Kavli IPMU, 18 June 2013 [arXiv:1306.0884]

Green Function Approach toSelf-force CalculationsBarry WardellUniversity College Dublinin collaboration with Marc Casals, Sam Dolan, Adrian Ottewill, Chad Galley and Anil Zenginoglu

ESA/SRE(2011)19December 2011

Assessment Study Report

NGORevealing a hidden Universe:

opening a new chapter of discovery

European Space Agency

EMRIs

✤ A major goal of the space-based Gravitational Wave programme is to study Extreme Mass Ratio Inspirals.

✤ Many orbits.✤ Expect generic (eccentric,

inclined) orbits.✤ Larger black hole generally

spinning.✤ Ultimate goal: ~104 accurate

evolved generic orbits in Kerr with gravitational self-force.

Image credit: eLISA/NGO Yellow book (ftp://ftp.rssd.esa.int/pub/ojennric/NGO_YB/NGO_YB.pdf)

✤ Model the system using black hole perturbation theory => perturbative parameter is the mass/charge (q, e, μ≡m/M)

✤ Solve the coupled system of equations for the motion of a point particle and its retarded field.

Self-force

⇤�ret = �4⇡q

Z�

4(x� z(⌧))p�g

d⌧

�R = �ret � �S

fa = ra�R

⇤Areta �Ra

bAretb =

�4⇡e

Zgaa0

u

a0p�g�4(x, z(⌧))d⌧

⇤hretab + 2Ca

cbdhret

cd =

�16⇡µ

Zga0(au

a0gb)b0u

b0p�g�4(x, z(⌧))d⌧

ARa = Aret

a �ASa hR

ab = hretab � hS

ab

fa = gabucAR[c,b] fa = kabcdhR

bc;d

dm

d⌧= u�f�a↵ = (g↵� + u↵u�)f�

Scalar Electromagnetic Gravitational

✤ System is coupled: retarded field depends on the entire past world-line and the world-line depends on field => delay differential equation.

✤ δ-function sources are difficult tohandle numerically.

✤ Retarded field diverges like 1/r nearthe world-line.

Practical considerations

Several considerations arise when trying to turn this formal prescription into a practical calculation scheme:

Approaches

✤ Several approaches have been developed for dealing with the numerical problems of point sources and singular fields.

✤ These broadly fall into three different categories

Mode-sum

5 10 15 200.04

0.06

0.08

0.10

0.12

0.14

Effective SourceGreen function

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 2 4 6 8 10 12

Dis

tant P

ast G

reen F

unction

T = t - t’ - !* - !*’

" = 1/6

# = $/2

Green function approach

Green function

✤ Solution of the wave equation with an impulsive source

✤ For self-force calculations, we work with the retarded Green function✤ Given the Green function, we can compute solutions of the sourced

wave equation by integrating the Green function against the source

✤ But, this diverges on the world-line,

Self-force via Green functions by worldline integration

Sam R.Dolan,1 Marc Casals,2 Chad R.Galley,3, 4 Adrian C. Ottewill,2 Barry Wardell,2 and Anıl Zenginoglu4

1School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom.2School of Mathematical Sciences and Complex & Adaptive Systems Laboratory,

University College Dublin, Belfield, Dublin 4, Ireland3Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California USA4Theoretical Astrophysics, California Institute of Technology, Pasadena, California USA

I. INTRODUCTION

The calculation of self-force (SF) in gravitationalphysics is an important problem both from a funda-mental and an astrophysical point of view (see reviews[1, 2]). This subject draws upon its deep roots in the his-tory of theoretical physics, and in particular, upon theconcept of radiation reaction in electromagnetism. TheAbraham-Lorentz-Dirac formula describes how an accel-erated charge experiences a breaking force, as a conse-quence of loss of energy through radiation. As Diracshowed [3], the force may be interpreted as arising froma point-particle’s interaction with its own radiative field,if a symmetric but formally divergent part of the field isremoved. In a somewhat similar way, a compact massmoving in a curved spacetime will experience a self-forcedue to interaction with its own gravitational field. Pre-cise knowledge of self-force can be used to understandthe evolution of (and gravitational-wave signal generatedby) extreme mass-ratio inspirals (EMRIs): astrophysicalbinary systems in which a compact mass µ orbits a blackhole of mass M , in the limit µ/M ⌧ 1.

