells. (The number of colloca-tion points in the two outer shells, Noutr , was kept the samein all runs at Noutr ¼ 161.)

B. The conservative piece, Fr

The conservative piece of the self-force is really thecrucial quantity to compute. This is the part of the self-force that cannot be inferred from observations far away[unlike Ft, for example, which can be determined from the

3.66e-05

3.68e-05

3.7e-05

3.72e-05

3.74e-05

3.76e-05

3.78e-05

3.8e-05

200 250 300 350 400 450 500 550 600

Ft

t/M

40x4060x60

correct Ft

FIG. 5 (color online). Ft computed at different angular reso-lutions of the multiblock code. The resolutions are described bythe number of angular grid points per patch, so that 40� 40corresponds to 40 grid points in both � and � directions withineach patch. The noise in this plot is due to the C0 nature of theeffective source. The frequency of this noise corresponds to theparticle travel time from one grid point to the next, and the low-frequency modulation corresponds to the particle travel timebetween patch boundaries.

3.66e-05

3.68e-05

3.7e-05

3.72e-05

3.74e-05

3.76e-05

3.78e-05

3.8e-05

200 250 300 350 400 450 500 550 600

Ft

t/M

Ft (multiblock)Ft (SGRID)

correct Ft

FIG. 6 (color online). Comparing Ft results from the multi-block and SGRID codes. The multiblock result was achieved with60� 60 angular resolution and �r ¼ M=10 radial resolution, asin Fig. 5. For the SGRID result shown here, the number ofcollocation points in the angular directions were N� ¼ 64 andN� ¼ 48. In the r direction, Ninr ¼ 53, for the three inner shells.The two outer shells were always set to use Noutr ¼ 161 collo-cation points.


084021-13

energy flux by using Eq. (22)]. For the case of circularorbits, this conservative piece shows up entirely as the rcomponent, Fr. In a mode-sum self-force calculation, thiswould be the quantity whose mode sum converges as l�n,where n > 1 is typically a small number depending on thenumber of regularization parameters one has access to. Inour approach, calculating Fr (where r is the Schwarzschildradial coordinate) amounts to taking derivatives of theregular field c R, which corresponds to taking simple alge-braic combinations of the interpolated values of theevolved fields at the location of the charge.

We present the results from the two codes together inFig. 7. The data from the multiblock code were computedusing runs at 40� 40 angular resolution and two radialresolutions �r ¼ M=10, M=15. For the SGRID code, wehave used data from the same run described in Fig. 6.

We immediately notice that all of the results eventuallysettle on a value slightly offset from the correct one. It isworth emphasizing though that for the SGRID data and themultiblock data calculated at radial resolution of �r ¼M=15, the final error is just

Fig. 6. The energy flux was calculated using an outerextraction radius of R ¼ 150M, and again converted tothe corresponding Ft. This is juxtaposed with the localcalculation of Ft and the frequency-domain result. Again,we observe that, except for early-time errors due to spu-rious initial data, the energy flux settles to within 1% of thecorrect answer.

One notable observation is that the energy flux improveswith increasing extraction radius. This is shown in Fig. 10.Knowing this, it is tempting to make the extraction radiusas large as possible. However, how far the extraction radiuscan be situated is limited by the fact that the flux taken atfarther radii naturally takes longer to equilibrate, since thebad initial data waves will have to propagate much farther.

Instead, one can use results from finite radii to extrapo-late the energy flux in the limit of an infinite extractionradius. This was done with results from the multiblockcode, and Fig. 11 shows the outcome. The results fromsix finite extraction radii, from R ¼ 50M to 300M, wereused to determine the energy flux in the limit of infiniteextraction radius. First though, one must account for thetime shifts in the fluxes. Obviously, emitted scalar radiationreaches 50M first and only after an interval of time, arrivesat the next extraction radius at R ¼ 100M.

Using the fact that outgoing null geodesics travel atcoordinate speed ðr� 2MÞ=ðrþ 2MÞ in Kerr-Schild coor-dinates, one can integrate and find that the time delaybetween the arrival at various radii are given in Table II.

