Constru

cting

Asy

mpto

tically

Constra

ined

Syste

ms

by

adju

sting

AD

M/B

SSN

equatio

ns

Hisa

-akiShin

kai

Com

putationalSci.

Div.,

RIK

EN

(The

Instituteof

Physical

andChem

icalResearch),

Japan

hshinkai@postm

an.riken.go.jp

work

with

Gen

Yoneda

Math.

Sci.

Dept.,

Waseda

Univ.,

Japan

Who

isH

S?

PhD

@W

asedaU

niv.(sup

ervisedby

Keiichi

Maeda)

PostD

oc@

Washington

Univ.

St.

Louis

Visiting

Assoc

@PennS

tateU

niv.(JS

PS

researcherin

abroad)

PostD

oc@

RIK

EN

Refs:

Ashtekar

variablesPRL

82

(1999)263,

PRD

60

(1999)101502,

IJMPD

9(2000)

13,

CQ

G17

(2000)4799,

CQ

G18

(2001)441

AD

Mvariables

PRD

63

(2001)120419,

CQ

G19

(2002)1027

BSSN

variablesgr-qc/0204002

(PRD

inprint)

generalgr-qc/0209106

andgr-qc/0209111

(review)

@Caltech,

Octob

er10,

2002

Outlin

e

•T

hre

eappro

ach

es:

AD

M/B

SSN

,hyperb

olic

form

ula

tion,attra

ctor

syste

ms

•P

roposa

ls:

Aunifi

ed

treatm

ent

as

Adju

sted

Syste

ms

Analy

ticSupport:

Constra

int

Pro

pagatio

neqs.

Som

epre

dictio

ns

and

Num

erica

lexperim

ents

Pla

nofth

eta

lk

1.In

troductio

n

2.T

hre

eappro

ach

es

(1)

Arn

ow

itt-Dese

r-Misn

er

/B

aum

garte

-Shapiro

-Shib

ata

-Nakam

ura

(2)

Hyperb

olic

form

ula

tions

(3)

Attra

ctor

syste

ms

–“A

dju

sted

Syste

ms”

3.A

dju

sted

AD

Msy

stem

s

Fla

tback

gro

und

Sch

warzsch

ildback

gro

und

4.A

dju

sted

BSSN

syste

ms

Fla

tback

gro

und

5.Sum

mary

1N

um

erica

lR

ela

tivity

and

“Form

ula

tion”

Pro

ble

m

Num

ericalRelativity

–N

ecessaryfor

unveilingthe

natureof

stronggravity

–G

ravitationalWave

fromcolliding

Black

Holes,

Neutron

Stars,

Sup

ernovae,...

–Relativistic

Phenom

enalike

Cosm

ology,Active

Galactic

Nuclei,

...

–M

athematical

feedbacksto

Singularity,

Exact

Solutions,

Chaotic

behaviors,

...

–Lab

oratoryof

Gravitational

theories,H

igherdim

ensionalm

odels,...

Neutron Stars / Neutron Stars / Black HolesBlack Holes

Gravitational WavesGravitational Waves

LIGO/VIRGO/GEO/TAMA, ...LIGO/VIRGO/GEO/TAMA, ...

Best

formulation

ofthe

Einstein

eqs.for

long-termstable

&accurate

simulation?

Many

(toom

any)trials

anderrors,

notyet

adefinit

recipe.

time

time

errorerrorB

low up

Blow

up

t=0

Co

nstra

ine

d S

urfa

ce(sa

tisfies

E

inste

in's

con

strain

ts)

Best

formulation

ofthe

Einstein

eqs.for

long-termstable

&accurate

simulation?

Many

(toom

any)trials

anderrors,

notyet

adefinit

recipe.

time

time

errorerrorB

low up

Blow

upB

low up

Blow

up

AD

MA

DM

BSSN

BSSN

Mathem

aticallyequivalent

formulations,

butdiff

erin

itsstability!

strategy0:

Arnow

itt-Deser-M

isnerform

ulation

strategy1:

Shibata-N

akamura’s

(Baum

garte-Shapiro’s)

modifications

tothe

standardAD

M

strategy2:

Apply

aform

ulationw

hichreveals

ahyp

erbolicity

explicitly

strategy3:

Form

ulatea

systemw

hichis

“asymptotically

constrained”against

aviolation

ofconstraints

By

addingconstraints

inRH

S,we

cankill

error-growing

modes

⇒H

owcan

we

understandthe

featuressystem

atically?

strategy0

The

standardapproach

::Arnow

itt-Deser-M

isner(A

DM

)form

ulation(1962)

3+1

decomposition

ofthe

spacetime.

Evolve

12variables

(γij ,K

ij )

with

achoice

ofgauge

condition.coord

inate constant linesurface norm

al linesurface norm

al lineN

i

lapse function, N

shift vector, Nshift vector, N

i

t = constant hypersurface

t = constant hypersurface

Maxw

elleqs.

AD

MEinstein

eq.

constraintsdiv

E=

4πρ

divB

=0

(3)R+

(trK)2−

Kij K

ij=

2κρ

H+

2Λ

Dj K

ji −D

i trK=

κJ

i

evolutioneqs.

1c∂

t E=

rotB

−4πc

j

1c∂

t B=−

rotE

∂t γ

ij=−

2NK

ij+

Dj N

i +D

i Nj ,

∂t K

ij=

N(

(3)Rij

+trK

Kij )−

2NK

il Klj −

Di D

j N

+(D

j Nm

)Km

i +(D

i Nm

)Km

j+

NmD

mK

ij −N

γij Λ

−κα{S

ij+

12 γij (ρ

H−

trS)}

strategy1

Shibata-N

akamura’s

(Baum

garte-Shapiro’s)

modifications

tothe

standardAD

M

–define

newvariables

(φ,γ̃

ij ,K,Ã

ij ,Γ̃i),

insteadof

theAD

M’s

(γij ,K

ij )w

here

γ̃ij ≡

e −4φγ

ij ,Ã

ij ≡e −

4φ(Kij −

(1/3)γij K

),Γ̃

i≡Γ̃

ijk γ̃jk,

usem

omentum

constraintin

Γi-eq.,

andim

pose

detγ̃

ij=

1during

theevolutions.

–T

heset

ofevolution

equationsbecom

e

(∂t −

Lβ )φ

=−

(1/6)αK

,

(∂t −

Lβ )γ̃

ij=

−2α

Ãij ,

(∂t −

Lβ )K

=αÃ

ij Ãij

+(1/3)α

K2−

γij(∇

i ∇j α

),

(∂t −

Lβ )Ã

ij=

−e −

4φ(∇i ∇

j α)T

F+

e −4φα

R(3)ij

−e −

4φα(1/3)γ

ij R(3)

+α

(KÃ

ij −2Ã

ik Ãkj )

∂t Γ̃

i=

−2(∂

j α)Ã

ij−(4/3)α

(∂j K

)γ̃ij

+12α

Ãji(∂

j φ)−

2αÃ

kj(∂

j γ̃ik)−

2αΓ̃

klj Ã

jk γ̃

il

−∂

j (βk∂

k γ̃ij−

γ̃kj(∂

k βi)−

γ̃ki(∂

k βj)

+(2/3)γ̃

ij(∂k β

k) )

Rij

=∂

k Γkij −

∂i Γ

kkj+

Γmij Γ

kmk −

Γmkj Γ

kmi=

:R̃

ij+

Rφij

Rφij

=−

2D̃i D̃

j φ−

2g̃ij D̃

lD̃l φ

+4(D̃

i φ)(D̃

j φ)−

4g̃ij (D̃

lφ)(D̃

l φ)

R̃ij

=−

(1/2)g̃lm

∂lm

g̃ij

+g̃

k(i ∂

j) Γ̃k

+Γ̃

kΓ̃(ij)k

+2g̃

lmΓ̃

kl(i Γ̃j)k

m+

g̃lm

Γ̃kim

Γ̃klj

–N

oexplicit

explanationsw

hythis

formulation

works

better.

AEIgroup

(2000):the

replacement

bym

omentum

constraintis

essential.

strategy2

Apply

aform

ulationw

hichreveals

ahyp

erbolicity

explicitly.

For

afirst

orderpartial

differential

equationson

avector

u,

∂t

u1

u2...

=

A

∂

x

u1

u2...

︸

︷︷︸

characteristicpart

+B

u1

u2...

︸

︷︷︸

lower

orderpart

ifthe

eigenvaluesof

Aare

weakly

hyperb

olicall

real.

stronglyhyp

erbolic

allreal

and∃

acom

pleteset

ofeigenvalues.

symm

etrichyp

erbolic

ifA

isreal

andsym

metric

(Herm

itian).

Symm

etric hyp.Sym

metric hyp.

Strongly hyp.Strongly hyp.

Weakly hyp.

Weakly hyp.

Exp

ectations

–W

ellposed

behaviour

symm

etrichyp

erbolic

system=⇒

WELL-P

OSED

,||u

(t)||≤eκt||u

(0)||

–Better

boundary

treatments⇐

=∃

characteristicfield.