Gravitational self-force (GSF) was studied in 1997 byMino, Sasaki and Tanaka [4], and Quinn and Wald [5].Working independently, they obtained the so-called MiS-aTaQuWa equation: an expression for the force at first-order in the the mass ratio µ/M (in recent years, the rig-orous foundation for this formula has been established,and second-order (in µ/M) extensions have been pro-posed). The MiSaTaQuWa formula closely resemblesthe expression obtained in earlier decades by DeWitt &Brehme [6] for the electromagnetic (EM) self-force oncurved spacetime, in which the force may be split into‘local’ and ‘tail’ terms. The ‘local’ terms describe theinteraction between the source and the local spacetimegeometry, and the ‘tail’ terms show the dependence onthe past history of the source’s motion.

Consider a scalar field toy model. Such a model avoidstechnical complications due to gauge degrees of freedombut captures many essential di�culties, such as the de-pendence of tail terms on history. It was shown by Quinn[7] that the self-force experienced by a particle of scalar-charge q may be written as F

↵

= F

loc.

↵

+F

tail

↵

where F loc.

↵

is the ‘local’ force (discussed in Eq. ) and the tail termis given by

F

tail

↵

= q

2 lim✏!0

+

Z⌧�✏

�1r

↵

G(z(⌧), z(⌧ 0))d⌧ 0. (1)

Here, z(⌧) describes the particle worldline, i.e., the space-time coordinates z↵(⌧) of an arbitrary timelike trajectoryparameterized by proper time ⌧ , and r denotes (covari-ant) di↵erentiation. Note that the integral extends intothe infinite past but is truncated just before coincidence,at ⌧ � ⌧

0 = ✏ ! 0+. The Green function is defined by

2x

G(x, x0) =4⇡p�g

�

4(x� x

0) , (2)

along with appropriate causal boundary conditions.Here, g is the determinant of the spacetime metric g

µ⌫

(with signature (�+++)), and �

4(x�x

0) is the productof Dirac delta distributions in the four spacetime coordi-nates.The Green function plays not only an essential role in

the definition of self-force but also in the understandingof wave propagation. In linear theories, a complete math-ematical description is provided by a Green function, alsoknown as the fundamental solution in the mathematicalliterature. When the fundamental solution to a linearpartial di↵erential operator is known, any concrete solu-tion with arbitrary initial data and source can be con-structed by a simple convolution. From a physical pointof view, the Green function G(x, x0) can be interpreted asa correlation function or an expression for the amplitudeof propagation between any two points x and x

0. Natu-rally, much e↵ort has been devoted to the study and thecalculation of Green functions, but a su�ciently accuratedescription remained an open problem.There are various di�culties regarding the calculation

of Green functions in curved spacetimes. In flat space-time the retarded Green function (RGF) is supportedon the lightcone only, whereas in curved spacetimes theRGF has extended support also within the lightcone(here ‘lightcone’ refers the set of points x connected tox

0 via null geodesics). Further, the light-cone intersectsitself generically along caustics due to focussing causedby curvature. Finally, a source may encounter ‘echoes’ ofits own field from earlier epochs due the trapping (thiscan happen also due to topological trapping without cur-vature).These features of the RGF make it di�cult to evalu-

ate the integral in (1) because the amplitude at a pointdepends on the entire past history of the source. Earlyattempts were founded upon the Hadamard parametrix[8], which gives the RGF in the form

G(x, x0) = ⇥�(x, x0) (U(x, x0)�(�)� V (x, x0)⇥(��)) .

(3)

�(x) = q

Z

�Gret(x, z(⌧

0))d⌧ 0

x = z(⌧)

Green function regularization

✤ Mino, Sasaki and Tanaka and Quinn and Wald derived an equation (MiSaTaQuWa) for the self-force in terms of a tail integral of the retarded Green function over the past world-line

✤ The tail integral appears only in curved spacetime and contains information about the non-locality of the self-force.

