Lecture note on 3+1

1

Lecture note on 3+1

formalism of numerical relativity

Masaru Shibata

(Yukawa Institute, Kyoto U)

2010/06/09

Basis equations for Numerical Relativity

 

 

4

[ ]

8

0

0

4 0

l l

G G T

c T

u

u Y Q

F j

F

p f p f S

p

 

 

 

 

 



 

 

 











   

 

 

 

 

   

 

  

 

   

 

     

 

  

 

Einstein equation

Matter fields:

Next time if any

3

• Imposing gauge conditions

• Extracting gravitational waves

• Finding black holes (finding apparent horizon)

• Adaptive mesh refinement

Others in Numerical Relativity

Contents

1. Structure of Einstein‟s equation (briefly) 2. 3+1 formalism of Einstein‟s equation

3. BSSN formalism 4. Gauge conditions

5. Initial value problem

6. Implementation of finite differencing 7. Extracting gravitational waves

8. Adaptive mesh refinement

9. Apparent horizon finder

5

Solution of Einstein’s equation for dynamical phenomena

• We have to solve Einstein‟s equations as an initial value problem

• Einstein‟s equations, G

= 8Gc

T

, are equations for space and time

 Space and time coordinates appear in a mixed way; “time coordinate” does not

always have the property of time.

E.g., Schwarzschild coordinates;

t is time for r > 2M, but is not for r < 2M

 Special formalism is necessary to follow

dynamics of a variety of spacetimes

Schwarzschild spacetime

t is the time coordinate.

t is a radial coordinate

 

1

2 2 2 2 2 2 2 2

2

2 2

1 sin

GM GM

ds c dt dr r d d

r rc   

   

              

Time has to be always “time”

in numerical relativity

t = const

Coordinate

singularity

7

Several formulations

1. 3+1 (N+1) formalism

2. Formulation based on a special (harmonic) coordinates

(e.g., Pretorius formalism; also often used in Post-Newtonian theory)

3. Hyperbolic formalism

(e.g., Kidder-Scheel-Teukolsky formalism)

• Others …..

In this lecture, I focus on the first one

8

Einstein’s equation = hyperbolic equations

 

      ^{ }

4

: Einstein tensor : Ricci tensor

1 8

2

1

2

: P

G R

G R g R G T

c R

g g g g

g G g g g g g g t

t





   

     

        

 

      

     

 



 ^

   



           

      

 

          

 

 

  ^{ }

2

2

seudo tensor of Landau-Lifshitz

de-Donder Gauge: 0

O g

g g

G g g g g O g



 

  

  

  

 

  

 

       

 

Wave equation

9

Einstein’s equation is similar to scalar wave equation

2

or

t

t t

t i i

i

t i

 

 

 

 

 

  

   

       

  

 

  

  

 

3+1 way

Post-Newtonian way

Hyperbolic way

  ,    K _ij , g _ij 

①

②

③

Similar: However, spacetime is not

a priori given in general relativity

Section I: 3+1 (ADM) formalism

Concept

1. Foliate spacetime by spacelike surfaces

2. “Choose” time coordinates for a direction of time at each location

3. Follow dynamics of spacelike surfaces

Spacelike

hypersurfaces

Time axes

11

2 Draw timelike unit normal wrt spacelike surface, and choose lapse 1 Choose a

spacelike surface

& give space-metric

3+1 decomposition (N+1)

4 Evolve g

& K

n t

Time

Space

t

t + t 

t

3 Choose time axes at each point

 ^, 

~ : Talk later

ij ij

ij ij

K K

g

g 

nn = -1

12

Definition of variables

g

= space metric

K

= extrinsic curvature

 = lapse function



= shift vector

 

: 0

1 2 1

2

; 0

i

ij i j n ij

t ij i j j i

n

K n

D D

g n n

n t n

  



   

 





 



g

g g g

g  



  

g 

    

      

 

 

 

 

 L

Lie derivative

g

 ( g

,  , 

) Dynamics

Gauge

13

Line element

t

n

Time

Space

t

t + t

i i

x

t d

 

t

 

  

2 i i j j

ds    dt  g

dx   dt dx   dt

Structure of variables

   

2 2 2

2

1 , , , 0 ; cf. 1, 0 : lapse function, : shift vector;

, 1/ , /

* , , * , / Cf. ,

i

i j

i ij

k i

k i

ij i j

ij k

k i

n n t

g g

 



 



 

 

   g 

      

g g   

   g

 

      

 



      

            

 0 , 0

,

* , * ,

, 0 , 0

,

* , * ,

ij ij

j j j

ij ij

ij ij

K K

K K

K K



 

g g g

  

   

    

   

 

   