Shifting the data by these appropriate time delays, weassume a form for the flux _EðRÞ at finite extraction radius Rgiven by

_EðRÞ ¼ _Eð1Þ þ XNn¼1

CnRn

þOð1=RNþ1Þ: (62)

Truncating at N ¼ 5 the constants C1; . . . ; CN and _Eð1Þcan then be determined from the six sets of flux data at the

different extraction radii. The resulting _Eð1Þ from thisprocedure is plotted in Fig. 11. For reference, we alsoinclude the frequency-domain result for Ft expressed as aflux with Eq. (22). As expected, the agreement is signifi-cantly improved. Extrapolating to infinite extraction ra-dius, the flux matches to �0:0005%. We take this resultas further validation that our effective source is a good C0

representation for a point charge that would otherwise havebeen represented with a � function.

VII. SUMMARYAND DISCUSSION

For our flux values to achieve the reported accuracies isnoteworthy. This indicates that our effective source Seff is agood representation for point particles, in place of � func-tions that are difficult to handle on a grid. Narrow Gaussianfunctions centered around the location of the particle havepreviously been employed for this task. Reference [48]uses a time-domain calculation of the gravitational energyflux for a point mass orbiting a Kerr black hole whichresults in errors �10%. But by optimizing the number ofgrid points used to sample the narrow Gaussian, one canactually get results of �1% [49]. Recent work bySundararajan et al. [50] has done even better than this,with a novel discrete representation of the � function.Errors of

with a (3þ 1) simulation are already at a comparableaccuracy, albeit only for the scalar energy flux. It is difficultto speculate on how narrow Gaussians and discrete repre-sentations will perform in a (3þ 1) context.

The main highlight, however, has to be the accuracy ofour self-force results, which have errors& 1%. Unlike anyother self-force calculations thus far, these values werecalculated by merely taking a derivative of the regular fieldat the location of the point charge. These results are prom-ising as a first attempt at doing a (3þ 1) self-forcecalculation.

The judicious placement of collocation points by theSGRID code close to the location of the charge appears to

enable it to represent the effective source better (and toachieve slightly more accurate results) as opposed to theuniform grid that the multiblock code uses. This seems tosuggest that devoting more resources to resolving theregion around the charge (like what would be done withadaptive mesh refinement) is the right strategy.

Both codes show convergence with respect to increasesin radial and angular resolution. It is clear that the finitedifferentiability of the source reduces the order of conver-gence of the codes relative to what it would be if one had asmooth source. For instance, exponential convergenceought to be observed in the SGRID code, and the non-smoothness of the source is significant for the multiblockfinite difference code, where only second order conver-gence is achieved while much higher order operators areactually used. Modifications of either code aimed at treat-ing sources with limited differentiability would be likely toimprove the order of convergence. Such a modificationmight take the form of an adjustment to a stencil or aspectral function to anticipate the location of the charge.

An important issue has been made apparent by theseinitial results. Since the self-force is a very small quantity,the effects of imperfect boundary conditions become acause of concern. In both the SGRID and multiblock codesthe outer boundary, conditions were implemented in a waythat ignored the back scattering off the curvature of thespacetime outside the computational domain. Since theself-force contains a tail effect, any such boundary condi-tion will, when the boundary comes into causal contactwith the particle location, affect the calculation of the self-force. In practice it will seem like the outer boundarypartially reflects the outgoing waves. In order to avoidsuch effects, we have to place the outer boundaries farenough out that they remain out of causal contact with theparticle (or the sphere where the energy flux is measured)for the duration of the run. This of course makes the runsmore computationally expensive both in terms of memoryand cpu usage and limits the number of orbits that can besimulated. Accurate self-force analyses require carefultreatment of the boundary conditions. One way to do thiswould be to use the nonlocal radiation boundary conditionsdeveloped by Lau [51] and used in practice for calculating

the metric perturbations of an extreme-mass-ratio binarywith a (1þ 1) discontinuous Galerkin code [52].In summary, with this preliminary study, we have dem-

onstrated how it is possible to compute self-forces withexisting (3þ 1) codes—in fact one of our implementationsruns on a laptop! Moreover, it has been shown that even inthe (3þ 1) context, the effective source is a good smeared-out alternative to standard �-function representations ofpoint sources. The flux resulting from the effective sourcematches that due to a point charge with very good accuracy.Some questions do remain to be explored, like the benefitsof optimizing the codes, the reduction of the convergenceorder due to the finite differentiability of the effectivesource, and the limitations set by the effects of boundaryreflections. As this is merely a first cut analysis, we shallleave these issues for future work.A goal of this project is to raise interest within the

numerical relativity community in self-force analyses ofthe EMRI problem. Thus the C++ code, which evaluates theeffective source for a �-function scalar charge, has beenplaced in the public domain via a website http://www.phy-s.ufl.edu/~det/effsource. Our initial expectation is to ex-tend this work by allowing for a generic world line. Ourlong term goal is to have code for an effective source whichrepresents a point mass orbiting a rotating black hole. Ateach step as the project progresses, we will continue to putin the public domain all of our codes necessary for evalu-ating the effective source.