✤ This can be understood geometrically in terms of null geodesics wrapping around the black hole and re-intersecting the world-line.

f

a= (local terms) + lim

✏!0q

2

Z ⌧�✏

�1ra

Gret(x, x0)d⌧

0

Tail contribution to the self-force

Timelike

r = 3M

Null H1LNull H2LNull H3LNull H4L

-5 0 5

-5

0

5

xêM

yêM

Null geodesics intersecting a circular geodesic orbit

x

x1

x2

x3

x4

Timelike

r ! 3M

Null !1"

Null !2"

Null !3"

Null !4"

"10 "5 0 5 10"15

"10

"5

0

5

10

x#M

y#M

Null geodesics intersecting an eccentric geodesic orbit

x

x1

x2

x3x4

Tail contribution to the self-force

11

FIG. 7. Illustration of the singular structure of the Greenfunction. This plot shows a ‘snapshot’ of the scalar-fieldGreen function (GF) in the xy plane at t = t

1

associatedwith the initial point at x

0 = 8M, y

0 = 0, t0 = 0. Note thatt

1

⇡ 41.837M is the coordinate time it takes for part of theinitial wavefront to orbit the BH once and return to x = x

0,y = 0. On the outermost wavefront [pink line], which liesoutside the first caustic point at x ⇡ �24.36M, y = 0, theGF is proportional to a positive delta function [Eq. (50)]. Onthe innermost wavefront [blue line], inside the second caus-tic point (at x = 8M, y = 0), the GF is proportional to anegative delta function. Across the intermediate wavefront[green line] which joins the two caustics, the GF has antisym-metric ‘wings’ approximated by 1/� (where � is the Syngeworldfunction). The dotted black line shows the position ofthe wavefront estimated from the lowest-order asymptotics ofthe QNM sum (see also Fig. 5). The red (positive) and blue(negative) shadings indicate the value of the GF (computedvia the asymptotic approximations [Eq. (55)]). The coloureddisk indicates the area of spacetime in which the sum overQNM overtones [Eq. (36)] is convergent.

Schwarzschild QN modes and their excitation factors.We used the asymptotic results to investigate the QNMcontribution to the scalar-field retarded Green function.This led to insight into the singular structure of the GFnear the lightcone. We showed that the form of the GFchanges every time a caustic is encountered, and thatit undergoes a four-fold repeating pattern [see Eq. (50),(52), (55) and Fig. 7].

A four-fold singular structure for a 4D spherically-symmetric spacetime (like the Schwarzschild black hole)has been anticipated (e.g. [32, 33]); however, we believethis is the first time it has been demonstrated for a blackhole spacetime using a QN mode representation. As wasdiscussed in Sec. VE of Ref. [33], an alternative way tomake sense of the four-fold pattern is with a Hadamard

ansatz [30] for the ‘direct’ part of the GF, i.e.

Gdir.

ret

(x, x0) ⇠ lim✏!0

+Re

i

�1/2

⇡� + i✏

�

⇠ Re

�1/2

✓�(�) +

i

⇡�

◆�. (56)

Here � = �(x, x0) is the Synge world function [48] (halfthe squared interval along the geodesic joining x to x0),and � = �(x, x0) is the van Vleck determinant, a mea-sure of the degree of focussing of neighbouring geodesics[55]. The determinant is singular at the caustic. In pass-ing through a caustic on the Schwarzschild spacetime, �changes sign. If we presume that this leads to an addi-tional factor of �i in the square bracket of (56) each timea caustic is traversed, we recover once more the four-foldpattern �(�), 1/(⇡�), ��(�), �1/(⇡�), �(�), etc.The e↵ect of caustics upon wave propagation has been

studied with a variety of methods (e.g. WKB methods,path integrals, functional integration, etc.) in a rangeof disciplines including optics [50], acoustics [51], seis-mology [44], symplectic geometry [52], and quantum me-chanics [53, 54]. The formation of caustics in black holespacetimes has also been studied extensively, in the con-text of strong gravitational lensing [26–28]. Some furtherinvestigation of the e↵ect of caustics on the details ofwave propagation (i.e. on the Green function) in curvedspacetimes now seems to be warranted. In particular,analysis of the singular structure of the Green functionnear a rotating (Kerr) black hole, an axisymmetric space-time, would be of great interest [49]. For this challenge,the utility of the QN expansion method appears ratherlimited [15]. Instead, let us focus (for now) on what morecan be done in Schwarzschild with our methods.In Paper I [13] we introduced the QNM expansion

