gr-qc/9702002 31 Jan 1997 z August Short PACS numbers University Park PA USA The Present title eth address The eth formalism in numerical relativity allows new in the nonlinear regime RobinsonTrautman equation and reveals some new featureson of gravitational top ologically waveforms spherical manifolds dimensions and the calculation of the curvature scalar of arbitrarily curved geometries Abstract USA Department which spherical with singularities R GomezL Lehner P Papadopoulos isolated formalism x Cb and Department interpolations use source eective of We of The metho d employs two overlapping stereographic co ordinate patches tensor Physics co ordinates We present of computational to ol b etween test elds Astronomy in and the a numerical in are natural such nite Astronomy the formalism with spherical and patches dierence The formalism is applied to the Astrophysics for co ordinates University dealing in the the version as z relativity and the evolution of regions with in generation of radiation from an Pennsylvania of J Winicour of a a Pittsburgh manner the wide of overlap eth waves variety which formalism State Pittsburgh in solution of the It of three avoids University systems provides spatial which p olar PA in a

Introduction

The inspiral and merger of a binary black hole system is anticipated to b e the prime

source of gravitational waves for future wave detectors Calculation of the emitted

waveform by means of emerging sup ercomputer technology is the goal of the Binary

Black Hole Grand Challenge In this pap er we present a formalism which plays a

strategic part in extracting the waveform pro duced in this three dimensional problem

At large distances from a compact source the wavefronts of any radiation eld

b ecome spherical this leads naturally to the use of a spherical co ordinate system

Indeed spherical co ordinates and spherical harmonics are standard analytic to ols in

the description of radiation But the use of spherical co ordinates in numerical work

leads invariably to the vexing problem of co ordinate singularities at the origin and

along the p olar axis Finite dierence approximations suer particularly b ecause they

have no natural way of enforcing the correct b oundary conditions on their solutions

As a result spherical co ordinates have mainly b een used in axisymmetric systems

where the p olar singularities may b e regularized by standard tricks In the absence

of symmetry these techniques do not easily generalize and they would b e esp ecially

prohibitive to develop for tensor elds

This pap er presents an approach that should b e of interest to researchers working

in numerical relativity in problems where spherical co ordinates are a natural to ol

The eth formalism and the asso ciated spinweighted spherical harmonics

allow a simple and unied description of vector and tensor elds without the undue

complexity of traditional vector and tensor harmonics Unfortunately these techniques

have remained obscure and unfamiliar to many researchers who would b enet from

them

We present here a nite dierence version of the eth formalism with applications

to numerical relativity The framework for the computational application of the eth

formalism is developed in Sec Although our presentation is on the algorithmic

level we present enough detail to direct a well dened transition from algorithm to

co de development In Sec we present tests and applications of the formalism which

demonstrate b oth its accuracy and usefulness The rst application the evolution of

scalar waves is illustrative of the techniques necessary to carry out a dimensional

evolution using spherical co ordinates The second application the numerical calculation

of the curvature scalar of a top ologically spherical manifold illustrates the global and

tensorial features of the metho d This calculation is at the heart of several imp ortant

problems such as the computation of the Hawking mass The third application

solution of the RobinsonTrautman equation reveals new and unexp ected nonlinear

prop erties of the gravitational waveform This work illustrates how computational

relativity can greatly b enet from approaches based up on the b est analytic to ols

The Eth Formalism

We present here a nite dierence treatment of the sphere based up on i the standard

metho d of describing the global dierentiability of functions by means of lo cal co ordinate

patches and iithe eth formalism for describing dierentiation of elds on the sphere