ACKNOWLEDGMENTS

I. V. acknowledges support from the University ofFlorida and acknowledges the Institute for FundamentalTheory (http://www.phys.ufl.edu/ift) for financial supportthrough the course of this work. S. D. and I. V. acknowl-edge the University of Florida High-PerformanceComputing Center (http://hpc.ufl.edu) for providing com-putational resources. This work was supported in part bythe National Science Foundation, Grants No. PHY-0652874 and No. PHY-0855315 with Florida AtlanticUniversity (W. T.), Grant No. PHY-0555484 with theUniversity of Florida (S. D. and I. V.), and throughTeraGrid resources provided by LONI and in part byLONI through the use of Oliver, Poseidon, Louie, Eric,and Painter. We also employed the resources of the Centerfor Computation and Technology at Louisiana StateUniversity, which is supported by funding from theLouisiana legislature’s Information Technology Initiative(P. D.).

APPENDIX A: EFFECT OFA C0 SOURCE ON AFINITE DIFFERENCE CODE

In order to better understand the convergence propertiesof a finite difference code for the scalar wave equation witha C0 source, we turned to the (1þ 1) case with unit speed


084021-16

in flat space

� @2c

@t2þ @

2c

@x2¼ Sðt; xÞ: (A1)

Similar to the (3þ 1) case, we can introduce the additionalvariables � ¼ @tc and � ¼ @xc and rewrite the waveequation in first order in time, first order in space form

@t� ¼ @x�� Sðt; xÞ; @t� ¼ @x�; @tc ¼ �:(A2)

If c ð0; xÞ ¼ 0, @tc ð0; xÞ ¼ 0 and if Sðt; xÞ ¼ 0 for t < 0,the solution to Eq. (A1) at a given coordinate (t; x) can beshown to be given in terms of an integral of the source overthe domain of dependence, i.e.

c ðt; xÞ ¼ �1=2Z t0

Z xþt�t0x�tþt0

Sðt0; x0Þdx0dt0: (A3)

The solutions for @tc ðx; tÞ and @xc ðx; tÞ are then given by

@tc ðt;xÞ¼�1=2Z t0ðSðt0;xþ t� t0ÞþSðt0;x� tþ t0ÞÞdt0;

(A4)

and

@xc ðt;xÞ¼�1=2Z t0ðSðt0;xþ t� t0Þ�Sðt0;x� tþ t0ÞÞdt0:

(A5)

These integrals can be evaluated numerically inMathematica to high accuracy for any given source thusyielding the exact solution to be compared with an ap-proximate finite difference solution.

A function of the form

fðt; xÞ ¼ exp��x� at

c

�2�tanhðx� atÞ (A6)

is negative for x < at and positive for x > at. Forming

sðt; xÞ ¼k fðt; xÞ k �fðt; xÞ (A7)thus results in a source that is positive for x < at and zerofor x > at. This source is then C0 at x ¼ at and C1 every-where else.

Solving the system of equations in Eq. (A2) with thesource in Eq. (A6) and (A7) using fourth order centeredfinite differencing and fourth order Runge-Kutta time in-tegrator, we can then use the exact solution in Eq. (A4) tocalculate the error in the numerically evaluated �. For the

specific choice of source parameters a ¼ ffiffiffi2p =2 and c ¼1:3, we calculated the solution for 3 different spatial res-olutions �x ¼ ð0:1; 0:05; 0:025Þ on the spatial interval x 2½�6; 6�. The time step was �t ¼ �x=4.

