Post on 30-Jan-2021
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.