Patching

Let and lab el the p oints on the sphere Then the real and imaginary parts of

i

the stereographic co ordinate tan e provide a smo oth co ordinatization of the

N

sphere excluding the p oint at the south p ole Similarly provides a

S N

smo oth co ordinatization except at the north p ole These two stereographic co ordinate

patches are sucient to cover the sphere It would b e p ossible in principle to cover

the sphere with a number of nonsingular patches by selecting several directions

on the sphere and asso ciating with them a lo cal co ordinate patch and then

i i

excluding from those patches the lo cal p olar region Stereographic co ordinates provide

the most economical choice where only two nonsingular patches are used

A smo oth scalar eld on the sphere may b e represented in terms of smo oth

functions of on the lower hemisphere j j and as smo oth functions of

S S S N

on the upp er hemisphere j j The continuity of is ensured by requiring

N N

at the equator but in order to ensure smo othness this equality must

S S N N

extend to a common overlap region ab out the equator Our rst task is to implement

this in a nite dierence treatment

1 2 1 2

In order to construct the grid let S S and N N b e the real and imaginary

i

parts of and resp ectively Let the grid consist of the p oints S s M and

S N i

i

N n M where s and n are each pairs of integers with M s M and

i i i i

M n M Thus the grid p oints at the intersection of the real and imaginary

i

axes with the equator corresp ond to s M and s M and n M

i i i

and n M The grid p oints at the p oles corresp ond to s and n Note

i i i

that our choice of rectangular grid domains is not unique Other strategies can b e

followed to economize the number of p oints in the overlap region b etween the S and

N grids but the rectangular choice lends itself to a simple implementation at the grid

b oundaries

Given a smo oth function on the sphere we now give instructions for representing

it on the grid Denote the values of at p oints on the S grid by and the

S S S s s

1 2

values of at p oints on the N grid by In the overlap region the grid

N N N n n

1 2

representation must b e consistent with the condition However

S S N S

only in exceptional cases will a p oint that lies on the S grid also lie on the N grid To

deal with this we introduce

n in M

1 2

S N n n

1 2

2 2

n n

1 2

which gives the value of at a p oint which lies on the N grid Similarly we dene

S

s is M

1 2

N S s s

1 2

2 2

s s

1 2

In the same way in the overlap region and determine the values and

S N S s s

1 2

at p oints which lie on the S grid and and at p oints on the N

N S s s N n n S N n n

1 2 1 2 1 2

grid However the grid values are weaker information than the smo oth function

S s s

1 2

and do not by themselves determine the value of at p oints on the N grid Similarly

S

do not by themselves determine at a p oint on the S grid In these the values

N n n

1 2

cases interpolation is necessary to evaluate functions on one grid at p oints lying on the

other grid

In order to see how this applies to dierentiation consider the gradient of with

comp onents i in the S patch and i in the N patch Then at p oints

S S N N

S N

which do not lie on the grid b oundary the partial derivatives can b e approximated by

i

centered nite dierences in the standard way For example at the p oint S s the

i

appropriate nite dierence approximation is

S s +1s S s 1s

1 2 1 2

2

1

O

S

S

1

For a p oint on the grid b oundary eg the p oint S s we use the same

2

approximation except that the eld at the virtual grid p oint s M s is

i 2

approximated by the value obtained from interpolating In order to

N

NS (M +1)s

2

achieve an error of second order in the derivative the interpolation error must b e of

third order

This provides second order accurate nite dierence representations of the gradient

of at all grid p oints and in fact two representations corresp onding to i

S S

S

i

at p oints which lie in the overlap b etween the two grids The analytic and

N N

N

expressions are related by the transformation

N

S S N N

S N

S

As a check on the nite dierence approximation the nite dierence equivalent of

must hold to second order accuracy Equivalently the gradient may b e converted into a

scalar eld by taking basis comp onents and the scalar eld compared b etween patches

This is the strategy of the eth formalism that we shall consider later

Second derivatives of a scalar eld with resp ect to the co ordinates may also b e

approximated to second order by central dierencing The interior p oints in each grid

patch are handled by the standard techniques and the p oints on the grid b oundary

are handled by introducing a virtual grid p oint as ab ove The following fourth order

interpolation scheme ensures that b oth the rst and second derivatives are approximated

to second order accuracy at b oundary p oints

First we extend the grid by one p oint on each co ordinate direction The p oints at

the S grid with co ordinates s M s will dene the computational rightmost

i

b oundary of the grid etc Centered derivatives in the S direction require the eld