The scaled errors (for second order convergence) in � att ¼ 3 can be seen in Fig. 12. As can be seen, the errors areas high frequency as can be allowed given the spatialresolution, i.e. the error varies dramatically from grid pointto grid point. Therefore it is impossible to talk about

pointwise convergence, since the error at a given grid pointmay be positive at one resolution but negative at another.However, the amplitude in the error can still be consideredsecond order convergent. In fact, calculating the discreteL2 norm of the error, we find that

k eð�x ¼ 0:1Þ k2 = k eð�x ¼ 0:05Þ k2¼ 4:21;and

k eð�x ¼ 0:05Þ k2 = k eð�x ¼ 0:025Þ k2¼ 4:12;showing that we have global second order convergence inthe L2 norm. The numerical methods used here are for-mally fourth order accurate, but because the source is C0,we are limited to only second order convergence.If instead we use the source

Sðt; xÞ ¼ �fðt; xÞ (A8)with the same values for a and c as before, we obtain the

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-6 -4 -2 0 2 4 6

Sca

led

erro

r in

ρ

x

e(∆x=0.1)4*e(∆x=0.05)

16*e(∆x=0.02)

FIG. 12 (color online). Scaled errors for second order conver-gence in � at t ¼ 3 with a C0 source.

-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

-6 -4 -2 0 2 4 6

Sca

led

erro

r in

ρ

x

e(∆x=0.1)16*e(∆x=0.05)

256*e(∆x=0.02)

FIG. 13 (color online). Scaled errors for fourth order conver-gence in � at t ¼ 3 with a C1 source.


084021-17

scaled errors shown in Fig. 13. Here, the errors are smoothand low frequency and the scaled errors from differentresolutions (scaled for fourth order convergence) agreevery well, i.e. we have pointwise convergence. For theL2 norms of the errors in this case, we get

k eð�x ¼ 0:1Þ k2 = k eð�x ¼ 0:05Þ k2¼ 15:74;and

k eð�x ¼ 0:05Þ k2 = k eð�x ¼ 0:025Þ k2¼ 15:92;clearly showing the expected reduction in errors by a factorof 16 with a doubling of the resolution.

APPENDIX B: SELF-FORCE AS BOUNDARYINTEGRALS IN KERR-SCHILD COORDINATES

In this Appendix, we derive the relationship between the(Kerr-Schild) time component of the self-force on thescalar charge and the energy flux at spatial infinity andacross the event horizon. We also derive explicit expres-sions for these fluxes in Kerr-Schild coordinates.

1. How Ft and the energy flux are related for charges incircular orbits

For a scalar charge going in a circular orbit around aSchwarzschild black hole, there exists a direct relationshipbetween the time component of the self-force on the scalarcharge and the energy flux at spatial infinity and across theevent horizon.

In the absence of external fields, the motion of a scalarparticle is governed by the self-force acting on it,

mab ¼ qðgbc þ ubucÞrcc R � Fb: (B1)The energy per unit mass (i.e. specific energy) of a particlealong a geodesic with a four velocity ub is just E ¼ �tbub,where tb is the time-translation Killing vector of theSchwarzschild spacetime. The rate of change in this spe-cific energy per unit proper time is _E � ucrcE ¼�ucubrctb � tbucrcub ¼ �tbab, since rðctbÞ ¼ 0. InKerr-Schild coordinates, this is just _E ¼ �at. Evaluatingthis on a particle moving in a circular orbit, Eq. (B1) givesus

_Ejp ¼ � qm ð@tcR þ utubrbc RÞjp (B2)

¼ � qm@tc

Rjp; (B3)where p signifies the location of the particle. The secondterm in the first equality vanishes for a circular orbit,

uarac Rjp ¼ ðut@tc R þ u�@�c RÞjp (B4)

¼ ðdt=d�Þð@tc R þ�@�c RÞjp (B5)

¼ ðdt=d�ÞLc Rjp (B6)

¼ 0; (B7)where a is the Killing vector associated with the helicalsymmetry, so that the third equality vanishes due toEq. (18). Thus

Ft ¼ �m _E ¼ q@tc R: (B8)The time component of the self-force in Kerr-Schild coor-dinates is then just the amount of energy lost by the particleper unit proper time.This energy loss must obviously be related to the scalar

energy flux. We shall now derive these relationships andalso the explicit expressions for the energy fluxes in Kerr-Schild coordinates.The scalar field produced by the charge is determined by

the field equation,

gabrarbc ¼ �4�qZ �ð4Þðx� zðtÞÞffiffiffiffiffiffiffi�gp d�: (B9)

Multiply both sides by tarac , integrate overV (which wetake to be the 4 volume bounded by constant Kerr-Schild tsurfaces t ¼ ti and t ¼ tf, the event horizon, and timelikehypersurface r ¼ R), and ta is the time-translation Killingvector of Schwarzschild. Simplify the integral over the �function to obtainZ

