Introduction to 3+1 and BSSN formalismyuichiro.sekiguchi/3+1.pdfSolving Einstein’s equations on computers 7 APCTP International school on NR and GW July 28-August 3, 2011 Einstein’s

Introduction to 3+1 and BSSN

formalism

Yuichiro Sekiguchi (YITP)

July 28-August 3, 2011 1 APCTP International school on NR and GW

Overview of numerical relativity

Setting ‘realistic’ or ‘physically motivated’

initial conditions

Locating BH (solving AH finder )

Extracting GWs

Solving Einstein’s equations

Solving source filed equations

Solving gauge conditions

Main loop

BH excision

Solving the constraint equations

GR-HD GR-MHD GR-Rad(M)HD Microphysics • EOS

• weak processes


Scope of this lecture


initial conditions


Extracting GWs




Main loop

BH excision




• weak processes

The Goal

July 28-August 3, 2011 APCTP International school on NR and GW 4

The main goal that we are aiming at:

“To Derive Einstein’s equations in BSSN formalism”

Notation and Convention


The signature of the metric: ( - + + + )

(We will use the abstract index notation)

e.g. Wald (1984); see also Penrose, R. and Rindler, W. Spinors and spacetime vol.1,

Cambridge Univ. Press (1987)

Geometrical unit c=G=1

symmetric and anti-symmetric notations

)(())(()

nnnn aaaaaaaa Tn

TTn

T ...]...[...)...( )sgn(!

1 ,

!

111

)(2

1 ),(

2

1][)( baababbaabab TTTTTT -+



initial conditions


Extracting GWs




Main loop

BH excision



• weak processes


Solving Einstein’s equations on computers


Einstein’s equations in full covariant form are a set of coupled partial differential equations The solution, metric g

ab, is not a dynamical object and

represents the full geometry of the spacetime just as the metric of a two-sphere does

To reveal the dynamical nature of Einstein’s equations, we must break the 4D covariance and exploit the special nature of time

One method is 3+1 decomposition in which spacetime manifold and its geometry (graviational fields) are divided into a sequence of ‘instants’ of time

Then, Einstein’s equations are posed as a Cauchy problem which can be solved numerically on computers

3+1 decomposition of spacetime manifold

Let us start to introduce foliation or slicing in the spacetime manifold M

Foliation {S } of M is a family of slices (spacelike hypersurfaces) which do not intersect each other and fill the whole of M

In a globally hyperbolic spacetime,

each S is a Cauchy surface which is

parameterized by a global time

function, t , as St

Foliation is characterized by the gradient one-form


0 , ][ baaa t　

　taa

The lapse function


The norm of a is related to a function called “lapse function”, a(xa) , as :

As we shall see later, the lapse function characterize the proper time between the slices

Also let us introduce the normalized one-form :

the negative sign is introduced so that the direction of n

corresponds to the direction to which t increases

na is the unit normal vector to S

2

1

a- ttgg ba

ab

ba

ab

1 , -- ba

ab

aa nngn a

The spatial metric of S : gab


The spatial metric gab induced by gab onto S is defined by

Using this ‘induced’ metric, a tensor on M is decomposed

into two parts: components tangent and normal to S

The tangent-projection operator is defined as

The normal-projection operator is

Then, projection of a tensor into S is defined by

It is easy to check

baab

cd

bdacab

baabab

nnggg

nng

+

+

gg

g

b

aa

b

a

b nn+

a

b

a

ba

ba

b nn --

s

rs

s

r

rs

rdd

ccb

d

b

d

a

c

a

cbbaa

TT ......

......

1

11

1

1

11

1 ......

abcd

d

b

c

aab gg g

Covariant derivative associated with gab


Covariant derivative acting on spatial tensors is defined by

The covariant derivative must satisfy the following conditions

It is a linear operator : (obviously holds from linearity of ∇)

Torsion free : DaDb f = DbDa f , (easy to check by direct calculation)

Compatible with the metric : Dc gab=0 , (easy to check also)

Leibnitz’s rule holds :

s

rs

s

r

r

s

r

s

r

ddcc

f

b

d

b

d

a

c

a

c

f

e

bbaa

ebbaa

e

T

TTD

......

......

......

1

11

1

1

1

1

1

1

1

......

0for

)(

)()(

)( )(

+

+++

+++

+

d

d

c

dc

d

d

adca

c

c

bd

d

ccb

dc

d

b

a

d

adca

c

c

b

d

c

d

cd

b

acb

c

d

c

d

db

a

c

b

d

cd

b

acb

c

d

db

ac

c

b

b

ac

c

a

wvvDwwDv

vwwvvDwwDv

vwwv

vwwvwvwvD

Intrinsic and extrinsic curvature for S


The Riemann tensor for the slice S is defined by

An other curvature tensor, the extrinsic curvature for S is

defined by

extrinsic curvature provides information on how much the normal direction changes and hence, how S is curved

the antisymmetric part vanish due to Frobenius’s theorem : “For unit normal na to a slice ”

d

abcdcabba RvvDDDD )( -

S- tonormalunit : , )(

a

baab nnK

Sbx

bb xx +

)( ba xn)(

bba xxn +

parallel transported b

aba xKn -

0][ cba nn

][][ )(3 0 cbcba

a nnnn -

The other expressions of Kab


First, note that

where ab is the acceleration of nb which is purely spatial

Because the extrinsic curvature is symmetric, we have

where L n is the Lie derivative with respect to n

Also, we simply have

Thus the extrinsic curvature is related to the “velocity” of the spatial metric gab

baabbaba

dc

d

b

d

b

c

a

c

adc

d

b

c

aba

anKann

nnn

---

++

))((

0)( b

ba

a

ba

ba

b

b nnnnnnan

abnbaabnbabaab nngannK g )()(2

1)(

2

1LL -+---

abnbaab gnK )(2

1L--

3+1 decomposition of 4D Riemann tensor


Geometry of a slice S is described by gab and Kab gab and Kab represent the “instantaneous” gravitational fields in S

In order that the foliation {S} to “fits” the spacetime manifold, gab and Kab must satisfy certain conditions known as Gauss, Codazzi, and Ricci relations

They are related to 3+1 decomposition of Einstein’s equations

These equations are obtained by taking the projections of the 4D Riemann tensor

db

abcd

d

abcdabcd

nnR

nRR

4

4

4 , ,

abKabgabcdR

abcdR4abg

Gauss relation: spatial projection to S


Let us calculate the spatial Riemann tensor

where we used (also note that n vanishes if n is uncontracted)

Then we obtain the Gauss relation

The contracted Gauss relations are

cd

d

ab

dd

baccba

d

bacd

d

cadbcba

cbacba

wnKwKKw

www

wwDD

--

++

))(( ))((

)(

)()()()( baab

dd

a

d

abba

dd

ba

d

ba anKnanKnnnnn +-+-+

bcadbdacabcdabcd

d

d

abcd

d

bacd

d

abcd

d

abccabba

KKKKRR

wKKwKKwRwRwDDDD

-+

+--

4

4 )(

b

cabacac

db

abcdac KKKKRnnRR -++ 44

ab

ab

ba

ab KKKRnnRR -++ 244 2

Codazzi relation : mixed projection to S and n


Next, let us consider the “mixed” projection

where the right hand side is calculated as

Then we obtain the Codazzi relation

The contracted Codazzi relation is

)( 4

cabcbad

d

abc nnnR -

abcbca

bacbcacbbcacba

KaKD

naKDanKn

+-

----

)(

bcaacbd

d

abc KDKDnR - 4

b

aba

b

ab KDKDnR -4

Gauss and Codazzi relations


Note that the Gauss and Codazzi relations depend only on the spatial metric gab, the extrinsic curvature Kab, and their spatial derivatives