1

value at the virtual grid p oint s M s Note that this virtual p oint maps into

i

a p oint on the N grid which has at least two neighbors on each direction who also fall

inside the computational b oundary of the N grid namely their grid co ordinates satisfy

the relations jn j jM j

i

We interpolate the value at the virtual p oint by the following pro cedure First we

evaluate the co ordinates n of the virtual p oint which according to are given by

i

2 2 2 2

n s M s s n s M s s We then compute the indices i j of

1 1 2 2 b b

1 2 1 2

the lower left corner of the grid cell on the N grid where the virtual p oint falls ie

i n i and j n j The nine grid cells whose corners

b 1 b b 2 b

are given by the p oints i j with i i i and j j j fall

b b b b

entirely inside the N grid We then t a cubic Lagrange p olynomial to each of the four

set of p oints obtained by keeping the j index xed and obtain the value of the eld

at the p oints s j j j j Another cubic Lagrange p olynomial is then

1 b b

t through these four p oints to yield the value at the virtual p oint s s Note that

1 2

all grid p oints at the b oundary can b e treated with the same algorithm This metho d

has the computational advantage that it vectorizes on machines with the appropriate

hardware such as Cray sup ercomputers

We have carried out checks of the interpolation error Even for very coarse grids

the measured convergence rate is in go o d agreement with the theoretical limit of

four

Eth

The eth g formalism gives a compact and ecient manner of treating vector and tensor

elds on the sphere as well as their covariant derivatives In addition the asso ciated

spinweighted spherical harmonics provide a very tidy alternative to vector and tensor

spherical harmonics We give here a brief description of the basic ideas and the details

of how it works in some simple cases

The line element for the distance b etween neighboring p oints on the sphere is given

by the quadratic form

2 a b

ds q dx dx

ab

where the comp onents q of the metric tensor dep end up on the choice of co ordinates ab

2

2 2 2

In standard spherical co ordinates this takes the form ds d sin d but here we

use stereographic co ordinates for which

2 2 1 2 2 2

ds dS dS

S S

in the North patch with a similar expression holding in the South patch The eth

formalism represents the metric in terms of a complex basis vector q

a

q q q q q

ab a b a b

a a

where q q and q q Note that this departs from other conventions to

a a

p

avoid unnecessary factors of which would b e awkward in numerical work Since the

real and imaginary parts of q are b oth vector elds of unit length this basis cannot b e

a

assigned in a smo oth manner over the entire sphere so that two patches are necessary In

a a a

the S patch we make the choice q i so that its real and imaginary

S S

S 1 2

a a a

i parts line up with the S axes Similarly in the N patch q

N N

2 1 N

i

Any two complex bases q and q are related by a unitary transformation q e q

a a a a

where the phase is a real valued function In particular in the overlap b etween the

a i a i

patches q e q with e

S S

N S

Introduction of this complex basis allows us to represent any vector eld on the

a

sphere with comp onents v in terms of the complex scalar eld v q v Here it is

a a

a a

implicit that v represents either v q v or v q v dep ending up on which patch

S a N a

S N

i

is b eing used with v e v in the overlap region A eld v with this transformation

N S

prop erty is called a spinweight eld

Similarly a tensor eld on the sphere with comp onents w can b e represented by

ab

scalar elds First we decomp ose w into its symmetrictracefree part its trace part

ab

and its antisymmetric part

p u

w t q i q q q q

ab ab ab a b a b

cd c d c d

where p q w and u iq q q q w The real scalar elds p and u are

cd cd

indep endent of choice of basis and are called spinweight zero elds The symmetric