–know

nnum

ericaltechniques

inN

ewtonian

hydrodynamics.

formulations

numerical

applications(0)

The

standardA

DM

formulation

AD

M1962

Arnow

itt-Deser-M

isner[12,

78]⇒

many

(1)T

heB

SSNform

ulationB

SSN1987

Nakam

uraet

al[62,

63,72]

⇒1987

Nakam

uraet

al[62,

63]⇒

1995Shibata-N

akamura

[72]⇒

2002Shibata-U

ryu[73]

etc1999

Baum

garte-Shapiro[15]

⇒1999

Baum

garte-Shapiro[15]

⇒2000

Alcubierre

etal

[5,7]

⇒2001

Alcubierre

etal

[6]etc

1999A

lcubierreet

al[8]

1999Frittelli-R

eula[41]

2002Laguna-Shoem

aker[54]

⇒2002

Laguna-Shoem

aker[54]

(2)T

hehyperbolic

formulations

BM

1989B

ona-Massó

[17,18,

19]⇒

1995B

onaet

al[19,

20,21]

⇒1997

Alcubierre,

Massó

[2,4]

1997B

onaet

al[20]

⇒2002

Bardeen-B

uchman

[16]1999

Arbona

etal

[11]C

B-Y

1995C

hoquet-Bruhat

andY

ork[31]

⇒1997

Scheelet

al[69]

1995A

brahams

etal

[1]⇒

1998Scheel

etal

[70]1999

Anderson-Y

ork[10]

⇒2002

Bardeen-B

uchman

[16]FR

1996Frittelli-R

eula[40]

⇒2000

Hern

[43]1996

Stewart

[79]K

ST2001

Kidder-Scheel-T

eukolsky[51]

⇒2001

Kidder-Scheel-T

eukolsky[51]

⇒2002

Calabrese

etal

[26]⇒

2002Lindblom

-Scheel[57]

2002Sarbach-T

iglio[68]

CFE

1981Friedrich[35]

⇒1998

Frauendiener[34]

⇒1999

Hübner

[45]tetrad

1995vanP

utten-Eardley[84]

⇒1997

vanPutten

[85]A

shtekar1986

Ashtekar

[13]⇒

2000Shinkai-Y

oneda[75]

1997Iriondo

etal

[47]1999

Yoneda-Shinkai

[90,91]

⇒2000

Shinkai-Yoneda

[75,92]

(3)A

symptotically

constrainedform

ulationsλ-system

toFR

1999B

rodbecket

al[23]

⇒2001

Siebel-Hübner

[77]to

Ashtekar

1999Shinkai-Y

oneda[74]

⇒2001

Yoneda-Shinkai

[92]adjusted

toA

DM

1987D

etweiler

[32]⇒

2001Y

oneda-Shinkai[93]

toA

DM

2001Shinkai-Y

oneda[93,

76]⇒

2002M

exicoN

RW

orkshop[58]

toB

SSN2002

Yoneda-Shinkai

[94]⇒

2002M

exicoN

RW

orkshop[58]

⇒2002

Yo-B

aumgarte-Shapiro

[88]

80s90s

2000s

ADM

Shibata-Nakamura95

Baumgarte-Shapiro99

Nakamura-Oohara87

Bona-Masso92

Anderson-York99

ChoquetBruhat-York95-97Frittelli-Reula

96

62

Ashtekar86

Yoneda-Shinkai99

Kidder-Scheel -Teukolsky

01

lambda-system99

Alcubierre97

Iriondo-Leguizamon-Reula

97

80s90s

2000s

ADM

Shibata-Nakamura95

Baumgarte-Shapiro99

Nakamura-Oohara87

Bona-Masso92

Anderson-York99

ChoquetBruhat-York95-97Frittelli-Reula

96

62

Ashtekar86

Yoneda-Shinkai99

Kidder-Scheel -Teukolsky

01

NCSA AEI

G-code H-code

BSSN-code

Cornell-Illinois

UWash

Hern

Caltech

PennState

lambda-system99

Shinkai-Yoneda

Alcubierre97

Nakamura-OoharaShibata

Iriondo-Leguizamon-Reula

97

LSU

80s90s

2000s

ADM

Shibata-Nakamura95

Baumgarte-Shapiro99

Nakamura-Oohara87

Bona-Masso92

Anderson-York99

ChoquetBruhat-York95-97Frittelli-Reula

96

62

Ashtekar86

Yoneda-Shinkai99

Kidder-Scheel -Teukolsky

01

NCSA AEI

G-code H-code

BSSN-code

Cornell-Illinois

UWash

Hern

Caltech

PennState

lambda-system99

adjusted-system01

Shinkai-Yoneda

Alcubierre97

Nakamura-OoharaShibata

Iriondo-Leguizamon-Reula

97

LSU

Illinois

strategy2

Apply

aform

ulationw

hichreveals

ahyp

erbolicity

explicitly.(cont.)

symm

etrichyp

erbolic⊂

stronglyhyp

erbolic⊂

weakly

hyperb

olicsystem

s,

•Are

theyactually

helpful?—

ifso,

which

levelof

hyperb

olicityis

necessary?

•U

nderw

hatconditions/situations

theadvantages

will

be

observed?

Unfortunately,

we

donot

haveconclusive

answers

tothem

yet.

•Several

numerical

experim

entsindicate

thatthe

directionis

NO

Ta

fullof

success .

–Earlier

numerical

comparisons

reported

theadvantages

ofhyp

erbolic

formulations,

but

theywere

againstto

thestandard

AD

Mform

ulation.[C

ornell-Illinois,N

CSA,...]

–N

umerical

evolutionsare

always

term

inate

dw

ithblo

w-u

ps.

–If

thegauge

functionsare

evolvedw

ithhyp

erbolic

equations,then

theirfinite

propagation

speeds

may

causea

pathologicalsh

ock

form

atio

ns

[Alcubierre].

–N

odrastic

numerical

differences

betw

een

threehyp

erbolic

levels[H

SYoneda,

Hern].

–Prop

osedsym

metric

hyperb

olicsystem

swere

notalw

aysthe

best

onefor

numerics.

Of

course,these

statements

onlycasted

ona

particularform

ulation,therefore

we

haveto

be

carefulnot

toover-announce

theresults.

strategy2

Apply

aform

ulationw

hichreveals

ahyp

erbolicity

explicitly.(cont.)

•Rem

arksto

hyperb

olicform

ulations

(a)Rigorous

mathem

aticalproofs

ofwell-p

osednessof

PD

Eare

mostly

fora

simple

sym-

metric

orstrongly

hyperb

olicsystem

s.Ifthe

matrix

components

orcoeffi

cientsdep

end

dynamical

variables(like

inany

versionsof

hyperb

olizedEinstein

equations),alm

ost

nothingwas

provedin

itsgeneral

situations.

(b)T

hestatem

entof

“stability”in

thediscussion

ofwell-p

osednessm

eansthe

bounded

growth

ofthe

norm,and

doesnot

mean

adecay

ofthe

normin

time

evolution.

(c)T

hediscussion

ofhyp

erbolicity

onlyuses

thecharacteristic

partof

theevolution

equations,and

ignorethe

rest.

cf.Recent

discussions

•K

ST

formulation

with

“kinematic”

parameters

which

enablesus

toreduce

non-principalpart.

•links

toIB

VP

approach.

•relations

betw

eenconvergence

behavior

andlevels

ofhyp

erbolicity.

strategy3

Form

ulatea

systemw

hichis

“asymptotically

constrained”against

aviolation

ofconstraints

“A

sym

pto

tically

Constra

ined

Syste

m”–

Constraint

Surface

asan

Attractor

t=0

Co

nstra

ine

d S

urfa

ce(sa

tisfies

E

inste

in's

con

strain

ts)

time

time

errorerror

Blow

upB

low up

Stabilize?Stabilize? ?

method

1:λ-system

(Brodb

ecket

al,2000)

–Add

aritificialforceto

reducethe

violationof

con-

straints

–To

be

guaranteedif

we

applythe

ideato

asym

-

metric

hyperb

olicsystem

.

method

2:Adjusted

system(Y

onedaH

S,2000,

2001)

–W

ecan

controltheviolation

ofconstraints

byad-

justingconstraints

toEoM

.

–Eigenvalue

analysisof

constraintpropagation

equationsm

ayprodict

theviolation

oferror.

–T

hisidea

isapplicable

evenif

thesystem

isnot

symm

etrichyp

erbolic.⇒

forthe

AD

M/B

SSN

formulation,

too!!

Ideaof

λ-system

Brodb

eck,Frittelli,

Hübner

andReula,

JMP40(99)909

We

expect

asystem

thatis

robustfor

controllingthe

violationof

constraints

Recip

e1.

Prepare

asym

metric

hyperb

olicevolution

system∂

t u=

J∂

i u+

K

2.Introduce

λas

anindicator

ofviolation

ofconstraint

which

obeys

dissipativeeqs.

ofm

otion∂

t λ=

αC−

βλ

(α�=

0,β>

0)

3.Take

aset

of(u

,λ)

asdynam

icalvariables

∂t

uλ

�

A

0

F0

∂

i uλ

4.M

odifyevolution

eqsso

asto

forma

symm

etrichyp

erbolic

system∂

t uλ

=

A

F̄

F0

∂

i uλ

Rem

arks

•BFH

Rused

asym

.hyp.

formulation

byFrittelli-R

eula[P

RL76(96)4667]

•T

heversion

forthe

Ashtekar

formulation

byH

S-Y

oneda[P

RD

60(99)101502]

forcontrolling

theconstraints

orreality

conditionsor

both.

•Succeeded

inevolution

ofG

Win

planarspacetim

eusing

Ashtekar

vars.[C

QG

18(2001)441]

•D

othe

recoveredsolutions

representtrue

evolution?by

Sieb

el-Hübner

[PRD

64(2001)024021]

Ideaof

“Adjusted

system”

andO

urConjecture

CQ

G18

(2001)441,

PRD

63(2001)

120419,CQ

G19

(2002)1027

General

Procedure

1.prepare

aset

ofevolution

eqs.∂

t ua

=f(u

a,∂b u

a,···)

2.add

constraintsin

RH

S∂

t ua

=f(u

a,∂b u

a,···)+

F(C

a,∂b C

a,···)︸

︷︷︸

3.choose

appropriateF

(Ca,∂

b Ca,···)

tom

akethe

systemstable

evolution

How

tosp

ecifyF

(Ca,∂

b Ca,···)

?

4.prepare

constraintpropagation

eqs.∂

t Ca

=g(C

a,∂b C

a,···)

5.and

itsadjusted

version∂

t Ca

=g(C

a,∂b C

a,···)+

G(C

a,∂b C

a,···)︸

︷︷︸

6.Fourier

transformand

evaluateeigenvalues

∂t Ĉ

k=

A(Ĉ

a)︸

︷︷︸Ĉ

k

Conje

cture

:Evaluate

eigenvaluesof

(Fourier-transform

ed)constraint

propagationeqs.

Iftheir

(1)real

partis

non-positive,

or(2)

imaginary

partis

non-zero,then

thesystem

ism

orestable.

The

Adju

sted

syste

m(e

ssentia

ls):

Purp

ose:Control

theviolation

ofconstraints

byreform

ulatingthe

systemso

asto

havea

constrainedsurface

anattractor.

Procedure:

Add

aparticular

combination

ofconstraints

tothe

evolutionequations,and

adjust

itsm

ultipliers.