This implies that the Gauss-Codazzi relations represent integrability conditions that gab and Kab must satisfy for any slice to be embedded in the spacetime manifold

The Gauss-Codazzi relations are directly associated with the constraint equations of Einstein’s equation

Ricci relation (1)


Let us start from the following equation

The Lie derivative of Kac is

Then we obtain the Ricci relation

acacb

b

ca

b

abc

cb

b

aacacb

b

ca

b

a

b

abc

cb

b

aacacb

b

ca

b

abc

caacbcbbca

b

cabcba

bdb

abcd

aaKnaDKK

annaaKnaanKK

annaaKnanK

anKanKn

nnnnnR

+++-

++++--

++++

+++-

-

])([

][

)]()([

)( 4

b

acb

b

cabacb

bb

acb

b

cabacb

b

acn KKKKKnnKnKKnK --++ )( L

acca

b

cabacn

db

abcd aaaDKKKnnR +++

4 L

Ricci relation (2)


Note that the Lie derivative of Kab

is purely spatial, as

Thus the Ricci relation is

0 +-++ c

bc

a

ab

c

abc

ac

bac

a

abc

ca

abn

a aKaKnKnnKnKnnKn L

acca

b

cabacn

db

abcd aaaDKKKnnR +++

4 L

Ricci relation (3)


The acceleration ab is related to the lapse function a, as

where we have used the fact that is closed one-form

Then the Ricci relation can be written as

Furthermore, using the contracted Gauss relation

we obtain

aaa

aaa

ln lnln

)(22 ][][

bc

c

bcb

c

bccb

c

bc

c

bc

c

bc

c

b

Dnn

nnnnnna

+

---

aa

aaa

ca

b

cabacn

aaca

b

cabacn

db

abcd

DDKKK

DDDDKKKnnR

1

lnlnln

4

++

+++

L

L

b

cabacac

db

abcdac KKKKRnnRR -++ 44

b

cabacaccaacnac KKKKRDDKR 21

4 -++-- aa

L

“Evolution vector” and a na


What is the natural “evolution” vector ?

As stated before, the foliation is characterized by the closed one-

form

Dual vectors ta to will be the evolution vector : ata =1

One simple candidate is ta=ana

Note that na is not the natural evolution vector because

This means that the Lie derivative with respect to na of a tensor

tangent to S is NOT a tensor tangent to S

On the other hand, and any tensor field tangent to S is Lie transported by ana to a tensor field tangent to S

0

))( )(

+-++-

b

a

a

b

a

b

a

bb

a

c

cc

b

a

c

a

c

c

b

a

bc

ca

bn

an

anKKnnnnnnL

0 a

bnaL

The shift vector


We have a degree of freedom to add any spatial vector, called “shift vector”, b a to ana because ab a = 0

Therefore the general evolution vector is : ta=ana+b a

This freedom in the definition of the evolution time vector

stems from the general covariance of Einstein’s equations

It is convenient to rewrite the Ricci relation in terms of the Lie derivative of the evolution time vector, as

where we have used

b

cabacaccaactac KKKKRDDKR 21

)(1

4 -++--- aaa

bLL

ababnab

c

abc

c

bacabc

c

ababnabt KKKnKnKKnKKK )()( bbba aaaa LLLLLL +++++

3+1 decomposition of Einstein’s equations (1) - Decomposition of Tab


Now we proceed 3+1 decomposition of Einstein’s equations

using the Gauss, Codazzi, and Ricci relations

To do it, let us decompose the stress-energy tensor as

where are the energy

density, momentum density/momentum flux, and stress tensor of the source field measured by the Eulerian observer

the trace is

We shall also use Einstein’s equations in the form of

abababab TRgRG 82

1 44 -

abbabaab SnPnEnT ++ )(2

ababab

b

aabba TSTnPTnnE - and ),( ,

EST -

- TgTR ababab

2

184

3+1 decomposition of Einstein’s equations (2) - Hamiltonian constraint


We first project Einstein’s equation into the direction perpendicular to S to obtain

For the left-hand-side, we use the contracted Gauss relation

We finally obtain the Hamiltonian constraint

This is a single elliptic equation which must be satisfied everywhere on the slice

ERnnR ba

ab 82

1 44 +

ab

ab

ba

ab KKKRnnRR -++ 244 2

EKKKR ab

ab 162 -+

3+1 decomposition of Einstein’s equations (3) - Momentum constraint


Similary, “mixed” projection of Einstein’s equations gives

Using the contracted Codazzi relation

We reach the momentum constraint

includes 3 elliptic equations

a

b

ab PnR 84 -

b

aba

b

ab KDKDnR -4

aa

b

ab PKDKD 8-

3+1 decomposition of Einstein’s equations (4) - Evolution equations


The evolution part of Einstein’s equations is given by the full projection onto S of Einstein’s equations : Using a version of the Ricci relation

We obtain the evolution equation for Kab

The evolution equation for gab is given by an expression of Kab

--

++--+ )(

2

18)(

2

1)(

2

128 )(

4 ESSESnnESPnSR ababbaabbaabab gg

b

cabacaccaactac KKKKRDDKR 21

)(1

4 -++--- aaa

bLL

----++-- )(

2

18]2)( ESSKKKKRDDK abab

c

bacababbaabt gaaabLL

ababt K 2)( agb --LL

Summary of 3+1 decomposition


Einstein’s equations are 3+1 decomposed as follows

3+1 decomposition of the stress-energy tensor

ba

ab

ba

ab nnTnnG 8

b

ab

b

ab nTnG 8

abab TG 8

EKKKR ab

ab 162 -+

aa

b

ab PKDKD 8-

( ))(24

]2)(

ESS

KKKKRDDK

abab

c

bacababbaabt

---

-++--

ga

aabLL

ababt K 2)( agb --LL

abbabaab SnPnEnT ++ )(2 ababab

b

aabba TSTnPTnnE - ),( ,

)( baab nK -

Gauss rel.

Codazzi rel.

Ricci rel.

Hamiltonian constraint

Momentum constraint

Evolution Eq. of Kab

Evolution Eq. of gab Definition of Kab

e

[ ]

4 4 ( )

0 * 0

ab a a a

b

ab

a bc a

F n J

F F

+

J

*

ab a b b a abc

c

ab a b b a abc

c

F n E n E B

F n B n B E

- +

- -

0, , 0a a a

a a aE n B n J n

(4 )ab a

a b an F n J

(4 )ab a

bF J

* 0ab

a bn F

* 0ab

b F

e4a

aD E

0a

aD B

( ) ( ) 4a abc a a

t b cE D B J KEb a a a - - +L

( ) ( )a abc a

t b cB D E KBb a a - - +L

Gauss’s law, No monopole no time derivatives of E, B

Constraint equations

Normal projection

Spatial projection

Faraday’s law, Ampere’s law

Evolution equations

Similarity with the Maxwell’s equations


Evolution of constraints


It can be shown that the “evolution” equations for the Hamiltonian (CH) and Momentum (CM) constraints becomes

Where Fij is the spatial projection : the evolution equation

The evolution equations for the constraints show that the constraints are “preserved” or “satisfied” , if They are satisfied initially (CH = CM = 0)