           

15

Geometric meaning of K _ij

0 u ^  _ n ^ 

n

u

i i 0

ij j i

u K   g u ^  _ n 

K

denotes the “curved” degree of chosen spatial hyper-surfaces

If space-like hyper-surface is curved, n is not parallel-transported.

n n

16

Next step is to rewrite Einstein’s equation by g _ij & K _ij

 As a first step, it is necessary to define covariant derivative associated with g

' ' ' ' ' ' '

' ' ' ' ' ' '

' ' ' '

' ' '

0 Required property : A spacetime tenso

Define

r

Then,

i jk ijk

lmn

ijk h i j k l m n i j k

h lmn h i j k l m n h l m n

i j k i

i jk i j k i j k i j

D T

D T T

D g

g g g g g g g g g g g g g g



 

  

' ' '

' ' ' ' ' '

( )

= 0

j k

k i j k j k

i j k

i j k j i k k i j

g n n

n n n n

g g g g

 

     

 

projection

OK

17

3+1 definition of K _ij

 

   

   

 

1 2

0

1 2 1

2 1

2

ij i j i j

i j i i j

i j i j

ij i j j i i j

j i i j ij

j i i j

K n n n

n g n n g n n n

n n n n

K n n n n n

n n n

n

   

     

     

     

 

 

  

  

  

g g g g

g g

g g g

g g

       

     

   

       

      

      

 

1

2

ij

t ij i j j i

n n

D D

 



g

g  



 

    

Usuful, often used relations

: : accelerat

l

ln ion

n No

K n n n n n

n n D

D te

n

   

          





 

      

We will use it for many times

19

3+1 decomposition of Einstein’s equation

• First two eqs = constraint eqs

・・ no second derivative of spatial metric

• Last one = evolution eqs

・・ hyperbolic eqs of spatial metric

• No time derivative for  & 

8 : Hamiltonian constraint

8 : Momentum constraint

8 : Evolution equation

k k

i j i j

G n n T n n

G n T n

G T

   

 

   

 

   

 



g  g

g g  g g







Similar to Maxwell’s eqs

 

 

4 0

4

i

i e

i i

i i i

t i i t

E B

E B j

B E





 

 

   

   

Constraint eqs

Evolution eqs

Step 1: Give an initial condition which satisfies constraints.

Step 2: Solve evolution equations.

21

All eqs have to be written by ( g , K)

• Method: Derive Gauss-Codacci equations

 

 

(3)

: 3 vector Using defintion of 3D covariant de

Start fr

rivative, om definition of 3D Riemann tensor

l

i j j i k ijk l l

a b c l m

i j k i j k a b c l m

a b c a b m

i j k a b c i j k a

D D D D R D

D D

  

 g g g g g 

g g g  g g g g

  

  

       ^{ }

  ^{ }

 

l

b l m

a l c m

i j k a c l m

a b c a m c

i j k a b c

n

K

K K



g g g g 

g g g   g 



  

     

   

  ^{ }

 

 

(3)

where we used

ln ln

0

Then, gives

=

c a b b

i

b b b b b b

c a c a a c a a c c a

b b b

a c c ac c a

l m l m m a

j l m j m l m j a

i j j

j

i k

l i

c a ij

jk l

g n n n n n n n n

n K n D n K n D

n n K n

D

n K

D D D R

g g g

 

g  g   





g

         

    



     



  

 

(3)

( ) 1

Note:

a b c d

ijkl i j k l ab

a b c l l l

i j k abc l l j ik l i jk

N a b c d N

ijkl i j k l a

cd il jk ik j

bcd il

l

jk ik jl

R K K K K

R R

R R K K K K

K K K K g g g g

g g g   

g g g g

 



 





  ^{Gauss Codacci}

eq.

23

Gauss-Codacci equations I, continued

 

 

(3)

(3) 2

3

3) 2

)

( (

Multiplying

Multiplying again 2

1

He re 2 2 16

6 ,

a b c d

ijkl i j k l abcd il jk ik j

b d a c k

jl j l bd abcd jk l jl

a c

a

il

ad il

i

c a c a c

ad

l

l ik

j

a b

i

b l

l

a

R R R n n K K KK

R R R n n K K K

R K K K

R R n n G n

R R K K K

n n

K

T n

g g g g g

g g

 g

   

   

  

  





 

Hamiltonian constraint

a c

T n n



This will be used later

1 component

• Derive Gauss-Codacci equations II

 

 

 

Start fr

om

l

k a b k c a b k d c e

i j i j c a b i j c a b e d

a b k c a b k d c

i j c a b i j c a b d

a b k c e

i j c a e b

a b k c k d c b k e

i j c a b c ij d j i e b

a b k c k

i j c a b

D D n D n n

n n

n

n K n n K n n

n D

g g g g g g g g

g g g g g g g

g g g g

g g g g g

g g g

    