method and applied it to investigate the QN frequencyspectrum. In this work – Paper II, let’s say – we haveexplored the geometric interpretation, using first-order(in L) expansions to bring qualitative understanding ofthe singular structure of the GF. It seems there is scopefor a future work (i.e. Paper III) in which higher-order(analytic) expansions are combined with numerical meth-ods [10, 24] in the small-l regime, to achieve quantitative

calculations of QNM sums [such as Eq. (14)].Let us suggest two areas where this work may find

further applications. The first is in the calculation of thegravitational self-force [33, 56], which is required to (e.g.)model extreme mass-ratio black hole inspirals [57], testthe cosmic censorship hypothesis [58], and calibrate e↵ec-tive one-body theories [59]. With complete (i.e. global)knowledge of the GF, it is straightforward (in principle)to compute the self-force; yet the former is not trivial toobtain. In the ‘method of matched expansions’ [60] oneseeks to match a ‘quasilocal’ expansion of the GF [61],valid inside a normal neighbourhood (approximately, upto the first caustic), to a ‘distant-past’ expression. TheQNM expansion makes up part of the latter; however,it must also be augmented by an accurate calculation ofthe branch cut integral. This remains to be achieved.

3

-4

-2

0

2

4

6

8

-4 -2 0 2 4 6 8

y /

M

x / M

(0)

(1)

(2)

FIG. 1. Left: Light rays orbiting a Schwarzschild black hole. This plot shows that two points at fixed spatial coordinates inthe vicinity of a black hole are linked by multiple null geodesics (i.e. light rays). Here we show the ‘direct’ ray (ray 0) [passingthrough �� = ⇡/2] and two orbiting rays [passing through �� = 3⇡/2 (ray 1) and �� = 5⇡/2 (ray 2)] in the xy plane.Right: The self-intersecting light cone (reproduced from Fig. 1.1 of [29]; see also [26]). The right plot shows the lightcone of aspacetime point (at the apex of the cone) in the vicinity of a black hole (the horizon of the BH is visible as the smaller circle).The coordinate time t runs vertically downwards, and one spatial dimension (z) has been suppressed. Note the formation ofcaustics (lines along which the lightcone intersects itself). See also Fig. 5 and 7.

-6

-5

-4

-3

-2

-1

0

-4 -3 -2 -1 0 1 2 3 4

Im(2

M!

ln)

Re(2M!ln)

l = 2 l = 10

n = 0

n = 11

FIG. 2. The gravitational quasinormal mode spectrum of theSchwarzschild black hole. The plot shows the real and imagi-nary parts of the gravitational (|s| = 2) QNM frequencies, forangular momenta l = 2 . . . 10 and overtones n = 0 . . . 11.

III. THE GREEN FUNCTION ANDQUASINORMAL MODE SUMS

The role of QNMs in wave propagation may be ap-preciated by examining the contribution of QNMs to theretarded Green function [20, 22]. Here, for simplicity, wewill consider the retarded Green function for the scalarfield on Schwarzschild spacetime, G

ret

(x, x0), which is de-

fined by

⇤x

Gret

(x, x0) = �4(x� x0) (8)

(where �4 is a covariant version of the Dirac deltadistribution) with appropriate causal conditions. TheGreen function may be used to solve two types of prob-lem. Firstly, if a field �(x) is generated by a sourceS(x) of compact support, i.e. ⇤�(x) = S(x), thenthe field is found from the spacetime integral �(x) =RG

ret

(x, x0)S(x0)d4x0. Secondly, if an unsourced field sat-isfies a Cauchy initial value problem then the field maybe found via an integral over a hypersurface. For exam-ple, if at t = 0 the field (in the exterior region of the BH)is given by �|

t=0

= �0

(x) and @t

�|t=0

= �0

, then thefield at later times t is given by

�(t,x) =

Z hG

ret

(t,x; 0,x0)�0

(x0)+

@t

Gret

(t,x; 0,x0)�0

(x0)] d3x0. (9)

The retarded Green function on Schwarzschild space-time may be expressed in terms of an inverse Laplacetransform,

Gret

(x, x0) =1

2⇡rr0

1X

l=0

(2l + 1)Pl

(cos �)⇥Z

+1+ic

�1+ic

Gl!