The evolution equation is solved correctly (Fab=0)

( )

( ) aaaaa

aaaa

b

b

i

H

ki

M

j

M

i

j

ij

j

i

Mt

ij

ij

Hk

k

M

k

MkHt

DHFCFDKCCKFDC

FKFCKDCCDC

)2()(2)(

)2()(

-+-+++--

+-+---

L

L

-- abababab TgTRF

2

18 4

Coordinate-basis vectors


Let us choose the coordinate basis vectors

First, we choose the evolution timelike vector t a as the

time-basis vector :

The spatial basis vectors are chosen such that

The spatial basis vectors are Lie transported along t a :

( e i )a remains purely spatial because

( em )a constitute the commutable coordinate basis

Then

We define the dual basis vectors by

aa et )( 0

0)( a

ia e

0] ,[)()()( - a

i

a

b

b

i

a

ib

ba

it etteeteL

0)(2))( (

))(()())(())((

][ +

-

a

iab

ba

i

b

abab

b

a

iat

a

ita

a

iat

a

iat

etett

eeee LLLL

tt L

a

aa e )()( :)( mmm

Components of geometrical quantities (1)


Now we have set the coordinate basis we proceed to calculate the components of geometrical quantities

Because the evolution time vector is the time-coordinate basis we have

From the property of the spatial basis, we have

Then, 0th contravariant components of spatial tensors vanish

From the definition of the time vector and normalization condition of na, we obtain

The normalization of na gives

] 0 0 0 1 [ )()( 0 m

m

m teett aaa

ii

a

iai nen )(0 ,0 a m

m

ijij

i

KK

0

0 0 ,

0

0 0 ], 0 [ mmm

ggbb

] [ iaaa nnt baaba

m -- -+

] 0 0 0 [ 1 am -- nnn a

a

Components of geometrical quantities (2)


From the definition of spatial metric, we have

We here note that from the spatial component of the following equation, we have

This means that the indices of spatial tensors can be lowered and raised by the spatial metric

Then, from the inverse of gab, we obtain

-

--

--

--

jiiji

i

baabab gnngbbagba

baamg

2

ijijbaabab gnng gg -

i

jkj

ikngnngg ggg mm

mm

m ++ )(

))(( ,

22

2

dtdxdtdxdtdsg jjii

ij

iji

i

k

kbbga

gb

bbbam +++-

-

+

An intuitive interpretation


an

ana

dtib

idx

at dtdx iib+

))(( 22 dtdxdtdxdtds jjii

ij bbga +++-

tS

dtt+S

The lapse function measures proper time between two adjacent slices

The shift vector gives relation of the spatial origin between slices

Conformal decomposition


The importance of the conformal decomposition in the time evolution problem was first noted by York (PRL 26, 1656 (1971); PRL

28, 1082 (1972)) He showed that the two degrees of freedom of the gravitational

field are carried by the conformal equivalence classes of 3-metric, which are related each other by the conformal transformation :

In the initial data problems, the conformal decomposition is a powerful tool to solve the constraint equations, as studied by York and O’Murchadha (J. Math. Phys. 14, 456 (1973); PRD 10, 428 (1974)) (see for reviews , e.g., Cook, G.B., Living Rev. Rel. 3, 5 (2000); Pfeiffer, H. P. gr-qc/0412002)

In the following, we shall derive conformal decomposition of Einstein’s equations

ijij gg ~4

“Conformal” decomposition of Ricci tensor (1)


The covariant derivative associated with the conformal metric is characterized by

The two covariant derivatives are related by (e.g. Wald )

where Cijk is a tensor defined by difference of Christoffel symbols

By a straightforward calculation, we can show (e.g Wald )

0~~abcD g

i

l

l

kj

l

j

i

kl

i

j

i

jk TCTCTDTD -+~

)ln~~lnln(2

)~~~

(2

1~

g

gggg

k

iji

k

jj

k

i

ijliljlji

klij

kij

kij

k

DDD

DDDC

-+

-+-

jl

kj

k

il

k

ij

l

lk

k

kji

k

ijk

j

ij

jl

kj

k

il

k

ij

l

lk

k

kji

k

ijk

j

jiij

j

jiij

j

ij

vCCCCCDCDvR

vCCCCCDCDvDDDDvDDDDvR

)~~

(~

)~~

()~~~~

()(

-+-+

-+-+--

“Conformal” decomposition of Ricci tensor (2)


Thus the Ricci tensor is decomposed into two parts, one which is the Ricci tensor associated with the conformal metric and one which contains the conformal factor

More explicitly one can show (see e.g. Wald (1984))

Then, the Ricci scalar is decomposed as

g

g

ijij

k

kijji

k

kijjiijij

RR

DDDD

DDDDRR

+

-+

--

~

)ln~

)(ln~

(~4)ln~

)(ln~

(4

ln~~~2ln

~~2

~

k

k

k

k

k

k

DDR

DDDDRR~~

8~

))]ln~

)(ln~

(ln~~

(8~

[

--

-

-

+-

Conformal decomposition of extrinsic curvature


The first step is to decompose Kij into trace (K) and traceless (Aij) parts as

Then, we perform the conformal decomposition of the traceless part as

Under these conformal decompositions of the spatial metric and the extrinsic curvature, let us consider the conformal decomposition of Einstein’s equation

KAKKAK ijijij

ijijij gg3

1 ,

3

1++

ijkl

jlikij

ijij AAAAA~

, ~ - gg

“Conformal” decomposition of the evolution equations (0) – an additional constraint


In the following, with BSSN reformulation in mind, we set the determinant of the conformal metric to be unity:

with this setting, the conformal factor becomes

In the BSSN formulation, the conformal factor is defined by = ln so that = lng/12

In the case that we do not impose the above condition to the

background conformal metric, the equations derived in the following are modified slightly (for this, see Gourgoulhon, E.,

gr-qc/0703035)

1~det~ ijgg

g ln12

1ln

“Conformal” decomposition of the evolution equations (1) : the conformal factor


Let us start from the evolution equation of the spatial metric g

ij :

Taking the trace of this equation, we have

Now we use an identity for any matrix A :

By setting gij = exp A and taking the Lie derivative, we obtain

Now we can derive the evolution equation for the conformal factor :

ababnabt K 2)( agg ab -- LLL

Kabn

ab 2agg a -L

]tr exp[]det[exp AA

ijn

ij

ijnijn gggggg aaa LLL ))(lntr ( )](lntr exp[

K

K

t

nnijn

ij

a

aggg

b

aaa

6

1ln)(

2ln12ln

--

-

LL

LLL


Again, we start from the evolution equation for gij :

Substituting the decomposition of gij and Kij, we obtain

Now, we shall use the evolution equation for the conformal factor, and finally, we get

“Conformal” decomposition of the evolution equations (2) : the conformal metric

ababnabt K 2)( agg ab -- LLL

+--+-

+--+-

KA

KA

ijijtijijt

ijijtijijt

gagg

gagg

bb

bb

~

3

1~2ln)(~4~)(

~

3

1~2)(~4~)( 43

LLLL

LLLL

ijijt A~

2~)( agb --LL


For the later purpose, let us derive the evolution equation for the inverse of the conformal metric

It is easily obtained from the evolution equation for the conformal metric, as

“Conformal” decomposition of the evolution equations (3) : the inverse conformal metric

ijij

t

ijjl

tkl

j

kt

ik

ij

klt

jlik

A

A

A

~2~)(

~2]~)(~)[(~

~2~)(~~

ag

aggg

aggg

b

bb

b

-

----

--

LL

LLLL

LL


We start from the evolution equation for Kij :