a b

trace free tensor eld t can b e represented by the complex scalar eld t t q q

ab ab

2i

Under change of basis we then have t e t and t is called a spinweight eld Thus

N S

an arbitrary tensor eld can b e represented by two real spinweight elds and a complex

spinweight eld These spinweighted elds are the irreducible representations of

the unitary group of basis transformations contained in the tensor eld t Note that

ab

a b

t t q q alternatively represents the tracefree symmetric part of w as a eld with

ab ab

spinweight minus

Derivatives of First and Second Order

In treating derivatives of tensor elds it is again convenient to represent them in terms

of spinweighted scalar elds This can b e accomplished by taking basis comp onents of

covariant derivatives r In the case of a scalar eld r which we represent

a a a

a

by the spinweight one quantity g q which represents the comp onents of the

a

gradient in terms of a complex eld Thus at the grid p oint s in the S patch centered

i

approximations to the derivatives give

2 2 2

s s

1 2

2

g i i O

S S S s +1s S s 1s S s s +1 S s s 1

1 2 1 2 1 2 1 2

i

The overlap condition b etween patches is g e g

N N S S

In the case of a vector eld the covariant derivative with resp ect to the unit sphere

metric is

c

r v v v

a b a b c

ab

where the connection co ecients are determined from the metric by

cd c

q q q q

a db b ad d ab

ab

a b

This is equivalent to the spinweight eld gv q q r v and the spinweight eld

a b

a b a

gv q q r v Here as ab ove we have v q v These complex elds contain all the

a b a

information ab out the gradient of v Finite dierence approximations can b e obtained

a

by rst reexpressing

a

gv q v v

a

and

a

gv q v v

a

where

a b

q q r q

a b

For a second order accurate nite dierence approximation the rst term in either

or may b e approximated by a centered dierence as in and the second term

evaluated using the grid values of in This leads to nite dierence expressions

2i

in b oth the S and N patches satisfying g v g v and g v e g v in the

N N S S N N S S

overlap

To treat dierentiation of twoindex tensor elds it suces to consider tracefree

a b

symmetric tensors represented by the pure spinweight eld t t q q Then

ab

the covariant derivative of t can b e expressed in terms of the spinweight eld

ab

a b c a b c

gt q q q r t and the spinweight eld gt q q q r t These can b e reexpressed

a bc a bc

as

a

gt q t t

a

and

a

gt q t t a

with given by As b efore these expressions have straightforward nite dierence

i

approximations in each patch with the overlap relationships g t e g t and

N N S S

3i

g t e g t This pro cedure generalizes to treat derivatives of tensor elds with

N N S S

an arbitrary number of indices

Second covariant derivatives can b e treated as pro ducts of rst derivatives However

this can lead to inecient nite dierence approximations For example the Laplacian

may b e written as

2 ab a b

D q r r q q r r gg

a b a b

A straightforward nite dierence approximation to g followed by an approximation to

g would involve next nearest neighbors whereas a central dierence approximation to

the second derivative would involve only nearest neighbors Both approximations are

second order accurate but the latter is clearly preferable There are various other second

derivative expressions in which the latter approach should also b e invoked

One such expression which commonly o ccurs in dealing with a spinweight eld t

2

is the spinweight zero eld g t We will use this as an example of how such expressions

can b e approximated to second order using only nearest neighbors In our choice of

stereographic co ordinates we have

2 a b

g t q q t gt t

a b

2

Thus calculation of g t reduces to the previous calculation of gt and the mixed

derivatives t which can all b e approximated to second order by centered dierences

a b

involving only nearest neighbors At the grid b oundary this involves values at virtual

grid p oints which can b e determined to fourth order accuracy by the interpolation

scheme

Two general op erators expressing arbitrary rst or second order derivatives acting

on any spinweighted eld are dened here Those op erators have as arguments the

eld values over the sphere the spin of the eld and a set of binary values that indicate