(r, r0)e�i!(t�t

0)d!. (10)

Here x, x0 are spacetime points at radii r, r0, separatedby coordinate time t � t0 and spatial angle �, [where

0 20 40 60 80 100 120 140-0.004

-0.002

0.000

0.002

0.004

DtêM

M2 G

ret

Retarded Green function along a circular geodesic orbit

x1 x2 x3 x4 x5 x6 x7

0 20 40 60 80 100 120 140-0.004

-0.002

0.000

0.002

0.004

DtêM

M2 G

ret

Retarded Green function along an eccentric geodesic orbit

x1 x2 x3 x4 x5 x6 x7

Green function regularization

✤ Mino, Sasaki and Tanaka and Quinn and Wald derived an equation (MiSaTaQuWa) for the self-force in terms of a tail integral of the retarded Green function over the past world-line

✤ If we can compute the Green function along the world-line, then we’re done: just integrate this to get the regularized self-force for any orbit.

✤ The difficulty is in developing a strategy for computing the Green function over a sufficiently large portion of the world-line.

f

a= (local terms) + lim

✏!0q

2

Z ⌧�✏

�1ra

Gret(x, x0)d⌧

0

Matched expansions calculation of the Green function

Matched expansions

✤ Compute the Green function using matched asymptotic expansions[Anderson and Wiseman, Class. Quantum Grav. 22, S783 (2005); M. Casals, S. R. Dolan, A. C. Ottewill, and B. Wardell, Phys. Rev. D 79, 124043 (2009) ]

✤ Separately compute expansions of the Green function valid in the recent past (Taylor series) and in the distant past (quasi-normal mode + branch cut/numerical time-domain evolution).

✤ Stitch together expansions in an overlapping matching region to give the full Green function.

Timelike worldline of the particle

Current location of the particle - z(τ)

Matching point - z(τm)

Integral in quasilocal region

Integral outside quasilocal region

Boundary of causal domain where Hadamard form is valid

Z ⌧m

�1raGret(z(⌧), z(⌧

0))d⌧ 0

First null geodesic which re-intersects the worldline

Z ⌧�

⌧m

r⌫Gret(z(⌧), z(⌧0))d⌧ 0

Expansion in quasilocal region

Early times - quasilocal expansion

✤ For early times, we can use the Hadamard form

✤ Only need V(x,x’) since tail integral is inside the light-cone. Compute this as a series expansion (WKB) valid for x and x’ close together; Padé re-summation to increase radius of convergence, accuracy.

G(x, x0) = ⇥�(x, x0) (U(x, x0)�(�)� V (x, x0)⇥(��))

V (x, x

0) =

1X

i,j,k=0

vijk(r) (t� t

0)

2i(1� cos �)

j(r � r

0)

k

Expansion in distant past region

Late times - spectral expansion

✤ Spectral decomposition of the Green into spherical harmonic and Fourier modes

✤ Sum over l and integration over omega can be rendered convergent provided x and x’ are far enough apart.

G

ret(x, x

0) =

1X

`=0

1

r r

0 (2`+ 1)P`(cos �)Gret` (r, r

0;�t)

Gret` (r, r0;�t) ⌘ 1

2⇡

Z 1+ic

�1+icd! G`(r, r

0;!)e�i!�t

d2

dr2⇤+ !2 � V (r)

�G`(r, r

0;!) = ��(r⇤ � r0⇤)

Late times - spectral expansion

✤ Integral over frequencies may be done by deforming the contour into the complex-frequency plane [Leaver (1988)].

✤ Residue theorem of complexanalysis dictates that one mustaccount for singularities of theintegrand.