We first simply take the trace of this equation

Here, let make use of the evolution equation for the inverse of the spatial metric,

then we obtain

Finally, using the Hamiltonian constraint, we obtain

“Conformal” decomposition of the evolution equations (4a) : the trace of the extrinsic curvature

)2)((4]2 ijij

k

jikijijjiijn SESKKKKRDDK --+-++- gaaaaL

)2)(3(4]22 SESKKKRDDKK ij

ij

i

i

ij

nijn --+-++-- aaagaa LL

abab

t K 2)( agb -LL

)3(4]2 ESKRDDK i

in -+++- aaaaL

)](4)( SEKKDDK ij

ij

i

it +++-- aabLL

EKKKR ab

ab 162 -+


For convenience, let us express the right-hand-side in terms of the conformal quantities, as well as give a suggestion how to evaluate the derivative term :

“Conformal” decomposition of the evolution equations (4b) : the trace of the extrinsic curvature

( ) ( ) ( )agagagg

a j

jk

kj

jk

k

k

k

k

k DDDD -- ~1 2

22

3

1~~

3

1KAAKAAKK ij

ij

ij

ij

ij

ij ++


We start from the Lie derivative of Kij :

Substituting the following equations into this yields

where TF denotes the trace free part : Tij

TF = Tij - (1/3)gij(trT)

The terms that involve K in the right-hand-side can be written as

“Conformal” decomposition of the evolution equations (5a) : the traceless part of the extrinsic curvature

)2)((4]2 ijij

k

jikijijjiijn SESKKKKRDDK --+-++- gaaaaL

ijnnijijnijn KKAK gg aaaa LLLL3

1

3

1++

)3(4]2 ESKRDDK i

in -+++- aaaaL ijijn K 2aga -L

--+-+- ij

k

jikij

TF

ijijjiijn KKKKKSRDDA gaaaa2TFTF

3

12

3

5)8()( L

---- k

jikij

k

jikijij

k

jikij AAAKAAKAKKKKK~~

2~

3

1 2

3

1

3

12

3

5 2 g


We further proceed to decompose the left-hand-side :

Combining all of the result, we finally reach

We here note that the second-order covariant derivative of the lapse function may be calculated as

NOTE: there is the same 2nd order derivative in Rij

“Conformal” decomposition of the evolution equations (5b) : the traceless part of the extrinsic curvature

-+ ijijnnijijnijn AKAAAA

~

3

2~ln

~4

~ 44 a aaaa LLLL

k

jikij

TF

ijijjiijt AAAKSRDDA~~

2~

)8()(~

)(

TFTF4 -+-+-- - aaabLL

aggaaa

aaaa

lk

kl

ijjik

k

ijji

k

k

ijjijiji CDDDD

---

-

ln~~ln22~

~

)(

Let us turn now to consider the conformal decomposition of the constraint equations

Hamiltonian constraint

Momentum constraint

“Conformal” decomposition of the constraint

equations


EKKKR ab

ab 162 -+

k

k DDRR~~

8~ -- -

3/~~ 2KAAKK ij

ij

ij

ij +

aaab

b PKDKD 8-

ln~~

6~~

ln~

10~

~

3/

j

ijij

j

j

ijij

ikj

jk

kji

jk

ij

j

ij

j

iij

j

ij

j

DAAD

DAAD

ACACADAD

KDADKD

+

+

++

+

-

0212

1~~~~~ 2

+-

+

-

EKAARDD ij

ij

i

i

ii

j

ijij

j PKDDAAD

-+ 8

~ln

~~6

~~

Summary of conformal decomposition


With the conformal decomposition defined by

The 3+1 decomposition (ADM formulation) of Einstein’s equations becomes

0212

1~~~~~ 2

+-

+

-

EKAARDD ij

ij

i

i

ii

j

ijij

j PKDDAAD

-+ 8

~ln

~~6

~~

k

jikij

TF

ijijjiijt AAAKSRDDA~~

2~

)8()(~

)(

TFTF4 -+-+-- - aaabLL

)](4)( SEKKDDK ij

ij

i

it +++-- aabLL

ijijt A~

2~)( agb --LL

Kt ab6

1ln)( --LL

Constraint equations

Evolution equations

ijij gg ~4 KAK ijijij g3

14 + 1~det~ ijgg

Lie derivatives of tensor density


A tensor density of weight w is a object which is a tensor times gw/2 :

One should be careful because the Lie derivative of a tensor density is different from that of a tensor, as

ij

w

ij T2/gT

k

k

aa

bbw

aa

bb

k

k

aa

bb

c

b

r

i

aa

bcb

a

c

s

i

aca

bb

aa

bbc

caa

bb

s

r

s

r

s

ri

s

r

is

r

s

r

s

r

w

w

b

bbbb

b

b

+

+

--

...

...0

...

...

...

...

1

...

.. ..

1

.. ..

...

...

...

...

...

1

1

1

1

1

1

1

1

1

1

1

1

1

1

TT

TTTTT

L

L

k

k

wi

j

k

j

i

k

i

k

k

j

i

jk

k

k

k

wi

j

k

j

i

k

i

k

k

jk

kwi

j

i

jk

k

k

k

wi

j

k

j

i

k

i

k

k

jk

kwi

j

i

jk

k

ij

ijwi

j

k

j

i

k

i

k

k

j

w

k

ki

j

i

j

w

k

k

wi

j

k

j

i

k

i

k

k

j

i

jk

kwi

j

wi

j

wT

wTwT

DwTwT

wTTT

wTTTTT

bgbbb

bgggbbgbgb

bgbbgbgb

ggggbbgbgb

ggbbbgg

b

bbb

++-

++--

++--

++--

++-

--

-

-

-

TTT

TTT

TTT

TT

T

)(

)2/()(

)2/()(

2/12/12/)1(

2/12/)1(

2/2/

2/2/

L

LLL

Lie derivatives in conformal decomposition


The weight factor of the conformal factor = g1/12 is 1/6

Thus the weight factor of the conformal metric and the conformal extrinsic curvature is -2/3, so that

Note that the Lie derivative along t a is equivalent to the

partial derivative along the time direction

Thus ( ) k

kk

k

tt bbb ---6

1lnln)( LL

( ) k

kij

k

ijk

k

jikijk

k

tijt bgbgbggbgb +---- ~

3

2~~~~)( LL

( ) k

kij

k

ijk

k

jikijk

k

tijt AAAAA bbbbb +----~

3

2~~~~)( LL

Evolution of constraints


It can be shown that the “evolution” equations for the Hamiltonian (CH) and Momentum (CM) constraints becomes

Where Fij is the spatial projection of the evolution equation

The evolution equations for the constraints show that the constraints are “preserved” or “satisfied” , if They are satisfied initially (CH = CM = 0)

The evolution equation is solved correctly (Fab=0)

( )

( ) aaaaa

aaaa

b

b

i

H

ki

M

j

M

i

j

ij

j

i

Mt

ij

ij

Hk

k

M

k

MkHt

DHFCFDKCCKFDC

FKFCKDCCDC

)2()(2)(

)2()(

-+-+++--

+-+---

L

L

-- abababab TgTRF

2

18 4

Numerical-relativity simulations based on

the 3+1 decomposition is unstable !!