Vðtarac Þrbrbc ffiffiffiffiffiffiffi�gp d4x

¼ �4�qZ tfti

ðtarac Þjpðdt=d�Þ�1dt: (B10)

We notice now that the integrand on the left can be ex-pressed as

taðrac Þrbrbc ¼ tagabrbcr2c ¼ 4�tagabrcTbc;(B11)

where Tbc is the stress-energy tensor for the scalar field,

Tbc ¼ 14�

ðrbcrcc � 12gbcrdcrdc Þ: (B12)

We then haveZVtarcTca ffiffiffiffiffiffiffi�gp d4x ¼ �Z tf

ti

Ftðdt=d�Þ�1dt; (B13)

where, specializing to circular orbits, Ft ¼ q@tc jp. Here,we have exploited the fact that @tc

ret requires no regulari-zation and thus equals @tc

R. Since ta is a Killing vector,rðctaÞ ¼ 0, and Tca is symmetric in its indices, we havetarcTca ¼ rcðtaTcaÞ. Thus,Z

VrcðtaTcaÞ ffiffiffiffiffiffiffi�gp d4x ¼ �Z tf

ti


I@V

taTcad�c ¼ �

Z tfti



084021-18

where d�c is the directional volume element of the bound-ary @V . The integrand of the left-hand side is essentiallythe conserved current for the scalar field, taTa

c.We recall again that for the case of a scalar charge in a

perpetual circular orbit of angular velocity�, there exists ahelical Killing vector a given by

a@

@xa¼ @

@tþ� @

@�: (B16)

We break up the left-hand side of Eq. (15) into the fourhypersurface integrals,

I@V

taTcad�c ¼

�Zr¼2M

�l̂cr2d�d�þZr¼R

r̂cr2dtd�

þZt¼tf

n̂cffiffiffih

pd3x�

Zt¼ti

n̂cffiffiffih

pd3x

�taTca:

(B17)

l̂a is the null generator of the event horizon, and the rest ofthe hatted quantities are the respective outward unit normalvectors to the other hypersurfaces making up @V . � is anarbitrary parameter on the null generators k̂a of the eventhorizon. From the helical symmetry of the problem, it iseasy to see that the last two integrals just cancel each otherout. This simply means that energy content in each con-stant t hypersurface is the same. But we can show thisexplicitly.

Consider the time evolution of the total energy in a thypersurface,

d

dt

Ztn̂ctaTac

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirðr� 2MÞ

pr2drd�

¼Zt

@

@tðn̂ctaTacÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirðr� 2MÞp r2drd�

¼ ��Zt

@

@�ðn̂ctaTacÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirðr� 2MÞp r2drd�

¼ ��Z

d�d

d�

�ZZn̂ctaTac

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirðr� 2MÞ

pr2 sin�drd�

�¼ 0; (B18)

where we have used the helical symmetry LF ¼ ð @@t þ� @@�ÞF ¼ 0, for any F. In other words, time evolution isreally just axial rotation.

Thus, for a circular orbit r ¼ ro, Eq. (B15) becomes�Zr¼2M

�l̂cr2d�d�þZr¼R

r̂cr2dtd�

�taTca

¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 3M

ro

s Z tfti

Ftdt: (B19)

For convenience, we may set the arbitrary parameter �on the horizon to be t. If we then differentiate both sideswith respect to t, we finally get

dE

dt

��r¼2MþdE

dt

��r¼R¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 3M

ro

sFt; (B20)

where

dE

dt

��r¼2M¼Ir¼2M

taTcað�l̂cÞr2d�; (B21)

dE

dt

��r¼R¼Ir¼R

taTcar̂cr2d�: (B22)

These are the general formulas for the energy flux atspatial infinity and the event horizon. In the next section,we write them out explicitly in terms of c and itsderivatives.Finally, Eq. (B20) can be written in the form

dE

dt

��r¼2MþdE

dt

��r¼R¼ mdEpdt

; (B23)

where Ep ¼ �taua is the specific energy of a particlemoving along a geodesic. This is just a statement of theconservation of energy: the energy lost by the charge isalso the energy flowing through r ¼ 2M and r ¼ R.