✤ Simple poles (quasi-normalmodes) and a branch cut down the negative-imaginary axis on the complex-frequency plane.

✤ High-frequency arc may be ignored; only contributes at “early” times.

ReHwL

ImHwL

HF

BC

QNMs

Distant past Green function

GBC + GQNM

GQNM

GBC

50 100 150 200 250 30010-12

10-10

10-8

10-6

10-4

0.01

DtêM

M2 »G»

Contribution of QNM and BC to Green function Hcircular orbitL

Distant past Green function

G, rBC + G, rQNM

G, rQNM

G, rBC

50 100 150 200 250 30010-12

10-10

10-8

10-6

10-4

0.01

DtêM

M3 »G ,

r»

Contribution of QNM and BC to Green function derivative Hcircular orbitL

Numerical calculation in distant past region

Numerical time-domain evolution

✤ Alternative option is to numerically compute the retarded Green function using time-domain evolution.

✤ Numerically evolve a wave equation for the Green function.

✤ Still need to make use of quasi-local expansion for the recent past, but late time calculation is much easier than with quasi-normal modes + branch cut.

Numerical time-domain evolution

✤ Given initial data on a spatial hyper-surface Σ and the full Green function, can determine the solution at an arbitrary point xʹ′ in the future of Σ (Kirchhoff theorem)

✤ Basic idea: choose as initial data

then in the limit σ → 0

3

0 5 10 15 20p

M0.0

0.2

0.4

0.6

0.8

1.0e

-20

-10

0

10

20

-10 0 10 20 30

FIG. 1: SD: This figure may not be particularly relevant -- we can remove it later. Knowledge of the Green func-tion G(x, x0) 8x0 enables the calculation of SF at x on any trajectory that passes through the base point x. The left plot showsthe 2D parameter space for the bound geodesic orbits (with semi-latus rectum p and eccentricity e). Stable bound orbits lieto the right of the separatrix [bold line] at p/M = 6 + 2e. The shaded area shows the stable bound orbits that pass throughthe point r0 = 10M . The right plot shows a sample of geodesics passing through r0 = 10M , indicated by red points in the leftplot.

contribution to the retarded Green function as requiredin the present context. It is also straightforward to takepartial derivatives of these expressions at either space-time point to obtain the derivative of the Green function.

As proposed in Ref. [23], we have not used the quasi-local expansion in the specific form of Eq. (4). Instead,we used a Pade resummation in order to increase theaccuracy and extend the domain of validity of the quasi-local expansion. While not essential, this proved veryuseful for increasing the region of overlap between thequasi-local and numerical Gaussian domains.

2. Late-time behavior

B. Numerical approximations to the retardedGreen function

We use two closely related prescriptions for computinga numerical approximation to the Green function. Inboth methods a delta distribution is approximated bya Gaussian of finite width �, and we obtain the Greenfunction by taking the limit � ! 0.

1. Kircho↵ representation with Gaussian initial data

Given initial data on a spatial hypersurface ⌃ and theGreen function for the wave equation, Kirchho↵’s theo-rem may be used to determine the solution at an arbi-trary point x0 in the future of ⌃,

�(x0) = � 1

4⇡

Z

⌃

[G(x, x0)r↵�(x)��(x)r↵

G(x, x0)]d⌃↵

,

(5)

where d⌃↵

= �n

↵

p�d

3

x is the surface element on ⌃,n

↵

= [�↵, 0, 0, 0] is the unit normal to the surface, ↵ isthe lapse and � is the determinant of the induced metric.In the case of the Schwarzschild spacetime in Kerr-

Schild coordinates,

↵ =

✓1 +

2M

r

◆�1/2

,

p� =

✓1 +

2M

r

◆1/2

r

2 sin ✓ (6)

and [for �(x) = 0] the Kirchho↵ integral takes the form

�(x0) =1

4⇡

Z

⌃

G(x, x0)@t

�(x)

✓1 +

2M

r

◆r

2 sin ✓drd✓d�.