Theoretical

support:

Eigenvalue

analysisof

theconstraint

propagationequations.

Advantages:

Available

evenif

thebase

systemis

nota

symm

etrichyp

erbolic.

Advantages:

Keep

thenum

ber

ofthe

variablesam

ew

iththe

originalsystem

.

Conje

cture

on

Constra

int

Am

plifi

catio

nFacto

rs(C

AFs):

(A)

IfCAF

hasa

negativereal-part

(theconstraints

areforced

tobe

diminished),

thenwe

seem

ore

stableevolution

thana

systemw

hichhas

positive

CAF.

(B)

IfCAF

hasa

non-zeroim

aginary-part(the

constraintsare

propagatingaw

ay),then

we

seem

ore

stableevolution

thana

systemw

hichhas

zeroCAF.

Exam

ple

:th

eM

axw

ell

equatio

ns

Yoneda

HS,CQ

G18

(2001)441

Maxw

ellevolution

equations.

∂t E

i=

c�i jk∂

j Bk

+P

i CE

+Q

i CB,

∂t B

i=

−c�

i jk∂j E

k+

Ri C

E+

Si C

B,

CE

=∂

i Ei≈

0,C

B=

∂i B

i≈0,

sym.

hyp⇔

Pi=

Qi=

Ri=

Si=

0,

stronglyhyp

⇔(P

i −S

i )2+

4Ri Q

i>

0,

weakly

hyp⇔

(Pi −

Si )

2+

4Ri Q

i ≥0

Constraint

propagationequations

∂t C

E=

(∂i P

i)CE

+P

i(∂i C

E)+

(∂i Q

i)CB

+Q

i(∂i C

B),

∂t C

B=

(∂i R

i)CE

+R

i(∂i C

E)+

(∂i S

i)CB

+S

i(∂i C

B),

sym.

hyp⇔

Qi=

Ri ,

stronglyhyp

⇔(P

i −S

i )2+

4Ri Q

i>

0,

weakly

hyp⇔

(Pi −

Si )

2+

4Ri Q

i ≥0

CAFs?

∂t

ĈE

ĈB

=

∂

i Pi+

Pik

i∂

i Qi+

Qik

i

∂i R

i+R

iki

∂i S

i+S

iki

∂

l Ĉ

E

ĈB

≈

P

iki

Qik

i

Rik

iS

iki

Ĉ

E

ĈB

=

:T

Ĉ

E

ĈB

⇒CAFs

=(P

iki +

Sik

i ±√(P

iki +

Sik

i )2+

4(Qik

i Rjk

j −P

iki S

jkj ))/2

Therefore

CAFs

becom

enegative-real

when

Pik

i +S

iki<

0,and

Qik

i Rjk

j −P

iki S

jkj<

0

Exam

ple

:th

eA

shte

kar

equatio

ns

HS

Yoneda,

CQ

G17

(2000)4799

Adjusted

dynamical

equations:

∂t Ẽ

ia=

−iD

j (�cba N∼

Ẽjc Ẽ

ib )+

2Dj (N

[jẼi]a )

+iA

b0 �c

abẼ

ic +X

ia CH

+Y

ijaC

Mj+

Piba C

Gb

︸︷︷

︸adju

st

∂t A

ai=

−i�

abc N∼

Ẽjb F

cij+

NjF

aji +D

i Aa0

+ΛN∼

Ẽai+

Qai C

H+

Raj

i CM

j+

Zab

i CG

b︸

︷︷︸

adju

st

Adjusted

andlinearized:

X=

Y=

Z=

0,P

iab

=κ

1 (iNiδ

ab ),Q

ai=

κ2 (e −

2N∼Ẽ

ai ),R

aji=

κ3 (−

ie −2N∼

�acd Ẽ

di Ẽjc )

Fourier

transformand

extract0th

orderof

thecharacteristic

matrix:

∂t

ĈH

ĈM

i

ĈG

a

=

0i(1

+2κ

3 )kj

0

i(1−2κ

2 )ki

κ3 �

kji k

k0

02κ

3 δja

0

ĈH

ĈM

j

ĈG

b

Eigenvalues:

(0,0,0,±κ

3 √−kx

2−ky

2−kz

2,±√(−

1+

2κ2 )(1

+2κ

3 )(kx

2+

ky

2+

kz

2) )

Inorder

toobtain

non-positive

realeigenvalues:

(−1

+2κ

2 )(1+

2κ3 )

<0

AC

lassifi

catio

nofC

onstra

int

Pro

pagatio

ns

(C1)

Asy

mpto

tically

constra

ined

:

Vio

latio

nofco

nstra

ints

deca

ys

(converg

es

toze

ro).

(C2)

Asy

mpto

tically

bounded

:

Vio

latio

nofco

nstra

ints

isbounded

at

ace

rtain

valu

e.

(C3)

Div

erg

e:

At

least

one

constra

int

will

div

erg

e.

Note

that

(C1)⊂

(C2).

(C1)

Decay

(C2) B

ounded

(C3)

Dive

rge

time

time

errorerror

Diverge

Diverge

Constrained

, C

onstrained,

or Decay

or Decay

AC

lassifi

catio

nofC

onstra

int

Pro

pagatio

ns

(cont.)

gr-q

c/0209106

(C1)A

sym

pto

tically

constra

ined

:V

iola

tion

ofco

nstra

ints

deca

ys

(converg

es

toze

ro).

⇔A

llth

ere

alparts

ofC

AFs

are

negativ

e.

(C2)A

sym

pto

tically

bounded

:V

iola

tion

ofco

nstra

ints

isbounded

at

ace

rtain

valu

e.

⇔(a)A

llth

ere

alparts

ofC

AFs

are

not

positiv

e,and

(b1)th

eC

Pm

atrix

Mαβ

isdia

gonaliza

ble

,or

(b2)th

ere

alpart

ofth

edegenera

ted

CA

Fs

isnot

zero

.

(C3)D

iverg

e:

At

least

one

constra

int

will

div

erg

e.

The

nece

ssary

and

suffi

cient

conditio

ns

for

(C1)

and

(C2)?

Pre

para

tion

With

out

loss

ofgenera

lity,th

eC

Pm

atrix

Mca

nbe

assu

med

tobe

atria

ngula

rm

atrix

.Suppose

we

have

an

expre

ssion,

∂t

Cn...

C1

=

λn

∗∗

0...

∗0

0λ

1

Cn...

C1

,

(1)

where

λs

are

the

eig

envalu

es

of

M,

and

the

indice

sare

form

ally

labele

din

this

ord

er.

Pro

positio

n1

The

solu

tion

of(1

)ca

nbe

expre

ssed

form

ally

as

Cj (t)

=j∑i=1

exp(λ

i t)n

i −1

∑k=0 (a

(i)k

tk)

,(2)

where

λiis

the

i-theig

envalu

eof

M,and

niis

the

multip

licityof

λiup

toi≤

j.

Pro

positio

n1

The

solu

tion

of(1

)ca

nbe

expre

ssed

form

ally

as

Cj (t)

=j∑i=1

exp(λ

i t)n

i −1

∑k=0 (a

(i)k

tk)

,(2)

where

λiis

the

i-theig

envalu

eof

M,and

niis

the

multip

licityof

λiup

toi≤

j.

For

exam

ple

:λ

1<

λ2

=λ

3=

λ4<

λ5

=λ

6<

···,C

1=

exp(λ

1 t)(@)

C2

=exp

(λ1 t)(@

)+

exp(λ

2 t)(@)

C3

=exp

(λ1 t)(@

)+

exp(λ

2 t)(@+

@t)

C4

=exp

(λ1 t)(@

)+

exp(λ

2 t)(@+

@t+

@t2)

C5

=exp

(λ1 t)(@

)+

exp(λ

2 t)(@+

@t+

@t2)

+exp

(λ5 t)(@

)

C6

=exp

(λ1 t)(@

)+

exp(λ

2 t)(@+

@t+

@t2)

+exp

(λ5 t)(@

+@

t)

The

hig

hest

pow

er

Nin

all

constra

ints

isbounded

by

N≤

max

1≤i≤

n (multip

licityof

λi )−

1.(3)

Asy

mpto

tically

Constra

ined

CP

–(C

1)

–

Theore

m1

Asy

mpto

tically

constra

ined

evolu

tion

(vio

latio

nof

constra

ints

converg

es

toze

ro)

⇔A

llth

ere

alparts

ofC

AFs

are

negativ

e.

pro

ofof⇐

)W

euse

the

expre

ssion

(2).

If�

e(λi )

<0

for∀i,

then

Cw

illco

n-

verg

eto

zero

at

t→∞

no

matte

rw

hat

t−poly

nom

ialte

rms

are

.

pro

ofof⇒

)W

esh

ow

the

contra

positiv

e.

Suppose

there

exists

an

eig

envalu

eλ

1su

chas

which

real-p

art

isnon-n

egativ

e.

By

settin

gλ

1at

the

low

er-e

nd

ofth

etria

ngula

rm

atrix

Min

(1),

then

we

get

∂t C

1=

λ1 C

1w

hich

solu

tion

isC

1=

C1 (0)

exp(λ

1 t).C

1does

not

converg

eto

zero

.

Asy

mpto

tically

Bounded

CP

–(C

2)

–

Theore

m2

Asy

mpto

tically

bounded

evolu

tion

(all

the

constra

ints

are

bounded

at

ace

rtain

valu

e)

⇔(a

)A

llth

ere

alparts

ofC

AFs

are

not

positiv

e,and

(b1)th

eC

Pm

atrix

Mαβ

isdia

gonaliza

ble

,or

(b2)th

ere

alpart

ofth

edegenera

ted

CA

Fs

isnot

zero

.

pro

of

of⇐

for

the

case

(a+

b1):

By

adia

gonaliza

tion,w

eobta

in∂

t Ci=

λi C

i ,w

hich

solu

tion

isC

i=

Ci (0)

exp(λ

i t).T

his

isbounded

since

�e(λ

i )≤0.

pro

ofof⇐

for

the

case

(a+

b2):

We

use

the

expre

ssion

(2).

When

λis

degenera

ted,

the

t-poly

nom

ials

have

non-ze

ropow

er.