     

  

      

     ⁿ 

where ln , 0

ij

d c c e

d e b

K

n n D n n





   

25

• Derive Gauss-Codacci equations II

 

 

 

 

c

ln

ln

ln

,

k a b k c k

i j i j c a b ij

i a b c i

i j c j a b ij

i a b c i

j i j c a b ij

i i a b c

i j j i c j a b b a

a b d

c j abd

i i

i j

D D n n D K

D D n n D K

D D n n D K

D D n D D n n

R n D n K D n

g g g 

g g 

g g 

g g g g

   

    

  

   



       

 

  

8 Mo

,

mentum constraint

i b d b d

i j j bd

i a b d b d

j c j abd bd j

j bd j

K R

D K D K

n n

R n n

R T

g g g

g  g

  

    



3 components

(i,k)

(j,k)

Derive evolution equation

 

 

 

(3)

2

Start from contracted Gauss-Codacci eq. I

contains or Let us calculate

ln

a c

abcd a

b d i

jl j l bd il j jl

t ij t ij

d

abcd a b b a c

a b c a bc b c

c c

bcd

a b

a

R R K K KK

K

R n n

n K

R n

n D K

n R n n

g g

g



   

  

     

     

   ^ln  ^ln

Remem

l

ber: ln

n

ab a b

b

c

b a c

c bc b c

n K n

K n D D

n D

D

 





   

 

 

27

 

 

 

 

From

ln ln ln

ln

d

a b b a c abc d

a c b d a c b

j l abcd j l a bc b ac

j l j l

a c b a b

j l a bc j bl a

a b b k

j bl a a kl j

a c b a c

j l b ac j l n ac ab c

n R n

R n n n K K

D D D D

n K K n

K K n D K K

n K K K

g g g g

  

g g g

g 

g g g g

      

   

 

   

    

   L   

1

2

b b

bc a

k

a c b d k

j l abcd

n

n jl jk l j l

jl jk l

R n n K

n K n

K K K

K K D D

g g  



 

 

 



L

L

28

 

(3) 1

(3) 1

Thus, the G-C equation become

2

s

Her

2

e

,

b d

jl j l bd n jl j l

k

kl j

k k k

n i

jl

k

t ij ij i j ij ik

j t ij k ij ik j j

j

k i

R R K D D

K K KK

K

K R D D KK

K D K K D

K

K D K

  

g g  

  







 

      

  

 

    



L

L

N

ote: 8

2

k k k b d

k ij

b d b d ij

i j bd

ik j jk i i j

d

d

i j b

D K K D K D

R T T

    R

g g  g

g g

g

 g 

   



  





Evolution equations = 6 components

29

 

(3) 2

(3)

16 8

2

8 1 ( )

2 2

ij ij

i

i j j j

l

ij ij ij il j i j

l l l

il j jl i l ij

i j ij

ij ij i j j i

R K K K T n n

D K D K T n

K R KK K K D D

K D K D D K

T n n

K D D

  

  

    





 g

 

  

 g g g g

g   

  

  

   

  

 

      

   

Summary of 3+ 1 formalism

Constraints

Evolution

,

Gauge condition

  

Constrained system

• ADM equations seem to have too many components: Constraints seem to be

redundant equations, because g

& K

are determined by solving evolution equations

• Constraints are guaranteed to be satisfied if evolution equations are solved correctly

 No inconsistency

31

Evolution of constraints

 

 

0 0

0 0

0

8

0, 0 H & M Constraints 0 Evolution eqs.

0

2

i i i i

i i ij

i ij

l i i ij

t l i i ij

l

t l i i i k

A G T

H n n H n H n H

H H

H A

H KH H D D H H K

H H D KH H

  

       

 



g g g g

    

   

 

   

 



  

      

      

^{ D}

 ^{ H}



If constrains are zero at t = 0 and evolution equations are satisfied for any t,

constraints are always satisfied.