It is known that simulations based on the 3+1 decomposition (ADM formulation), unfortunately crash in a rather short time

This crucial limitation may be captured in terms of notions of hyperbolicity (e.g. see textbook by Alcubierre (2008) ) Consider the following first-order system The system is called

Strongly Hyperbolic ,if a matrix representation of A has real eigenvalues and complete set of eigenvectors

Weakly Hyperbolic , if A has real eigenvalues but not a complete set of eigenvectors

The key property of strongly and weakly hyperbolic systems : Strongly hyperbolic system is well-posed, and hence, the solution

for the finite-time evolution is bounded

Weakly hyperbolic system is ill-posed and the solution can be unbounded

0+ UAU i

i

t

Numerical-relativity simulations based on

the 3+1 decomposition is unstable !!


It is known that the ADM formulation is only weakly hyperbolic ( Alcubierre (2008) )

Consequently, the ADM formulation is ill-posed and the numerical solution can be unbounded, leading to termination of the simulation

We need formulations for the Einstein’s equation which is (at least) strongly hyperbolic

Let us consider Maxwell’s equations in flat spacetime to capture what we should do to obtain a more stable system

kj

ijkit

i

kj

ijkit

i

i

e

i

i

EB

jBE

B

E

-

-

4

0

4kj

ijki

k

AB

AA

,

m )(

--

--

iiit

iij

j

j

j

iit

e

i

i

DEA

jADDADDE

E

4

4

Note the similarity of these equations to those in the ADM formulation

Consideration in Maxwell’s equations


First of all, let us note the similarity of the Maxwell’s equations with the ADM equations (for simplicity in vacuum)

Second, the Maxwell’s equations are ‘almost’ wave equation

Recall that in the Coulomb gauge DjAj=0, the longitudinal part

(associated with divergence part) of the electric field E does not

obey a wave equation but is described by a Poisson equation

(see a standard textbook, e.g., Jakson)

--

-

iiit

ij

j

j

j

iit

i

i

DEA

ADDADDE

E eq.)t (constrain 0

( )

ijijn

ijlk

kl

ilkj

kl

kjil

kl

ijijn

K

RK

ag

gggggga

a

a

a

2

22

eqs.)t (constrain

-

+-+

+-

L

L wave-like part

“mixed-derivative” part

-+- tij

j

iik

k

it DADDADDA2

Reformulating Maxwell’s equations (1) - Introducing auxiliary variables


A simple but viable approach is to introduce independent auxiliary variables to the system

Let us introduce a new independent variable defined by

The evolution equation for this is

Then, the Maxwell’s equations for the vector potential become a wave equation in the form :

k

k ADF

-- k

ki

i

k

k

tt DDEDADF

FDDADDA itiik

k

it ++- 2


A second approach is to impose a good gauge condition

In the Lorenz gauge, the Maxwell’s equations in the flat spacetime are wave equations

Alternatively, by introducing a source function, one may “generalize” the Coulomb gauge condition so that Poisson-like equations do not appear

Recall again, that in the Coulomb gauge DjAj=0, the longitudinal

part (associated with divergence part) of the electric field E is

described by a Poisson-type equation

Reformulating Maxwell’s equations (2) - Imposing a better gauge

0 m

m A

)( mxHAD k

k


A third approach is to use the constraint equations

To see this, let us back to the example considered in “introducing auxiliary variables”

The constraint equation can be used to rewrite the evolution equation for the auxiliary variable

Seen as the first-order system, the hyperbolic properties of the two system is different: the hyperbolicity could be changed !

It is important and sometimes even crucial to use the constraint equations to change the hyperbolic properties of the system

Reformulating Maxwell’s equations (3) - Using the constraint equations

--

--

--

k

ki

i

t

iiit

iik

k

iit

e

i

i

DDEDF

DEA

jADDFDE

ED

4

4

--

--

--

k

ket

iiit

iik

k

iit

e

i

i

DDF

DEA

jADDFDE

ED

4

4

4

Reformulating Einstein’s equations


The lessons learned from the Maxwell’s equations are Introducing new, independent variables

BSSN ( Shibata & Nakamura PRD 52, 5428 (1995);

Baumgarte & Shapiro PRD 59, 024007 (1999) )

see also Nakamura et al. Prog. Theor. Phys. Suppl. 90, 1 (1987)

Kidder-Scheel-Teukolsky ( Kidder et al. PRD 64, 064017 (2001) )

Bona-Masso ( Bona et al. PRD 56, 3405 (1997) )

Nagy-Ortiz-Reula ( Nagy et al. PRD 70, 044012 (2004) )

Choosing a better gauge Generalized harmonic gauge ( Pretorius, CQG 22, 425 (2005) )

Z4 formalism ( Bona et al. PRD 67, 104005 (2003) )

Using the constraint equations to improve the hyperbolicity adjusted ADM/BSSN ( Shinkai & Yoneda, gr-qc/0209111 )

BSSN outperforms ( Alcubierre (2008) ) ! Exact reason is not clear

BSSN formalism (1)


Let first analyze the conformal Ricci tensor

By noting that the conformal Ricci tensor is

If we divide the conformal metric formally as , we have

Thus we can eliminate the “mixed derivative” terms by introducing new auxiliary variable ( Shibata & Nakamura (1995) )

g~ln~

2 i

k

ik

( ) ) with terms(~~~~

2

1

~~~~~~~

gggggg +---

-+-

ikljjkliijlk

kl

l

kj

k

il

l

kl

k

ij

k

ikj

k

ijkijR

ijijij f+ g~

ijW

( ) ) with terms(~ ffW ki

k

jkj

k

iij

k

kij ++- ggg

ij

j

ijk

jk

iF gg ~~

( ) ) with terms(~ ffFFW ijjiij

k

kij ++- g

BSSN formalism (2)


Baumgarte and Shapiro introduced the slightly different auxiliary variables

In this case, the mixed-second-derivative terms are encompassed as

In linear regime, SN and BS are equivalent

ij

j

i g~-

( ) ) with terms(~~~~

2

1~ gggggg +--- k

ijk

k

jikijlk

kl

ijR

BSSN formalism (3)


Finally let us consider the evolution equation for the auxiliary variables (giving only a rough sketch of derivation) Let us start from the momentum constraint equation

Substituting the evolution equation for the conformal extrinsic curvature

We obtain the evolution equations for Fi and i, respectively

It can be seen from the above sketch of derivation, the evolution equation for i is slightly simpler

ii

j

ijij

j PKDDAAD

-+ 8

~ln

~~6

~~

ijij

t A~

2~)( agb -LL

iik

jk

ijij

jk

k PKDDAAD

-+ gg 8

~ln

~~~6)

~~(~

ijijt A~

2~)( agb --LL

BSSN formalism (4)


The explicit forms of the evolution equations are

( )

-++)(+-+

-+-++--

l

lij

l

ijl

l

jilkijl

l

kkij

jk

ij

j

ijki

jkjk

kijijk

jk

iik

k

t

A

KAAAAfPF

bgbgbggba

ggaab

~

3

2~~)~(~

2

3

2ln

~6~~

2

1~~~216

( )

i

kj

jkk

kj

ijj

j

ii

j

ji

j

j

j

ij

j

ij

j

ijjki

jk

ii

k

k

t AKAAP

bgbgbbb

agaab

+++-+

-

-++--

~~

3

1

3

2

~2~

3

2ln

~6

~~216

BSSN formalism : summary (1)