How

-ever,

the

assu

mptio

n,�

e(λ)<

0,in

dica

tes

exp(λ

t)(t-poly

nom

ials)

will

converg

eto

zero

.W

hen

λis

not

degenera

ted,th

ere

isonly

aco

n-

stant

term

rath

er

than

t-poly

nom

ials.

So

that

(2)

rem

ain

sfinite

for

�e(λ

)≤0.

pro

ofof⇒

)W

esh

ow

the

contra

positiv

e.

(a)

and

{(b

1)

or

(b2)

}⇔

(a)

or{(a

)and{(b

1)

and

(b2)}}

(a)⇒

div

erg

e::

trivia

l.(a

)and{(b

1)

and

(b2)}

⇒div

erg

e::

By

triangu

lating

the

matrix

,w

ecan

setth

edegen

eratedC

AFs

λw

hich

real-part

iszero.

Let

us

consid

ern

=3

case,

M=

λ

ia

b0

λc

00

λ

,

a,b,c

=con

stant.

Then

we

getfirst

C1

=C

1 (0)ex

p(λ

t)w

hich

isa

constan

tor

atrigon

alfu

nction

,an

d

∂t C

2=

λC

2+

cC

1=

λC

2+

cC

1 (0)ex

p(λ

t)

⇒C

2=

C2 (0)

exp(λ

t)+

cC

1 (0)ex

p(λ

t)t.

Therefore

C2

will

diverge

when

c�=0,

and

remain

finite

when

c=

0.Sin

cew

eare

assum

ing

the

matrix

isnot

diagon

alizable,

the

min

imal

poly

nom

ialdoes

not

taketh

eform

asth

epro

duct

of(M

−λ

i E)

fordiff

erent

eigenvalu

esλ

i .W

hen

there

exists

λi �=

λ,w

esee

that

(M−

λE

)(M−

λi E

)=

λ

i −λ

ab

00

c0

00

0

ab

0λ−

λi

c0

0λ−

λi

=

0

0a

c0

0c(λ

−λ

i )0

00

,

which

shou

ldnot

equal

tozero

matrix

,th

atin

dicates

c�=0.

Therefore

C2

will

diverge.

When

λ=

λi ,

some

of

a,b,c

isnon

-zeroin

order

not

tovan

ish(M

−λE

).T

herefore

relatedC

iw

illdiverge.

Aflow

chart

tocla

ssifyth

efa

teofco

nstra

int

pro

pagatio

n.

Q1: Is th

ere a CA

F wh

ich real p

art is positive?

NO / YES

Q2: A

re all the real p

arts of CA

Fs negative?

Q3: Is th

e constrain

t prop

agation m

atrix diagon

alizable?

Q4: Is a real p

art of the d

egenerated

CA

Fs is zero?

NO / YES

NO / YES

NO / YES

Diverge

Asym

ptotically C

onstrain

ed

Asym

ptotically B

ounded

Diverge

Asym

ptotically Bounded

Constru

cting

Asy

mpto

tically

Constra

ined

Syste

ms

Hisa

akiShin

kai

1.In

troductio

n

2.T

hre

eappro

ach

es

(1)A

rnow

itt-Dese

r-Misn

er/

Baum

garte

-Shapiro

-Shib

ata

-Nakam

ura

(2)

Hyperb

olic

form

ula

tions

(3)

Attra

ctor

syste

ms

–“A

dju

sted

Syste

ms”

3.A

dju

sted

AD

Msy

stem

s

4.A

dju

sted

BSSN

syste

ms

5.Sum

mary

3A

dju

sted

AD

Msy

stem

s

We

adjustthe

standardAD

Msystem

usingconstraints

as:

∂t γ

ij=

−2α

Kij

+∇

i βj+∇

j βi ,

(1)

+P

ij H+

Qkij M

k+

pkij (∇

k H)+

qklij (∇

k Ml ),

(2)

∂t K

ij=

αR

(3)ij

+αK

Kij −

2αK

ik Kkj −

∇i ∇

j α+

(∇i β

k)Kkj+

(∇j β

k)Kki +

βk∇

k Kij ,(3)

+R

ij H+

Skij M

k+

rkij (∇

k H)+

sklij (∇

k Ml ),

(4)

with

constraintequations

H:=

R(3)

+K

2−K

ij Kij,

(5)

Mi

:=∇

j Kji −

∇i K

.(6)

We

canw

ritethe

adjustedconstraint

propagationequations

as

∂t H

=(original

terms)

+H

mn

1[(2)]+

Him

n2

∂i [(2)]+

Hijm

n3

∂i ∂

j [(2)]+H

mn

4[(4)],

(7)

∂t M

i=

(originalterm

s)+

M1i m

n[(2)]+M

2i jm

n∂j [(2)]+

M3i m

n[(4)]+M

4i jm

n∂j [(4)].

(8)

Orig

inalA

DM

The

originalconstruction

byAD

Muses

thepair

of(h

ij ,πij).

L=

√−

gR

=√

hN

[ (3)R−

K2+

Kij K

ij],w

hereK

ij=

12£

n hij

thenπ

ij=

∂L∂ḣ

ij

=√

h(K

ij−K

hij),

The

Ham

iltoniandensity

givesus

constraintsand

evolutioneqs.

H=

πijḣ

ij −L

=√

h{N

H(h

,π)−

2Nj M

j(h,π

)+

2Di (h

−1/2N

j πij) }

,

∂t h

ij=

δHδπij

=2

N√h

(πij −

12h

ij π)+

2D(i N

j) ,

∂t π

ij=

−δHδh

ij=

−√

hN

((3)R

ij−12

(3)Rh

ij)+

12

N√hh

ij(πm

n πm

n−12π

2)−2

N√h

(πinπ

nj−

12ππ

ij)

+ √h(D

iDjN

−h

ijDmD

mN

)+√

hD

m(h

−1/2N

mπ

ij)−2π

m(iD

mN

j)

Sta

ndard

AD

M(b

yY

ork

)N

Rists

referAD

Mas

theone

byYork

with

apair

of(h

ij ,Kij ).

∂

t hij

=−

2NK

ij+

Dj N

i +D

i Nj ,

∂t K

ij=

N(

(3)Rij

+K

Kij )−

2NK

il Klj −

Di D

j N+

(Dj N

m)K

mi +

(Di N

m)K

mj+

NmD

mK

ij

Inthe

processof

converting,Hwas

used,i.e.

thestandard

AD

Mhas

alreadyadjusted.

3C

onstra

int

pro

pagatio

nofA

DM

syste

ms

3.1

Orig

inalA

DM

vs

Sta

ndard

AD

M

Try

theadjustm

entR

ij=

κ1 α

γij

andother

multiplier

zero,w

hereκ

1=

0

thestandard

AD

M

−1/4

theoriginal

AD

M

•T

heconstraint

propagationeqs

keepthe

first-orderform

(cfFrittelli,

PRD

55(97)5992):

∂t

HMi

�

βl

−2α

γjl

−(1/2)α

δli +

Rli −

δli R

βlδ

ji

∂

l HM

j .

(5)

The

eigenvaluesof

thecharacteristic

matrix:

λl

=(β

l,βl,β

l±√α

2γll(1

+4κ

1 ))

The

hyperb

olicityof

(5):

symm

etrichyp

erbolic

when

κ1

=3/2

stronglyhyp

erbolic

when

α2γ

ll(1+

4κ1 )

>0

weakly

hyperb

olicw

henα

2γll(1

+4κ

1 )≥0

•O

nthe

Minkow

skiibackground

metric,

thelinear

orderterm

sof

theFourier-transform

ed

constraintpropagation

equationsgives

theeigenvalues

Λl=

(0,0,±√−

k2(1

+4κ

1 )).

That

is, (tw

o0s,

two

pureim

aginary)for

thestandard

AD

MBET

TER

STABIL

ITY

(four0s)

forthe

originalAD

M

4C

onstra

int

pro

pagatio

ns

insp

herica

llysy

mm

etric

space

time

4.1

The

pro

cedure

The

discussionbecom

esclear

ifwe

expandthe

constraintC

µ:=

(H,M

i )T

usingvector

harmonics.

Cµ

=∑l,m

(Alm

(t,r)alm

(θ,ϕ)+

Blm

blm

+C

lmclm

+D

lmd

lm

),

(1)

where

we

choosethe

basisof

thevector

harmonics

as

alm

=

Ylm000

,b

lm=

0Ylm00

,c

lm=

r√l(l

+1)

00

∂θ Y

lm

∂ϕ Y

lm

,d

lm=

r√l(l

+1)

00

−1

sinθ ∂

ϕ Ylm

sinθ∂

θ Ylm

.

The

basisare

normalized

sothat

theysatisfy

〈Cµ ,C

ν 〉=

∫2π

0dϕ

∫π0

C∗µ C

ρη

νρsin

θdθ,

where

ηνρ

isM

inkowskii

metric

andthe

asteriskdenotes

thecom

plexconjugate.

Therefore

Alm

=〈a

lm(ν) ,C

ν 〉,∂

t Alm

=〈a

lm(ν) ,∂

t Cν 〉,

etc.

We

alsoexpress

theseevolution

equationsusing

theFourier

expansionon

theradial

coordinate,

Alm

=∑k

Âlm(k

) (t)eik

retc.

(2)

So

thatwe

will

be

ableto

obtainthe

RH

Sof

theevolution

equationsfor

(Âlm(k

) (t),···,D̂lm(k

) (t))T

ina

homogeneous

form.

Exam

ple

1:

standard

AD

Mvs

orig

inalA

DM

(inSch

warzsch

ildco

ord

inate

)

-1

-0.5 0

0.5 1

05

10

15

20

no adjustments (standard A

DM

)

Real / Imaginary parts of Eigenvalues (AF)

rsch

(a)

-0.5 0

0.5

05

10

15

20

original AD

M (κ

F = - 1

/4)

rsch

(b)

Real / Imaginary parts of Eigenvalues (AF)

Figu

re1:

Am

plifi

cationfactors

(AFs,

eigenvalu

esof

hom

ogenized

constrain

tprop

agationeq

uation

s)are

show

nfor

the

standard

Sch

warzsch

ildco

ordin

ate,w

ith(a)

no

adju

stmen

ts,i.e.,

standard

AD

M,

(b)

original

AD

M(κ

F=

−1/4).