the nature of the op erator These op erators allow for easy and versatile numerical

implementation and cover the vast ma jority of expressions we exp ect to encounter in

numerical relativistic calculations

The rst order op erator is dened as

2 2

D s x y i sx iy

1 x y

where x iy s is the spinweight of and gives the g op erator while

gives the g op erator

Similarly the second order op erator is dened by

2 2 2 2 2

x y i D s f

1 2 1 2 x y 2 1 2

x y

2 2

x y s x is y

1 2 1 2 1 2 x

s x i s y i

1 2 1 2 1 2 1 2 y

2 2

s s x y

2 1 1 2 1 2 1 2

s ixy g

1 2 1 2

The dierent combinations of generate the following second order op erators

1 2

2 2

D g D gg D gg D g

2 2 2 2

Applications and Tests

Wave Evolution

An imp ortant rst application and test of the approximation scheme developed in

the previous section is the numerical solution of the D wave equation in spherical

co ordinates We start with the the discretization of the Laplace op erator on the sphere

In terms of the complex co ordinate we have

2 2

D D



2

2

Since D is a scalar op erator gives the same value in the overlap region when

evaluated in either the S or N co ordinates

In retarded time co ordinates the wave equation 2 takes the form

2

D g

g g

ur r r

2

r

where g r We solve by an explicit marching algorithm from r to null

innity

The convergence and stability of the marching algorithm is dictated by the

CourantFriedrichsLewy CFL condition which requires that the numerical domain

of dep endence includes the physical domain of dep endence In the present case this

requirement is strongest at the vertex of the outgoing cones It limits the timestep by

u k r s s

1 2

where k is a number of order one related to the details of the startup pro cedure near

the vertex r To go o d accuracy our numerical investigations give the value k

for stable evolution

A go o d test of the patching scheme fo cuses on the accuracy of the numerical

evolution The linearity of the problem provides a useful family of exact solutions

l+1

r

Y G u r

lm lm

l+1 l+1

u u r

Expressions for the standard and spinweighted spherical harmonics in stereographic

co ordinates can b e found in The characteristic scheme evolves a global solution

to the wave equation and thus it is p ossible to check the global numerical error The l

2

norm of the error for the evolution of initial data given by an l m harmonic

converges to zero with a measured convergence rate

Figure shows a sequence of snapshots of the radiation patterns as seen at null

innity for initial data which consist of a lo calized pulse in the North patch traveling

towards the origin Notice the smo oth propagation of the eld across the co ordinate

patches as the evolution progresses

Calculation of Scalar Curvature

As a tensorial illustration of the forgoing metho ds we consider a problem which arises

in many dierent contexts in general relativity Given the metric h of a top ological

ab

sphere calculate the scalar curvature There are many ways to formulate this problem

analytically Here we present a treatment in terms of an auxiliary unit sphere metric

which is exible enough to b e of broad usefulness in numerical applications

The metric is uniquely determined by its unit sphere dyad comp onents K

a b a b

h q q and J h q q which are of spin weight zero and two resp ectively The

ab ab

ab ab

dyad comp onents of the inverse metric are h q q K H and h q q J H where

a b a b

2

H K J J deth detq Also note that in our conventions the alternating

ab ab

tensor of the unit sphere is iq q and gg gg s for a spinweight s eld

ab [a b]

Let D represent the covariant derivative asso ciated with h Then its curvature

a ab

determines the commutator

ef

D D D D q h R q R q

a b b a cd abcf ed abdf ce

where in two dimensions R Rh h We let the tensor eld

abcd

a[c d]b

c cd

C h r h r h r h

a db b ad d ab

ab

represent the dierence b etween the connection asso ciated with D and the unit sphere

a

a b c d c

v Then by contracting with q q q q we connection eg D r v C

c a a b

ab

c

obtain an expression for the curvature scalar in terms of the dyad comp onents of C