0212

1~~~~~ 2

+-

+

-

EKAARDD ij

ij

i

i

ii

j

ijij

j PKDDAAD

-+ 8

~ln

~~6

~~

( ) k

kij

k

ijk

k

jik

k

jikij

TF

ijijjiijk

k

t

AAA

AAAKSRDDA

bbb

aaab

-++

-+-+-- -

~

3

2~~

~~2

~)8()(

~

TFTF4

( ) )](4 SEKKDDK ij

ij

i

ik

k

t +++-- aab

( ) k

kij

k

ijk

k

jikijijk

k

t A bgbgbgagb -++-- ~

3

2~~~2~

( ) k

kk

k

t K bab +--6

1

6

1ln

Hamiltonian

constraint is used


Momentum

constraint is used

( )

-++)(+-+

-+-++--

l

lij

l

ijl

l

jilkijl

l

kkij

jk

ij

j

ijki

jkjk

kijijk

jk

iik

k

t

A

KAAAAfPF

bgbgbggba

ggaab

~

3

2~~)~(~

2

3

2ln

~6~~

2

1~~~216

( )

i

kj

jkk

kj

ijj

j

ii

j

ji

j

j

j

ij

j

ij

j

ijjki

jk

ii

k

k

t AKAAP

bgbgbbb

agaab

+++-+

-

-++--

~~

3

1

3

2

~2~

3

2ln

~6

~~216

BSSN formalism : summary (2)



initial conditions


Extracting GWs




Main loop

BH excision



• weak processes


Gauge conditions


Associated directly with the general covariance in general relativity, there are degrees of freedom in choosing coordinates (gauge freedom) Slicing condition is a prescription of choosing the lapse function

Shift condition is that of choosing the shift vector

Einstein’s equations say nothing about how the gauge conditions should be imposed

As we have seen in the reformulation of the ADM system, choosing “good” gauge conditions are very important to achieve stable and robust numerical simulations An improper slicing conditions in a stellar-collapse problem will

lead to appearance of (coordinate and physical) singularities

Also, the shift vector is important in resolving the frame dragging

effect in simulations of e.g. compact binary merger

Preliminary – decomposition of covariant derivative of na –


The covariant derivative of a timelike unit vector za can be decomposed as

Where, the deformation of the

congruence of the timelike vector

is characterized by these tensors

For the unit normal vector to S, na we have

The expansion is –K

The shear is –Aab

The twist vanishes

baabababba zhz -++3

1

ion)(accelerat ,

)(expansion ,

(shear) ,

(twist) ,

metric) (induced ,

TF

)(

][

a

c

ca

c

c

baab

baab

baabab

zz

z

z

z

zzgh

+

baababba anKAn --- g3

1

Geodesic slicing a=1, bi=0


In the geodesic slicing, the evolution equation of the trace of the extrinsic curvature is

For normal matter (which satisfies the strong energy condition),

the right-hand-side is positive

Thus the expansion of time coordinate ( -K ) decreases monotonically in time

In terms of the volume element g1/2 , this means that the volume element goes to zero, as

This behavior results in a coordinate singularity

As can be seen in this example, how to impose a slicing condition is closely related to the trace of the extrinsic curvature

)+(+ SEKKK ij

ijt 34

KDK k

kijt

ij

t -+- 2

1ln baggg

Maximal slicing


Because the decrease in time of the volume element results in a coordinate singularity, let us maximize the volume element

We take the volume of a 3D-domain S : and consider a variation along the time vector

At the boundary of S, we set a=1, b i=0

Thus if K=0 on a slice, the volume is extremal (maximal)

We shall demand that this maximal slicing condition holds for all slices and set

The maximal slicing has a strong singularity avoidance property (E.g. Estabrook & Wahlquist PRD 7, 2814 (1973); Smarr & York, PRD 17, 1945/2529 (1978) )

However this is a elliptic equation and is computationally expensive

S

xdSV 3][ gaaa nt ba +

-+-SS

i

it xdKKxdSV 33 )(][ gabggaL

)](4)(0 SEKKDDK ij

ij

i

it +++-- aabLL

(K-driver)/(approximate maximal) condition


As a generalization of the maximal slicing condition, let us consider the following condition with a positive constant c

This (elliptic) condition drives K back to zero even when K

deviates from zero due to some error or insufficient convergence

Balakrishna et al. ( CQG 13, L135 (1996) ) and Shibata ( Prog. Theor.

Phys. 101, 251 (1999) ) converted this equation into a parabolic one by adding a “time” derivative of the lapse :

If a certain degree of the “convergence” is achieved and the lapse

relaxes to a “stationary state”, it suggests

This condition is called K-driver or approximate maximal slicing condition

cKKt -

cKKDSEKKDD i

iij

ij

i

i +-++- baaa )](4

cKKt -

( ) tcKKtt , a +-

Harmonic slicing


The harmonic gauge condition have played an important role in theoretical developments (Choquet-Bruhat’s textbook)

Existence and uniqueness of the solution of the Cauchy problem of

Einstein’s equations (somewhat similar to Lorenz gauge in EM)

The harmonic slicing condition is defined by

Note that

The harmonic slicing condition can be written as

This is an evolution equation

It is known that the harmonic slicing condition has some singularity avoidance property, although weaker than that of the maximal slicing ( e.g. Cook & Scheel PRD 56, 4775 (1997), Alcubierre’s textbook )

0 ac

c x

( ) 0 0 - mm ggtc

c

ga- g

( ) Kk

kt

-- aab

Generalized harmonic slicing


Bona et al. ( PRL 75, 600 (1995) ) generalized the harmonic slicing condition to

This family of slicing includes the geodesic slicing ( f=0 ), the

harmonic slicing ( f=1 ), and formally the maximal slicing ( f= )

The choice f(a)=2/a , which is called 1+log slicing , has stronger singularity avoidance properties than the harmonic slicing ( Anninos et al. PRD 52, 2059 (1995) )

The 1+log slicing has been widely used and has proven to be a

successful and robust slicing condition

( ) Kfk

kt )( aaab --

Minimal distortion (shift) condition


Smarr and York (PRD 17, 1945/2529 (1978)) proposed a well motivated shift condition called the minimal distortion condition

As seen in the preliminary, the “distortion” part of the congruence is contained in the shear tensor

They define a distortion functional by and take a variation in terms of the shift

here the distortion tensor is defined by

The resulting shift condition is Beautiful and physical but vector elliptic equations (computationally

expensive)

TF

)(

TFTF ~

2

1 baababtab nK -S gL

SS 3dxI ab

ab g

0Sab

aD

+

+++ ab

ab

b

ab

ab

bb

ab

c

ca

ac

c PKDDAADRDDDD gaaabbb 1622

-Freezing and

approximate minimal condition


With some calculations, one can show that the minimal distortion condition is written as The conformal factor is coupled !

Modifications of the minimal distortion condition are proposed by Nakamura et al. (Prog. Theor. Phys. Suppl. 128, 183 (1997)) and Shibata (Prog. Theor. Phys. 101, 1199 (1999))

E.g., Nakamura et al. proposed instead to solve the decoupled pseudo-

minimal distortion condition :

Alcubierre and Brugmann (PRD 63, 104006 (2001)) proposed an approximate minimal distortion condition called Gamma-Freezing :

Anyway, these conditions are elliptic-type !