The

solidlin

esan

dth

edotted

lines

with

circlesare

realparts

and

imagin

aryparts,

respectively.

They

arefou

rlin

eseach

,but

actually

the

two

eigenvalu

esare

zerofor

allcases.

Plottin

gran

geis

2<

r≤

20usin

gSch

warzsch

ildrad

ialco

ordin

ate.W

eset

k=

1,l=

2,an

dm

=2

throu

ghou

tth

earticle.

∂t γ

ij=

−2α

Kij

+∇

i βj+∇

j βi ,

∂t K

ij=

αR

(3)ij

+αK

Kij −

2αK

ik Kkj −

∇i ∇

j α+

(∇i β

k)Kkj+

(∇j β

k)Kki +

βk∇

k Kij

+κ

Fαγ

ij H,

Exam

ple

2:

Detw

eile

r-type

adju

sted

(inSch

warzsch

ildco

ord

.)

-1

-0.5 0

0.5 1

05

10

15

20

Detw

eiler type, κL =

+ 1/2

(b)

Real / Imaginary parts of Eigenvalues (AF)

rsch

-1

-0.5 0

0.5 1

05

10

15

20

Detw

eiler type, κL =

- 1/2

Real / Imaginary parts of Eigenvalues (AF) (c)

rsch

Figu

re2:

Am

plifi

cationfactors

ofth

estan

dard

Sch

warzsch

ildco

ordin

ate,w

ithD

etweiler

type

adju

stmen

ts.M

ultip

liersused

inth

eplot

are(b

)κ

L=

+1/2,

and

(c)κ

L=

−1/2.

∂t γ

ij=

(originalterm

s)+

Pij H

,

∂t K

ij=

(originalterm

s)+

Rij H

+S

kij M

k+

sklij (∇

k Ml ),

where

Pij

=−

κL α

3γij ,

Rij

=κ

L α3(K

ij −(1/3)K

γij ),

Skij

=κ

L α2[3(∂

(i α)δ

kj) −(∂

l α)γ

ij γkl],

sklij

=κ

L α3[δ

k(i δlj) −

(1/3)γij γ

kl],

Exam

ple

3:

standard

AD

M(in

isotro

pic/

iEF

coord

.)

-1

-0.5 0

0.5 1

05

10

15

20

isotropic coordinate, no adjustments (standard A

DM

)

�

(a)

Real / Imaginary parts of Eigenvalues (AF)

riso

-1

-0.5 0

0.5 1

1.5 2

05

10

15

20

iEF

coordinate, no adjustments (standard A

DM

)(b

)

Real / Imaginary parts of Eigenvalues (AF)

rsch

Figu

re3:

Com

parison

ofam

plifi

cationfactors

betw

eendiff

erent

coord

inate

expression

sfor

the

standard

AD

Mform

ulation

(i.e.no

adju

stmen

ts).Fig.

(a)is

forth

eisotrop

icco

ordin

ate(1),

and

the

plottin

gran

geis

1/2≤

riso .

Fig.

(b)

isfor

the

iEF

coord

inate

(1)an

dw

eplot

lines

onth

et=

0slice

foreach

expression

.T

he

solidfou

rlin

esan

dth

edotted

four

lines

with

circlesare

realparts

and

imagin

aryparts,

respectively.

Exam

ple

4:

Detw

eile

r-type

adju

sted

(iniE

F/P

Gco

ord

.)

-1

-0.5 0

0.5 1

1.5 2

05

10

15

20

iEF

coordinate, Detw

eiler type κL =

+0

.5(b

)Real / Imaginary parts of Eigenvalues (AF)

rsch

-1

-0.5 0

0.5 1

1.5 2

05

10

15

20

PG

coordinate, Detw

eiler type κL =

+0

.5(c

)

Real / Imaginary parts of Eigenvalues (AF)

rsch

Figu

re4:

Sim

ilarcom

parison

forD

etweiler

adju

stmen

ts.κ

L=

+1/2

forall

plots.

No.

No.

inadjustm

ent1st?

Sch/isocoords.

iEF/P

Gcoords.

Table.??

TR

Sreal.

imag.

real.im

ag.0

0–

noadjustm

entsyes

––

––

–P

-12-P

Pij

−κ

Lα

3γij

nono

makes

2N

eg.not

apparentm

akes2

Neg.

notapparent

P-2

3P

ij−

κLαγ

ijno

nom

akes2

Neg.

notapparent

makes

2N

eg.not

apparentP

-3-

Pij

Prr

=−

κor

Prr

=−

κα

nono

slightlyenl.N

eg.not

apparentslightly

enl.Neg.

notapparent

P-4

-P

ij−

κγ

ijno

nom

akes2

Neg.

notapparent

makes

2N

eg.not

apparentP

-5-

Pij

−κγ

rr

nono

red.Pos./enl.N

eg.not

apparentred.P

os./enl.Neg.

notapparent

Q-1

-Q

kij

καβ

kγij

nono

N/A

N/A

κ∼

1.35m

in.vals.

notapparent

Q-2

-Q

kij

Qrrr

=κ

noyes

red.abs

vals.not

apparentred.

absvals.

notapparent

Q-3

-Q

kij

Qrij

=κγ

ijor

Qrij

=καγ

ijno

yesred.

absvals.

notapparent

enl.Neg.

enl.vals.

Q-4

-Q

kij

Qrrr

=κγ

rr

noyes

red.abs

vals.not

apparentred.

absvals.

notapparent

R-1

1R

ijκ

Fαγ

ijyes

yesκ

F=

−1/4

min.

absvals.

κF

=−

1/4m

in.vals.

R-2

4R

ijR

rr

=−

κµα

orR

rr

=−

κµ

yesno

notapparent

notapparent

red.Pos./enl.N

eg.enl.

vals.R

-3-

Rij

Rrr

=−

κγ

rr

yesno

enl.vals.

notapparent

red.Pos./enl.N

eg.enl.

vals.S-1

2-SS

kij

κLα

2[3(∂(i α

)δkj) −

(∂l α

)γij γ

kl]

yesno

notapparent

notapparent

notapparent

notapparent

S-2-

Skij

καγ

lk(∂l γ

ij )yes

nom

akes2

Neg.

notapparent

makes

2N

eg.not

apparentp-1

-p

kij

prij

=−

καγ

ijno

nored.

Pos.

red.vals.

red.Pos.

enl.vals.

p-2-

pkij

prrr

=κα

nono

red.Pos.

red.vals.

red.Pos/enl.N

eg.enl.

vals.p-3

-p

kij

prrr

=καγ

rr

nono

makes

2N

eg.enl.

vals.red.

Pos.

vals.red.

vals.q-1

-qklij

qrrij

=καγ

ijno

noκ

=1/2

min.

vals.red.

vals.not

apparentenl.

vals.q-2

-qklij

qrrrr

=−

καγ

rr

noyes

red.abs

vals.not

apparentnot

apparentnot

apparentr-1

-rkij

rrij

=καγ

ijno

yesnot

apparentnot

apparentnot

apparentenl.

vals.r-2

-rkij

rrrr

=−

κα

noyes

red.abs

vals.enl.

vals.red.

absvals.

enl.vals.

r-3-

rkij

rrrr

=−

καγ

rr

noyes

red.abs

vals.enl.

vals.red.

absvals.

enl.vals.

s-12-s

sklij

κLα

3[δk(i δ

lj) −(1/3)γ

ij γkl]

nono

makes

4N

eg.not

apparentm

akes4

Neg.

notapparent

s-2-

sklij

srrij

=−

καγ

ijno

nom

akes2

Neg.

red.vals.

makes

2N

eg.red.

vals.s-3

-sklij

srrrr

=−

καγ

rr

nono

makes

2N

eg.red.

vals.m

akes2

Neg.

red.vals.

Tab

le1:

List

ofad

justm

ents

we

testedin

the

Sch

warzsch

ildsp

acetime.

The

colum

nof

adju

stmen

tsare

non

zerom

ultip

liers.T

he

effects

toam

plifi

cationfactors

(when

κ>

0)are

comm

ented

foreach

coord

inate

system

and

forreal/im

aginary

parts

ofA

Fs,

respectively.

The

‘N/A

’m

eans

that

there

isno

effect

due

toth

eco

ordin

ateprop

erties;‘n

otap

paren

t’m

eans

the

adju

stmen

tdoes

not

chan

geth

eA

Fs

effectively

accordin

gto

our

conjectu

re;‘en

l./red./m

in.’

mean

sen

large/reduce/m

inim

ize,an

d‘P

os./Neg.’

mean

spositive/n

egative,resp

ectively.T

hese

judgem

ents

arem

ade

atth

er∼

O(10M

)region

onth

eirt=

0slice.

3.2

.2N

um

erica

ldem

onstra

tion

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.00.00.5

1.01.5

2.0

Detw

eiler's adju

stmen

ts on

Min

kow

skii sp

acetime

L=

-0.01L

=0.0

L=

+0.01

L=

+0.02

L=

+0.03

L=

+0.35

time

Log1 0

(L2 norm of constraints )

L=

0 (standard AD

M)

L=

+ 0.01

L=

+ 0.03

L=

- 0.01

L=

+ 0.035

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.00.00.5

1.01.5

2.0

Sim

plified

Detw

eiler's adju

stmen

ts on

Min

kow

skii sp

acetime

L=

-0.01L

=0.0

L=

+0.01

L=

+0.02

L=

+0.04

L=

+0.06

L=

+0.08

time

Log1 0

(L2 norm of constraints )

L=

+ 0.08

L=

- 0.01

L=

0 (standard AD

M)

L=

+ 0.01

L=

+ 0.06

Figu

re1:

We

confirm

ednum

erically,usin

gM

inkow

skii

pertu

rbation

,th

atD

etweiler’s

system

presen

tsbetter

accuracy

than

the

standard

AD

M,but

only

formsm

allpositive

L.