ab

2

RJ gC gB C A AB B BC

a b c a b c a b c

where A q q q C B q q q C and C q q q C

c c c

ab ab ab

Next we use to relate these to the metric comp onents

K gK K gJ J gJ A H

B K gJ J gJ

H

K gJ J gK J gJ C

H

Substitution into then yields the expression for the curvature scalar after

canceling a common factor of J from b oth sides

2

fK ggK g J gK K gK K gJ J gJ R

H H

K K

gJ J gJ gJ gJ J gJ gJ g

cc

2

where H K J J The derivation of is quite lab orious and it would

b e extremely useful to develop symbolic techniques to carry out general dyad

decomp ositions and reduce covariant derivatives to eth op erations In the form the

curvature scalar may b e computed to second order accuracy using the nite dierence

techniques presented ab ove

In the particular case where deth detq as in the Bondi formalism

ab ab

H and the metric is uniquely determined by J In this choice of gauge the curvature

formula simplies considerably

2 2

R K ggK g J g J gJ gJ gJ gJ

K

Note that in the complex conjugation has b een explicitly carried out

2

As a test we choose J to b e a simple spinweight harmonic eg J c g Y which

20

facilitates the calculation of exact expressions for the curvature Scaling the metric data

with the parameter c generates arbitrarily non linear curvature expressions which are

used to check the numerical implementation of

An interesting check of the numerical pro cedure utilizes the global geometrical

prop erties of curvature on spherical surfaces The GaussBonnet theorem for spherical

top ology implies

I

I RdS

A global test of the computation of the scalar curvature can thus b e p erformed by

integrating the numerically evaluated curvature over the entire sphere The surface

12 2

area element is dS i H d d The integration must take into account

the overlap b etween co ordinate patches A simple and natural choice is to use the

equator as the smo oth and symmetric b oundary of the integration within each

patch A second order integration scheme in two dimensions is the simple four p oint

center average metho d We must pay careful attention to the b oundary cells ie those

which overlap the equator Their number scales like and if we were to omit their

contribution to the integral the order of accuracy would b e reduced by one To take

them into account the interior area of each b oundary cell is approximated assuming

that the equator lo oks like a straight grid line lo cally The centered four p oint average

2

of the cell is then weighted by its interior area divided by

Figure presents the GaussBonnet global test and conrms the convergence rate

of

RobinsonTrautman Solutions

RobinsonTrautman spacetimes contain purely outgoing gravitational radiation

which decays to leave a Schwarzschildlike horizon The Bondi news function

for these spacetimes is given by

1 2

N W g W

where W u satises the RobinsonTrautman equation

2 3 2 2 2

g W W W g W g W W g

u

The real and imaginary parts of the news function represent the two indep endent

p olarization mo des of the radiation The Bondi mass is given by the solid angle integral

H

3

M W d The Schwarzschild spacetime for a unit mass black hole

corresp onds to the solution W The linearized p erturbation W AuY

m

2

u (+1)( +2)12

decays exp onentially according to A Ae

Accurate numerical evolution of a fourth order parab olic equation such as by

means of an explicit nite dierence scheme is a challenge b ecause the CFL condition

requires that the time step u scale as the fourth p ower of the spatial grid size

Nevertheless we have applied the g formalism to obtain an ecient secondorder

accurate evolution algorithm based up on a three time level AdamsBashford scheme

with predictor W given by

u

W u u W u W u W u u

u

and corrector

u

3

W u u W u W u W u u O u

u

Here the terms are to b e considered reexpressed in terms of g op erators using

u

The second order convergence of numerical solutions was conrmed in the p erturbative

regime using solutions of the linearized equation

This algorithm reveals some new and unanticipated features of the gravitational

waveforms in the nonlinear regime Figures and illustrate the Bondi news function

for the RobinsonTrautman spacetime with initial data in the o dd parity m

mo de

W j Y

u=0 33

for the case In order to provide a physically intrinsic time dep endence the

news function is plotted versus Bondi time u which is related to the co ordinate time