32

Nature of standard 3+1 formalism

• Evolution equations are wave equations of 6 components, but it is not simple one:

Many additional terms even in linear order

 

(3)

, ,

~ 1

2

~ 1 2

. Maxwell's equation

0

ij ij

ij ij

ij i

kl

ik lj jk li ij k

j

l

K R

cf

A A

 

   

g

g

g g

g g  g  g



   

 

   

    





33

Linearized equations

, , ,

, ,

Linearized Einstein equations with =1 & 0

; | | 1 Evolution eq. :

Constraint H : 0

M :

i

ij ij ij ij

ij ij ik kj jk ki kk ij

ii ik ki

ij i

h h

h h h h h

h h h h

 

g 



 

    

  



 0

This causes a problem

34

Stability analysis

TT

, ( , )

TT TT

, ,

,

TT

,

Decomposition:

2

, ,

definition: 0 , : scala

, 0

Trace 3 ,

Divergen

: vec

c

t or : t

e

ensor r

ij ij ij i j ij

i i

ij

ij j ii

ii

ij j i

A C

h A C B h

B h h

h A C

h

h A

B



   

  

   



substitute

, , ,

, , ,

( )

0, 0

i i

ij ij ik kj jk ki kk ij

ii ik ki ij i ii j

C B

h h h h h

h h h h

  

      

    

35

3+1 Equations





TT TT

( , )

H: 0

Constraints :

M: 2 0

Evolution eqs. : 0

t i i

ij ij

i j

A

A B

h h

B

A A C A

  

     



  

 

 

  

   ^Strange

forms Wave

equation

36

Solutions I

1 Equations for

=Wave equations

True degree of GWs: No problem 2 Constraint (H) : 0

& Evolution equation for = wave eq.

Violated constraint will propagate a h

A

A







   

   

    ^{ }

,

way.

3 Constraint (M) : 2 For 0,

Evolution equation for gives 0 &

i

i i

i

i Bi

i

i i B

i

i Bi

A B F x

A B F x

B B F x

B F x t F x

   

 

 

  Perhaps no problem

Numerical integration

of zero is zero

37

Solutions II

2 1

,

is not constrainted by constraints, but determined by ,

If constraint is violated and 0 initially,

( ) ( )

( ) ( )

lm lm

l m

lm

C

C A A A A

f r t g r t

A Y

r

f r t g r t

C Y C

r C t

  



  



  

   





Constraint violation is serious in this case

Small error in A results in serious error in C

To summarize

• In the original 3+1 (N+1) formalism, if constraints are violated even slightly,

the error increases with time even in a nearly flat spacetime with no limit

• Namely,

it is unsuitable for numerical relativity

• Source:

, , ,

ik k

ij ij

h

h

h   h    h

First, realized by T. Nakamura (1987)

39

Section II: BSSN formalism

Essence

• Need reformulate of 3+1 formalism

• At least, in the linear level, constraint violation mode must not appear

, ,

,

,

Define new variables

and rewrite as

i ij j

ii

ij ij i j j i ij

F h

F

h F

h h

 

  



     

Evolution eqs.

for F

and  ?

Reformulation using constraint equations

,

, , ,

, ,

,

,

Momentum constraint: 0

: Evolution eq for Trace of

2 2

Hamiltonian constrain 0

0 0 0

t: 0

i i

i j j i i

ij j j

j

i

ij i

j i

i

ij ij

i i

i i

j

F

h h

F

h h

F

F

F F

F

h h

  

  

 



 

    





     





  

 



No problem !!

41

Reformulation increasing the variables and using constraints appears to be

robust

• Similar definition of new variables F

&  is possible in the non-

linear case

First, derived by T. Nakamura (1987)

Subsequently modified by Shibata (1995),

Baumgarate and Shapiro (1998)

Original BSSN formalism

(Shibata-Nakamura 1995)

 

 

2 2 2 4

4

First of all, write the line element 2

Here, det 1.

As conjugat

( corr

es for and , define

= 1 and

3

esponds to .)

i i i j

i i ij

ij

ij

ij ij ij

ds dt dx dt e dx dx

A e K K





  



 g

g

g 

g



    



  

 

 



 

trace

K  K  g K Up to here, we increase 2 variables (  , K)

and two constraints, ^det   ^g

^{ 1 and A}

^g

^{ 0}

43

Then, the equations are

 

 

, , ,

,

4 4

( ) 2 2

3

( ) 1

6

1 1

( )

3 3

+ 2 2

3

l l l l

t l ij ij il j jl i ij l

l l

t l l

l

t l ij ij ij i j ij

l l l

ij il j il j jl i

A K

A e R R e D D

KA A A A A

 

 g  g  g  g 

   

  g  g 

   

 

       

     

   

               

     

 

, 4

2

8 1

3

( ) 1 4

3

l

l ij

i j ij

l ij

t l ij

A e T

K A A K T n n

   



  



 g g g g

    g



 

     

 

            

Not sufficient in this stage !

Used

H-constraint

Note

• The linear analysis for the simple

conformally-decomposed formalism shows

“System is even more unstable”

• An exponentially growing mode appear

 

, ,

, ,

ij ij

ij

ik kj jk ki

ij i j ij

h h

h C B h

h h

C C

 

  

 