Com

pariso

ns

ofA

dju

sted

AD

Msy

stem

s(lin

ear

wave)

Mexico

NR

2002

Work

shop

particip

ants

0

0.002

0.004

0.006

0.008

0.01

050

100150

200

originalAD

MstandardA

DM

Detw

eiler_0.02D

etweiler_0.04

Sim

plifiedDetw

eiler_0.02S

implifiedD

etweiler_0.04

L2 norm of Hamiltonian constraint

time

Stan

dard

AD

M

Orig

inal A

DM

Ad

justed

AD

M (S

D)

Ad

justed

AD

M (D

)

Figu

re1:

Violation

ofH

amilton

iancon

straints

versus

time:

Adju

stedA

DM

system

sap

plied

forTeu

kolsky

wave

initial

data

evolution

with

harm

onic

slicing,

and

with

perio

dic

bou

ndary

condition

.C

actus/C

actusE

instein

/AD

Mco

de

was

used

.G

rid=

243,

∆x

=0.25,

iterativeC

rank-N

icholson

meth

od.

“Einstein

equations”are

time-reversal

invariant.So

...

Why

allnegative

amplification

factors(A

Fs)

areavailable?

Expla

natio

nby

the

time-re

versa

lin

varia

nce

(TR

I)

•the

adjustment

ofthe

systemI,

adjustterm

to∂

t︸︷︷︸(−

) Kij

︸︷︷︸(−

)

=κ

1α︸︷︷︸(+

) γij

︸︷︷︸(+

) H︸︷︷︸(+

)

preservesT

RI.

...so

theAFs

remain

zero(unchange).

•the

adjustment

by(a

partof)

Detw

eiler

adjustterm

to∂

t︸︷︷︸(−

) γij

︸︷︷︸(+

)

=−

Lα︸︷︷︸(+

) γij

︸︷︷︸(+

) H︸︷︷︸(+

)

violatesT

RI.

...so

theAFs

canbecom

enegative.

Therefore

We

canbreak

thetim

e-reversalinvariant

featureof

the“A

DM

equations”.

Constru

cting

Asy

mpto

tically

Constra

ined

Syste

ms

Hisa

akiShin

kai

1.In

troductio

n

2.T

hre

eappro

ach

es

(1)A

rnow

itt-Dese

r-Misn

er/

Baum

garte

-Shapiro

-Shib

ata

-Nakam

ura

(2)

Hyperb

olic

form

ula

tions

(3)

Attra

ctor

syste

ms

–“A

dju

sted

Syste

ms”

3.A

dju

sted

AD

Msy

stem

s

4.A

dju

sted

BSSN

syste

ms

5.Sum

mary

strategy1

Shibata-N

akamura’s

(Baum

garte-Shapiro’s)

modifications

tothe

standardAD

M

–define

newvariables

(φ,γ̃

ij ,K,Ã

ij ,Γ̃i),

insteadof

theAD

M’s

(γij ,K

ij )w

here

γ̃ij ≡

e −4φγ

ij ,Ã

ij ≡e −

4φ(Kij −

(1/3)γij K

),Γ̃

i≡Γ̃

ijk γ̃jk,

usem

omentum

constraintin

Γi-eq.,

andim

pose

detγ̃

ij=

1during

theevolutions.

–T

heset

ofevolution

equationsbecom

e

(∂t −

Lβ )φ

=−

(1/6)αK

,

(∂t −

Lβ )γ̃

ij=

−2α

Ãij ,

(∂t −

Lβ )K

=αÃ

ij Ãij

+(1/3)α

K2−

γij(∇

i ∇j α

),

(∂t −

Lβ )Ã

ij=

−e −

4φ(∇i ∇

j α)T

F+

e −4φα

R(3)ij

−e −

4φα(1/3)γ

ij R(3)

+α

(KÃ

ij −2Ã

ik Ãkj )

∂t Γ̃

i=

−2(∂

j α)Ã

ij−(4/3)α

(∂j K

)γ̃ij

+12α

Ãji(∂

j φ)−

2αÃ

kj(∂

j γ̃ik)−

2αΓ̃

klj Ã

jk γ̃

il

−∂

j (βk∂

k γ̃ij−

γ̃kj(∂

k βi)−

γ̃ki(∂

k βj)

+(2/3)γ̃

ij(∂k β

k) )

Rij

=∂

k Γkij −

∂i Γ

kkj+

Γmij Γ

kmk −

Γmkj Γ

kmi=

:R̃

ij+

Rφij

Rφij

=−

2D̃i D̃

j φ−

2g̃ij D̃

lD̃l φ

+4(D̃

i φ)(D̃

j φ)−

4g̃ij (D̃

lφ)(D̃

l φ)

R̃ij

=−

(1/2)g̃lm

∂lm

g̃ij

+g̃

k(i ∂

j) Γ̃k

+Γ̃

kΓ̃(ij)k

+2g̃

lmΓ̃

kl(i Γ̃j)k

m+

g̃lm

Γ̃kim

Γ̃klj

–N

oexplicit

explanationsw

hythis

formulation

works

better.

AEIgroup

(2000):the

replacement

bym

omentum

constraintis

essential.

Constra

ints

inB

SSN

syste

m

The

normal

Ham

iltonianand

mom

entumconstraints

HB

SSN

=R

BSSN

+K

2−K

ij Kij,

(1)

MB

SSN

i=

MA

DM

i,

(2)

Additionally,

we

regardthe

following

threeas

theconstraints:

Gi

=Γ̃

i−γ̃

jkΓ̃ijk ,

(3)

A=

Ãij γ̃

ij,(4)

S=

γ̃−

1,(5)

Adju

stments

inevolu

tion

equatio

ns

∂Btϕ

=∂

Atϕ

+(1/6)αA

−(1/12)γ̃

−1(∂

j S)β

j,(6)

∂Btγ̃

ij=

∂Atγ̃

ij −(2/3)α

γ̃ij A

+(1/3)γ̃

−1(∂

k S)β

kγ̃ij ,

(7)

∂BtK

=∂

AtK

−(2/3)α

KA

−αH

BSSN

+αe −

4ϕ(D̃j G

j),(8)

∂BtÃ

ij=

∂AtÃ

ij+

((1/3)αγ̃

ij K−

(2/3)αÃ

ij )A+

((1/2)αe −

4ϕ(∂k γ̃

ij )−(1/6)α

e −4ϕγ̃

ij γ̃−

1(∂k S

))Gk

+αe −

4ϕγ̃k(i (∂

j) Gk)−

(1/3)αe −

4ϕγ̃ij (∂

k Gk)

(9)

∂BtΓ̃

i=

∂AtΓ̃

i−((2/3)(∂

j α)γ̃

ji+(2/3)α

(∂j γ̃

ji)+

(1/3)αγ̃

jiγ̃−

1(∂j S

)−4α

γ̃ij(∂

j ϕ))A

−(2/3)α

γ̃ji(∂

j A)

+2α

γ̃ijM

j −(1/2)(∂

k βi)γ̃

kjγ̃

−1(∂

j S)+

(1/6)(∂j β

k)γ̃ijγ̃

−1(∂

k S)+

(1/3)(∂k β

k)γ̃ijγ̃

−1(∂

j S)

+(5/6)β

kγ̃−

2γ̃ij(∂

k S)(∂

j S)+

(1/2)βkγ̃

−1(∂

k γ̃ij)(∂

j S)+

(1/3)βkγ̃

−1(∂

j γ̃ji)(∂

k S).

(10)

AFull

set

ofB

SSN

constra

int

pro

pagatio

neqs.

∂B

St

HB

S

Mi

Gi

SA

=

A11

A12

A13

A14

A15

−(1/3)(∂

i α)+

(1/6)∂i

αK

A23

0A

25

0αγ̃

ij0

A34

A35

00

0β

k(∂k S

)−

2αγ̃

00

00

αK

+β

k∂k

HB

S

Mj

Gj

SA

A11

=+

(2/3)αK

+(2/3)αA

+β

k∂k

A12

=−

4e −4ϕα(∂

k ϕ)γ̃

kj−

2e −4ϕ(∂

k α)γ̃

jk

A13

=−

2αe −

4ϕÃ

kj ∂

k −αe −

4ϕ(∂

j Ãkl )γ̃

kl−

e −4ϕ(∂

j α)A

−e −

4ϕβ

k∂k ∂

j −(1/2)e −

4ϕβ

kγ̃−

1(∂j S

)∂k

+(1/6)e −

4ϕγ̃−

1(∂j β

k)(∂k S

)−(2/3)e −

4ϕ(∂

k βk)∂

j

A14

=2α

e −4ϕγ̃−

1γ̃lk(∂

l ϕ)A

∂k+

(1/2)αe −

4ϕγ̃−

1(∂l A

)γ̃lk∂

k+

(1/2)e −4ϕγ̃−

1(∂l α

)γ̃lkA

∂k+

(1/2)e −4ϕγ̃−

1βm

γ̃lk∂

m∂

l ∂k

−(5/4)e −

4ϕγ̃−

2βm

γ̃lk(∂

m S)∂

l ∂k+

e −4ϕγ̃−

1βm

(∂m

γ̃lk)∂

l ∂k+

(1/2)e −4ϕγ̃−

1βi(∂

j ∂i γ̃

jk)∂k

+(3/4)e −

4ϕγ̃−

3βiγ̃

jk(∂i S

)(∂j S

)∂k −

(3/4)e −4ϕγ̃−

2βi(∂

i γ̃jk)(∂

j S)∂

k+

(1/3)e −4ϕγ̃−

1γ̃pj(∂

j βk)∂

p ∂k

−(5/12)e −

4ϕγ̃−

2γ̃jk(∂

k βi)(∂

i S)∂

j+

(1/3)e −4ϕγ̃−

1(∂k γ̃

ij)(∂j β

k)∂i −

(1/6)e −4ϕγ̃−

1γ̃m

k(∂k ∂

l βl)∂

m

A15

=(4/9)α

KA

−(8/9)α

K2+

(4/3)αe −

4ϕ(∂

i ∂j ϕ

)γ̃ij

+(8/3)α

e −4ϕ(∂

k ϕ)(∂

l γ̃lk)

+αe −

4ϕ(∂

j γ̃jk)∂

k

+8α

e −4ϕγ̃

jk(∂j ϕ

)∂k+

αe −

4ϕγ̃

jk∂j ∂

k+

8e −4ϕ(∂

l α)(∂

k ϕ)γ̃

lk+

e −4ϕ(∂

l α)(∂

k γ̃lk)