(7)Choosing as initial data

@

t

�(x) =4⇡

(2⇡�2)3/2

✓1 +

2M

r

◆�1

e

� (x�x0)2+(y�y0)2+(z�z0)2

2�2,

(8)in the limit � ! 0 we find that Eq. (7) becomes

�(x0) =

Z

⌃

G(x, x0)�3

(x� x

0

)r2 sin ✓drd✓d�

= G(x0

, x

0). (9)

In this way, we can recover the Green function with onepoint fixed at x

0

by numerically evolving initial data ofthe form (8) for a sequence of values of � and extrapo-lating to � = 0.A similar approach may be used to compute derivatives

of the Green function. In that case, choosing initial data

@

t

�(x) =4⇡

r

2(2⇡�2)3/2

✓1 +

2M

r

◆�1

@

r

✓r

2

e

� (x�x0)2+(y�y0)2+(z�z0)2

2�2

◆

(10)

@t�(x) =4⇡

(2⇡�2)3/2e

� |x�x0|2

2�2�(x) = 0

3

0 5 10 15 20p

M0.0

0.2

0.4

0.6

0.8

1.0e

-20

-10

0

10

20

-10 0 10 20 30

FIG. 1: SD: This figure may not be particularly relevant -- we can remove it later. Knowledge of the Green func-tion G(x, x0) 8x0 enables the calculation of SF at x on any trajectory that passes through the base point x. The left plot showsthe 2D parameter space for the bound geodesic orbits (with semi-latus rectum p and eccentricity e). Stable bound orbits lieto the right of the separatrix [bold line] at p/M = 6 + 2e. The shaded area shows the stable bound orbits that pass throughthe point r0 = 10M . The right plot shows a sample of geodesics passing through r0 = 10M , indicated by red points in the leftplot.

contribution to the retarded Green function as requiredin the present context. It is also straightforward to takepartial derivatives of these expressions at either space-time point to obtain the derivative of the Green function.

As proposed in Ref. [23], we have not used the quasi-local expansion in the specific form of Eq. (4). Instead,we used a Pade resummation in order to increase theaccuracy and extend the domain of validity of the quasi-local expansion. While not essential, this proved veryuseful for increasing the region of overlap between thequasi-local and numerical Gaussian domains.

2. Late-time behavior

B. Numerical approximations to the retardedGreen function

We use two closely related prescriptions for computinga numerical approximation to the Green function. Inboth methods a delta distribution is approximated bya Gaussian of finite width �, and we obtain the Greenfunction by taking the limit � ! 0.

1. Kircho↵ representation with Gaussian initial data

Given initial data on a spatial hypersurface ⌃ and theGreen function for the wave equation, Kirchho↵’s theo-rem may be used to determine the solution at an arbi-trary point x0 in the future of ⌃,

�(x0) = � 1

4⇡

Z

⌃

[G(x, x0)r↵�(x)��(x)r↵

G(x, x0)]d⌃↵

,

(5)

where d⌃↵

= �n

↵

p�d

3

x is the surface element on ⌃,n

↵

= [�↵, 0, 0, 0] is the unit normal to the surface, ↵ isthe lapse and � is the determinant of the induced metric.In the case of the Schwarzschild spacetime in Kerr-

Schild coordinates,

↵ =

✓1 +

2M

r

◆�1/2

,

p� =

✓1 +

2M

r

◆1/2

r

2 sin ✓ (6)

and [for �(x) = 0] the Kirchho↵ integral takes the form

�(x0) =1

4⇡

Z

⌃

G(x, x0)@t

�(x)

✓1 +

2M

r

◆r

2 sin ✓drd✓d�.

(7)Choosing as initial data

@

t

�(x) =4⇡

(2⇡�2)3/2

✓1 +

2M

r

◆�1

e

� (x�x0)2+(y�y0)2+(z�z0)2

2�2,

(8)in the limit � ! 0 we find that Eq. (7) becomes

�(x0) =

Z

⌃

G(x, x0)�3

(x� x

0

)r2 sin ✓drd✓d�

= G(x0

, x

0). (9)

In this way, we can recover the Green function with onepoint fixed at x

0

by numerically evolving initial data ofthe form (8) for a sequence of values of � and extrapo-lating to � = 0.A similar approach may be used to compute derivatives

of the Green function. In that case, choosing initial data

@

t

�(x) =4⇡

r

2(2⇡�2)3/2

✓1 +

2M

r

◆�1

@

r

✓r

2

e

� (x�x0)2+(y�y0)2+(z�z0)2

2�2

◆

(10)

Numerical time-domain evolution

✤ So, we evolve the homogeneous wave equation with initial data

for a sequence of values of σ then extrapolate to σ → 0 to get the Green function. Equivalent to smoothly cutting off the divergent sum over spherical-harmonic l modes, rendering it convergent.