B

u by du du W

B

Figure shows surface plots at representative times of the nonlinear contribution

to the news function ie N N N j The plots show the real part of

=0

N in the south patch The smo othness of the plots indicate the eectiveness of the

algorithm in handling the high spherical harmonics generated by the nonlinearity The

highest harmonics damp quickly and the system asymptotically approaches a monop ole

dip ole combination at late times

Figure displays the time evolution of the news function at a representative p oint

on the equator The graphs reveal an oscillatory b ehaviour qualitatively quite dierent

from the pure exp onential decay of the linearized solution In addition the b ehavior of

the two p olarization mo des is quite dierent Approximately of the initial mass is

radiated away during this simulation which ends at the time of black hole formation

This is in the range of energy exp ected to b e radiated during the inspiral of a binary

black hole system In this regime we have found by varying the value of that these

oscillatory waveforms cannot b e well approximated even by second order p erturbation

theory which emphasizes the imp ortance of an accurate computational treatment

Acknowledgments

We are thankful for research supp ort under NSF Grants PHY and PHY

ASC ARPA supplemented and for computer time made available by

the Pittsburgh Sup ercomputing Center

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0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.4 -0.4 -0.5 -0.5 -0.6 -0.6 -0.7 -0.7

0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.4 -0.4 -0.5 -0.5 -0.6 -0.6

-0.7 -0.7

Figure A sequence of radiation patterns at innity for initial data consisting of

a lo calized pulse which is traveling towards the origin The left column represents

the north patch the right column the south patch The evolution pro ceeds from

top to b ottom The rst pair is at retarded time u and displays the initial

radiation resulting from scattering of the pulse o the centrifugal barrier created by

its high angular momentum l multipole values This radiation shows up in the same

hemisphere as the initial data At the next two retarded times u and u

radiation app ears in all directions as the pulse crosses the origin dening retarded time

Finally at u the pulse has completely passed by the origin and the radiation

ceases The grid size is M ie x p oints p er patch

c = 0.001 4.0 c = 1. c = 30. c = 100. c = 1000.

2.15 2.0

π) 2.01 Rds-8 ∧ 0.0 2.00 log(

-2.0 1.98

-4.0 -2.1 -1.9 -1.7 -1.5 -1.3

log(∆)

Figure The global GaussBonnet convergence test The gure displays the

H

convergence of the numerically evaluated integral I Rds to the exact value of

The seed for the metric data is an l m harmonic and it is scaled over

six orders of magnitude The convergence is uniformly second order Note that for

very low amplitudes the error is dominated by the surface integration pro cedure and is indep endent of the data

0.3 0.3

-0.3 -0.3

a b

0.3 0.3

-0.3 -0.3

c d

Figure Comparison of linear versus nonlinear evolution of outgoing radiation

elds The graphs show the dierence b etween the Bondi News for the p erturbative

and the numerically evolved spacetimes for a u b u c u

B B B

and d u At early times u a complex angular structure reveals

B B

the presence of high spherical harmonics which damp in a short p erio d of time At

later times u the system approaches a monop oledip ole combination B

0.00

-0.10

0.00 -0.20

Re[N] -0.10

-0.20 -0.30 -0.30

-0.40 0.000 0.005 0.010 0.015 0.020 -0.40 0.0 0.2 0.4 0.6 0.8

u a

0.020

0.0181

0.015

0.0173

0.010 Im[N] 0.0165 0.000 0.002 0.004 0.006

0.005

0.000 0.0 0.2 0.4 0.6

u

b

Figure The Bondi News function at the p oint for initial

data corresp onding to Y Both the real a and imaginary part b are shown A

33

p erturbative calculation to rst order predicts that the real part at this p oint is zero

with the imaginary part decaying exp onentially In contrast the real part of the

nonp erturbative News is markedly dierent from zero and b oth comp onents show oscillations at early times