+2e −

4ϕ(∂

l α)γ̃

lk∂k

+e −

4ϕγ̃

lk(∂l ∂

k α)

A23

=αe −

4ϕγ̃

km

(∂k ϕ

)(∂j γ̃

mi )−

(1/2)αe −

4ϕΓ̃

mkl γ̃

kl(∂

j γ̃m

i )

+(1/2)α

e −4ϕγ̃

mk(∂

k ∂j γ̃

mi )

+(1/2)α

e −4ϕγ̃−

2(∂i S

)(∂j S

)−(1/4)α

e −4ϕ(∂

i γ̃kl )(∂

j γ̃kl)

+αe −

4ϕγ̃

km

(∂k ϕ

)γ̃ji ∂

m

+αe −

4ϕ(∂

j ϕ)∂

i −(1/2)α

e −4ϕΓ̃

mkl γ̃

klγ̃

ji ∂m

+αe −

4ϕγ̃

mkΓ̃

ijk ∂m

+(1/2)α

e −4ϕγ̃

lkγ̃ji ∂

k ∂l

+(1/2)e −

4ϕγ̃

mk(∂

j γ̃im

)(∂k α

)+

(1/2)e −4ϕ(∂

j α)∂

i+

(1/2)e −4ϕγ̃

mkγ̃

ji (∂k α

)∂m

A25

=−

Ãki (∂

k α)+

(1/9)(∂i α

)K+

(4/9)α(∂

i K)+

(1/9)αK

∂i −

αÃ

ki ∂

k

A34

=−

(1/2)βkγ̃

ilγ̃−

2(∂l S

)∂k −

(1/2)(∂l β

i)γ̃lkγ̃

−1∂

k+

(1/3)(∂l β

l)γ̃ikγ̃

−1∂

k −(1/2)β

lγ̃in(∂

l γ̃m

n )γ̃m

kγ̃−

1∂k

+(1/2)β

kγ̃ilγ̃

−1∂

l ∂k

A35

=−

(∂k α

)γ̃ik

+4α

γ̃ik(∂

k ϕ)−

αγ̃

ik∂k

BSSN

Constra

int

pro

pagatio

nanaly

sisin

flat

space

time

•T

heset

ofthe

constraintpropagation

equations,∂

t (HB

SSN,M

i ,Gi,A

,S)T

?

•For

theflat

backgroundm

etricg

µν

=η

µν ,

thefirst

orderperturbation

equationsof

(6)-(10):

∂t (1)ϕ

=−

(1/6) (1)K+

(1/6)κϕ(1)A

(11)

∂t (1)γ̃

ij=

−2 (1)Ã

ij −(2/3)κ

γ̃ δij (1)A

(12)

∂t (1)K

=−

(∂j ∂

j (1)α)+

κK

1 ∂j (1)G

j−κ

K2 (1)H

BSSN

(13)

∂t (1)Ã

ij=

(1)(RB

SSN

ij)T

F−

(1)(D̃i D̃

j α)T

F+

κA

1 δk(i (∂

j) (1)Gk)−

(1/3)κA

2 δij (∂

k (1)Gk)

(14)

∂t (1)̃Γ

i=

−(4/3)(∂

i (1)K)−

(2/3)κΓ̃1 (∂

i (1)A)+

2κΓ̃2 (1)M

i(15)

We

expressthe

adjustements

as

κadj

:=(κ

ϕ ,κγ̃ ,κ

K1 ,κ

K2 ,κ

A1 ,κ

A2 ,κ

Γ̃1 ,κ

Γ̃2 ).

(16)

•Constraint

propagationequations

atthe

firstorder

inthe

flatspacetim

e:

∂t (1)H

BSSN

=(κ

γ̃ −(2/3)κ

Γ̃1 −

(4/3)κϕ

+2)

∂j ∂

j (1)A+

2(κΓ̃2 −

1)(∂j (1)M

j ),(17)

∂t (1)M

i=

(−(2/3)κ

K1+

(1/2)κA

1 −(1/3)κ

A2+

(1/2))∂

i ∂j (1)G

j

+(1/2)κ

A1 ∂

j ∂j (1)G

i+((2/3)κ

K2 −

(1/2))∂

i (1)HB

SSN,

(18)

∂t (1)G

i=

2κΓ̃2 (1)M

i +(−

(2/3)κΓ̃1 −

(1/3)κγ̃ )(∂

i (1)A),

(19)

∂t (1)S

=−

2κγ̃ (1)A

,(20)

∂t (1)A

=(κ

A1 −

κA

2 )(∂j (1)G

j).(21)

Effect

ofadju

stments

No.

Constraints

(number

ofcom

ponents)

diag?Constr.

Am

p.Factors

H(1)

Mi(3)

Gi(3)

A(1)

S(1)

inM

inkowskii

background

0.standard

AD

Muse

use-

--

yes(0,0,�

,�)

1.BSSN

noadjustm

entuse

useuse

useuse

yes(0,0,0,0,0,0,0,�

,�)

2.the

BSSN

use+adj

use+adj

use+adj

use+adj

use+adj

no(0,0,0,�

,�,�

,�,�

,�)

3.no

Sadjustm

entuse+

adjuse+

adjuse+

adjuse+

adjuse

nono

difference

inflat

background4.

noA

adjustment

use+adj

use+adj

use+adj

useuse+

adjno

(0,0,0,�,�

,�,�

,�,�

)5.

noG

iadjustm

entuse+

adjuse+

adjuse

use+adj

use+adj

no(0,0,0,0,0,0,0,�

,�)

6.no

Miadjustm

entuse+

adjuse

use+adj

use+adj

use+adj

no(0,0,0,0,0,0,0,�

,�)

Grow

ingm

odes

7.no

Hadjustm

entuse

use+adj

use+adj

use+adj

use+adj

no(0,0,0,�

,�,�

,�,�

,�)

8.ignore

Gi,A

,Suse+

adjuse+

adj-

--

no(0,0,0,0)

9.ignore

Gi,A

use+adj

use+adj

use+adj

--

yes(0,�

,�,�

,�,�

,�)

10.ignore

Gi

use+adj

use+adj

-use+

adjuse+

adjno

(0,0,0,0,0,0)11.

ignoreA

use+adj

use+adj

use+adj

-use+

adjyes

(0,0,�,�

,�,�

,�,�

)12.

ignoreS

use+adj

use+adj

use+adj

use+adj

-yes

(0,0,�,�

,�,�

,�,�

)

New

Pro

posa

ls::

Impro

ved

(adju

sted)

BSSN

syste

ms

TR

Sbre

akin

gadju

stments

Inorder

tobreak

time

reversalsym

metry

(TRS)

ofthe

evolutioneqs,

toadjust

∂t φ

,∂t γ̃

ij ,∂t Γ̃

iusing

S,G

i,or

toadjust

∂t K

,∂t Ã

ijusing

Ã.

∂t φ

=∂

BS

tφ

+κ

φHαH

BS

+κ

φG αD̃

k Gk

+κ

φS1 αS

+κ

φS2 α

D̃jD̃

j S∂

t γ̃ij

=∂

BS

tγ̃

ij+

κγ̃H

αγ̃

ij HB

S+

κγ̃G

1 αγ̃

ij D̃k G

k+

κγ̃G

2 αγ̃

k(i D̃

j) Gk

+κ

γ̃S1 α

γ̃ij S

+κ

γ̃S2 α

D̃i D̃

j S∂

t K=

∂B

St

K+

κKM

αγ̃

jk(D̃j M

k )+

κKÃ

1 αÃ+

κKÃ

2 αD̃

jD̃j Ã

∂t Ã

ij=

∂B

St

Ãij

+κ

AM1 α

γ̃ij (D̃

kMk )

+κ

AM2 α

(D̃(i M

j) )+

κAÃ

1 αγ̃

ij Ã+

κAÃ

2 αD̃

i D̃j Ã

∂t Γ̃

i=

∂B

St

Γ̃i+

κΓ̃H

αD̃

iHB

S+

κΓ̃G

1 αGi+

κΓ̃G

2 αD̃

jD̃j G

i+κ

Γ̃G3 α

D̃iD̃

j Gj+

κΓ̃S

αD̃

iHB

S

orin

theflat

background

∂A

DJ

t(1)φ

=+

κφH

(1)HB

S+

κφG ∂

k (1)Gk

+κ

φS1 (1)S

+κ

φS2 ∂

j ∂j (1)S

∂A

DJ

t(1)γ̃

ij=

+κ

γ̃Hδij (1)H

BS

+κ

γ̃G1 δ

ij ∂k (1)G

k+

(1/2)κγ̃G

2 (∂j (1)G

i+∂

i (1)Gj)

+κ

γ̃S1 δ

ij (1)S+

κγ̃S

2 ∂i ∂

j (1)S∂

AD

Jt

(1)K=

+κ

KM

∂j (1)M

j+

κKÃ

1 (1)Ã+

κKÃ

2 ∂j ∂

j (1)̶

AD

Jt

(1)Ãij

=+

κAM

1 δij ∂

k (1)Mk

+(1/2)κ

AM2 (∂

i Mj+

∂j M

i )+

κAÃ

1 δij Ã

+κ

AÃ2 ∂

i ∂j Ã

∂A

DJ

t(1)̃Γ

i=

+κ

Γ̃H∂

i (1)HB

S+

κΓ̃G

1 (1)Gi+

κΓ̃G

2 ∂j ∂

j (1)Gi+

κΓ̃G

3 ∂i ∂

j (1)Gj+

κΓ̃S

∂i (1)S

Constra

int

Am

plifi

catio

nFacto

rsw

itheach

adju

stment

adjustment

CAFs

diag?eff

ectof

theadjustm

ent

∂t φ

κφH

αH(0,0,±

√−

k2(∗3),8κ

φHk

2)no

κφH

<0

makes

1N

eg.

∂t φ

κφG

αD̃

k Gk

(0,0,±√−

k2(∗2),

longexpressions)

yesκ

φG<

0m

akes2

Neg.

1Pos.

∂t γ̃

ijκ

SD

αγ̃

ij H(0,0,±

√−

k2(∗3),(3/2)κ

SDk

2)yes

κS

D<

0m

akes1

Neg.