0)~(~

ijt

jD g

0)~(~

ijt

jD g

0)~(~

i

t

ij

tjD g

-Driver condition


Alcubierre and Brugmann (PRD 63, 104006 (2001)) converted the -freezing elliptic condition into a parabolic one by adding a time derivative of the shift (somewhat similar to the K-driver)

Alcubierre et al. (PRD 67, 084023 (2003)) and others (Lindblom & Scheel

PRD 67, 124005 (2003); Bona et al. PRD 72, 104009 (2005)) extended the -freezing condition to hyperbolic conditions

There are several alternative conditions

Shibata (ApJ 595, 992 (2003)) proposed a hyperbolic shift condition

To date, the above two families of shift conditions are known to be robust

i

tt k b

ii

t

i

t

ii

t

BB

kB

b

-

( ) simulationin used step- time: ,~stepstep tFtF iti

iji

t + gb



initial conditions


Extracting GWs




Main loop

BH excision



• weak processes


3+1 decomposition of - Energy Conservation Equation (1)


First, substitute the 3+1 decomposition of Tab to obtain

Then, let us project it onto normal direction to S.

Noting that Pa, Kab, and ab is purely spatial, we obtain

Because naSab=0, we have

Similarly,

The divergence term of Pa is

0 ab

aT

b

abaab

bb

baba

b

aab

b

a

b

a

b

aa

b

a

b

b

b

ab

SKPPnPnKPKEnEaEnn

SnPnPnEnT

+-++--+

+++

)(0

0++-+- b

ab

a

ab

baa

ab

b SnPnnPKEEn

ab

aba

b

a

b

b

a

a

b

b

a

b

ab

a KSanKSnSSn +- )(

b

b

a

b

b

aab

ba aPnnPPnn --

a

a

a

a

a

b

b

a

b

a

a

b

b

a

a

a aPPPnnPPD -+ )(


Combining altogether, we reach the energy conservation equation

where we have used

The last equation will be used to derive the conservative forms

of the energy equation

3+1 decomposition of - Energy Conservation Equation (2)

0 ab

aT

0)()(

02])(

02

+--+-+

+--+-

--++

aaabba

aab

b

bab

ab

k

k

kk

kt

b

bab

ab

b

bk

k

t

ab

abb

bb

bb

b

DPSKKDEEPDE

DPSKKEPDED

SKKEaPPDEn

EDEEEn k

k

ttnb

b )() baa b -- --L(LL

aln bb Da


To this turn, let us project the equation onto S to obtain

The spacetime-divergence term of Sbc can be replaced by

the spatial-divergence by

The projection term with the covariant derivative of Pc is

Note that (an)-Lie derivative of any spatial tensor is spatial, and

so that

3+1 decomposition of - Momentum Conservation Equation (1)

0 ab

aT

0+-+- b

cb

c

aacb

bc

aba

b

a SKPPnKPEa

d

d

c

b

eb

e

c

b

ed

d

b

d

b

e

c

b

ed

e

c

d

b

b

cb aSSSnnSSD -+ )(

))(()( 11 d

cdcn

c

acb

bc

acb

bc

a nPPPnPn aaaa a - --L

)+(-+ a

b

a

bb

aa

bb

aa

b anKnnnn aaaaa )(

b

abancb

bc

a PKPPn + -

aa L


Combining altogether, we obtain the momentum conservation equation :

Where we have expressed the Lie derivative by spatial covariant derivative

The last equation will be used in conservative reformulation

NOTE: In York (1979), because he used Pa instead of Pa, a extra

term appear in the equation.

3+1 decomposition of - Momentum Conservation Equation (2)

0 ab

aT

0)()(

0)()(

0][)(

0][)(

+--+-+

++-++-

++-+-

+-++-

ababba

aaabab

aaa

a

b

b

a

c

aca

c

cc

cc

acat

ab

b

aa

c

c

b

abac

c

t

ab

b

aa

b

abat

aab

b

a

b

abat

EDDPPKDPSDP

EDDSPKDSDPD

EDDSKPSDP

EaKPaSSDP

L

LL


Now we will show the energy and momentum conservation equations can be recast to conservative form

First, by taking the trace of evolution eq. of gab, we get

Second, note that for any rank-(1,1) spatial tensor,

3+1 decomposition of - Conservative Formulation (1)

0 ab

aT

0)()(

0)()(

+--+-+

+--+-+

ababba

aaabba

a

c

aca

c

ca

cc

acat

b

bab

ab

c

c

cc

ct

EDDPPKDPSDP

DPSKEKDEPDE

gg

ggababbgg tijt

ija

aabbaabt

ab KDKDD ----1

2

1 2)(

( ) k

j

j

ik

k

ik

k

j

j

ik

j

ij

k

ik

k

j

j

ik

j

i

k

jk

k

ik

k

ik

TT

TTTTTTTD

1

)ln(

-

-+-+

gg

g


Using the equations derived the above, we can finally reach the conservative forms of the energy and momentum equations

For the perfect fluid, for instance, these equations may be solved by high resolution shock capturing schemes

3+1 decomposition of - Conservative Formulation (2)

0 ab

aT

( ) ( )( ) ( ) )()(

)()(

k

jj

k

k

iji

k

iki

kk

ikit

k

kij

ij

kk

kt

PSEDDPPSP

DPSKEPE

baabgbagg

aagbagg

-+--+

--+



initial conditions


Extracting GWs




Main loop

BH excision



• weak processes


Locating the apparent horizon (1)


Apparent horizon (e.g. Wald (1984)): the apparent horizon is the boundary of the (total) trapped region

Trapped region: the trapped region is collections of points where the

expansion of the null geodesics is negative or zero

Thus, to locate the apparent horizon, we must calculate the expansion of the null geodesics and determine the points where the expansion vanishes

Recall that the expansion is related to the trace of the extrinsic curvature : K expansion

So that let us first define the extrinsic curvature of a null surface N generated by an outgoing null vector on a slice S:



Let S to be an intersection of the slice S and the null surface N We denote the unit normal of S in S, as sa

Then the outgoing ( ka ) and ingoing ( la ) null vectors on S are

Using ka and la , the metric on S induced by gab is given by Thus we can define the projection operator to S :

( ) ( )aaaaaa snlsnk -+2

1 ,

2

1

babaab

abbaabab

ssnng

lklkg

-+

++

b

a

b

aa

b

a

b ssnnP -+



Using the projection operator, the extrinsic curvature for N is defined by

Because ka is the outgoing null vector on S, the 2D-surface S

is the apparent horizon if tr[k] = kaa = k = 0

This condition can be written in terms of sa as

This is a single equation for the three unknown “functions” sk !