2. Scalar energy flux in Kerr-Schild coordinates

For convenience, wewrite Eqs. (B21) and (B22) in termsof c and its derivatives, in Kerr-Schild coordinates. Theseformulas are essentially the same except for their unitnormals, where one is null and the other is spacelike.We first note that in Kerr-Schild coordinates, the

Schwarzschild metric and its inverse are simply

gab ¼ �ab þ 2Mr kakb; (B24)

gab ¼ �ab � 2Mr

kakb; (B25)

ka ¼�1;xir

�; ka ¼

�1;� x

i

r

�; (B26)

where r2 ¼ x2 þ y2 þ z2 and �ab ¼ diagð�1; 1; 1; 1Þ.We begin first with the energy flux through the event

horizon. The event horizon is essentially a surface ofconstant retarded time u¼ tðSÞ�rðSÞ�2M lnðrðSÞ=2M�1Þ,where the subscript S means that these are Schwarzschildcoordinates. In Kerr-Schild coordinates, these surfaces ofconstant u are

t ¼ rþ 4M lnðr=2M� 1Þ þ C; (B27)where C is just a constant. Any particular surface in thisfamily can be defined parametrically by the equations

t ¼ �; (B28)

x ¼ rð�Þ sin� cos�; (B29)


084021-19

y ¼ rð�Þ sin� sin�; (B30)

z ¼ rð�Þ cos�; (B31)where rð�Þ is defined implicitly by the relation

� ¼ rþ 4M lnðr=2M� 1Þ: (B32)With this, the null generator of the surface (which is alsonormal to it) is

l̂ a � @xa

@�¼

�1;

�r� 2Mrþ 2M

�xi

r

�: (B33)

With the stress-energy tensor given by Eq. (B12) andusing the expressions given in Eqs. (B24)–(B26), a smallamount of algebra yields

Tabtal̂b ¼ _c 2 þ

�r� 2Mrþ 2M

�_c ni@ic þ 12

��r� 2Mrþ 2M

�@cc @

cc ; (B34)

where the overdot means a derivative with respect to t. Atr ¼ 2M, the energy flux is then simply just

dE

dt

��r¼2M¼ �4M2Ir¼2M

_c 2d�: (B35)

The normal oneform associated with the hypersurfacer ¼ R is a � @ar ¼ ð0; xi=rÞ. The corresponding normal-ized vector is then

r̂ a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r

r� 2Mr �

2M

r;

�1� 2M

r

�xi

r

�: (B36)

This leads to the following:

Tabtar̂b ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir

r� 2Mr �

2M

r_c 2 þ

�1� 2M

r

�_c @rc

�:

(B37)

Thus, the flux through r ¼ R is justdE

dt

��r¼R ¼ R2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R

R� 2M

s

�Ir¼R

�2M

R_c 2 þ

�1� 2M

R

�_c @rc

�d�:

(B38)

Taking the limit r ! 1, this reduces to the more familiarflat spacetime case,

Tabtar̂b ¼ _c @rc : (B39)

And so, we have for the energy flux at spatial infinity,

dE

dt

��r¼1¼ limR!1R2IR

_c @rc d�: (B40)

One of the internal checks we perform is to verify that

Eq. (B20) holds by computing the t component of the self-force and the fluxes given in Eqs. (B35) and (B38).

APPENDIX C: A TOY ILLUSTRATION OF OURMETHOD

Partial differential equations with two dramatically dif-ferent length scales are difficult to solve with numericalanalysis. Consider the example of a scalar field ’ of aspherical object at rest, centered at ~r ¼ 0 and with a smallradius ro. And, the small object has a scalar charge density�ðrÞ, with �ðrÞ being constant for r � ro and �ðrÞ beingzero for r > ro. The small object is inside a much largerodd-shaped box, and ’ ¼ 0 on the surface of the box is theDirichlet boundary condition for ’. For simplicity, assumethat spacetime is flat and, with the object at rest, there is noradiation and the field equation is elliptic. Then

r2’ ¼ �4��; (C1)where ~r is the usual three-dimensional flat-space gradientoperator.The challenge is to numerically determine the actual

field ’act as a function of ~r everywhere inside the box,subject to the field Eq. (C1) and the boundary condition,and then to find the total force on the small object whichresults from its interaction with ’act.On the one hand, a very fine numerical grid is necessary

to resolve ’ in and around the object particularly forobtaining the force acting on the object. On the otherhand, a coarse grid would suit the boundary conditionwhile spe

physical review d 084021 (2009) self-force with (3þ1) codes:...

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