Case

(B)

∂t γ̃

ijκ

γ̃G1αγ̃

ij D̃k G

k(0,0,±

√−

k2(∗2),

longexpressions)

yesκ

γ̃G1>

0m

akes1

Neg.

∂t γ̃

ijκ

γ̃G2αγ̃

k(i D̃j) G

k(0,0,

(1/4)k2κ

γ̃G2 ±

√k

2(−1

+k

2κγ̃G

2 /16)(∗2),long

expressions)yes

κγ̃G

2<

0m

akes6

Neg.

1Pos.

Case

(E1)

∂t γ̃

ijκ

γ̃S1αγ̃

ij S(0,0,±

√−

k2(∗3),3κ

γ̃S1 )

noκ

γ̃S1<

0m

akes1

Neg.

∂t γ̃

ijκ

γ̃S2αD̃

i D̃j S

(0,0,±√−

k2(∗3),−

κγ̃S

2 k2)

noκ

γ̃S2>

0m

akes1

Neg.

∂t K

κKM

αγ̃

jk(D̃j M

k )(0,0,0,±

√−

k2(∗2),

(1/3)κKM

k2±

(1/3) √k

2(−9

+k

2κ2KM

))no

κKM

<0

makes

2N

eg.

∂t Ã

ijκ

AM1αγ̃

ij (D̃kM

k )(0,0,±

√−

k2(∗3),−

κAM

1 k2)

yesκ

AM1>

0m

akes1

Neg.

∂t Ã

ijκ

AM2α(D̃

(i Mj) )

(0,0,−k

2κAM

2 /4±√

k2(−

1+

k2κ

AM

2 /16)(∗2),

longexpressions)

yesκ

AM2>

0m

akes7

Neg

Case

(D)

∂t Ã

ijκ

AA1αγ̃

ij A(0,0,±

√−

k2(∗3),3κ

AA1 )

yesκ

AA1<

0m

akes1

Neg.

∂t Ã

ijκ

AA2αD̃

i D̃j A

(0,0,±√−

k2(∗3),−

κAA

2 k2)

yesκ

AA2>

0m

akes1

Neg.

∂t Γ̃

iκ

Γ̃HαD̃

iH(0,0,±

√−

k2(∗3),−

κAA

2 k2)

noκ

Γ̃H>

0m

akes1

Neg.

∂t Γ̃

iκ

Γ̃G1αG

i(0,0,(1/2)κ

Γ̃G1 ±

√−k

2+

κ2Γ̃G

1 (∗2),long.)

yesκ

Γ̃G1<

0m

akes6

Neg.

1Pos.

Case

(E2)

∂t Γ̃

iκ

Γ̃G2αD̃

jD̃j G

i(0,0,−

(1/2)κΓ̃G

2 ±√−

k2+

κ2Γ̃G

2 (∗2),long.)

yesκ

Γ̃G2>

0m

akes2

Neg.

1Pos.

∂t Γ̃

iκ

Γ̃G3αD̃

iD̃j G

j(0,0,−

(1/2)κΓ̃G

3 ±√−

k2+

κ2Γ̃G

3 (∗2),long.)

yesκ

Γ̃G3>

0m

akes2

Neg.

1Pos.

gr-q

c/0204002

(PR

Din

prin

t)

Com

pariso

ns

ofA

dju

sted

BSSN

syste

ms

(linear

wave)

Mexico

NR

2002

Work

shop

particip

ants

Figu

re2:

Violation

ofH

amilton

iancon

straints

versus

time:

Adju

stedB

SSN

system

sap

plied

forTeu

kolsky

wave

initial

data

evolution

with

harm

onic

slicing,

and

with

perio

dic

bou

ndary

condition

.C

actus/A

EIT

horn

s/BSSN

code

was

used

.G

rid=

243,

∆x

=0.25,

iterativeC

rank-N

icholson

meth

od.

Cou

rtesyof

N.D

orban

dan

dD

.Polln

ey(A

EI).

An

Evolu

tion

ofA

dju

sted

BSSN

Form

ula

tion

by

Yo-B

aum

garte

-Shapiro

,gr-q

c/0209066

02000

40006000

t/M

1018

1016

1014

1012

1010

108

106

104

∆Krms

A3

A4

A5

A6

A7

A8

02000

40006000

t/M

0.88

0.9

0.92

0.94

0.96A

ngular mom

entum/M

2

Innersurface + volum

eO

utersurface

0.89

0.9

0.91

0.92

0.93

0.94M

ass/M

Innersurface + volum

eO

utersurface

02000

40006000

t/M

1018

1016

1014

1012

1010

108

106

104

rms of ∆

fαtrK

0.1

0.2

0.3

0.4

0.5

0.6

0.7C

onstraint residual

Ham

.M

om.

Kerr-S

child

BH

(0.9

J/M

),excisio

nw

ithcu

be,

1+

log-lapse

,Γ-d

river

shift.

∂t Γ̃

i=

(···)+

23Γ̃

iβi,j −

(χ+

23)G

iβj,j

χ=

2/3fo

r(A

4)-(A

8)

∂t γ̃

ij=

(···)−καγ̃

ij Hκ

=0.1∼

0.2fo

r(A

5),

(A6)

and

(A8)

Sum

mary

Tow

ard

sa

stable

and

accu

rate

form

ula

tion

for

num

erica

lre

lativ

ity

•Severa

lre

ports

say

num

erica

lsta

bilitie

sdepend

on

the

form

ula

tions

toapply,

alth

ough

they

are

math

em

atica

llyequiv

ale

nt.

•sta

tus

=ch

aotic,

many

trials

and

erro

rs

We

tried

toundersta

nd

the

back

gro

und

inan

unifi

ed

way.

•O

ur

pro

posa

l=

“Evalu

ate

eig

envalu

es

ofco

nstra

int

pro

pagatio

neqns”

We

giv

esa

tisfacto

ryco

nditio

ns

for

stable

evolu

tions.

Fourie

rtra

nsfo

rmatio

nallo

ws

todiscu

sslo

wer-o

rder

term

s.

•O

ur

Obse

rvatio

n=

“Sta

bility

will

change

by

addin

gco

nstra

ints

inR

HS”

We

nam

ed

“A

dju

sted

Syste

m”.

Num

erica

llyco

nfirm

ed

inth

eM

axw

ell

syste

mand

Ash

tekar

syste

m.

•O

ur

Obse

rvatio

n2=

The

idea

work

seven

for

the

AD

Mfo

rmula

tion

We

expla

inth

eeffectiv

epara

mete

rra

nge

ofD

etw

eile

r’ssy

stem

(1987).

We

pro

pose

dvarie

tyofadju

stments.

Severa

lnum

erica

lco

nfirm

atio

ns.

•O

ur

Obse

rvatio

n3=

The

idea

work

salso

for

the

BSSN

form

ula

tion

We

expla

inw

hy

adju

sting

mom

entu

mco

nstra

ints

impro

ves

the

stability.

We

pro

pose

dvarie

tyofadju

stments.

Severa

lnum

erica

lco

nfirm

atio

ns.

Evalu

atio

nofC

AFsm

ay

be

an

alte

rnativ

eguid

elin

eto

hyperb

oliza

tion

ofth

esy

stem

.

Next

Ste

ps?

•G

enera

lizeth

epro

cedure

toco

nstru

ctadju

sted

syste

ms

–dynam

icaland

auto

matica

ldete

rmin

atio

nof

κunder

asu

itable

prin

ciple

.

–ta

rget

toco

ntro

leach

constra

int

vio

latio

nby

adju

sting

multip

liers.

–cla

rifyth

ere

aso

ns

ofnon-lin

ear

vio

latio

nin

curre

nt

test

evolu

tions.

•M

ore

on

hyperb

olic

form

ula

tions

–effects

ofnon-p

rincip

alpart?

–cla

rifyth

ere

aso

ns

ofadvanta

ges

ofkin

em

atic

para

mete

rs(in

KST

)m

ixed-

form

varia

ble

s,and/or

densitize

dla

pse

?

–lin

ks

toth

ein

itial-b

oundary

valu

epro

ble

m(IB

VP

).

•A

ltern

ativ

enew

ideas?

–co

ntro

lth

eorie

s,optim

izatio

nm

eth

ods

(convex

functio

nalth

eorie

s),m

ath

-

em

atica

lpro

gra

mm

ing

meth

ods,

or

....

•N

um

erica

lco

mpariso

ns

offo

rmula

tions

–“C

om

pariso

ns

of

Form

ula

tions

of

Ein

stein

’sequatio

ns

for

Num

erica

lR

el-

ativ

ity”

(Mexico

NR

work

shop,2002)

inpro

gre

ss

Kidder-S

cheel-Teukolsky

hyperb

olicform

ulation(A

nderson-York

+Frittelli-R

eula)

Phys.

Rev.

D.64

(2001)064017

•Construct

aFirst-order

formusing

variables(K

ij ,gij ,d

kij )

where

dkij ≡

∂k g

ij

Constraints

are(H

,Mi ,C

kij ,C

klij )

whereC

kij ≡

dkij −

∂k g

ij ,and

Cklij ≡

∂[k d

l]ij

•D

ensitizethe

lapse,Q

=log(N

g −σ)

•Adjust

equationsw

ithconstraints

∂̂0 g

ij=

−2N

Kij

∂̂0 K

ij=

(···)+

γN

gij H

+ζN

gabC

a(ij)b

∂̂0 d

kij

=(···)

+ηN

gk(i M

j)+

χN

gij M

k

•Re-deining

thevariables

(Pij ,g

ij ,Mkij )

Pij

≡K

ij+

ẑgij K

,

Mkij

≡(1/2)[k̂

dkij

+êd

(ij)k+

gij (â

dk

+b̂b

k )+

gk(i (ĉd

j)+

d̂bj) )],

dk

=g

abd

kab ,b

k=

gabd

abk

The

redefinitionparam

eters

–do

notchange

theeigenvalues

ofevolution

eqs.

–do

noteff

ecton

theprincipal

partof

theconstraint

evolutioneqs.

–do

affect

theeigenvectors

ofevolution

system.

–do

affect

nonlinearterm

sof

evolutioneqs/constraint

evolutioneqs.