However, the condition that S is closed 2-sphere and that sa is a unit normal vector bring two additional relation to sk

For detail, see ( e.g. Bowen, J. M. & York, J. W., PRD 21, 2047 (1980);

Gundlach, C. PRD 57, 863 (1998) )

)( dc

d

b

c

aab kPP -k

0+- ji

ij

k

k ssKKsD

Energy and Momentums


Canonical formulation (1)


The Lagrangian density of gravitational field in General Relativity is (e.g. Wald (1984))

Because the 4D Ricci scalar is written as

Noting that the extrinsic curvature is

The conjugate momentum ab is defined by

RgG

4-L

) termsDivergence ( )(

)(2

2

44

+-+

-

KKKR

nnRnnGR

ab

ab

ba

ab

ba

ab

ga

)(2

1 a

b

b

aabab DDK bbga

--

)( abab

ab

Gab KK ggg

-

L

Canonical Formulation (2)


Now we obtain the Hamiltonian density as

The Hamiltonian is defined by

The constraint equations are derived by taking the variations with

respect to the lapse and the shift, respectively, as

where we have dropped the surface term

( ) ( ) termsDivergence 22

1 +

-

-+-

-

- ab

abab

ab

Gab

ab

G

DR gbgg

ag

g LH

3dxH GG H

( ) constraint Momentum : 0

constraintn Hamiltonia : 02

1

2/1

-+-

-

--

ab

a

b

M

ab

abH

DC

RC

g

gg

Canonical Formulation (3)


The evolution equations are derived by taking the variations with respect to the canonical variables (e.g. Wald (1984)) :

again, we here dropped the divergence terms

( ) ( ) abb

c

acabc

c

c

c

abba

abb

c

ac

cd

cdababGab

abbaababab

Gab

ADDDDDD

RRH

BDH

-+-+

--

-+

--

+

-

-

--

-

2

2

12

2

1

2

1

2

1

22

12

)(2/1

2/122/1

)(

2/1

bbggagag

aggaggagg

bgag

g

ab

Energy for Asymptotically Flat spacetime (1)


Let us consider the energy of gravitational field in the asymptotically flat spacetime Although there is no unique definition of ‘local’ gravitational

energy in General Relativity, we can consider the total energy in the asymptotically flat spacetime

Asymptotically flat spacetime represent ideally isolated spacetime, and hence, there will be the conserved energy

A simple consideration based on the Hamiltonian density,

may lead to a conclusion that the energy of any spacetime is zero when the constraint equations are satisfied !

This “contradiction” stems from the wrong treatment of the divergence (surface) terms (which we have dropped)

( ) termsDivergence 2 +- b

MbHG CC bagH



The boundary conditions to be imposed are not fixed ones Q|boundary = 0 where Q denotes relevant geometrical variables, but the “asymptotic flatness” :

Keeping the divergence terms, the variation of the Hamiltonian now becomes (Regge & Teitelboim, Ann. Phys 88. 286 (1974))

where we have assumed that the constraint equations and the evolution equations are satisfied, dl is the volume element of the

boundary sphere and Mijkl is defined as

)( ),( ),( ),(1 2111 ---- -+ rOrOrOrO ij

ijij

i gba

-+-

--

ljk

jkljlkkl

k

lijkijk

ijkl

G

d

dDDMH

gbbb

gaga

)2(2

)()(

klijjkiljlikijklM ggggggg 22

1-+



Under the boundary conditions of the asymptotic flatness, the non-zero contribution of the surface terms is,

Thus, we define the Hamiltonian of the asymptotically flat spacetime as

Then, the energy of the gravitational fields is not zero but E[gij]

The overall factor is determined by the requirement that the energy of an asymptotically flat spacetime is M

( ) --- lijkikj

klij

lijk

ijkl ddDM gggggg )(

( ) -

+

lijkikj

klij

ijG

ijGGG

dE

EHH

ggggg

g

g

16

1][

][16

tasympt.fla

jiji

ij dxdxr

xMxdt

r

Mds

++

--

3

2222

1

Momentums for Asymptotically Flat spacetime (1)


The action for the region V (t1<t<t2) is

Taking the variation, we obtain (Regge & Teitelboim, Ann. Phys 88. 286 (1974))

When there is a Killing vector a , the action is invariant under the Lie transport by a

Making use of , we obtain

Note that the second term in the integrand vanished thanks to the

momentum constraint

- Gij

ijt

tVGG xddtdxS HL g 34 2

1

EOMby vanishingterms32

1 + ij

ijt

tG xd

dt

ddtS g

)( ijjiijij DD gg +-- L

( ) 02232

1

+- ij

jij

ij

i

t

tG DDxd

dt

ddtS


Finally the variation of the action is reduced to

Because the Killing vector approaches at the boundary (spacelike

infinity) to a constant translation vector field ta , we have

This equation means that PGk represent the total linear momentum

Similarly, the generator of the rotational Lie transport approaches ijk j j xk (j is a constant vector field, is the totally anti-symetric tensor) , we may define the total angular momentum by

Momentums for Asymptotically Flat spacetime (2)

( ) 022

1

-

t

tk

kl

lG dS

-- kl

lij

k

G

k

G

k

Gk dPtPtP

gt8

1][ ,0)()( 12

- kjl

ijklij

G

k

G

k

G

k

k xdLtLtL

gj8

1][ ,0)()( 12

To summarize, we define the energy, the linear momentum, and the angular momentum in the asymptotically flat spacetime by

A number of examples of the actual calculation will be found in a textbook (Baumgarte & Shapiro (2010))

Energy and Momentums : summary


( ) - lijkikj

klij

ijG dE ggggg

g

16

1][

- kl

lij

k

G dP

g8

1][

kjl

ijklij

G

k xdL

g8

1][



initial conditions


Extracting GWs




Main loop

BH excision



• weak processes


References


General Reference on General Relativity

Wald R. M., General Relativiy, Chicago University Press (1984)

The classics on the basis of Numerical Relativity

Arnowitt R., Deser S., & Misner C. W., The Dynamics of General Relativity,

in Gravitation, ed. L. Witten, Wiley (1962), (Gen. Rel. Grav. 40, 1997 (2008))

York J. W., Kinematics and dynamics of general relativity,

in Sources of Gravitational Radiation, ed. L. Smarr,

Cambridge University Press (1979)

References


Recent Textbooks on Numerical Relativity

Baumgarte T. & Shapiro S. L., Numerical Relativity.

Cambridge University Press (2010)

Results of Numerical Relativity Simulations

Alcubierre M., Intrduction to 3+1 Numerical Relativity,

Oxford University Press (2008)

Boundary Conditions

Gourgoulhon E., 3+1 Formalism and Bases of Numerical Relativity,

to be published in Lecture Notes in Physics, (gr-qc/0703035)

Einstein’s equations as the Cauchy Problem

Choquet-Bruhat Y., General Relativity and Einstein Equations,

Oxford University Press (2009)

References


Locating the Horizons

Thornburg, J. , Event and Apparent Horizon Finders for 3+1 Numerical

Relativity, Living Rev. Relativ. 10, 3 (2007)

ADM mass, and Some useful calculations

Poisson E. , A Relativist’s Toolkit, Cambridge University Press (2004)

Initial Data Problems

Cook, G. B. , Initial Data for Numerical Relativity,

Living Rev. Relativ. 3, 5 (2000)

Pfeiffer, H. P. , Initial value problem in numerical relativity, gr-qc/0412002

References


General Relativistic Numerical Hydrodynamics

Font, J. A. , Numerical Hydrodynamics and Magnetohydrodynamics in General

Relativity, Living Rev. Relativ. 11, 7 (2008)

Marti, J. M. & Muller, E. , Numerical Hydrodynamics in Special Relativity,

Living Rev. Relativ. 6, 7(2003)

Toro, E. F. , Riemann Solvers and Numerical Methods for Fluid Dynamics:

A Practical Introduction , Springer (2009)

Extracting Gravitational Waves

Reisswig, C., Ott, C. D., Sperhake, & U. & Schnetter, E. ,

Gravitational wave extraction in simulations of rotating stellar core collapse,

Phys. Rev. D 83, 064008 (2011)

Documents

Introduction to 3+1 and BSSN formalismyuichiro.sekiguchi/3+1.pdfSolving Einstein’s equations on computers 7 APCTP International school on NR and GW July 28-August 3, 2011 Einstein’s