✤ Somewhat surprisingly, this works very well for computing the self-force, even for quite large σ/M ~ 0.1 - 1.

✤ Narrower Gaussian improves resolution of spikes at null-geodesic crossings. Between crossings, even a large σ is sufficient.

@t�(x) =4⇡

(2⇡�2)3/2e

� |x�x0|2

2�2�(x) = 0

G(x0, x’) for x0 = {0, 6M, 0, π/2} in Schwarzschild spacetime

Distant past Green function

0 50 100 150 200 250 300

-6

-4

-2

0

t

GHx,x

'L

Numerical Green function along a circular orbit, r=6M

Matching early- and late-time expansions

Matching early- and late-time expansions

18 20 22 24 26

0.001

0.01

0.1

DtêM

»1-GQLêG

DP»

16 18 20 22 24 26 28

10-4

0.001

0.01

0.1

DtêM

M2 »G»

Green function along circular orbit

Early times - quasilocal expansion

0 50 100 150 200 250 300-8

-7

-6

-5

-4

-3

-2

t

GHx,x

'L

Green function along a circular orbit, r=6M

Late times - branch cut tail

✤ At very late times, Green function dominated by branch cut because quasi-normal modes decay much faster. Use analytic expressions.

0 500 1000 1500 2000-10

-8

-6

-4

-2

t

GHx,x

'L

Green function along a circular orbit, r=6M

Matched Green function

0 100 200 300 400 500-8

-7

-6

-5

-4

-3

-2

t

GHx,x

'L

Green function along a circular orbit, r=6M

Computing the self-force

Computing the self-force

✤ Integrate matched Green function to get regularized field/self-force

18 20 22 24 260.001

0.01

0.1

1

tm

»1-FrêF r

exact»

Partial self-force

Exact value

0 50 100 150 200 250 300-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

DtêM

M2 êq2

F r

Partial self-force for circular orbit

Generic orbits

✤ Green function approach works equally well with all types of orbit.

0 50 100 150

10-11

10-9

10-7

10-5

0.001

0.1

t

dGêdr

Radial derivative of Green Function along eccentric orbit with p=7.2, e=0.5 at r=6M

Generic orbits

✤ Green function approach works equally well with all types of orbit.

18 20 22 240.001

0.01

0.1

1

tm

»1-FrêF r

exact»

Partial self-force

Exact value

0 50 100 150 200-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

DtêM

M2 êq2

F r

Partial self-force for eccentric orbit

Conclusions

✤ Advantages:✤ Compute the Green function

once, get the self-force for all orbits (including unbound, highly-eccentric, zoom-whirl, ultra-relativistic, which are difficult or inaccessible with existing methods).

✤ Avoids numerical cancellation by directly computing regularized field.

✤ May yield geometric insight.✤ Green function can be

applied to other problems.

✤ Disadvantages:✤ Computing the Green

function can be hard.✤ Have to compute the Green

function for all pairs of points x and x’.

✤ Not naturally suited to self-consistent evolution.

✤ Second order not so well understood.

Conclusions and prospects

✤ Green functions are a flexible approach to self-force calculations.✤ Compute Green function once, get all orbits through that base point.✤ Need a separate calculation for each point on the orbit.✤ Gives insight into how much of the past matters for the self-force.✤ Interesting orbits not accessible by other means?✤ Schwarzschild case now complete [arXiv:1306.0884].✤ Application to Kerr spacetime.✤ Extension to gravitational case.✤ Self-force as a test of alternative theories of gravity?✤ Other applications beyond self-force.